Reporting Structures: Category Theory, Algebraic and Topological Properties

REPORTING STRUCTURES: CATEGORY THEORY, ALGEBRAIC AND TOPOLOGICAL PROPERTIES

A THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 2019

Andrew Ian Macfarlane M.A (Auck)., M.Sc.(N.S.W.) School of Information Systems Science and Engineering Faculty Queensland University of Technology Brisbane, Australia. i

DEDICATION To those who taught me about system analysis. Ngaire Miriama Murphy (nee Tukaki) who taught me the most by her persistent, forensic use of the system we built. Dave Robinson for the best project management course never taught. The QCOM Team for their Quality Assurance Processes. The un-named for their bad examples . . . . The American Mathematical Society and its authors as a conduit for modern research. ii

ABSTRACT

A reporting structure is a mathematical framework for capturing data about related events and calculating properties about sets of data that can be defined in terms of a given set of relations. Reporting structures include specifications and outcomes of information systems and control panels for factories or large engineering constructions. Re- porting structures can be represented by a class of finite categories, herein called “relational landscapes”. Relational landscapes are categories of nested relations connecting a large number of data-types; as categories they have attracted little attention except, perhaps, as models of type theories. This thesis demonstrates that relational landscapes have many algebraic, homological and topological properties that bear on actual activities in organizations. These properties reflect the nature of reporting structures and the underlying information systems but are not properties expressed in current software engineering techniques. Reporting structures have a long history, predating current corporate information systems. They arose from a need to know the state of the enterprise. They are now crucial to what decision makers can know about a national or corporate enterprise. The ways in which the state of the enterprise involves many classes of connections and levels of coordination in both space and time is an interwoven, multidimensional problem that has not been been adequately studied. Reporting structures, as defined here, are an abstract description of the way an organization seeks to know itself. By giving reporting structures a mathematical definition and a representation as a relational landscape, abstract properties that have little, if any, formal definition can be brought to light and be made explicit. There are many ways to give a high-level, possibly specialized, description of the reporting structure (or information system) but there is no mathematical study of how these descriptions can be rigorously derived from the original reporting structure. A mathematical theory of logically based, high-level descriptions is absent from the literature and needs to be developed so as to communicate about these constructions with “legitimate” concepts. The problem is to define and investigate a sequence of connective structures that expresses properties of the entire relational landscape not just components. Calling these “concordant properties," they include the propagation of changes in data, how coordination is expressed across the system, how concordant properties of an initial system affect iii an important class of future evolutions of the system and finally how the properties of the reporting structures are seen through the lens of high-level descriptions for specialized audiences. The aim of the research is to give reporting structures algebraic, category theoretic and topological properties that ultimately have meaning for enterprises. These properties measure a reporting structure’s capability to organize and evaluate the data it collects, to inform its users, not only on operational decisions, but how to improve the reporting structure itself. These mathematical properties have to provide significant additions to the way we can think about the state of large enterprises and engineering systems and so must have clear interpretations in terms of activities within an enterprise. The approach is inspired by functors of algebraic topology and geometry. These functors are developed anew for relational landscapes. Each functor “measures” a concordant structure in the reporting structure. The first functor concerns the chains of interactions involving data, the second new functor defines large-scale correlations that coordinate information across multiple relations. Extensions to reporting structures that conserve correlations gives the category of “elaborations” the first of a number of category valued functors. Exploiting natural (categorical) topologies of relational landscapes leads to both stacks and phase-space structures. The high-level description of reporting structures proceeds with the introduction of “scaled viewpoints” combining a “scaling” functor and “viewpoint” category with its homology, all of which parameterize the connected “library” of specialized high-level descriptions of a reporting structure. The outcome of this work is a precise mathematical description of the concordant (connective and coordinating) properties of the entire reporting structure. These are all functors that can be used to characterize and compare the specifications of information systems. By studying reporting structures we obtain a new application of category theory that reveals properties of reporting structures that lie beyond normal software engineering tools. The mathematics produces new perspectives on the way we attempt to “know” our largest, economically valuable, yet abstract, constructions.

Keywords: Reporting structures, Categories and Functors, Coupling Ring, Reductions, Coordination, Correlation Homology, Elaborations, Topology on Categories, Viewpoints, Category of Scaled Viewpoints. iv

Australian and New Zealand Standard Research Classiﬁcations (ANZSRC): 010103 Category Theory, K Theory, Homological Algebra 010112 Topology 080611 Information Systems Theory v

STATEMENT OF ORIGINALITY OF AUTHORSHIP ———————————————————————- I certify that this thesis is entirely my own work. It does not contain any material from any previous qualiﬁcation. To the best of my knowledge all the concepts introduced in this work have not been publish before in any form unless duly referenced.

QUT Verified Signature ————————— Andrew Ian Macfarlane Date: 12th August 2019

Acknowledgments ———————-

I wish to thank my supervisor Dr. Kirsty Kitto, now at University of Technol- ogy Sydney, for accepting me as a student for who efforts, along with those of my associate supervisor Dr. Greg Timbrell of shepherding me through the Ph.D. process. Also, at the last stretch, my thanks also to Professor Peter Bruza for his support. Also, of course, for all their support and friendship and the wide ranging discussions during this work. My thanks also to Kirsty for en- couraging me to apply for an Australian Postgraduate Award. This has made the whole Ph.D. journey much more rewarding and allowed me to concentrate on research free of other encumbrances.

I also want to thank Dr. Paul Gandar of Wellington, New Zealand, and Dr. Phil Watson who have been fully supportive of this project and have been additional sounding boards for many of the ideas in this work.

This research was supported by an Australian Postgraduate Award Scholar- ship over the period 2015 to mid 2017. Contents

xiii

1 INTRODUCTION AND OUTLINE 1 1.1 Introduction ...... 1 1.1.1 Research Topics ...... 6 1.2 Related Areas ...... 8 1.2.1 Representation of a Reporting Structure, Type theory and Category theory ...... 9 1.2.2 Reporting structures and Information Systems ...... 10 1.2.3 Reporting Structures, Cybernetics and Operations Research . . . . . 13 1.3 Outline of the Thesis ...... 13 1.3.1 Representing Reporting Structures as Categories ...... 13 1.3.2 The Coupling Ring ...... 14 1.3.3 The Lattice of Ideals ...... 15 1.3.4 Coordination Structures ...... 15 1.3.5 The Elaboration Functor ...... 16 1.3.6 High-level descriptions: analysis with viewpoints ...... 17 1.3.7 Beyond the Research Topics ...... 17 1.4 Why is This Signiﬁcant? ...... 18 1.5 Contribution to the Subject Area ...... 19

2 RELATIONAL LANDSCAPES AND THE COUPLING RING 21 2.1 Introduction ...... 21 2.2 The Concept of a Relation ...... 22 2.3 The Category L(S) ...... 23

vii CONTENTS viii

2.4 The Category Sys of Relational Landscapes ...... 27 2.5 The Coupling Ring ...... 28 2.5.1 Couplings ...... 28 2.5.2 Analysis of L by Ideals ...... 31 2.5.3 Ideals of Nested Relations ...... 37 2.6 The Coupling Ring and Integration of L ...... 41 2.6.1 The Application of C(L) to Testing ...... 43 2.7 Properties of Relational Translations ...... 44 2.7.1 Introduction: Comparing Reporting Structures ...... 44 2.7.2 Obstructions to Relational Translations ...... 46 2.7.3 Epimorphic Relational Translations and Equivalence Classes . . . . . 48 2.7.4 The Reduction of Separated Subcategories of L ...... 49 2.7.5 A Geometric Picture ...... 54 2.8 The Reduction Monoid ...... 56 2.8.1 Preliminaries ...... 56 2.8.2 Deﬁnition of the Reduction Monoid ...... 56 2.8.3 Calculating Red(L) versus Calculating C(L) ...... 60

3 CORRELATION HOMOLOGY 62 3.1 Introduction ...... 62 3.2 Adjoint Relations ...... 66 3.2.1 Level 1 Adjoint Relations ...... 66 3.2.2 Level 2 Adjoint Relations ...... 66 3.2.3 Level 3 Adjoint Relations ...... 69 3.2.4 The Hierarchy of Level p Adjoint Relations ...... 71 3.2.5 Example of a Level Three Adjoint Relation ...... 71 3.2.6 Transforming R[n] to an Adjoint Relation ...... 74 3.3 Correlation Homology of L ...... 75 3.3.1 The Correlation Complex: Faces and Boundary Maps ...... 76 3.3.2 The Meaning of δ: Low Dimension Examples ...... 78 3.3.3 Sums of Relations ...... 83 3.4 Calculation of the Homology for a Simple Relational Landscape ...... 83 !p 3.4.1 Hk(L) for L = Homogeneous R ...... 85 CONTENTS ix

3.4.2 Long Exact Sequence and Excision ...... 87 3.4.3 Excision Theorem ...... 88 3.5 Note on the Formal Development of Correlation Complex ...... 89

4 THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 93 4.1 Elaborations of L ...... 93

4.1.1 The Category E(L0) and consistent families ...... 94

4.1.2 Topologies on D(L0) and the Sheaf Z(L0) ...... 97

4.1.3 The Cohomology of Z(L0) ...... 101

4.2 The Homology Tableau for Z(L0) ...... 104 4.2.1 Distinguishing Elaborations ...... 104 4.2.2 The Homology Tableau ...... 105 4.2.3 The Homology Tableau as a Functor ...... 110 4.2.4 Homology Tableau and Sheaf Cohomology ...... 112 4.3 The “Global Theory” of Elaborations ...... 114

4.3.1 Topologies on E(L0) ...... 114 ¯ 4.3.2 The Category EN (L0) ...... 114

4.3.3 Topologies on L0 and E(L0) ...... 116 4.3.4 Convergence in Eˆ ...... 120 4.4 Sections and Speciﬁcations; the Dynamics of Change ...... 122 4.4.1 Operators of Elaboration Dynamics ...... 123 4.5 On Stacks ...... 124 4.5.1 What is a Stack? ...... 124

4.5.2 The Stack of EcN (L0) over the Conditional Topology of L0 ...... 126 4.5.3 The Category of Sections to htb ...... 127 4.5.4 A Note on Abstraction ...... 131

5 HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 133 5.1 Introduction: Adopting Different Scales of Understanding ...... 133 5.1.1 Relations among the Variables of Λ(L) ...... 134 5.2 Viewpoints ...... 137 5.3 Viewpoint Operations: Faces, Boundaries and Cycles...... 139 5.4 Deﬁnition of Viewpoint Homology ...... 141 5.4.1 The Boundary of a Relation ...... 142 CONTENTS x

5.4.2 Characterizing Cycles and Acyclic Viewpoints ...... 146

5.4.3 What is the Interpretation of Bn?...... 148 5.5 The Category of Scaled Viewpoints as A Functor ...... 149 5.5.1 The Category V(L) of Scaled Viewpoints of L ...... 153 5.5.2 Scaling Relations from L to Λ(L) ...... 154 5.5.3 Product Viewpoints and Increasing Homological Cover ...... 155 5.5.4 Integration Relative to a Viewpoint ...... 157 5.5.5 Consistency Classes of Viewpoints ...... 157 5.6 The Stack of Scaled Viewpoints Over L ...... 159 5.7 Conclusion: Scaled Viewpoints as Limited Epistemology ...... 164

6 CONCLUSION 166 6.1 Functors Resulting from the Research Topics ...... 166 6.2 Further Research ...... 174 6.3 Final Observation ...... 178

Bibliography 178

A EXAMPLES AND BACKGROUND 189 A.1 Example Reporting Structures ...... 189 A.1.1 Student Enrollments ...... 189 A.1.2 Manufacturing and Supply Networks ...... 190 A.2 Historical Antecedents ...... 197 bibliographies List of Figures

2.1 A tiny relational landscape ...... 27 2.2 The ideals containing the Enrol relation...... 36 2.3 The structure of a nested relation ...... 40 0 2.4 A picture of a reduction ϕ : L L ...... 55 2.5 A schematic of a more complicated reduction ...... 55

3.1 A level 3 adjoint relation expanded...... 69

4.1 An elaboration with few implications from split data-types ...... 104 4.2 An elaboration with many implications ﬂowing from split data-types. . . . 105

6.1 Functors of a relational landscape...... 175

xi List of Tables

1 Glossary of Symbols ...... xiii

6.1 Other systems with possible relational hierarchies...... 173

xii Table 1: Glossary of Symbols

Symbol or Notation Explanation Section

P(x) The defining property of a data-type x 2.3 D, D(L) The set of data-types defined in a re- 2.3 porting structure or its representation as the category L including defining calculated data-types.

R(x1, x2, . . . , xn) A relation among variables of data- 2.3

types x1, x2, ... , xn L(S) or L The relational landscape for a report- 2.3 ing structure system S or when S is implied. dom(R) The domain of a relation 2.3

R(x1, x2, . . . , xn) so the set

{x1, x2, . . . , xn} Data(R) The values of variables that (for a 2.3 given state of the system) satisfy the relation R.

ϕ : L1 → L2 A relational translation (functor be- Deﬁnition 4, 2.4 tween relational landscapes) Sys The category of relational landscapes 2.4 and relational translations

R1#R2 The coupling of R1 and R2 is the pull- 2.5

back over dom(R1) ∩ dom(R2). C(L) The coupling ring of L Deﬁnition 6, 2.5 Continued on next page

xiii xiv

Table 1: Glossary of Symbols continued from previous page Symbol or Notation Explanation Section J, J(R) J is an arbitrary ideal. J(R) = 2.5.2 {R#R0 | R0 is any other relation } Ann(J) The annihilator of the ideal J. 2.5.2 Ann(J) = {R0 | R#R0 = 0} J = J(C(L)) The lattice of ideals of C(L) 2.5.2

Px = t Pi and A data-type x is partitioned among n 2.5.3 i=1,2,...n R = t Ri sub-cases. Relation R is partitioned i=1,2,...n among n sub-cases spt(ϕ), cospt(ϕ) and Support, cosupport and the reduced 2.7.3 Z(ϕ) relations of a system morphism ϕ

Sysepi The subcategory of Sys where the 2.8.2 only relational translations are epimorphisms (surjective or onto) Red(L), u) The reduction monoid of L. u is the 2.8.2 addition ψ1 u ψ2 is the pushout of two reductions ψ1, ψ2 ω The null pushout. If ψ1 u ψ2 = ω then 2.8.2 ψ1 u ψ2 is a single equivalence class or a contradiction.

Hi, Hik Maximal submonoids of Red(L) 2.8.2 ! !! !p !p !p−1 !p R , R , R adjoint relations. R (R1 ,...,Rp+1) 3.2 is high-level relation expressed in terms of adjoint functors. p span(R! ) The span of an adjoint relation. The set 3.2.1

of all its L0 relations p J(R! ) The principal ideal of an adjoint rela- 3.2.2 tion to be interpreted as a category p Cat[R! ] The category with objects the queries 3.3 p (hence couplings) of J(R! ) with the

morphism the ordering q1 ≤ q2 if q1

is a specialization on the query q2. Continued on next page xv

Table 1: Glossary of Symbols continued from previous page Symbol or Notation Explanation Section

L1, L0, L1, ... , Lk(L) The levels of relations and adjoint re- 3.2.4 !p lations with R being in Lk(L).

Cp(L) The chain complex of p level relations. 3.3.1 The Z/2.Z module generated by Lp adjoint relations.

Fk The k-th face of an adjoint relation 3.3.1

δp : Cp(L) → Cp−1(L) The boundary map of an adjoint re- 3.3.1 lation being the alternate sums of !p 0p−1 0p−1 the faces. δ(R (R1 ,...,Rp+1 )) = p+1 k Σk=1(−1) Fk Zp, Bp and Hp The cycle group Zp = ker(δp), The 3.3.1

boundary group Bp = im(δp+1) and

the homology group Hp = Zp/Bp. p p CCl(R! ) The adjoint closure of R! 3.4.2 0 0 H∗(L, L ) The relative homology of L for L ⊂ L 3.4.2

E(L0) The category of elaborations of L0 4.1.1 ˜ ¯ ˜ φ : D(L0) → N For an elaboration φ : L L0, φ(y) = 4.1.2 ( #φ {y}) − 1 for y ∈ D(L0)

Z(L0) The sheaf of φ˜ functions on D(L0) Dp(U, Z) The p-th cochain module 4.1.3 ∂ : Dp → Dp+1 coboundary operator 4.1.3 Zp, Bp and Hp The cocycle group Zp = ker(∂p), the 4.1.3 coboundary group Bp = im(∂p−1) and the cohomology group Hp = Zp/Bp. h H∗(L¯) −→Z(L0) The homology tableau map 4.2.2 ht The homology tableau category with 4.2.2 objects any homology tableau for for

an arbitrary E(L0) HT The homology tableaux category 4.2.3 with objects the colimit of homology

tableau for E(L0) Continued on next page xvi

Table 1: Glossary of Symbols continued from previous page Symbol or Notation Explanation Section H The homology tableau functor Proposition 4.2.3, H Sysepi −→HT . 4.2.3

E¯N (L0) The extended category of elaborations Deﬁnition 28,

of L0 with largest dimension of homol- 4.3.2 ogy of the elaborations ≤ N ¯ ¯ ¯ Φ The projection Φ: EN (L0) L0 Deﬁnition 28, 4.3.3 s : L0 → EcN (L0) Sections (so functors) embedding L0 Deﬁnition 28, to an image (a slice) through EcN (L0) 4.3.3

The conditional topol- The topology on L0 of basic sets N(R) Deﬁnitions 29 ogy on L0 and the of relations dependent on R and their and 30, 4.3.3

Elaboration topology liftings by sections to E¯N (L0) on E¯N (L0) ¯ EcN (L0) The topological space of EN (L0) 4.3.3

σ : R0 ← R2 ← · · · ← A process or history of dependency 5.1

Rn connecting a set of relations R/L and L/R R/L is the category with objects maps 5.1. R → R0 ∈ L; the subsystem required to deﬁne R. L/R is the category of objects R0 → R; all relations dependent on the deﬁnition of R. Λ(L) A functor that selects subcategories of 5.1.1 L [n] R (σ1, σ2, ..., σn) A relation among n variables (process, 5.1.1 subcategories, scenarios) in the logic of Λ(L) Continued on next page xvii

Table 1: Glossary of Symbols continued from previous page Symbol or Notation Explanation Section V, Vm a viewpoint which has properties 5.1.1 and 5.2

{P1,P2,...,PN }, and operation ∗ linking properties and n projections tak-

ing Pi1 ∗ Pi2 ∗ · · · ∗ Pin to its n faces

Pi1 ∗ Pi2 ∗ · · · ∗ Pik−1 ∗ Pik+1 · · · ∗ Pin , k = 1, 2, . . . , n. Vm is the ∗ expressions with m ∗ symbols so V0 is the set of properties. ι : Λ(L) → V0 An interpretation of relations in a Deﬁnition 35, 5.2 viewpoint µ(σ), µ(R) The minimum models of σ and R in L 5.2

Logical equivalence of R1 and R2 are logical equivalent if and Deﬁnition 36, 5.2 relations R1 and R2 only if ∀p ∈ µ(R2)∃{s1, s2, . . . sm} ⊂ V µ(R1) such that si =⇒ p and vice i versa with R2 and R1.

Order of relations in R1(x1, x2, . . . xn) < 5.2

Λ(L) R2(x1, x2, . . . , xn, xn+1, xn+2, . . . , xn+r)

The k-face of a relation the largest n−1 relation without xk de- Deﬁnition 37, 5.2 [n] [n] R (x1, x2, . . . xn) ∈ ﬁned in Λ(L) that is less than R Λ(L(S))

δ(Pi1 ∗ Pi2 ∗ · · · ∗ The boundary map for a viewpoint Equation 5.6, 5.2

Pim (R(σi1 , σi2 , . . . , σim )) chain complex

Q(ι)m The sequence of modules called the Deﬁnition 38, 5.4 Q(ι) chain complex

Zn = Zn(Λ(L), V) The n-th module of cycles =defn 5.4

ker(δn)

Bn = Bn(Λ(L), V) The n-th module of boundaries =defn 5.4

im(δn+1)

Hn = Hn(Λ(L), V) The n-homology class =defn Zn/Bn. 5.4 Cyc(R) The non-empty “logical enclosure” of 5.4.2

µ (F Rk) (= µ (F Rk) in a cycle.) k odd k odd Continued on next page xviii

Table 1: Glossary of Symbols continued from previous page Symbol or Notation Explanation Section V, V(L) The category of scaled viewpoints Definition 42, 5.5 (Λ(L), V) and the category of scaled viewpoints for a fixed L in the category V, the morphisms given by commuting diagram (η,υ) (Λ1(L), V1) −−−→ A general translation in V(L) where Definition 43, 5.5 η (Λ2(L), V2) Λ1 −→ Λ2 is a natural transformation

between functors Λ1 and Λ2 deﬁned on Sys and υ a viewpoint morphism Γ(U, (Λ(L), V)) The category of sections from an open 5.5 and 5.6 set U (conditional topology) of L to a scaled viewpoint in V(L). Chapter 1

INTRODUCTION AND OUTLINE

1.1 Introduction

A number of areas of pure mathematics, particularly algebraic geometry, algebraic and analyti- cal number theory, topology and differential geometry, have developed techniques to understand high-dimensional concepts.These techniques have a special contribution to make to problems with a large number of parameters; problems that give rise to dynamics involving many dimensions as is the case of phase spaces or, similarly, problems involving intricately interwoven interactions. These are problems that cannot be easily rendered by computer graphics or other visualization techniques. The contribution of pure mathematics is that recasting properties of high dimensional surfaces into algebraic structures (groups and rings) can be useful; doing things algebraically can redefine properties in a way that makes it possible to calculate other properties that are new or difficult to make precise. This recasting has been formalized in terms of categories and functors [Eilenberg and MacLane, 1945, Eilenberg and Steenrod, 1952]. There are many examples. Ones that come to mind are discoveries of multiple ways to do calculus in high dimensional spheres [Milnor, 2011]. Here the sequence of ring-valued functors, the cohomology of the 7-dimensional sphere, was the crucial step. In algebra, the recasting of Galois theory using cohomology by Artin and Tate [1967] enabled the development of profinite groups [Shatz, 1972, Segal, 2008]. Profinite groups aggregate many finite field extensions into a single infinite “completion”. In algebraic geometry, Grothendieck defined K theory and established its connection with cohomology to give a more general setting for the important Riemann-Roch theorem [Macdonald, 1968, Ch. 10]. In global analysis, Atiyah and Hirzebruch [1961] developed the relationship between K theory and cohomology and Atiyah and Singer [1968] established the Index theorem that further generalized Riemann Roch. These are now old (perhaps “classical”) but very significant results and illustrate

1 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION the lesson of “thinking (co)homologically”. Or, more generally, expressing structure functorially. This is the mathematical lesson for the work here. Studying the way large organizations attempt to understand their own workings and control them over time reveals an area full of intri- cate connections. Organizations connect the many activities that occur daily in their offices and in their operational units. These activities coordinate the movement of goods and services across the world or implement government policies across a state or nation. This suggests organizations have structures that connect and coordinate many degrees of freedom, many options for doing something else in different circumstances. Although there are many ways in which these organizations have been represented, especially in the literature of systems and process analysis, there seems to be no literature that applies modern functorial methods to model the seemingly “high dimensional” aspects of large organizations such as large manufacturing, construction and banking enterprises. The approach here is to look at how organizations might model their own “metabolism;” how they use inputs, including information and expertise, to produce the products or services for which they are known. An organization’s own metabolism, its own inner workings, are known to its managers and many of its employees through the reports or computer displays that provide the information for decisions. The total set of reports and displays makes up what I shall call a reporting structure. The decisions in response to the information can be humble (for example, what is my next assignment?), or momentous, such as a large reinvestment decision. The decisions should be supported by the information contained in the organization’s reporting structure. The reporting structure is the end product of an information system and, sometimes, the specification of the information system. In its role as a specification, a reporting structure is a hypothesis about what is important, what needs to be known so that decisions can be well founded. I have used the term “reporting structure” to differentiate it from a “reporting system” which, historically, concerns the way data is gathered, who is responsible for collecting what data, and who is responsible for generating reports. (Reporting systems are discussed, in passing, in the references in Appendix A.2). Organizational reporting systems are now closely associated with information systems that have automated much of the data gathering. Reporting structures can be studied without reference to the underlying information system. The concern is not how the information was generated but what it is and what it tells its users. The size of modern organizations enforces a uniformity of business terms. This is especially important for an organization that spans many locations whether they are towns, states or across the globe. There must be a common set of terms that have the same meaning or measure throughout the organizational. This gives us the primary data that is the subject of the listings, graphs or various displays used throughout the organization. Of course, in multinational organizations,

2 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION terms have their language translations but what they refer to must be the same throughout the organization. In this thesis, the set of terms will be referred to as data-types and are deﬁned as follows.

Definition 1. A data-type x for an organization has a defining predicate P(x)) that identifies things, states or activities of relevance to the organization and identifiable by the organization by a value which can be a name, a number or a boolean value. The set of values up to a particular time is denoted Data(P(x)).

I shall refer to the various listings, graphs and displays as “reports.” Data(P(x)) is not always parametrized by time but, as a practical matter, reports should make clear the span of time to which they refer. That is, a history of changes. Central to this thesis is that information is delivered in relations. Relations give the business context for information. Relations can be as simple as “this number is less than that” or they can be the family of sub-assemblies needed to build a car. Typically, relations connect people or things relevant to the purposes of the organization. For example, students enrolled in the course Byzantium history, taught by the history department, towards a postgraduate diploma in European Culture awarded by the faculty of arts will part of a general list:

Enrol(semester, student, course, departmentqualification, faculty). that defines the relation Enrol for a given semester, linking students, courses, departments, qualifications and faculties, all of which are data-types. This list is the source of the data used for charging students. Administrators will use the enrollment report to compare the revenue of departments and faculties. Relations are used to define other relations such as which students still owe money or whether anyone cares about Byzan- tium history. In other cases, administrators or researchers might use this data to compare whether Byzantium history enrollments rise and fall with other specialist histories such as Scandinavian history. Consequently, organization reports contain functions of sets of data types. Indeed functions define a hierarchy of reports by calculating data-types from lower level reports, for example wages, taxes, monthly mortgage payments, quantities of materials to be ordered, estimated shipping rates and so on, that are the variables for higher-level reports. To be useful, functions must be a specified part of the definition of new listings and hence of reports. Reporting structures have a simple mathematical definition. The definition is effectively defined by its models, its examples and that is determined by an organization.

Deﬁnition 2. A reporting structure S is a triple (D, R, F) for which

1. D is a set of data-types that are of interest to staff and managers and auditors of an enterprise.

3 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION

2. The elements of R are the relations and are subsets of

Data(P ) × Data(P ) × · · · × Data(P ), m < #D. (xi1 ) (xi2 ) (xim )

The set R is speciﬁc to the organization.

3. Functions are deﬁned on overlapping listings R1 ∩ R2 ∩ · · · ∩ Rm where Ri and Ri+1 share at least one data-type with the same value. The range of these functions is a calculated data-type.

Comments

1. The set of relations R is determined by operational needs, governance policies and legal and certiﬁcation requirements. These are, in turn, determined by global business expectations as described in Chew and Gillan [2009]

2. Functions including maximum, minimum and any rational function of values listed in the domain of the function (such as sample statistics).

3. Reporting structures make no mention of the source of any data and the rate at which values

are accumulated beyond the speciﬁed deﬁnition, P(x), so the sources can be streamed data from satellites or from factory processes.

Reporting structures cover a wide range of mathematical objects including matroids [Cameron, 1998, p. 203] and aspects of topology. If D has four elements, and four relations linking all combinations of three elements of D, so defining faces, and a function is defined on each face we have a tetrahedron with values assigned to faces. It is then a short step to cocomplexes. This approach has been used to give homology models to measure data complexity [Degtiarev, 2000] and in finite element analysis of electromagnetic fields using de Rham cohomology [Gross and Kotiuga, 2001]. The emphasis here is organizational or enterprise reporting structures and an example is outlined in Appendix A. The mathematics that will be developed in this thesis comes from the following observation. Suppose D has many data-types, say a thousand (see comments at the end of Appendix A), with, perhaps, 50 relations. The domains of relations must contain on average 20 data-types. Only a small number of data-types will be in perhaps a dozen relations. Finally suppose there are only 20 functions. The domains of the functions are limited to three relations but usually two or one relation. Compare this with a reporting structure that has the same number of data-types but 160 relations and two hundred functions which can have domains drawn from up to ten relations. In the first case, data-types are mainly in the domain of a single relation; some data-types are in a small number of relations. The functions are typically measures of the history of aspects of the

4 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION organization and are restricted to at most three relations. This system can be approximated, in some sense, by subsystems that have little to do with each other; a system of “silos.” In contrast, the second reporting structure can have data-types sharing relations with many other data-types. The influence of a single data-type can spread from one relation to another. The abundance of functions can gather information from the many relations and compare with functions from other areas. Functions can calculate how sets of values from the history of activities in one part of the organization compare with resources allocated in another part of the organization. Functions that compare sets of activities in one area with resources in another are the basis for coordinating activities. These observations introduce a number of properties of reporting structures. The data-types can be connected in many ways. If a large set of data-types is divided among a comparatively small number of relations they are likely to form isolated regions while an abundance of relations allows any pair of data-types to be joined by a chain of relations. The structure of the relations define the property of “connectivity” or “connectedness.” This also affects what the functions can calculate, the classes of activities and events they can compare and how comparisons among many sets of relations can be used in statistical reports. Query languages are now a common part of a flexible reporting structure, but query languages, especially the now standard Structured Query language (SQL), rely on the way relations have intersecting domains. One can ask whether the first reporting structure, with few relations and functions, can “evolve” by incremental changes to the reporting structure with 160 relations and 200 functions? To answer this question there are many things that need to be made precise. What kind of changes are we going to allow? Which structures are going to be conserved? Organizations frequently want to divide data-types into sub-classes so they can better understand their interactions with, for example, subsets of students, customers or patients. This is a common type of evolution that drives the growth of data, relations and functions. Again, what is conserved in this process? There are many ways to approach reporting structures as there are many aspects to any system that manages streams of information. As shall be argued, they are designed to understand how an organization or engineering construction should manage its environments. They can be approximated by starting at some subsystem level and asking what relations hold among subsystems. This would be high-level description of the reporting structure by scaling up, looking at it above some level of detail. These observations and discussions lead to a number of Research Topics.

5 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION

1.1.1 Research Topics

The questions above suggest the following research topics or research areas. Research Topic 1. Investigate the way a reporting structure is connected. What is an appropriate concept that describes the way relations connect the data-types of a reporting structure and what is the role of functions in the concept of connective structure? Why might we want such information? The practical reasons here lie with testing and debugging the underlying information system. The first is testing. Black box, user and end to end or integration testing does not consider the mechanism behind the reports (Majchrzak [2012], Perry [1995]). Information systems have to present information in the way the user can understand: what is displayed or listed should have a documented specification. The fact that companies such as Microsoft, Oracle and SAP compete to sell their enterprise systems to the same customers is evidence that the underlying architecture and languages is less the concern of enterprises. Testers check the phenomena of the system by entering data (or controlling the source of data) to check outcomes on displays (so reports) are as intended by the specifications. Designing such tests is aided by knowing what has been specified and what affects what: the connections in the system. In the day to day operation of systems there are anomalies and mistakes in data that only show up long after the original mistakes are made, for example, during trial balance listings of money or materials when totals do not match. Tracing those anomalies to their source is aided by good knowledge of the way the reports are connected. When one enterprise system (the “legacy” system) is replaced by a commercial (off-the-shelf) enterprise system it is normal to convert valuable current and historical data held by the old system to the new system. This data-migration project might require data captured in one report in the new system to be aggregated from multiple reports in the older system. Data-migration planning needs to know how these reports and functions are connected in both systems. This planning is greatly aided by comparison of the way reports and components of reports are connected. Each part of the data migration accrues its own test plans as well; each part with their need to understand how reports are connected. In both the old and new applications we can work with the documentation of systems analysts (sometimes going back decades) and work out the connections but this is a “one at a time” exercise and not very efficient. Is there another way to understand these connections? Research Topic 2. On characterizing the coordinating mechanisms of a reporting structure. The description above of a function that compares the results of previously defined functions to show that activities in one area are matched to resources in another, is an example of a coordinating property. Such coordinating properties are a class of mathematical properties that are likely

6 CHAPTER 1. INTRODUCTION AND OUTLINE 1.1. INTRODUCTION to range from relatively simple reporting on activities and resources to much more complicated (“multidimensional”) comparisons involving classes of relations that are to be aligned in some way. What type of structures give rise to coordination properties? How do they characterize a reporting structure? Governance of enterprises requires audit, performance and certification reports, all of which have legal ramifications. These reports are used by share-market analysts to give the market assessment of the company in terms of profitability and survivability as an efficient operating business (Chew and Gillan [2009]). Making this happen requires a melding the many factors of balanc- ing resources. The coordinating role of an information system is not something that a group of uses see on their screens. Reports seen by one group of people do not indicate how their actions and inputs affect others in the enterprise. An equipment failure in one area can cause delays that are transmitted to logistics staff. The logistics staff will reschedule the specialized companies that move the products that required the failed equipment. Inventory and purchasing staff respond to production scheduling by reserving existing and incoming stock for scheduled sales or especially large orders that are not seen by the inventory staff. Indeed global companies rely on networks of coordination that happen as backgrounds to daily work without being obvious to any one group. Here the research asks for the mathematical nature of such networks of intersecting coordinating mechanisms. Indeed, the special knowledge of enterprises has been the development of coordination of resources and activities to a global scale; any study of connections in the reporting structure must include coordination. Research Topic 3. Investigate the dynamics of evolution. Develop a framework to study the possible evolution of reporting structures that is conservative, that maintains or enriches the properties of the original reporting structure. What can be said about collections of possible developments of a reporting structure that are consistent, or form a family? Research into topics 1 and 2 is intended to highlight the way reporting structures are connected. In database terms, these might be considered issues of integration (although the term is not precise). The dynamics of organizations, especially commercial enterprises and their reporting structures, are intertwined with, and cannot be separated from, legal restrictions and investors’ requirement of disclosure and audit. At the same time the need to avoid losses in money, time and resources impels refining the reporting structure to show an increasing number of details. Legal requirements, the need to increase efficiency and to conserve the brand of a company constrain the development of an initial reporting structure. The possible directions of evolution are not arbitrary, they must retain information already in the initial set of reports. This suggests that, whatever the possibilities for the evolution of a reporting structure, they reflect the initial structures elucidated

7 CHAPTER 1. INTRODUCTION AND OUTLINE 1.2. RELATED AREAS in research questions 1 and 2. Thus the possibilities for evolution tell us a great deal about a reporting structure: in practice; what can be, is constrained by what is. But by how much? For long-term planning this is a significant question. Research Topic 4. Develop a framework of high-level descriptions. Just as we do not think of a parts of a ship in terms of its bill of materials but, rather, the purpose for various structures, so we often ask for descriptions of reporting structures in terms of the reasons for particular subsystems and how they are related. These high-level descriptions involve “scaling- up”, selecting a minimum level of objects or subsystems, and interpreting their importance relative to the intent of the description. There are many high-level descriptions each for a specialist audience. These high-level descriptions are important for communicating the purpose of a structure for a given audience. Can the aspects of high-level, specialist descriptions be formalized? What can be taken as a high-level description or “outline,” or a re-scaling, of a reporting structure? If there are many such descriptions, what is the category? Enterprise reporting structures are large multipurpose, intricately connected, logical entities and are not easily captured in a single description without some inaccuracy or omission of features. We would like a high-level description that is focused on certain purposes, such as coordination and control of special resources, to be framed in terms that are well defined. Although high-level, we want some assurance that the description is meaningful and accurate. How might we define such descriptions? Are they limited to certain topics? Do they overlap? What can be said of the class of such descriptions? The first three questions investigate structures within a reporting structure. The answers to questions 1 and 2 would not make sense if there was no sense of comparisons. Question 3 implies comparison, otherwise how do we know when one reporting structure could evolve to another? Similarly, if two reporting structures have a number of high-level descriptions that are the same, are we in a position to say they are the same? If not, what is the standard of comparison between reporting structures? This implies that there must be an underlying category of reporting structures. It also suggests, for all these questions, the answers should be in terms of functors. In that way comparisons are built into the definition of structures.

1.2 Related Areas

The Research Topics and the criterion that structure should be expressed in functors, locates the subject matter as overlapping category theory, some aspects of topology and even algebraic geometry. The main applications will be in corporate reporting. Reporting structures, although predating computers, are now strongly associated with information systems. Category theory and computer

8 CHAPTER 1. INTRODUCTION AND OUTLINE 1.2. RELATED AREAS science now also overlap, especially in the study of type theories, and studying reporting structures in terms of type theories would also connect them with research in this area. This section will review these connections for their potential overlap with the Research Topics.

1.2.1 Representation of a Reporting Structure, Type theory and Cate- gory theory

Each of the areas, mentioned above, that overlap with reporting structures suggest an approach to representing a reporting structure in order to study its properties. What then guides the choice of the representation of reporting structures? This is a central question of methodology. A reporting structure, as a specification, can be represented in terms of logical specifications as in, say, the Z notation [Spivey, 2001], or in terms of a type theory such as typed lambda calculus [Barendregt, 1990]. Relational databases and data-models now underlie enterprise reporting structures and category theory has been used in formulating relational databases concepts [Spivak, 2010a,b] and integrating data models [Dampney and Johnson, 2002]. A reporting structure is, indeed, an example of a type theory common in computer science (these are surveyed in Mitchell [1990]). The expression of types, relations and functions or nested relations are all part of a type theory. A general approach might be to select a type theory that gives a set of terms and then construct the types, perhaps sets, functions and relations of the terms. Although type theories go back to Bertrand Russell at the beginning of the 1900s [Russell, 1903], more recent versions such as Martin-Lof’s type theory [Nordstrom et al., 2001] bring together type theory and proof theory [Statman, 1977]. Allied with type theories is the question of how expressive they are: can they express all that we want to say in a computer system [Immerman and Kolaitis, 1997]. In fact typed lambda calculus is as expressive as any theory of computation, anything that gives an algorithmic or rule-based decision theory, so as much as any corporation could need. This prompts questions about the approach I have taken. Why not use type theories? The answer concerns the difference between a type theory and its (category of) models. The first book devoted to category theory, [Freyd, 1964], discussed the link between model theory, the study of interpretations of first-order systems, and category theory. It was recognized that the models of systems finitely axiomatized in first-order predicate calculus, including type theories that can be so expressed, formed a category. In the development of elementary toposes by Lawvere and Tierney in the late 1960s, Lawvere [1972] showed that an axiomatization of set theory in category theory was not only a first-order theory but also included Grothendieck’s class of topoi: set-valued sheaves over a topological space. This provided a non-Boolean or intuitionist (or constructivist) class of models of the axioms of set theory. Previously, models of set theory had

9 CHAPTER 1. INTRODUCTION AND OUTLINE 1.2. RELATED AREAS been limited to forcing models [Cohen, 1966], and Boolean models [Rosser, 1969]. The intuitionist interpretations were of interest for theories of computation (and so computer science) and for intuitionist type theories. In this way category theory became significant in these areas of research. Categories also have their own version of proof theory called coherence theory, [Street and Kelly, 1974]. Coherence theory grew out of the study of closed categories by Eilenberg and Kelly [1965]. In coherence theory, rules for replacing one set of propositions with another become rules for transforming from one set of morphisms, usually a commutative diagram, to another commutative diagram. This is an example of concepts in type theory and computer science being translated into category theory. This work continues with current research of category theoretic semantics of type theories. Works by van den Berg and Garner [2012] and Garner [2009] interpret type theories as categories with extra structure. The result of this line of research is that if we adopt a category theory representation of a reporting structure, what can be proved in the type theory can be proved in a general class of category models by using coherence rules. What we gain is the 2-category of all the possible representations and a natural way to define functors starting with left and right representable functors [Freyd, 1964, Dieudonne, 1982]. Adopting a type theoretic approach commits to an syntactic framework to discuss formulations of reporting structures. Such an approach has significant limitations for a range of mathematics. For example, to say the theory of groups is about a simple first-order predicate calculus system misses the point. The models of such a simple axiomatization are where the mathematical action lies. This is the Abelian category of groups, but even this gives little of the depth of finite group theory, group representations, Lie groups and so on. Similarly, Definition 2 is a simple type theory but conveys little that is found in the example in Appendix A. It is individual models that drive questions that refine mathematical ideas.

1.2.2 Reporting structures and Information Systems

The deﬁnition of reporting structures given here owes more to the historical antecedents or business agendas than to software engineering. The information systems historian James W. Cortada argues that the digital revolution has not changed the rationale of business, and that the origins of the creation of reporting predate the era of computer information systems.

“To be sure how work was done in 2000 differs in many ways from how it was done in the 1950 but business agendas had not.” Cortada [2006, p. ix]

Reporting structures are now allied with information systems but have their own, longer, history. Chandler [1995, p. 315] describes how the nascent reporting structures were central to the debates about organizational structures and the effort to improve the ﬂow of unbiased, objective

10 CHAPTER 1. INTRODUCTION AND OUTLINE 1.2. RELATED AREAS information. The term a “single source of the truth” arose in the 1980s as a way to promote a single database to cover an entire enterprise system [Kosur, 2015]. Such a system is “integrated” in the sense of being consistent. This is merely a modern twist on the concerns of those defining the pre- information systems reporting structures. Various “top-down” techniques for designing corporate reporting structures to make them consistent and comprehensive have been worked out. Strate- gic Information System Planning (SISP) [Lederer and Sethi, 1988, Teubner, 2007] and its various refinements [Kandjani et al., 2014] concentrate on reporting structures that are derived from the way the enterprise wants to be perceived by all who interact with it. It works out “what has to go right,” for the goals to be achieved and how to measure deviations from the ideal: the “key performance indicators.” These “KPIs” are often calculated from operational statistics such as machine efficiency and maintenance scheduling [Lee, 1995, Cooper, 2009]. Without functions, reporting structures are just lists of data. The functions give the reporting structure their impact. Functions arise from the needs of corporate specialists. In corporations, audit and financial transparency requirements as in the Generally Accepted Accounting Practices (GAAP), [Harrison et al., 1987], provide the basis for financial reporting. This has become increasingly mathematical as finance itself has become more mathematically complex [Peirson et al., 1997]. Business operations and quality assurance best practices have been the bulk of operations reporting in manufacturing, giving rise to increasingly sophisticated functions for scheduling and supply chain and logistics management, [Ferreira and Otley, 2009, Knolmayer, Mertens, and Zeier, 2002]; these books also discusses the evolution of ideas that are part of the same stream of ideas that underpins the SISP technique. The connection between the specification of a reporting system, whether defined by techniques such as SISP or process modelling [White, 2004, Brookshier, 2011] or other techniques, and its implementation as a (strategic) information system is not straightforward. According to Robert Glass [2003, Fact 26] the design requirements for the supporting information system can be up to ten times longer than the business description. The detailed specifications, technical design and development can be shared across many teams of information systems specialists using system engineering tools. These tools have been part of the training of programmers and analysts over decades and include data-base design [Singh, 2009], business process modeling [White, 2004, Brookshier, 2011], unified modeling language [Rumbaugh et al., 1999], state diagrams [Yourdon, 1988] and system architecture [Giachetti, 2010]. None of these topics require knowledge of computer science. There is a divergence in concepts; the mathematical approach so ubiquitous in computer science literature such as [J. van Leeuwen, 1990], gives way to diagrammatic descriptions of information systems. Both sets of concepts aim at producing programs or systems. What seems to be lacking is a vocabulary of comparison beyond measures such as function points [Fischman, 2001]. As noted

11 CHAPTER 1. INTRODUCTION AND OUTLINE 1.2. RELATED AREAS before, once we start to compare structure we move to comparing reporting structures in terms of other properties such as how well they provide for coordination or how easy they are to enhance. Techniques, such as SISP or business process modeling, are hypotheses that assert that by following the technique we arrive at a reporting structure that has certain desired properties such as being integrated in the above sense. These techniques, part of business systems analysis, lead to information systems that produce reporting structures. Comparing reporting structures produced from these information systems is hampered by the lack of definition of, even the idea of, a reporting structure. Beyond that, concepts of comparison are lacking. Either we need a class of maps between the reporting structures or we could compare properties if we know these properties are invariants. But these concepts all need to be well defined. These are all gaps in the literature and lead to the Research Topics. While I do not address aspects of organizations beyond reporting structures, the very novelty of the approach in this thesis should have some contribution to make on a more general scale. In a survey and review of enterprise engineering1 [Bernus et al., 2016], it is noted that current information systems tools (including SISP) go from the abstract enterprise level to the concrete. This is echoed in the main example of a reporting structure in Appendix A which starts with an abstract structure and produces the concrete reporting structure. But the thesis goes in the opposite direction from concrete (or less abstract) to higher levels of abstraction. Bernus et al. [2016] see a new discipline of Enterprise Architecture as bringing together frameworks of complexity, communica- tions and limited controllability that are characteristics of large-scale organizations. The reporting structure reflects the enterprise as, indeed, it is the background of the SISP technique. The reporting structure should reflect what the staff of the enterprise find useful. But, also, over time, the reporting structure defines how many staff see the enterprise. These frameworks outlined in Bernus et al. [2016] are discussed in terms of “grand challenges”. One of the problems is to have modes of language to integrate the many frameworks or perspectives needed to establish “the systems science of the enterprise” (p. 88). One proposal is to use the expanded logic of “situation logic,” [Devlin, 2006]. Goranson and Cardier [2013] have developed situation logic as a two-sorted logic to handle hierarchies of structure and context. In this thesis a functorial approach is used to provide a category of viewpoints to look at different scales of reporting structures. The two approaches are discussed more in Chapter 5, Section 5.7.

1“Enterprise” being taken as the larger more complex organizations, private and public

12 CHAPTER 1. INTRODUCTION AND OUTLINE 1.3. OUTLINE OF THE THESIS

1.2.3 Reporting Structures, Cybernetics and Operations Research

A final area that might be seen to overlap reporting structures is cybernetics, control systems and operations research. Reporting structures can appear in all areas of business and engineering. They differ from the full feedback or cybernetic systems that have their mathematical basis in Von Bertalanffy’s 1973 work on General Systems and Norbert Wiener’s work on Cybernetics [Man- drekar and Masani, 1994]. (The relation between the two approaches, one dynamical systems and the other in terms of differential equations is explored in Drack and Pouvreau [2015]). The reporting structures defined here owe little to these views as they are designed for human control. Even in the case of control room screens displaying the ongoing states of factory systems, those systems are designed to be controlled by trained operators [Friedland, 2005]. Although operations research has had considerable influence on industry and administration its effect is to change the configuration of factories or to optimize processes [Ravindran, 2007]. It has reshaped many aspects of commerce and government. However, while the reporting of those processes might have changed by adding to the data collected (possible automatically), the data is still required and needs to be put in a way that is useful to decision makers.

1.3 Outline of the Thesis

The following outline of the research previews the way Research Topics listed in Section 1.1.1 have been addressed. The combination of connective and coordinating properties I shall call “concordant properties”. Classes of enhancements can be deﬁned in terms of conservation of concordance properties. Concordance properties have an effect when we want to approximate a reporting structure by changing the scale of how we view it. This outline clariﬁes how the treatment of reporting structures given here differs from the work described in Section 1.2.

1.3.1 Representing Reporting Structures as Categories

The description of a reporting structure at the start of this work begins with variables of different types (so data-types) clustered in various ways by relations. Functions are defined on these sets of relations and they calculate new data-types. Other relations can then be defined to include newly calculated data-types. If we tweak this definition by changing functions into relations we reduce the number of entities to two: data-types and relations. If we require the data-types to satisfy some definition so a property or a unary relation we have a single class of objects: nested relations. The structure

13 CHAPTER 1. INTRODUCTION AND OUTLINE 1.3. OUTLINE OF THE THESIS of functions defined on the result of other functions is replaced by a hierarchy of relations with a sense of dependence: what needs to be defined before a nested relation is defined. This gives us a formal definition of a reporting structure as a category of nested relations and morphisms that are dependency relations such as R(x1, x2, . . . xn) depends on (cannot be defined without) x1, x2, ... , xn. In other words, morphisms are defined by entailment. There are two overall axioms:

1. if a relation is defined then it implies that the defining properties of its domain variables, the data it relates, are also defined, and

2. every type of data (identified with its defining properties or name if it is defined by a relation) is in the system because it takes part in relations.

As we need a name for this class of categories, I have chosen the term “relational landscape” for the reporting structure over a set of data-types and this is formally defined in Section 2.3. (The term “relational landscape” is a slightly picturesque term for a class of categories that have relatively few properties. It coveys the idea that work in modern corporate office is a landscape of screens, each a view, or glimpse, of the overall activities and events in the corporation.) Morphisms between relational landscapes, or relational translations, are defined as functors taking variables to variables, relation to relations and entailments to entailments. This makes the category of relational landscapes into a 2-category Sys. As soon as we do this time is abolished - all past, present and future reporting structures exist in Sys. We are now in a position to address the first Research Topic: How is everything connected; how can we measure the connective structure?

1.3.2 The Coupling Ring

The coupling ring, deﬁned in Section 2.5, is the mathematical expression of the way changes propagate through a relational landscape L and so is the subject of Research Topic 1 of Section 1.1.1. Here the mathematics is standard ring theory but the context is new. What is of interest is how much of the reporting structure logic can be recovered by this functor. A coupling of two relations is a pullback over the generic maps to their domain data-types. In a modern database system it is a particular type of inner-join in the database. Coupling generalizes this: two relations are coupled if there is a sequence of inner-joined relations that connect them. (This is also standard in the many “NoSQL” databases2.) Coupling models propagation as each coupling implies the possibility that changes in the data-values of one relation can spread to coupled relations through the shared data-types. Iterating this traces webs of propagation.

2https://en.wikipedia.org/wiki/NoSQL

14 CHAPTER 1. INTRODUCTION AND OUTLINE 1.3. OUTLINE OF THE THESIS

The set of all couplings of a relational landscape L is denoted C(L) which is given the structure of a ring. This allows us to deﬁne a ring-valued functor C and provides a test for an arbitrary map f : L1 → L2 to be a relational translation. If we can take a subset of C(L1) and show C(f) cannot be a ring homomorphism then f cannot be a well deﬁned relational translation. This concept of propagation as a generalized coupling tells us a great deal about L but it is best seen through the ideals.

1.3.3 The Lattice of Ideals

Associated with any ring is its set of ideals. Principal ideals play a special role as the possible narratives or collections of queries that follow from the relation that generates the ideal. A narrative is a possible future. As time goes on the future narratives are generated by the principal ideal of the intersection of the past principal ideals. This gives a sense of dynamics in L. I demonstrate in Section 2.5 that a great deal of the business logic can be recovered from the coupling ring and its ideals. As one would expect, the lattice of ideals is a major tool in investigating and characterizing maps from one relational landscape L1 onto L2. In most cases this means L1 is locally ﬁbered over

L2 and this will be expanded upon in Chapter 4. We obtain propagation information about a larger system from its mapping to a simpler system. The lattice of ideals also gives a sense of dimension. The Krull dimension of the ring C(L) becomes the longest sequence of new data-types calculated from the primary (event level) data or new functions of that data. This is often a good measure of the sophistication of the reporting structure as ﬁnancial and operations analysis becomes increasingly mathematical [Peirson, Bird, Brown, and Howard, 1997, Hopp, 2007].

1.3.4 Coordination Structures

Research Topic 2 asks how is coordination manifested in a reporting structure? Coordination is a piecemeal phenomena spread out in space and time with teams often working in relative isolation, so what is the structure of coordination; what is its functorial expression? Chapter 3 introduces the concepts of adjoint relations and the correlation homology. An adjoint relation is a nested relation that exploits the natural partial order of principal ideals of the domains of the nested relation. These become categories and hence we have the possibility of adjoint functors between these categories. These adjoint functors allow for matching queries about one domain in relation with another. With this observation we deﬁne the hierarchy of “adjoint relations.” Each level of the hierarchy includes relations from the previous level giving adjoint relations a sense of dimension.

15 CHAPTER 1. INTRODUCTION AND OUTLINE 1.3. OUTLINE OF THE THESIS

Adjoint functors can provide rather coarse matchings and a homology functor is defined on the hierarchy of adjoint relations to ensure these matchings are well defined. Homology functors, one for each hierarchy level, count especially well-defined adjoint relations or “correlations” in each level. Correlations arise whenever we match the range of products or services with the expertise or resources to provide them. The matching goes both ways. Capabilities should be matched with what is to be provided. Such correlations are created in work or project schedules. If an enterprise has many intermediate specialties, each making different parts of products, there can be a network of correlations.

1.3.5 The Elaboration Functor

Research Topic 3 asks for a framework for change that helps us understand the possibilities of development for a reporting structure. This is the topic of Chapter 4. The framework must move from a static or moment in time representation to one that contains “futures”. It is therefore a test of how much can be obtained from the concepts already established. An important preliminary concept is that of a “reduction”. A reduction is an relational translation (in the category Sys) from one relational landscape onto another relational landscape: ϕ : L1 L2. By the deﬁnition of a relational translation, and discounting bijective translations, L2 has fewer data-types than L1. The term “reduction” refers to the fact that L2 has a reduced capability to make all the distinctions available in L1. In section 2.8.2 I introduce the reduction monoid which is a way of describing all possible reductions of a relational landscape L.

Starting with a reporting structure, represented as a relational landscape L0, there can be many ways it can be enhanced. The enhancements investigated in Chapter 4 are the class of relational landscapes L¯ that enhance L0 by splitting classes of data-types in D0 into sub-cases so that L0 is a reduction of L¯. L¯ is an elaboration of L0 if, in addition, L¯ has at least as many correlations as

L0. The correlation homology distinguishes elaborations from arbitrary expansions or cosmetic changes such as reworking the arrangement of fields on a screen or a reminder prompt for a staff member. Elaborations form a category which is itself a functor of L0. It is a very rich mathematical structure of L0 having a range of associated functors including a topological space of all the possible enhancement specifications. In particular, the clustering of these specifications, essentially possible futures for L0, reflects the structure of L0 itself. The topology of these futures is relevant to whether an enhancement must be achieved in a coordinated set of changes, a well specified (software) project, or can be approached by many small changes. The topology of elaborations is part of the class of concordant structures as it is an amalgam of the many ways dependency,

16 CHAPTER 1. INTRODUCTION AND OUTLINE 1.3. OUTLINE OF THE THESIS connection and correlation form a reporting structure and constrain its future possibilities.

1.3.6 High-level descriptions: analysis with viewpoints

A scaling functor Λ defined on a subcategory of relational landscapes gives a type-theory. The terms of Λ(L) are typically subcategories of L and the types are relations of these terms, so relations of subcategories. This gives a type-theory that is a high-level view of the original reporting structures. It is not a category but a type-theory. We interpret the type theory through a filter called a “viewpoint”. Viewpoints are a special logic of classification (Section 5.2) and they classify (or “interpret”) the terms of Λ(L) so they classify subcategories of L by (say) job classes or requiring certain resources. They are independent of L but linked by an interpretation of processes or subcategories of L. Viewpoints “place” the types of Λ(L), so the high-level relations, over products of classifications. A scaled viewpoint of L is then the combination of Λ(L) interpreted in a viewpoint. This mimics the description of reporting structures in terms of the job people do or areas that provide particular classes of information or even the security levels of a reporting structure. The use of viewpoints brings the advantage that a viewpoint always has an associated sequence of homology functors. The homology functors give a spectrum of high-level relations of sets of processes or subcategories of L that are optimal or significant by criteria interpretable in the viewpoint. The category of scaled viewpoints of a relational landscape is a category-valued functor which leads to the stack of scaled viewpoints over the relational landscape equipped with the conditional topology (Section 4.3.3, Definition 29). Although the concepts of viewpoints and their homology have been published, [Macfarlane, 2017], the development here generalizes the foundations of the theory and emphasizes the viewpoint approach to scaling.

1.3.7 Beyond the Research Topics

As enterprises become more interconnected, particularly global ﬁnancial and trading networks [Gorod, White, Ireland, Gandhi, and Sauser, 2015, Knolmayer, Mertens, and Zeier, 2002], their resulting reporting structures are assumed to knit together to give a useful, consistent picture. This suggests expanding the study of reporting structures to include very large reporting structures that are so big that they can only be known in part (“locally”) but can be aggregated to a consistent global construction. This “known locally, (assumed) consistent globally” suggests using the idea of stacks [Canonaco, 2004, Gomez, 1999, Olsson, 2016]. Stacks are well known in algebraic geometry, and there are similar constructions here. There are, at least, three examples of stacks, two from the chapter on elaborations and one constructed from scaled viewpoints. The stack associated with

17 CHAPTER 1. INTRODUCTION AND OUTLINE 1.4. WHY IS THIS SIGNIFICANT? scaled viewpoints deepens the link between the relational landscape and scaled viewpoints. The stacks constructed from elaborations and the homology tableaux are described in Section 4.5 in Chapter 4 and in Section 5.6 in Chapter 5. These incorporate many aspects of a reporting structure into category valued functors and position this subject in an area of active mathematical research.

1.4 Why is This Signiﬁcant?

Reporting structures are a neglected area of research. We have a tendency to study achievements without studying what it took to produce those achievements. That is one of the messages of Thomas Haigh in his review of Information Science and Technology [Haigh, 2011] and is discussed more fully in Appendix A.2. Throughout history, merchants and bureaucrats have developed ways to keep accounts of payments and debts, (the influence of these activities on early mathematics, particular the influence of commerce for arithmetic, is noted in Boyer [1991, p. 279]). People charged with managing building projects must also have developed reporting structures to report back to their sponsors. The works by the historians Alfred Chandler [1995] and Richard Overy [1996] (discussed in more detail in Appendix A.2) have little detail on reporting structures, but it is clear from those writings that the management of productive resources, especially on the scale discussed in those books, are ineffective unless well managed. Management needs data and data that can be assimilated. It needs reporting structures. The significance of reporting structures is that they are what makes the difference between the capacity for production and the ability to use it effectively. They are the key to economic and social efficiency. Of course data might be available in abundance but poorly used. The presentation of data might obscure, or be blind to, cause and effect across the enterprise; there might be few structures to monitor the effectiveness of attempts to coordinate. Attempts to improve the system might build on its faults or weaken its strong points. Attempts to convey the system in a summary form might be inaccurate or otherwise misleading. Herein lies the significance of this work. The problems listed in the previous paragraph correspond to the questions raised by the Research Topics. Cause and effect: what might influence what? What levels of structure are required to produce effective coordination? What are the possibilities of improvement in terms of more “resolution” of data while conserving other important structures? What are the possibilities of a description of the system that avoid being bogged down in the details? Reporting structures are ubiquitous and socially and economically important. For all their importance, they have stayed in the background as exciting technological changes have transformed the way we collect and store data. Accounting with its legal requirements is, perhaps, the one

18 CHAPTER 1. INTRODUCTION AND OUTLINE 1.5. CONTRIBUTION TO THE SUBJECT AREA subject that has a standard for reporting structures [Harrison et al., 1987]. In operations, the details of what data to report and what to report about it, are speciﬁc to individual corporations and so one can only write about general principles as in [Lewis and Slack, 2003, Fogarty et al., 1994, Knol- mayer et al., 2002]. The more trade oriented publications of the American Production Inventory and Control Society are designed to train inventory managers and so are more detailed (see Ap- pendix A). In all this there is no hint that the reporting could be part of a mathematical subject. The innovation here is to see corporate reporting as a new mathematical subject and to demonstrate its depth of analysis while maintaining its relevance to the problems of our evolving information (and reporting) society.

1.5 Contribution to the Subject Area

Reporting structures do not have a high profile in the history of industrial enterprises nor in the history of information technology (Appendix A.2). They do not represent any of the areas of mathematics listed in the extensive index of Iyanaga and Kawada [1977], nor those of computer science topics as listed in the index of [J. van Leeuwen, 1990]. If they are considered at all, they are part of information systems. Yet many reporting structures can exist on one computer system, even on one database as shown by the world-wide trade in packaged systems that have to be configured for each customer3. That organizations are really interested in their reporting structures (rather than the details of the information technology) is shown by the growth of the sales of ERP software to be run over the Internet ( “the cloud”). The value of sales in 2016 for such “hosted” software have been estimated at 17.6bn USD4 with a faster growth rate than software sales generally. (While these figures vary considerable from source to source, they give an order of magnitude size of the global enterprise market for large systems.) Definition 2 of a reporting structure, along with the example of a tetrahedron with functions (given after the definition) as well as the example given in the Appendix A, shows that the domain of application for a rigorous definition of this important concept. The Research Topics in Section 1.1.1 provide a set of topics about structure that introduce new areas to study from a mathematical perspective. The outline of the approach, Section 1.3, shows that answering these questions requires a number of new concepts of relevance to the operation of organizations, especially large

3Enterprise resource planning (ERP) software is reported in various web-sites to have global sales in tens of billions of (US) dollars. For example, https://www.alliedmarketresearch.com/press- release/global-ERP-software-market-is-expected-to-reach-41-69-billion-by-2020.HTML, https://www.statista.com/topics/1823/business-software/ gives the “2017 Forecasted global ERP software revenue as 34.36bn USD”. 4https://www.statista.com/topics/1823/business-software/

19 CHAPTER 1. INTRODUCTION AND OUTLINE 1.5. CONTRIBUTION TO THE SUBJECT AREA organizations. The contribution of this thesis is, ﬁrstly, to separate reporting structures as a mathematical entity from being “just” the requirements speciﬁcations for an enterprise information system or, worse, epiphenomena of studies of information systems. The second contribution is to show that they are a serious mathematical topic linked to many concepts that have entered the mainstream of current mathematical research in the last 70 years. Above all they establish reporting structures as “interesting structures”.

“Mathematics is about “interesting structures”. What makes an interesting structure is an abundance of interesting problems: we study a structure by solving these problems.” Mikhail Gromov [2013]

20 Chapter 2

RELATIONAL LANDSCAPES AND THE COUPLING RING

2.1 Introduction

The introduction has made the case that a reporting structure S = (D, R, F), (Definition 2), of the set of data-types, relations among the data-types and functions of relations, is an object of economic interest. It is a model of what is to be known about a economically productive enterprise. In the next four chapters the mathematics of S will be developed, almost as a pure mathematics but with a theme of interpretations in the world of corporate activities. In this chapter I introduce a number of foundational concepts. The first is the concept of the relational landscape which is defined in detail in Section 2.3. This is the representation of a corporate reporting structure in terms of a category. As a category has two classes of things, the objects, essentially the subject matter of the category, and the morphisms, the way the objects themselves might be compared, the choice of the objects is the key to all that follows. The choice is based on the description of pre-computer enterprise reporting structures: lists of related data just as one has in bank statements. Today the information can be on paper or on screen. Here the nature of the storage and retrieval of data is of no concern. The logic is built from the relations that make the columns of data useful; the way the data-types are part of the relations and the relations part of the domains of functions and dependencies that arise from business rules. Morphisms are defined as following from dependency: how the definition of one relation is dependent on another. This dependence can arise either in terms of definition, a logical dependency; or, in terms of business requirements, a “conditional” dependency. In this way the structure of the relational landscape reflects both the sequence of summarizing data or calculating important parameters of business

21 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.2. THE CONCEPT OF A RELATION health. The next concept is the coupling ring, it is the first functor that represents a structure of relational landscapes. The coupling ring models the way changes in data can propagate through the system. This is a measure of how well the system, as a whole, is connected. The rest of the chapter studies the coupling ring, its lattice of ideals and the Krull dimension. The mathematics is all part of commutative rings but in the entirely new context of relations rather than algebraic geometry. Much of this chapter is devoted to showing how much of the inherent logic in the reporting structure, represented in a relational landscape, can be recovered from the coupling ring. Section 2.7 examines how the propagation structure is reflected in maps to other relational landscapes. A relation and dependency preserving map onto another relational landscape can be thought of as simplified representation of the original system. Section 2.8 points out the study of all such representations, something common in the way category theory is used, produces a functor that should be seen as expressing a structure of the logic of the system: what substitution of data-types can be made under various conditions. This very abstract concept models something as practical as questions asked maintaining equipment in factories and kitchens on a daily basis: what can be used as a substitute for something else.

2.2 The Concept of a Relation

The representation of a reporting system is built on a collection of relations among data-types. Data-types define lists that change with time and can have specific properties. For this reason, the term is more specific than the term “variable”. In an enterprise, such data-types are to be clearly identifiable or measurable or else are logical (Boolean) variables such as switches or options. Here we give relations a mathematical property that distinguishes them from arbitrary sequences of data. A consequence is that relations “propagate” changes in the value of data. An example, that will be used frequently, is enrollment data at the beginning of a semester x. Enrol(x, s, c, d, q, f) is a list of the students enrolled in a course given by a particular department, the course is for a qualification that is conferred by a faculty. The form of such a report predates computer listings. Enrol is a set of related data which is the template for the term “relation”. If I change a student all the other values of course, department, qualification, and faculty might change. Or, only the course might change. If I change the faculty all else will change but the student; the student might be doing conjoint degrees. This is commonplace for relations but we now formalize it.

From now on sequences (x1, x2, ..., xn) of data-types will be represented by a vector ~x when there is no ambiguity. A relation among n data-types {x1, x2, ..., xn } is written R(x1, x2, ..., xn)

22 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.3. THE CATEGORY L(S)

with {x1, x2, ..., xn } being the domain of the R:

dom(R) =defn {x1, x2, ..., xn }.

The phrase “R is a relation among n data-types”, implies a certain “coupling” of the domain data-types. We assume that each data-type xi can take on a set of values Data(xi). Sup- pose a1 in Data(xi) occurs with b1 in Data(xj) in the list of values of R(x1, x2, ....xn). That is

R(. . . , a1, . . . b1 ... ) is in the list of R values, a1 in the i-th place, b1 in the j-th place. A change of value of a1 to a2 might require a change in b1 to b2 or some other value of x2.

If π(xi) is the set of changes a1 7→ a2 of Data(xi) then for all pairs xi, xj in {x1, x2, ....xn } there is a function. p(ij) π(xi) −−−→ π(xj). (2.1) that takes a change in values a1 7→ a2 in Data(xi) to a change b1 7→ b2. This can be trivial; in long lists p(ij) maps many interchanges π(xi) to the identity. If this is always the case there is no need for one of the data-types xi or xj. To avoid this redundancy I shall assume the relation is always well deﬁned; there is another data-type xk such that the composed transformation

p(ik) p(kj) π(xi) −−−→ π(xk) −−−→ π(xj) (2.2) is not the identity; the composition is not trivial. This is an indication of size and the information content of a relation. If there is only one faculty and all the students go through the same qualiﬁ- cation in cohorts, the constraint 2.2 can fail. This links information in relations to transitions of the permutation group of the values of the relation.

The definition of a data-type x, which will be denoted by P(x), needs to include the way it accommodates data; its elements Data(x). Data(x) will always be a pointed set with a special element . The state of Data(x) is the data at some specific time; it always includes as well as a normal set X of items such as students currently enrolled in courses. X is the current state. There are operations on Data(x). If X is the state and a∈ / X, [a]: 7→ a ∈ Data(x) so the current state becomes {a} ∪ X. The reverse operation of deleting a is denoted ¬[a] and finally a ∈ Data(x) can be corrected or updated by replacing a by a0 or [a0] ◦ ¬[a]. (These operations are known as the “CRUD” operations: create, read (list, display), update and delete).

2.3 The Category L(S)

Deﬁnition 3. The representation of the reporting structure S = (D, R, F) is deﬁned as a category L = L(S) having the objects:

23 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.3. THE CATEGORY L(S)

1. A ﬁnite set of data-types D = D(L), each data-type x in D has a corresponding ﬁrst-order predicate

P(x) deﬁning x in S. D contains “primary” or “raw” data-types as well as any calculated data-type relevant for reporting.

2. A ﬁnite set of relations R such that ∀R(x1, x2, . . . , xn) ∈ R every domain data-type of R is found in D and: ^ R(x1, x2, ...., xn) ⇒ P(xi) (Relations are well deﬁned). (2.3) xi∈dom(R) and every data-type in D is used in some relation:

∀x P(x) ⇒ ∃R ∈ R ∃~yR(x, ~y) (All data-types have a role to play). (2.4)

Given R(~x) = R(x1, x2, . . . , xn) is dependent on the P(xi) in its domain, R reﬂects changes in data that come in two ways; these are expressed in the commuting diagrams:

g¯ p(ij) Data(R(~x)) / Data(R(~x)) Data(R(~x)) / Data(R(~x)) (2.5)

g p(ij) / / Data(P(xi)) Data(P(xi)) π(xi) π(xj)

3. Nested relations of previously deﬁned relations. These are expressions

R(R1,R2, ...Rm, ~z) where Ri is an object of L and ~z is a vector of data-types. (2.6)

and

dom(R(R1,R2, ...Rm, ~z)) =defn ∪ dom(Ri) ∪ ~z i=1,...m

(identifying ~z with its set of components).

4. The morphisms between objects of the category L take two forms:

(a) Logical entailment that come from relations being well deﬁned. These include equations 2.3, 2.4 and 2.6. (b) System rules that come from S.

i. If a change in R2 implies there must have been a change in R1 we write R2 → R1. Here

changes in R2 cannot take place without a prior change in R1. In this ways all morphisms in this category arise from entailment.

24 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.3. THE CATEGORY L(S)

ii. We shall assume that the rules of reporting structure mean that R1 → R2 implies dom(R1)∩

dom(R2) 6= ∅.

We make the following observations on the items in this deﬁnition. (The numbering relates to that in the deﬁnition.)

1. P(x) is a deﬁnition with sufﬁcient information to identify each example of an x and distin-

guish it from other possible x’s. If a is a student so P(student)(a) is true, then a has sufficient 0 0 attributes to distinguish it from any other a with P(student)(a ). We assume the definition of these data-types uses ordinary predicate calculus although, wherever possible, these data- types are finite so that existential quantification can be replaced by disjunctions (so quantifier elimination).

2. The condition 2.4 might also be called the “No isolated data-types” condition. Obvious examples include the type “customer”: a customer is a customer because they have bought a product from us; equally, a student, to be a student, has to be enrolled in a course.

3. In the diagram 2.5 g¯ and p(ij) are induce by changes in data. They are changes in the list of data that are cause by the additions, changes and deletions of items of data. For example, a late enrollment, a spelling correction in the name, or when a student withdraws from a course.

4. When relations are functions, R(R1,R2, ...Rm, ~z) is a function of functions. Nested relations

allow the expression of networks of functions with each Ri providing input to a high level

report, R. R(R1,R2, ...Rm, ~z) can also be a complex data-type. If the domains relations Ri,

i = 1, 2 . . . m represent optional components in a subassembly then R(R1,R2, ...Rm, ~z) can represent the higher level assembly. Many systems that cater for complicated products and services with multiple options have "conﬁgurators" to check that the products are actually available (Forzaa and Salvadora [2002], Haug et al. [2012] and Trentin et al. [2012]). Relations of relations are a way of constructing consistent sets of options when the options selected in

Ri can limit those in another domain relation Rj.

5. The rule R1 → R2 implies dom(R1) ∩ dom(R2) 6= ∅ is a practical consideration. For example,

if R1 means a customer has been notiﬁed by email of an invoice then R2 means the invoice data, date of notiﬁcation, list of outstanding payments has been sent to the customer. The

invoice number, notification date and the customer connect the relations R1 and R2. Before queries became available in information systems this would have been verified by checking lists of invoices and notifications (sent by mail).

25 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.3. THE CATEGORY L(S)

6. Note that a conditional morphism R1 → R2 that covers a wide range of inputs and responses

might be best captured by a relation of relations: R(R1,R2, ~z) where a set of options, ~z, indexes the way the relations are linked.

7. As a ﬁnal observation we note that by reversing the arrows in L, the opposite category L(S)op represents the processes of the enterprise. Following these (process) arrows gives the process mapping down to the detail of changes in reports notifying responses that result in further report changes. Indeed, the L(S)op representation includes what data is to be in front of a staff member before deciding on the next step in a process. The process modeling software such as Arisr or BizAgir does not require this to be explicit. This apparently more abstract approach is more explicit than is treated in recent software.

An image of what a tiny relational landscape might look like is in Figure 2.1. The arrows are dependencies. The dotted arrows are dependencies independent of implied (composed) dependencies. In fact an enterprise system is more likely to be made of up many hundreds, if not thousands of these types of diagrams with arrows connecting relations and data-in a highly non-planar arrangement. Figure 2.1 gives the barest indication of the multiple connections that exist in enterprise systems. The budget for training can affect how long a delay in production takes to fix because a certain expertise is not available. Long delays in production might cause shipping schedules to be missed and so on. Reporting structures tend to grow as these connections become apparent. The categories L are simple to describe in terms of a type theory. The terms are the data-types of D(L) and the types are relations of terms and relations of previously defined types. The equations 2.3, 2.4, and 2.6 are the basis of proof rules. Each model of this type theory corresponds to a category L with its objects corresponding to the terms and types and the dependency morphisms being the proof rules. But again, as mentioned in Section 1.3.1 and 1.2.1, for the same reasons that algebraist seldom use formal logic, we work directly in the category of models, that is with the categories L. L is not a second-order theory as there is no quantification of relations. It is a first order theory insofar as the formulation of the rules of the system can use the full first-order predicate logic with quantification over primary data-types. In all other areas we use category theory (even though universal properties are defined using ∀ over morphisms this is not expressed in the system). One can also require that all relations are recursive hence their truth (or not) is decidable.

26 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.4. THE CATEGORY SYS OF RELATIONAL LANDSCAPES

R¯(x, y, z, R, R¯) Planned Work

* ¯ R(u, v, R2,R3) Cash rpt.

R(w, p, R1, R2) Ongoing work

z ' R(R1,R2) R2(R2,R3) Capacity rpt.

* × ) R1(x, v, p) R2(y, u, w) R3(x, v, p) Current State

w z % ' r , ) - P1(x) P1(y) P3(z) P4(u) P5(v) P6(w) P7(p) Data-types

Figure 2.1: A tiny relational landscape

2.4 The Category Sys of Relational Landscapes

From the point of view of category theory, categories themselves are perfectly good objects for a category of some class of categories. The category of relational landscapes is fundamental to be able to compare reporting structures. Deﬁne Sys as the category with objects the categories L(S) for arbitrary S and morphisms given as follows. Let L1 = L(S1) and L2 = L(S2).

Deﬁnition 4. A functor ϕ : L1 → L2 is a relational translation if

1. ϕ : D(L1) → D(L2) and for all x ∈ D(L1), ϕ(P(x)) is an existing data-type P(ϕ(x)) in D(L2).

2. ϕ is a ﬁnitely complete functor hence takes relations to relations, preserves all entailments and preserves products and pullbacks.

3. ϕ(x) = ϕ(y) ⇒ ∃R ∈ L1 and {x, y} ⊂ dom(R). This is to ensure that ϕ does not identify unrelated data-types.

The deﬁnition of relational translations emphasizes that they are conservative: preserving logic but not creating new entailments nor deleting or destroying data-types or relations. This category

27 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

Sys was first used in Macfarlane [2017] in which a relational translation was called a system morphism. Sys is our universe of discourse. As a category of categories it is a 2-category [Street and Kelly, 1974]. The left and right representable functors can be defined for any category and, in the case of Sys, right representable functors will be used as the starting point for the reduction monoid (defined in Section 2.8) and left representable functors the starting the category of elaborations (defined in Chapter 4).

2.5 The Coupling Ring

In this section I introduce the functor that captures the connective structure of a relational landscape L. It builds on the property of relations to propagate changes in data (section 2.2). The functor, the coupling ring and an associated contravariant functor, the lattice of ideals, are the main tools to investigate the relational translations. The definition of the coupling ring proceeds by assuming that L has pullbacks of relations which would make L finitely complete. The product in L is the conjunction of relations and the equalizer of two dependencies is also their conjunction in L. A conjunction R1 ∧ R2 can be seen as a restriction, or specialization of R1 to the conditions of R2 and vice versa. For L to be finite complete is equivalent to requiring relations can be restricted (or specialized) by other relations; something that would have been assumed in any paper based reporting structure.

2.5.1 Couplings

In this section we deﬁne an algebraic structure that captures the propagation of change throughout the category L. Section 2.2 deﬁnes the sense in which relations are used in this work; changing the values of one data-type of the list of items satisfying a relation can change the values of other data-types in the data. Here we examine how this spreads beyond a given relation.

Deﬁnition 5. Given R1 and R2 with domains dom(R1) and dom(R2), if dom(R1) ∩ dom(R2) 6= ∅ deﬁne the coupling of R1 and R2, denoted R1#R2, as the pullback

28 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

R1#R2

z $ R1 R2

" | V P(x) x∈dom(R1)∩dom(R2)

This is the pullback over the entire set of common data-types.

Using the usual notation for pullbacks note that

R1#R2 6= R1 ×dom(R1)∩dom(R2) R2 as relations are logical constructs not sets and usually the Cartesian product is not deﬁned in L.

If dom(R1) ∩ dom(R2) = ∅ then it is still possible for R1#R2 6= ∅ as intermediate data-types produced couplings as in the case:

0 0 0 R1#R2 R2#R3 ... Rn−1#R2 (2.7)

Ô Ô 0 0 0 R1 R2 R3...Rn−1 R2

0 0 where Ri = Ri(xi, yi). By taking the limit of this diagram, effectively a sequence of iterated pull- 0 0 backs, we obtain a limit R1#R2#...#Rn−1#R2 which is a coupling as changes from R1 can propagate to cause changes in R2. If dom(R1) ∩ dom(R2) = ∅ and there is no intermediate relation or couplings of relations R¯ such that R1#R¯#R2 is non-zero, only then is R1#R2 deﬁned to be 0.

Relations R1(~x,t) and R2(~y, t) have the common data-type t and can have common data-types among ~x and ~y but have R1#R2 = 0. This occurs when R1(~x,t) is true only when t < 0 and

R2(~y, t) is true only when t > 0. Thus the coupling contains additional details of logic in the relations. This will be the basis of equation (2.8) below.

Deﬁnition 6. The coupling ring of the category L, C(L), is the Z/2.Z module with addition freely generated by all the relations and couplings and with the multiplication given by the coupling operation. The W multiplicative unit of the coupling ring, U, is deﬁned as P(x). x∈X(S) Furthermore, the following condition is imposed:

R1#R2 6= 0 and R2#R3 6= 0 ⇒ R1#R2 + R2#R3 = R1#R2#R3. (2.8)

29 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

A Z/2.Z module over a set of symbols {α1, α2, . . . αm} is the set of expressions

Σ ai.αi. ai∈Z/2.Z;i=1,2...m

The only values ai can take are 0 and 1 so each sum corresponds to a set of αi: those with the coefficient 1. The addition of two or more sets is the union of the sets less everything that appears an even number of times in the combined sets. Such modules are sufficient for the accounting that we wish to do. A consequence is that the addition in the coupling ring is as simple as possible: if R1 and R2 are in the ring we also have R1 + R2 but R + R = 0 (characteristic 2 addition) and so in sums of relations there are no coefficients in front of the relations. This allows us to define distributivity as a matter of notation. Equation 2.8 is the algebraic version of the diagram 2.7.

For each relation, R, R(x1, x2, ...xn) → P(xi) is defined for i = 1, 2..n. From propositional calculus p ⇒ p ∨ q so R → U. Pulling back over the common data-types yields R, so U#R = −1 R#U = R. There is a group of units [Atiyah and Macdonald, 1969, p.2]. u is a unit if ∃ u and −1 −1 u#u = U. If u = tiP(xi) and u = tjP(yj ) with one of the xi = yj and the set of xi and yj −1 = D(L) then u#u = U. If L is not complete then R1#R2 belongs to the finite completion of L. It is possible that in a given implementation or point in the history of an organization, the set of data in the system means that the pullback R1#R2 has no data but the potential for coupling lies in the logical possibility of shared data. In particular, the equation (2.8) might seem to depend on what data is in the relations. But couplings are defined in category theory terms, independently of sets of data.

Consider R1(x, y, z), R2(y, u, v) and R3(v, w), the point is not that by deﬁnition R1#R3 is empty but that a change in values of the data-types in R1 can propagate through to change values in R3. Equation (2.8) makes the coupling ring into an algebraic model of propagation of data changes. Some comments.

1. The coupling product # is commutative and has null products.

2. In the case of organizational database systems R1#R2 corresponds to the "tightest" inner join

between tables R1 and R2: the inner join that makes all the common data-types the same.

3. If R1 and R2 are functions R1#R2 = R2#R1 implies interdependence but does not give

information on whether the shared data-types are input for R2 and output for R1.

4. Concentration of information or interactions relies on coupling of relations. Where relations have many shared data-types is where reporting activity is most developed. Equation (2.8) ampliﬁes this concentration.

5. The rule that R → R¯ implies a shared variable further implies that

30 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

R0#R¯ 6= ∅ implies R#R¯ + R¯#R0 = R#R¯#R0.

A fundamental result is

ϕ Lemma 2.5.1. If L1 −→L2 is an epimorphism and if R1#R2 6= 0 in C(L1) then ϕ(R1#R2) = ϕ(R1)#ϕ(R2) 6=

0 in C(L2)

0 0 0 0 Proof. If R1#R2 6= 0, there is a sequence R1#R1 6= 0, R1#R2 6= 0, ...Rk#R2 6= 0 where in each 0 0 0 0 case dom(Ri) ∩ dom(Ri+1) 6= ∅. Also Ri#Ri+1 6= 0 requires that there are no contrary conditions 0 0 such as a parameter t with Ri true for t > 0 and Ri+1 only true for t < 0. ϕ cannot make a 0 0 0 non-empty set of data-types empty so the chain dom(Ri) ∩ dom(Ri+1) 6= ∅ maps to dom(ϕ(Ri)) ∩ 0 dom(ϕ(Ri+1)) 6= ∅. We can now repeat the chain of couplings in L2 to get ϕ(R1)#ϕ(R2) 6= 0.

Proposition 2.5.1. Given ϕ : L1 → L2, both ﬁnitely complete and ϕ preserves pullbacks then there is an induced homomorphism C(ϕ): C(L1) → C(L2).

Proof. ϕ takes relations to relations and so the Z/2.Z addition is clearly preserved. By lemma 2.5.1 ϕ takes couplings to couplings so preserves ring multiplication. As both addition and multiplicative are preserved ϕ induces a ring homomorphism.

If ϕ does not preserve pullbacks we still have maps ϕ(R1#R2) → ϕ(Ri) → P(ϕ(x)) but these need not be a pullback as universality of pullbacks is not guaranteed.

Corollary 2.5.2. C deﬁnes a functor from Sys to the category of rings and ring homomorphisms.

C deﬁnes the propagation structure of the relational landscape. This is a property of the entire logic of the reporting structure S. C is therefore an existence proof that properties of the entire reporting structure can be deﬁned, are functors and so transform in a way that can be calculated from relational translations.

2.5.2 Analysis of L by Ideals

We review a set of standard concepts before continuing. These can be found in any text on Ring Theory or Commutative Algebra such as Atiyah and Macdonald [1969].

1. An ideal J of C = C(L) is a subset closed under addition (which also includes 0 as R + R = 0 by Z/2.Z addition) and, if R¯ is a coupling in J, then for all other r in C, r#R¯ is in J.

Note that R¯1#R + R#R¯2 = R¯1#R#R¯2 applies in ideals.

31 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

2. The annihilator of an ideal J, denoted Ann(J), is the set r ∈ C | ∀R¯ ∈ J, r#R¯ = 0}. Ann(J) is also an ideal of C and can be thought of as the ideal “orthogonal” to J.

3. J is a prime ideal if and only if

R¯1#R¯2 ∈ J ⇒ (R¯1 ∈ J or R¯2 ∈ J).

4. If R is a relation, we write J(R) for the ideal generated by the couplings R#R0. J(R) is the “principal ideal” generated by R. A principal ideal is a theme: everything to do with its gen-

erator R. If x is the data-type of customer, J(P(x)) is everything that mentions a customer, hence sub-systems most closely associated with “customer relationship management” systems.

5. An ideal J 6= C(L) is maximal if it is not contained in any other ideal.

Proposition 2.5.2. J(P(x)) is prime. ¯ ¯ ¯ ¯ ¯ Proof. Suppose R1#R2 ∈ J(P(x)) but neither of dom(Ri), i = 1, 2, includes x then R1#R2 = r1#r2 + r2#r3 in J(P(x)) with one of dom(r1) or dom(r2) containing x along with other common data-types. This implies R¯1#R¯2 = r1#r2#r3 with at least one of the factors containing x. Assume 1 this is the least factorized form of R¯1#R¯2 , this means at least one of R¯1 or R¯2 contains x hence is in J(P(x)).

Deﬁnition 7. A data-type x is partitioned into sub-types x1, x2, . . . xn if

P(x) ⇔ P(x1) t P(x2) t ... t P(xn)

(where t denotes exclusive “or”).

This means P(xi) → P(x). This allows us to treat R(xi, y1, ...) → P(xi) as R(x, y1, ...) ∧ P(xi) →

P(x) ∧ P(xi) = P(xi).

Corollary 2.5.3. If x is not partitioned (so does not have sub-types) J(P(x)) is a maximal prime ideal.

Comments

1. The relation J(P(x) ∧ P(y)) is seldom prime. For example, given R1(x, z), R2(z, y), neither of

which have domains with both x and y, but R1#R2 → P(x) ∧ P(y) so R1#R2 ∈ J(P(x) ∧ P(y)).

2. Prime ideals can be generated by a single relation, say R0(x, y, z), as long as L has the following property: for any relation R(x, y, z, ...)

1The options for factorization are ﬁnite in a ﬁnite set of relations.

32 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

R(x, y, z, ...) → P(x) ∧ P(y) ∧ P(z) = R(x, y, z, ...) → R0(x, y, z) → P(x) ∧ P(y) ∧ P(z).

That is, if dom(R) contains {x, y, z } the necessary entailments factor through R0. This can

happen if the only couplings R1#R2 in J(R0) have the form R1(R0, ...) or R2(R0, ...). In such

a case J(R0) is a prime ideal. The proof that J(P(x)) is a prime ideal adapts to proving J(R0)

is prime. In this case P(x) ∧ P(y) ∧ P(z) is forced to be an “inseparable set”. An example is the rule that a student, to be a student must be enrolled in a course which as been invoiced or paid for.

Proposition 2.5.3. If R1 and R2 are relations with overlapping domains but R1 ⇒ ¬R2 then R1#R2 = 0 and Ann(J(R1)) contains J(R2) and vice versa.

For example R1(x, y, z) is true when z > 0 and R2(x, y, z) when z ≤ 0. More generally this can occur when one or more data-types partition or splits a relation among multiple cases.

Proof. The pullback of R1 and R2 depends for its deﬁnition on R1 and R2 so jointly on R1 ∧ R2 and the equalizer of those maps which is empty so R1#R2 = 0. This is also true for any R¯ ∈ C(L) which shares variables with R1 and R2: R1#R¯#R2 ⇒ R1#R2 = 0. This means R2, and anything of the form R¯#R2 hence J(R2), is in Ann(J(R1)).

0 As part of the deﬁnition of business rules expressed as R (y1, y2, .., ym) → R(x1, x2, ...xn) there is a common variable in the sets {y1, y2, .., ym } and {x1, x2, ...xn } so there are p, q with xp = yq. In particular

0 R (y1, y2, .., ym) is not an annihilator of J(R(x1, x2, ...xn)) so 0 J(R(x1, x2, ...xn)#R (y1, y2, .., ym)) ⊂ J(R(x1, x2, ...xn)).

Consequently a necessary condition for a business rule is that there are principal ideals that intersect J(R0) ⊕ J(R).

The lattice of ideals

Ideals of C(L) form a modular lattice, denoted J = J(C(L)) ([Birkhoff, 1967, Davey and Priestley, 2002] and for the special case of lattices of ideals Atiyah and Macdonald [1969]). The greatest lower bound of J1 and J2 is J1 ∩ J2 while the least upper bound is J1 ⊕ J2. Modularity implies

J1 ≤ J3 ⇒ J1 ⊕ (J2 ∩ J3) = (J1 ⊕ J2) ∩ J3.

In many cases J3 is a maximal ideal J(P(x)) (which will often be abbreviated to J(x)) so many applications concern sub-ideals within some J(x).

33 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

J(C(L)) as a contravariant functor

ϕ C(ϕ) A relational translation L1 −→L2 induces a homomorphism C(L1) −−−→ C(L2). Any homomorphism between rings induces a contravariant functor of the lattice of ideals by pulling back ideals in J(C(L2) to those in J(C(L1) as in the diagram:

/ 0 J J

C(ϕ) C(L1) / C(L2)

∗ 0 Denote J as J(ϕ) (J ). When ϕ is onto, deﬁne J(ϕ)∗(J) =defn im(C(ϕ)(J)). In this case J(ϕ)∗(J) is an ideal. This is the proof that

ϕ J(ϕ)∗ Proposition 2.5.4. J(C(L)) is a contravariant functor from Sys to lattices; L1 −→L2 deﬁnes J(C(L2)) −−−→

J(C(L1)).

J(C(L)) is an important contravariant functor in our approach. If L has thousands of relations the same is true for the generators of C(L). J(C(L)) still has principal ideals for each relation but by using ideals we have a tractable way to simplify the logic of L into equations among ideals. The following is a standard result.

ϕ ∗ Proposition 2.5.5. If L1 −→L2 is onto then J(C(ϕ)) and J(C(ϕ))∗ are adjoint functors.

(This follows from elementary proofs, or exercises, from any course in ring theory or commu- f 0 tative algebra. If A1, A2 are rings, A1 −→ A2 is a homomorphism, J is a prime ideal in a ring A2, −1 0 then f (J ) is prime [Lang, 1969, p. 63]. Going the other way, if f is onto and J is prime in A1 0 0 −1 0 and u.v ∈ f(J) = J then ∃x, y ∈ A1, f(x) = u, f(y) = v and so f(xy) ∈ f(J) = J . As f (J ) is prime one of x or y belongs to f −1(J 0) and so one of u or v is in f(f −1(J 0)) = f(J).) This can be interpreted as a “weak equivalence” at the structural level of ideals. In particular, prime ideals J(L1) are mapped to prime ideals in J(L2). J(C(L)) is another way to look at the propagation structure of L and so is a contravariant invariant of the reporting structure. Much of the structure of L can be recovered from J especially from the equations of the form:

J(R1(x1, x2, ...xn)) ∩ J(R2(y1, y2, ..., ym)) = ⊕J(Ri(zi1 , zi2 , ...zi(p))), i where {x1, x2, ...xn, y1, y2, ..., ym } ∩ (∪ zi , zi , ...zi(p) }) 6= ∅. i 1 2 A straightforward application that mixes category theory concepts with logic and lattice theory is an application of proposition 2.5.3 on partitions and the propagation of partitions.

34 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

Deﬁnition 8. Partitioning of relations.

1. A relation R is partitioned into sub-relations, if R = t Ri. i=1,2,...n

2. The partition propagates if for any R¯ ∈ C(L), R¯#R = t R¯#Ri. i=1,2,...n

Proposition 2.5.6. If R = t Ri is a partition that propagates then i=1,2,...n

J(R) = ⊕ J(Ri). i=1,2,...n ¯ ¯ Proof. Suppose dom(R) ∩ dom(R) 6= ∅ and R#R ∈ J(R). We have Ri R so the restriction of R¯#R to Ri is given as R¯#R / R i i

pullback

R¯#R / R and each R¯#Ri ∈ J(Ri). R¯#R ⇔ t R¯#Ri. As each pair of the Ri ⇒ ¬Rj and the partitioning i=1,2,...n of the J(R) follows from proposition 2.5.3.

As an enterprises grows it needs to split relations into special cases; customers are divided into classes as are products and suppliers. These different classes are of interest in many areas of the enterprise so splitting the original simple relations propagates. This proposition shows how the splitting affects the lattice of ideals. This also reﬂects how the lattice of ideals keeps pace with the logic of relations. Examples We can illustrate these ideas with our standard example Enrol(s, c, d, q, f), the relation of student s enrolled in course c taught by department d for qualiﬁcation q awarded by faculty f. In the Figure 2.2 the size of the ideal increases up the page. There might be no relation between student s and faculty f except through intermediate variables.

1. In the Figure 2.2 the hooked arrows signify subset and J(s, c) is the everything that follows from the student enrolled in the course. The course might be a service course required by many qualiﬁcations and their awarding faculties so course is no predictor of qualification or faculty.

35 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

J(c) J(s) J(q) J(f) e O i 5 O 8

3 S O 6 V ( Ø O + J(c) ∩ J(s) J(s) ∩ J(q) J(d) ∩ J(f) O = J(s, c) ⊕ J(q, f) 5 i

( Ø 6 V J(s, c) J(d, f) i 5

6 V ( Ø J(Enrol(s, c, d, q, f))

Figure 2.2: The ideals containing the Enrol relation.

2. The center of interest is J(s, c) ⊕ J(q, f) = J(s) ∩ J(q). It means that anything that relates to students and qualifications decomposes into the specific course the student is doing and the qualification and its awarding faculty. This follows from links between courses and departments and departments and faculties (these come from “has-to-have” rules: a course has to be managed by a department and a department has to be in a faculty.)

In the ﬁgure the smallest ideal is J(Enrol(s, c, d, q, f)). What are the minimal principal ideals? T These would be J(xi) 6= ∅ where X cannot be increased. xi∈X⊂X(S) ¯ ¯ Proposition 2.5.7. If R¯ = R1#R2#...#Rn gives an minimal ideal J(R¯) (so it has no proper sub-ideals), then

1. R¯ is a maximal coupling.

2. the annihilator of J(R¯) is the compliment of J(R¯) as a subset of C(L). ¯ Proof. For R¯ = R1#R2#...#Rn to give a minimal principal ideal any r ∈ C(L) that is not a sub- ¯ ¯ ¯ expression of R1#R2#...#Rn gives r#R¯ = 0. Otherwise J(r#R¯) would be a sub-ideal of J(R¯). Consequently R¯ is a maximal coupling. Furthermore, an arbitrary r that is not in a sub-coupling R¯ will contain other domain variables not in the domain of R¯ and, as above, gives r#R¯ = 0. Thus the annihilator of J(R¯) is the set of relations not contain in J(R¯).

36 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

J(C(L)) is ﬁnite so the set of principal ideals is ﬁnite hence there will always be minimal principal ideals. They are the most specialized reports that gather up all related information about some subject. Of course they not need actually occur in a given system as an existing report or display but they can be created if the need is there. The set of minimal principal ideals partition data- types into coupled clusters. These maximal sets of coupled data-types are a distinctive functorial property of L that come from the lattice of ideals.

2.5.3 Ideals of Nested Relations

Nested relations connect a number of relations. This complicates the way their principal ideals are related. Given R(R1,R2, ..., Rm, ~z), for each domain relation Ri, i = 1, 2, ..., m, dom(Ri) ( dom(R) so J(R) is necessarily contained in J(Ri) yet, because R might have many conditions, its overall couplings with relations that share its domain can be different from the individual unconstrained

Ri. For example, Ri might be a set of parameters such as hourly rates for a types of specialist work only some of which are relevant to defining R. Indeed, the relations of the ideals J(R) and J(Ri) can reflect the way R filters data-type values into included and discarded values. The values that are not used can be characterized by relations R˜ ∈ Ann(J(Ri)) that do not annihilate J(R).

From this we decompose J(R) in terms of J(Ri) ∩ J(R) and Ann(J(Ri) ∩ J(R)) over all the ˘ domain relations Ri. Deﬁne Ri[+1] to contain the sub-relations of Ri that are to be included in R ˘ and Ri[−1] are those that are excluded.

Theorem 2.5.4. Given R(R1,R2, ...Rm) as above, then

˘ J(R) = ⊕ Ri[±1] i[±1],i=1,2...m where

˘ Ri[−1] = Ann(J(Ri)) ∩ J(R) and ˘ Ri[+1] = J(Ri) ∩ J(R)

Proof. If R¯ is in J(R), R¯#R ∈ J(R) has a “projection” R¯#Ri in J(Ri). If this is zero R¯ ∈ Ann(J(Ri))∩ ˘ ¯ ˘ J(R) = Ri[−1]. If not, then R ∈ J(Ri) ∩ J(R) is non empty and is Ri[+1].

The effect of a function

As mention in Section 2.3, functions can be represented as nested relations. A payroll calculation can be represented as a nested relation R that takes input relations R1, the employee contract rates for types of work, R2, the employees’ paid tasks and times for the pay period, R3, the tax table, R4,

37 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

the employees’ superannuation deduction, R5, the employees’ special allowances or deductions,

R6 the employees’ contracted holiday loading, and then calculates each employee’s gross and nett pay and other required numbers for the pay period. Theorem 2.5.4 also applies to functions as relations connecting “incoming” relations (such as tables or other structured data) and output data. Output data is always a relation by virtue of having been sifted or calculated from the inputs. Writing this as

Rfunction(Input(R1,R2, ...Ri); Ri+1,Ri+2, ..., Rm) the decomposition of Rfunction gives a speciﬁcation of the function as a mechanism for building relations. In this high level perspective, the “effect” of the function Rfunction is what couples to its creations (including updates) so the effect of Rfunction is

J(Ri+1) ⊕ J(Ri+2) ⊕ ... ⊕ J(Rm).

The details of ideals as contained in the lattice J(C(L)) perhaps come closest to a “global overview” of the information system implementation of a reporting structure S. The term “global overview” was coined by Fred Brooks who was the project leader for the IBM 360r operating system, the OS360r, the biggest software project at its time (mid 1960s). Brooks coined the phrase “intricately interlocked software elephant”.

“... software is very difficult to visualize. Whether we diagram control flow, variable scope nesting, variable cross references, data flow, hierarchical data structures, or whatever, we feel only one dimension of the intricately interlocked software elephant” [Brooks, 1986, p. 4].

In the same article Brooks comes to a conceptual gap.

. . . If we superimpose all the diagrams generated by the many relevant views, it is difﬁcult to extract any global overview. ” [Brooks, 1986, p. 10]

Such a global overview must summarize the many other views or diagramming representations so must be more abstract than the individual classes of diagrams. It must also contain sufﬁ- cient information to deduce important aspects conveyed by the various classes of diagrams. This suggests that something of a global overview emerges from the ring C(L) and the lattice J(C(L)). The global overview that Brooks wanted for computer systems applies just as well to their outcome: their reporting structures.

38 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

The Krull dimension of C(L)

The Krull dimension of C(L) is the length of longest chain of prime ideals [Hartshorne, 1977, p. 6].

Call a relation R(x1, x2, ...xn) prime if the ideal J(R(x1, x2, ...xn)) is a prime ideal. A prime

R(x1, x2, ...xn) is not a pullback or coupling of relations with fewer variables as such individual smaller relations will decompose J(R(x1, x2, ...xn)) into direct sums. The Krull dimension therefore gives the maximum tower of prime relations. Prime relations are the “givens;” they are definitions in L or distinguish subsystems that are particularly tightly woven set of conditions. For example, if a qualification q is defined in terms of a combination of a minimum of ni subjects in class Ci then Qual(n1,C1, n2,C2, ...n2,Cn, q) is given as a definition of the qualification. The values in Qual(n1,C1, n2,C2, ...n2,Cn, q) might be different for different students but they must reach the minimums. This type of relation occurs in numerous places from the definition of assemblies in manufacturing (then called a specification) to allowable compositions of freight, to the mix of staffing on a shift with a requirement for a mix of skills. The Krull dimension is a measure of the existence of complicated definitions in L. It indicates the level of layered concepts that occur in the business of L.

¯ 0 0 0 Deﬁnition 9. R(R1,R2,...,Rm, y) deﬁnes y if for every R with y ∈ dom(R ), R → P(y) factors through R¯.

That is we can factorize:

} R¯

! P(y).

Lemma 2.5.5. J(y) = J(R¯).

Proof. J(y) ⊇ {R0 | {y} ⊂ dom(R0)}. R¯ ∈ J(y) therefore J(R¯) ⊆ J(y) as anything that couples with R¯, couples through R¯ to y. J(y) ⊆ J(R¯). If R0 ∈ J(y) it is coupled to some R00 with y ∈ dom(R00). By the factorization of 00 00 ¯ 00 ¯ 0 ¯ ¯ R → P(y), R maps to R and so R couples with R. Hence R couples with R and so is in J(R) giving J(y) ⊆ J(R¯). Therefore J(y) = J(R¯).

Proposition 2.5.8. R¯(R1,R2,...,Rm, y) deﬁnes y then J(R¯) is a prime ideal.

39 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.5. THE COUPLING RING

R1 R2 ... Rn

Figure 2.3: The structure of a nested relation

Proof. J(y) is a prime ideal as, if R1#R2 ∈ J(y) implies one of the factors must contain y. Hence J(R¯) is also prime.

Corollary 2.5.6. J(y) is not a maximal ideal.

T Proof. J(y) = J(R¯(R1,R2,...,Rm, y)) and J(R¯(R1,R2,...,Rm, y)) ⊆ J(x) which x∈dom(Ri),i=1,2,...m is not maximal.

Deﬁnition 10. Height of a data-type.

1. If x ∈ D(L) and J(x) is a maximal ideal then ht(x) = 0.

2. Deﬁne the height of y in general as:

ht(y) = min {max{ht(x) | x ∈ (dom(R) \{y})} + 1}. R∈L∧y∈dom(R)

The reason behind this definition goes as follows. If we take R(x1, x2, . . . , xn, y) and for all 0 0 0 R ∈ L,(dom(R ) ⊆ {x1, x2, . . . , xn}) and R → R (entails) then R is the first relation to mention or define y in R/L. In this subsystem it would have a height larger than any of the {x1, x2, . . . , xn}. We then have to check that no other relation (or function) defines y with less height.

Suppose R(R1,R2,...Rn, ~z) deﬁnes the selection of relations Ri determined by the values of ~z.

If ~z has one of a number of possible values then choose Ri. For example, S is a logistics subsidiary of a infrastructure company and R is a decision system that takes ~z, a set of environmental and transport parameters, and decides what mix of transport options is best for the project. These are represented by relations Ri in the ﬁgure 2.3.

The queries to R are selections of values of ~z that can return one or a number of the Ri along with a sub-query of the type “for the following values of ~z do the chosen Ri satisfy the sub-query

... ?”. R is in J(~z) and also in each J(Ri) for which ~z is a deﬁning criteria for R to be Ri. Hence:

J(R) ⊆ J(~z) ∩ ⊕ J(Ri). i=1,2,...n

40 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.6. THE COUPLING RING AND INTEGRATION OF L

Proposition 2.5.9. If all queries to R must include environmental parameters and each of the Ri is a prime relation then J(R) is prime ideal.

Proof. Given R¯1, R¯2 with R¯1#R¯2 ∈ J(R). R¯1#R¯2 ∈ J(R) has to correspond to R#r for some r that has to select a set of ~z values that would select Ri and query the properties of Ri. That is r has the form: W r(~z) ∧ r |Ri . i where r(~z) selects the values from ~z and r |Ri is a constraint deﬁning a property that could be true ¯ ¯ of Ri. If Ri does have this property then r |Ri ∈ J(Ri). Thus, if R1#R2 = R#r has a component or ¯ ¯ 0 0 factor r |Ri (in J(Ri)) and so R1#R2 = r #r |Ri for some r that selects parameters relevant to Ri. ¯ ¯ ¯ ¯ Furthermore, if r |Ri belongs to one of the factors R1 or R2, say R1, so R1 becomes a coupling with 0 ¯ r |Ri and r which intersects R and so R1 is in J(Ri) ∩ J(R) ⊂ J(R). That is, J(R) is prime.

If one of the Ri has a diagram similar to R then that increases the depth of prime ideals. Hence high Krull dimension indicates recursive deﬁnition of products or expertise or assessment subsystems through many levels of components. This also suggests the existence of high-value products or services in L.

Corollary 2.5.7. The Krull dimension is at least as large as the greatest height of any data-type.

Proof. Follows from proposition 2.5.8

Higher-level relations R(R1(~x1),R2(~x2), ...Rm(~xm), ~z) can also play a major role in coordinating events and changes in L as they connect all the data-types in dom(Ri(~xi))∪~z over all i although ¯ these might be a small fraction of the total set D(L). When relations such as R¯(R1,R2, ...Rm, ~z) are prime and play a role as definitions we can see how heterogeneous the concepts of L can be. In the case of information systems, configurators (as mentioned in 2.3) act as complex definitions of what combination of options a product might have, or, perhaps more familiarly, what collections of courses constitute a qualification. More generally, the functions that encode these complex definitions act as axioms that define the legitimate concepts of L.

2.6 The Coupling Ring and Integration of L

The term “integration” and “integrated system” (for example in Hanseth and Ciborra [2007], Gorod et al. [2015]) and in Vernadat [1996]) is often used when describing information systems, particularly database systems, sometimes without clear deﬁnition. Here we present a deﬁnition

41 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.6. THE COUPLING RING AND INTEGRATION OF L that applies to reporting structures that might be called algebraic integration that uses the coupling ring to see integration in terms of the propagation of data-changes. The deﬁnition of L requires that every data-type is used in a relation. Therefore there must be smallest sub-ring of A ⊂ C(L) generated by a set of relations W for which dom(A) = S dom(R) = R∈A D(L). Here we are taking all sums and couplings of the relations in W to get A. A might be generated by a small number of relations, but they are coupled to everything else in L. [n1] [n2] [nm] Suppose W is the set {R1 ,R2 ,...,Rm } of prime relations. Let Im be an ideal

[n1] [n2] [nm] J(R1 ) ⊕ J(R2 ) ⊕ · · · ⊕ J(Rm ), which is maximal for a set of m relations: adding more relations does not give a larger ideal.

Suppose also that Im does not have an annihilator. This means every relation is coupled to at least [n1] [n2] [nm] one of the generating set {R1 ,R2 ,...,Rm }. The existence of Im without an annihilator is a necessary condition for “a single source of truth”: all relations are coupled in some way through Im and changes in data in the relations of W are propagated throughout the system. The system is not a set of isolated sources of information.

What is left out of the maximal ideal Im is the group of units of C(L) and this has no annihilator. From this we deﬁne

Deﬁnition 11. L is m-integrated if there exists a set of m prime relations that generate a maximal ideal Im that has zero as its annihilator. Furthermore there is no smaller set of relations.

By this definition m is the “degree" (or “rank”) of integration. Im need not be unique. ϕ How does the m change with L1 −→L2? If ϕ is an embedding and L1 is m-integrated and L2 is m0-integrated then nothing can be said about m0. It might even be smaller than m due to some large relations outside the image of L1. Assume that ϕ is an epimorphism in Sys and so an onto function in terms of objects and morphisms. C(L2) can be smaller than C(L1) as identification of data-types can cause the identification of relations. However, it is also possible that identification of data-types does not cause any changes in relations. For example, an epimorphism ϕ : L1 L2 can occur when L2 is a restriction of L1 defined to deal with a reduced range of data and so does not express all the distinctions built into L1.

ϕ 0 Proposition 2.6.1. If L1 −→L2 is an epimorphism in Sys and L1 is m integrated then L2 is m integrated for some m0 ≤ m. ϕ Proof. Let L1 −→L2 be an epimorphism then, by the proposition 2.5.5 and comments following, [n1] [n2] [nm] prime ideals in C(L1) map to prime ideals in C(L2). Let Im = J(R1 ,R2 ,...,Rm ) be the [ni] integrating ideal of L1 then φ(Ri ), i = 1, 2, . . . , m are prime ideals in C(L2).

42 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.6. THE COUPLING RING AND INTEGRATION OF L

−1 ϕ −1 J(L2) −−→ J(L1) so if ϕ(Im) is not maximal there is a ideal J˜ ⊃ ϕ(Im). Then ϕ (J˜) ⊃ Im which is a contradiction. Thus ϕ(Im)) is maximal in L2.

Suppose R˜ in L2 annihilates ϕ(Im) so J(R˜) is contained in the possible annihilator of ϕ(Im). 0 −1 0 But if R ∈ ϕ (R˜), R cannot annihilate everything in Im which has no annihilator, therefore [nk] 0 [nk] 0 [nk] ∃Rk and R #Rk 6= 0 and ϕ(R #Rk ) 6= 0 so giving a contradiction by lemma 2.5.1. Thus 0 ϕ(Im) is an integrating ideal for L2 and m is then the number of distinct images in

[n1] [n2] [nm] {ϕ(R1 ), ϕ(R2 ), . . . , ϕ(Rm )} which cannot exceed m.

0 [nj ] [nk] [nj ] [nk] The difference m − m is signiﬁcant as it means that ∃Rj ,Rk and ϕ(Rj ) = ϕ(Rk ). This can occur when we have

R[nj ](x , x , . . . , x ) 6= R[nk](x , x , . . . , x ) j j1 j2 jnj k k1 k2 knk and most of the variables are the same but with at least one variable being different. ϕ identiﬁes the different variables which might be conﬁguration variables parameterizing the conditions where [nj ] [nk] Rj 6= Rk . These conditions do not occur in L2. Where these special conditions do not apply we identify the different data-types to get L2. [n1] [n2] [nm] This leads to questions of how separated is the set {R1 ,R2 ,...,Rm }? Are there concepts of integration that require relations to be from different classes? This will be taken up in Chapter 5.

2.6.1 The Application of C(L) to Testing

In section 2.5.3 I suggested that the coupling ring and its associated lattice of ideals make a possible candidate for Fred Brooks’ global overview. These functors also have a role to play in defining a concept of integration so we should expect some significant benefits. If the calculation of the coupling ring was part of the documentation during system development it would transform the planning of system-wide testing or integration testing. This in not just testing of new software, these are considerations that are important in formulating policies and thinking through what needs reporting and how changes flow through the system. In large- scale enterprises this means changes in the information system that are to be checked by testing specialists. Typically, testing staff do not have the time to test everything and adopt an approach known as Risk-based testing [Redmill, 2004, Gerrard and Thompson, 2002, Felderer, Haisjackl, Pekar, and Breu, 2015]. Risk is rather crudely estimated and estimation must assume a very good understanding of the connections, or in our case, the couplings of the system. If the coupling ring

43 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS of the specification is available, test planning can proceed to analyze propagation of high risk areas "outwards" by tracing the overlaps of ideals. Testing often proceeds by testing “scenarios” or “use cases”. These terms are used by information system test professionals to indicate specific tests of the system (see [Kaner, 2003]). Suppose we want to test a dependency R1 → R0 and R0 encompasses several logical variations that affect dependencies that follow from those variations. The logical variations are inconsequential unless, 0 0 as specialized cases of R0, they can be defined as R0#R where R allows no further specialization, 0 effectively partitioning R0 into sub-cases. At the most detailed level of sub-cases J(R ) is minimal. 0 0 In this way J(R0) = ⊕J(Rj): every coupling with R0 will couple with one of the specializations R1, j 0 0 0 0 R2 ...Rj ... . These couplings specialize the implication R1 → R0 to the cases R1#Rj → R0#R 0 0 0 where R1#Rj is the obvious specialization of R1. If R1#Rj is zero then Rj is not a special case 0 that affects R1. Not all cases need pullback to R1 but some will by virtue of J(R0) = ⊕J(Rj). j As ideals can be thought of as a class of queries and scenarios can often represent the correct execution of queries we can identify classes of scenarios (collected into a set of test cases) with the couplings in an ideal. Access to a (“user-friendly”) version of the coupling ring would aid both the risk assessment and the consequent test planning. Graphical representations are much more limited in such cases.

2.7 Properties of Relational Translations

2.7.1 Introduction: Comparing Reporting Structures

Comparing systems occurs all the time as companies seek to replace some or all of their existing systems. Requirements, in effect specifications for reporting structures, themselves the specifications for the systems, are drawn up for the replacement system. These are then submitted to vendors with packages or software capabilities that might satisfy the requirements. (Much is at stake in these replacements and they can stumble on crucial areas of logic that present problems in transferring data from the older system to its replacements. A number of case studies of the such problems are documented in [Hanseth and Ciborra, 2007].) In this section we apply the concepts developed above to comparisons of reporting structures. In our context, reporting structures are comparable when there is a relational translation ϕ : L → L0. As this is, first of all, a mapping between sets of relations it factorizes as

ψ L / / L00 / incl / L0.

44 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

ϕ = incl ◦ ψ so L00 will be the subsystem of L0 that has all the modiﬁcations of L. All the important aspects of this map are in the epimorphism ψ. L00 will be the focus of data migration problems and the consequent testing. This section will examine how epimorphisms depend on the logic of L and the way they simplify that logic by reducing the number of data-types. Each reduction of data-types generalizes a set of data-types in D(L) to a single more abstract data-type in D(L0). This is a necessary step towards the reduction monoid, a contravariant functor to be introduced in the next section. The word “simplifying” has too many meanings in the context of reporting structures so, as 0 we shall be concentrating on epimorphisms ϕ : L L , we shall refer to such maps as reductions. Reductions reduce the capability to see differences deﬁned in L but invisible in L0. This section is a set of results that give meaning to the introduction of the reduction monoid. These results are summarized in the sections below as follows.

1. Section 2.7.2 looks at the structures that stop an arbitrary mapping from being a relational translation. Relational translations impose a number of conditions on an arbitrary mapping 0 0 f : L L and when f fails a condition we have an “obstruction” to f : L L .

0 2. A relational translation ϕ : L L has an associated set of equivalence classes in L which are indexed by a set of variables of L0. Section 2.7.3 introduces the concepts that will be used throughout the thesis.

0 3. If ϕ : L L gives rise to equivalence classes what happens when we have two or more epimorphisms? Are their associated equivalence classes always compatible? This is the subject of Section 2.7.4. This section also investigates when locally deﬁned relational translations can be patched together to form an overall morphism. This is straightforward when the local sub-systems are well separated. If there is an interface, so a relation shared between them, additional constraints are needed. Finally, incompatible reductions are deﬁned in Section 2.7.4 and characterized in terms of annihilating ideals.

4. Section 2.7.5 illustrates the concept of a reduction with a geometrical picture of an epimor- 0 phism ϕ : L L .

These topics are interwoven with ideals and demonstrate that the coupling ring is also the fundamental tool for understanding relational translations and simpliﬁed representations of reporting structures.

45 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

2.7.2 Obstructions to Relational Translations

0 What stops us creating relational translations ϕ : L L ? The properties of mappings are frequently clarified by what they cannot do; what violates their defined properties. This is significant when two reporting structures are compared in the expectation that there is a simple mapping that will guide translation of concepts. When no relational translation can be constructed there is no simple way to reconnect data and relations and concepts might end up being redefined. The term we use when we can prove there is no such mapping is an obstruction, a term originally used in algebraic topology [Spanier, 1966, Ch. 9]. The following definition recalls the comment in Section 2.4 that relational translations, and so reporting structure maps, are “conservative” and do not create new relations nor new implications.

0 Deﬁnition 12. An obstruction to a proposed Sys epimorphism ϕ : L L occurs when there is no map 0 R1 → R2 in L but ϕ(R1) → ϕ(R2) would exist in L .

Example 1

Suppose R(R1,R2, ...Rm) is an assembly of items, each one of which is, itself, a relation of a set of components. For example, R can be a car engine and R1 is the relation "necessary starter motor components" defining the components of the starter motor. Suppose we assume that the components or data-types needed to define each Ri are not used outside the definition of Ri. We seek ϕ 0 0 a relational translation L −→L that can reduce R(R1,R2, ...Rm) to a single data-type ξ ∈ D(L ): 0 00 ϕ(R) = P(ξ). This is prevented, (obstructed) by the existence of morphisms R → Ri or Rj → R in L,(Ri,Rj ∈ dom(R),) for which there is no corresponding morphisms to or from ϕ(R). This can be represented by the following diagram.

0 R / Ri

00 Rj / R

( o dom(R) / )

0 0 where dom(Ri) ∩ dom(Rj) = ∅. If ϕ : L L identiﬁes R(R1,R2, ...Rm) to ξ ∈ D(L ) then it produces maps ϕ(R0) to ϕ(R00) that have no corresponding morphisms in L. This can also force new couplings in L0 which is also an obstruction by the following results.

f C(f) Proposition 2.7.1. Given a function L −→L0 for which C(L) −−→ C(L0) is not a homomorphism then f is not a relational translation.

46 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

Proof. This is the “modus tollens” version of C being a functor of the category Sys.

0 Corollary 2.7.1. 1. If R1#R2 = 0 in C(L) and f(R1)#f(R2) 6= 0 in C(L ) then f cannot be an onto f relational translation L −→L0.

f 0 0 2. Given L −→L and J(R1) ∩ J(R2) = {0} in C(L) and there exists a R¯ in L with J(f(R1)) ∩

J(f(R2)) = J(f(R1)) ⊕ J(R¯) ⊕ J(f(R2)) then f cannot be an onto relational translation.

Proof. 1. Immediate

0 2. J(f(R1)) ∩ J(f(R2)) is non-empty then there exists a R¯ ∈ L with R¯ ∈ J(f(R1)) and

R¯ ∈ J(f(R2)) coupling with both f(R1) and f(R2); R¯#f(R1) 6= 0 and R¯#f(R2) 6= 0 and

so f(R1)#R¯#f(R2) 6= 0 but this has no corresponding coupling in C(L) so contradicting proposition 2.7.1.

We use these results to rule out possible epimorphisms. Example 2.

Suppose a reporting structure has a number of data-types X = {x1, x2, ...xn} that are part of the deﬁnition of the options of a product. They are connected in many ways so that ∩ J(xi) 6= ∅. x∈X Can they be mapped to a single partitioned data-type x in a new or modiﬁed system L0 with:

P(x) ⇔ t P(x )? i=1,2,...n i

ϕ 0 This requires a relational translation L −→L where ϕ(xi) = x. Such a ϕ maps (R(xi, ~y) → P(xi)) in

L to ϕ(R)(x, ϕ(~y)) → P(x). Suppose

∀xi, xj ∈ X (xi 6= xj) ⇒ (R(xi, ~y) ⇒ ¬R(xj, ~y)).

Following Proposition (2.5.3) we then have R(xi, ~y)#(R(xj, ~y) = 0. Any relational translation making ϕ(xi) = ϕ(xj) = x creates a coupling

ϕ(R)(ϕ(xi), ϕ(~y))#ϕ(R)(ϕ(xi), ϕ(~y)) 6= 0 in violation of proposition (2.7.1).

A simple example of P(x) ⇔ t P(x ) with n = 2 is x is a “person” (student, customer, pa- i=1,2,...n i tient, client) which has two cases: male or female. If ϕ drops the distinction then other distinctions, say mother/father, sister/brother, aunt / uncle, are dropped as well.

47 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

2.7.3 Epimorphic Relational Translations and Equivalence Classes ϕ Every epimorphism L −→L0 that is not an isomorphism reduces the information contained in L0 in comparison to that in L. The reduction is associated with the number of logically independent equivalence classes stemming from the map ϕ : D(L) → D(L0). We write #X for the cardinality of a set X.

ϕ Deﬁnition 13. Support, cosupport and reduced objects of a relational translation L −→L0.

1. The support of a relational translation:

−1 spt(ϕ) =defn {x ∈ D(L) | #ϕ (ϕ(x)) > 1)}.

2. The cosupport of a relational translation:

0 −1 cospt(ϕ) =defn {y ∈ D(L ) | #ϕ ({y}) > 1}.

3. The reduced objects. Given numbers n and m and ~x = (x1, x2, ...xn), ~y = (y1, y2, ...ym) with m < n,

0 0 0 Z(ϕ) =defn {R(~x) | ∃R (~y) ∈ L , (ϕ(R(~x)) = R (~y)) and (m < n)}

Deﬁne x1 ∼ϕ x2 ⇔ ϕ(x1) = ϕ(x2). This is an equivalence relation. Denote the ∼ϕ classes be Eq(ϕ). Equivalence classes among the data-types D(L) form a lattice with the least upper bound

E1 ∨ E2 corresponding to x1 ∼E1∨E2 x2 ⇔ (x1 ∼E1 x2 or x1 ∼E2 x2). Similarly, the greatest lower bound is x1 ∼E1∧E2 x2 ⇔ (x1 ∼E1 x2 and x1 ∼E2 x2). Eq(ϕ) is the indexing of spt(ϕ) by cospt(ϕ). The support and reduced objects are related as follows:

Proposition 2.7.2. If ϕ is an epimorphism, the set Z(ϕ) is the set of relations of data-types that intersect spt(ϕ).

0 Proof. Suppose R(x1, ..., xn) ∈ Z(ϕ) such that ϕ(R)(ϕ(x1), ϕ(x2), ..., ϕ(xn)) = R (y1, ...ym), m < n.

Then at least two elements of {x1, ..., xn } must belong to spt(ϕ). That is {x1, ..., xn } ∩ spt(ϕ) 6= ∅.

Corollary 2.7.2. Z(ϕ) ⊂ J(spt(ϕ)) =defn ⊕ J(x). x∈spt(ϕ)

Proposition 2.7.3. Given y1 and y2 in cospt(ϕ), if J(y1) ∩ J(y2) = {0} then the domains of the inverses are disjoint.

48 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

−1 −1 −1 Proof. We want to prove dom(ϕ (J(y1)) ∩ dom(ϕ (J(y2)) = ∅ where dom(ϕ (J(yi)) =defn −1 −1 −1 ϕ ({dom(R˜) | R˜ ∈ J(yi)}). Suppose to the contrary that ∃x ∈ dom(ϕ (J(y1)) ∩ dom(ϕ (J(y2)) then ϕ(x) is in the domain of a relation that is in J(y1) and J(y2) and so J(y1) ∩ J(y2) 6= ∅.

The proposition allows us to partition cospt(ϕ) into disjoint subsets {Y1,Y2, ...Ym}. The set of reduced objects that ϕ maps to Yk is denoted Z(Yk)(ϕ).

Corollary 2.7.3. J(Z(ϕ)) = ⊕ J(Z(Yk))(ϕ)). Yk,k=1,2,...,m

Proof. The direct sum is any coupling that is in one of the summands.

2.7.4 The Reduction of Separated Subcategories of L

In this section we consider a relational landscape L containing disjoint subcategories each of which have a “local” reduction to a relational landscape that is hoped to be a reduction of L. This is a common situation in which we have a number of reductions that are well deﬁned on subcategories that we take as evidence for an overall reduction. This is often prompted by the consideration of possible new reporting structures being claimed to “do everything the old reporting structure does”. The old structure is to be a reduction of a new structure (which anticipates the subject of Chapter 4). What are the conditions on interfaces or common relation, between pairs of subcategories so these separate reductions add up to a reduction of L?

Given ϕk : L → Lk, k = 1, 2 such that in L1 the data-types {x1, x2, x3, z } ⊂ D(L) are identiﬁed with ϕ(z). In L2, z plays a role distinct from {x1, x2, x3 } and x2 is identiﬁed with y1, y2 from D(L). A combination of equivalence classes means that data-types are forced into playing contrary roles.

For example, there is a class of products in which one option can have any of {x1, x2, x3, z } with one (only) of either y1 or y2; another has options any of {x2, y1, y2 } with one only of x1, x3 and z separate but there is no option allowing any ﬁve of {x1, x2, x3, z, y1, y2 }. A relational translation which forces such equivalence, such indifference to options, ends up contradicting a business (or physical or sociological) rule. Whether we can combine the two relational translations ϕk : L → Lk and so combine the equivalence classes Eq(ϕ1) and Eq(ϕ2) is equivalent to the existence of a pushout of ϕ1 and ϕ2:

ϕ1 L / L1

ϕ2 00 L2 / L .

49 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

When the pushout of the two maps ϕ1, ϕ2 is not admissible the pushout will be labeled as ω. In such a case "anything goes," and the reduction will be deemed “trivial”. A colimit (pushout) is trivial if it collapses all distinctions or forces the combination of equivalence relations that contradict other properties of the system that must be maintained.

Proposition 2.7.4. If, for i, j = 1, 2, ..n, J(spt(ϕi)) ∩ J(spt(ϕj)) = {0} then the colimit of the ϕi which is a mutual pushout is non trivial (not ω).

Proof. The areas where each ϕi is not the identity on D(L) do not intersect with those of any of the other ϕj. Consequently, for those data-types that are the equivalence classes created by each ϕi there is no overlap. The collective effect of the equivalences class is Eq(Colimit(ϕi)) = S i Eq(ϕi).

Given Li ( L we write L1 ∩ L2 = ∅ if and only if there is no relation R(x1, x2) ∈ L with ϕi 0 xi ∈ D(Li)) and so no common relation between L1 and L2. Let L −→L be epimorphisms deﬁned by ψi on Li and the identity outside Li. We can write this as ϕi = 1L1\Li t ψi. Under what 0 conditions do we have a single ϕ = 1L\(L1tL2) t ψ1 t ψ2 : L L combining both ϕi?

ϕ 0 Corollary 2.7.4. : If L −→L with ϕ given by 1L\(L1tL2) tψ1 tψ2 then ϕ is not trivial if J(L1)∩J(L2) =

( ⊕ J(P(x))) ∩ ( ⊕ J(P(y))) = {0} x∈D(L1) y∈D(L2)

Proof. This is a sufﬁcient condition and is little more than a recipe for deﬁning ϕ when there is no interface. It follows from proposition 2.7.4 above.

An interface between two subcategories L1 , L2 ( L exists when there is at least one relation R(~x,~y) with ~x ⊂ D(L1), ~y ⊂ D(L2). We note the necessary condition.

0 Proposition 2.7.5. If ψi : {xi1 , xi2 , ...xn1 } ⊂ D(L) 7→ ξi ∈ D(L ), i = 1, 2, there are data-types x1p = x2q and ϕ is non trivial, then there is an interface between L1 and L2.

Proof. Assume there is no common relation linking L1 and L2. ψi(x1p ) = ξ1 = ψi(x1q ) = ξ2 so

ϕ(x1p ) = 1L\(L1tL2) t ψ1 t ψ2(x1p ) = ϕ(x2q ). The deﬁnition of relational translations implies there is a relation R(xip , xiq ). R(xip , xiq ) acts as an interface between the two sub categories so L1∩L2 6= ∅ contradicting the assumption.

Example. x1, x2 ∈ D(L1) are two classes of certiﬁcates for machine maintenance or operation, typically the concern of human resources, and y1, y2 ∈ D(L2) the corresponding machines of concern in operations and maintenance. If ψ1(x1) = ψ1(x2), L1 would not be able to claim quality assurance certiﬁcation in those machines which might be crucial for operating oil wells or chemical

50 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

factories. Suppose now that y1 and y2 are related to certiﬁcation of products w1 and w2 in D(L1) that remain apart in L1. Merging x1 and x2 can be done without merging w1 and w2. But the requirement to distinguish between w1 and w2 can only be made when y1 and y2 are different.

Expressing this in terms of ideals: suppose J(x1) and J(x2) can be merged without forcing

J(w1) = J(w2). In this case J(w1) ∩ J(x1) ∩ J(x2) 6= J(w2) ∩ J(x1) ∩ J(x2): there are couplings containing w1 and both x data-types that are different from those containing w2. If J(w1) ∩ J(y1) ∩

J(y2) = J(w2)∩J(y1)∩J(y2) then every coupling containing w1, y1 and y2 is also one that includes w2, y1 and y2.

When does an interface block the creation of relational translations such as 1L\(L1tL2) tψ1 tψ2? In the case above, the existence of certiﬁcations for types of work does not actually affect any states in operations and maintenance (our L2) and R(~x,~y) only amounts to (say) xi pertains to yi. This allows the separate reductions of the subcategories L1 and L2. Suppose instead R(~x,~y) forces the simultaneous reduction of the subcategories L1 and L2 in the case when operations requires the existence of separate certiﬁcations when planning operations and maintenance. We can see this as R(~x,~y) = R(x1, x2, ...xn; y1, y2, ....ym), but to emphasize the functional aspect we require that this relation can be seen as a collection of functions ri,j(Xi,Yj) where Xi ⊂ {x1, x2, ...xn} and

Yj ⊂ {y1, y2, ....ym}. This is part of a general problem of what happens when the ability to discriminate among classes of data-types depends on data-types in a subcategory L¯ ⊂ L and L¯ includes spt(φ) for 0 φ : L L . The next proposition makes this precise.

¯ 0 Proposition 2.7.6. Given L ⊂ L that includes spt(φ) for φ : L L and φ satisﬁes the following conditions:

0 1. φ = 1L\L¯ t ψ : L L ,

2. {x1, x2, ...xn} ⊆ spt(ψ) ⊂ D(L),

3. {y1, y2, ....ym} ⊂ D(L) \ D(L¯),

4. a set of functions ri,j(Xi,Yj) where Xi ⊂ {x1, x2, ...xn} and Yj ⊂ {y1, y2, ....ym},(not necessarily injective and not onto) in which the values of the y data-types are determined by x data-types.

Then the image of φ(Y ) is indexed by cospt(ψ) ∩ ψ(L¯).

The last sentence says that the image of {x1, x2, ...xn} under ψ maps to the set of elements of the rest of the image of L\ L¯. The proposition can be interpreted as the way a reduction alters the “diffusion of logic”. Without knowing how information in L¯ determines values in other parts of the system one really does not know how far a “local” reduction (as given by ψ) really extends.

51 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

0 Proof. {x1, x2, ...xn} ⊆ spt(ψ), ψ : {x1, x2, ...xn} 7→ {η1, η2, ...ηp} in L where p ≤ m/2. Let Xi =

{xi1 , xi1 , ..., xn(i)} ⊂ {x1, x2, ..., xn} and Yj = {yj1 , yj2 , ...ym(j)},

φ : rij(Xi,Yj) 7→ φ(rij)(ψ(xi1 ), ψ(xi2 ), ..., ψ(xn(i)); yj1 , yj2 , ..., ym(j))

= φ(rij)(ηi1 , ηi2 , ..., ηq; yj1 , yj2 , ..., ym(j)) where q ≤ p < m/2 which reduces the discrimination of the xi by at least half. Given rij : ~xi 7→ ~yj, 0 0 0 0 and ri0j0 : ~xi 7→ ~yj, but ψ(~xi) = ψ(~xi) then the difference between ~yj and ~yj cannot be made in 0 L . Instead the differences given by the yj distinctions in φ(L\ L¯) are seen only in terms of their η classes in L0: ¯ yjk ∼η ylp ⇔ ∃φ(rrk): ~ηrl 7→ yjk , ylp in φ(L\ L) and where ~ηrl ∈ cospt(ψ)

This gives an indexing of each set Yj by the set cospt(ψ) ∩ φ(Xi).

0 The indexing of {y1, y2, ..., yn} by cospt(ψ) sets up a new indexing structure in L which is an abstraction of the relation R(x1, x2, ...xn; y1, y2, ...ym) that contains the functions rij. Translating this into the language of ideals we get.

Corollary 2.7.5. If X˜ ⊂ spt(ψ) and there exists a functional relation R : X˜ → L \ L¯ (so a relation that resolves into a set of functions) then

−1 φ (J(cospt(ψ))) ) J(spt(ψ)).

−1 Proof. As we have maps φ(rij): yj 7→ ηi in cospt(ψ) then rij(ψ ({ηj}, yj)) is a relation in J(spt(ψ)) −1 −1 so φ : J(cospt(ψ)) 7→ J(spt(ψ)) giving φ (J(cospt(ψ))) ) J(spt(ψ)).

Applying this to Li ( L; D(L1) ∩ D(L2) = ∅ and a set {y1, y2, ....ym} ⊂ L \ L2 which are dependent on {x1, x2, ...xn} ⊆ D(L1) for their values. The interface R(~x,~y) = R(x1, x2, ...xn; y1, y2, ....ym)), is then a set of functions ri,j(Xi,Yj) where Xi ⊂ {x1, x2, ...xn} and Yj ⊂ {y1, y2, ....ym}. 0 In this case we set L1 as L¯ and apply proposition (2.7.6). This shows that L can “accommo- date” this type of interface but distinctions in data-types outside L2 are indexed by equivalence classes not by single data-types.

Example: yj are states or milestones reach by student while {x1, x2, ...xn} are items of data that distinguish students and what counts for a given milestone in their qualiﬁcations.

52 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

Incompatible reductions

Proposition (2.7.4) gives conditions for combining reductions together. We now investigate when this should not happen, when two reductions ψ1 and ψ2 are “incompatible”.

Any data-type can belong to a range of equivalence classes. Suppose xi ∼ψ1 xj and xj ∼ψ2 xk that come from ψ1(xi) = ψ1(xj) and ψ2(xj) = ψ2(xk). If ψ1(xi) = ψ2(xj) then we can compose the equivalence relations and get xi ∼ψ2◦ψ1 xk. ∼ψ2◦ψ1 =∼ψ1◦ψ2 are the same if ψ1 and ψ2 are independent, both composite equivalence relations being the same as Eq(ψ1) ∨ Eq(ψ2). Incompatibility arises when this contradicts a business rule of L.

Deﬁnition 14. ψ1 and ψ2 are incompatible if there are data-types x ∈ spt(ψ1) and y ∈ spt(ψ2) with

ψ1(x) = ψ2(y) combining equivalence classes containing x and y and there exist relations R1(x, ~z) and 0 R2(y, ~w) with R1 ⇒ ¬R2 in L but ψ1(R1)(ψ1(x), ψ1(~z)) ∧ ψ2(R2)(ψ2(y), ψ2(~w)) is a proposition in L

0 0 That is ψ1(R1)(ψ1(x), ψ1(~z)), and ψ2(R2)(ψ2(y), ψ2(~w)) “coexist” in L making L an inconsistent representation of L.

From this deﬁnition, if ψ1 and ψ2 are incompatible as R1#R2 = 0 in L but ψ1(R1)#ψ2(R2) 6= 0 giving an obstruction. If ~z and ~w are logically independent and do not affect the coupling of the relations this is indeed an obstruction. Suppose R1 ⇒ ¬R2, equivalently ¬(R1 ∧R2); ψ1(x) = ψ2(y) implies that {x, y} ∈ spt(ψ1) ∩ spt(ψ2) and {ψ1(x), ψ2(y)} ∈ cospt(ψ1) ∩ cospt(ψ2). This means

J({x, y}) contains relations R1(x, ~z) and R2(y, ~w) so J({x, y}) splits into mutually annihilating sub-ideals and (perhaps) a residue. But L0 merges the mutually annihilating parts.

Proposition 2.7.7. Given X ⊂ D(L) and two ideals J1 and J2 contained in J(X) such that J1 and J2 mutually annihilate then there is a R0 with J1 ⊂ J(R0) and J2 ⊂ J(¬R0).

Proof. The proof is a generalization of the situation R1(x, y, t)#R2(y, z, t) = 0 because R1 is true only when t < 0 and R2 is true only when t > 0.

Let R1(~x,~z) ∈ J1 and R2(~y, ~w) ∈ J2 with ~x and ~y in X but ~v and ~w are not in X. Suppose

R1#R2 = 0. If ~v and ~w act as parameters that cause R1 and R2 to annihilate then there would have to be a subset of parameters common to all the relations in J1 and in J2 so there is a subset ~u ⊆ ~v ∩ ~w with the relations having the form:

R1 = R1(~x,~v \ ~u,R0(~u)) and

R2 = R2(~y, ~w \ ~u, ¬R0(~u)).

The negation can be a business rule or a logical rule but it covers all the relations in J1 and J2. Equally well the crucial set of data-types could be in ~u ⊆ ~x ∩ ~y to give

53 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

R1 = R1(~x \ ~u,~v, R0(~u)) and

R2 = R2(~y \ ~u,~w, ¬R0(~u)).

In either case we have J1(X) ⊆ J(R0) and J2(X) ⊆ J(¬R0).

Indeed, the existence of an R0 is a property which pinpoints reasons for such mutual annihila- tions.

Corollary 2.7.6. If J(spt(ψ1) ∩ spt(ψ2)), is not empty and J(spt(ψi)), i = 1, 2 mutually annihilate then

ψ1 and ψ2 are incompatible.

Proof. X = spt(ψ1) ∩ spt(ψ2) and Ji(spt(ψi), i = 1, 2. Then by proposition 2.7.7 these ideals are contained in either J(R0) or J(¬R0) and so couplings with R0 and ¬R0 are in the range of ψ1 and

ψ2 and so are incompatible.

2.7.5 A Geometric Picture

0 The results in Section 2.7 give us the algebraic properties of an epimorphism ϕ : L L . I now give a geometric picture of what a reduction can involve. In many cases ϕ also recalls the concept of a ﬁbration or covering space in topology (Spanier [1966]). If we think of L0 as a surface lying directly under L and ϕ a projection then every y ∈ cospt(ϕ) is a branching of L into #ϕ−1({y}) variations of L0 "in the neighbourhood" of y; that is in the domains of relations that contain y. By the corollary of proposition 2.7.7 the y ∈ cospt(ϕ) need to be appropriately separated. 0 0 0 Given ϕ : L L then every object P(y) or R in L can have many "versions" Pi and Rj in L that apply to the inverses ϕ−1(y), y ∈ D(L0). This is illustrated in Fig. 2.4 where the top three lines 00 0 are L and ϕ(~xi ) = ϕ(~xi) = ϕ(~xi) and ϕ is the identity on ~x5 and ~x6. To keep it simple the properties have been omitted but it is assumed that

00 00 / 00 Rk(~xk) Pi(xkj)

ϕ ϕ 00 / 00 Rk(ϕ(~xk)) Pi(ϕ(xkj)) = P(ykj ) where kj ranges over the components of the vector ~xk. The following diagram simpliﬁes this picture so that L can be considered in terms of local lay- erings of subcategories giving something similar to a covering space but with ﬁbers being different in different neighborhoods. This diagram indicates that L corresponds to two general versions of L0 but in certain areas of L0, L splits into 5 differently parametrized subsystems.

54 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.7. PROPERTIES OF RELATIONAL TRANSLATIONS

00 00 00 00 00 00 00 00 R (x~ 1) / R (x~ 2) / R (x~ 3) / R (x~ 4) 1 2 9 3 4

% % 0 0 0 0 0 L : R (~x2) / R (x~ 3) / R (x~ 4) 9 2 3 9 4

% # ϕ 0 R1(~x1) / R2(~x2) / R3(~x3) / R4(x~ 4) / R5(~x5) / R6(~x6)

0 L : R1(ϕ(~x1)) / R2(ϕ(~x2)) / R3(ϕ(~x3)) / R4(ϕ(~x4)) / R5(~x5) / R6(ϕ(~x6)

0 Figure 2.4: A picture of a reduction ϕ : L L 2 . / . / . 2 . / . ... / . / . / . /, . / . / . / . /, ... L ... / . / . / . 84/ . / . / . / . /74 ... φ * . / . / . * . / . , / & / . . . . L0 : ... / . / . / . / . / . / . / . / ...

Figure 2.5: A schematic of a more complicated reduction

L can be a reporting structure of a national organizations that has to have quality or safety certiﬁcations that allow permissible variations. In the following, we assume that these variations are parameterized by ~z = (z1, z2, ...zr). Some relations are standard, others have the form

R(x1, x2, ...xm, ~z) so that, if R(y1, y2, , ym) is the general standard underlying the possible variations R(x1, x2, ...xm, ~z), then the relational translation φ : R(x1, x2, ...xm, ~z) 7→ R(y1, , , ym) takes into account all variations of ~z and so φ reduces {x1, x2, ...xm, ~z} to {y1, y2, , ym}. cospt(ϕ) is then indexed by ~z. This section has given a number of results on the way reductions can be combined and what prevents them doing so. It has made extensive use of the ideals of the coupling ring so validating its use in the analysis of the way relational landscape is connected. This also reﬂects how reporting structures have multidimensional connective properties. In section 2.8 I bring all of the possible epimorphisms under one umbrella concept with the introduction of the reduction monoid. In 0 section 4.3, the study of embeddings: ρ : L L, becomes important. In this case the reduced version of L, L0, can be mapped into multiple versions inside L as suggested by Figure 2.4.

55 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

2.8 The Reduction Monoid

2.8.1 Preliminaries

The reduction monoid investigates the total set of epimorphisms of a given L. If an epimorphism 0 ϕ : L L is not one-to-one there will be data-types x1 and x2 for which ϕ(x1) = ϕ(x2). As ϕ has to preserve the logic, there is an implied equivalence class of data-types in L and L0 is the result of simplifying L by referring to the equivalence class not its elements. Equivalences classes themselves can have complicated definitions in terms of multiple parameters and often represent true insights produced by deep understanding of processes and data. For example, the conditions when two drug regimes are equally effective. In other cases, the introduction of new technology makes distinctions obsolete so that L0 represents streamlined version of L. Whatever their origin or expression, the existence of epimorphisms and their associated equivalence classes can be seen as a level of abstraction where individual cases are generalized to single class of items. The reduction monoid expresses the two sides of epimorphisms: abstraction and simplification, and is itself a sophisticated “measurement” of systems. This opens the prospect of concept discovery in a reporting structure by investigating the representation of L in other relational landscapes. The reduction monoid is also a quintessentially categorical construction as it starts with a left representable functor in a subcategory of Sys. These functors contain the entire set of maps to other objects: the entire set of ways one object can be represented in another object with fewer data-types. Such functors contain a wealth of analytic information. For this reason, the reduction monoid is much more difficult to calculate than the coupling ring. Yet interpretation of the reduction monoid relies on the results of section 2.7 so it is closely tied in with the coupling ring. Reductions are not designed to find statistical patterns in big data - abstractions that indicate correlations among events with all the attendant spurious correlations (see Calude and Longo [2016]). If we think of L as the syntax of histories then reductions express abstractions in the vocabulary of these histories.

2.8.2 Deﬁnition of the Reduction Monoid

Let Sysepi denote the subcategory of Sys where the only relational translations are epimorphisms.

Because the composition of epimorphisms is another epimorphism, Sysepi is a well deﬁned category. The left representable functor of L in Sysepi is: 0 Sysepi(L, _) =defn {ψ : L L | ψ is an epimorphism in Sys}.

56 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

Such functors have a long history in category theory (see [Freyd, 1964, pp. 16, 81]). Sysepi(L, _) is all the possible reductions of L. Let ψi : L Li, i = 1, 2 be two different reductions of Sysepi(L, _) and construct their pushout which we denote by ψ1 u ψ2.

L ? 1 ψ1

ψ ψ 1u 2 / 00 L L> ( pushout )

ψ2 L2

The pushout of epimorphisms is an epimorphism This gives us a monoid Red(L), u) with the underlying set Sysepi(L, _) and the operation of pushout. It has a unit 1L as pushout(1L, ψ) = ψ.

ψ1 and ψ2 can be incompatible (see sub-section 2.7.4). When this occurs, we use the convention ψ1 u ψ2 = ω. The adjoined element ω is idempotent with ω u ω = ω. In some cases, particularly when a monoid (such as the natural numbers) has the cancellation property, a + b = a + c =⇒ b = c, we can form a group, the Grothendieck group of the monoid [Lang, 1969, p. 43]. This avenue is not open to us here because of the following observations. Each epimorphism ψ is associated with an equivalence class Eq(ψ) deﬁned on D(L). These form a sub-lattice of the powerset of D(L) (see the discussion in section 2.7.3). We can create the associated lattice of epimorphisms by observing that Eq(ψ1) ∨ Eq(ψ2) corresponds to the pushout 0 of Eq(ψ1 uψ2). This creates a quotient L of L by introducing the equivalence class Eq(ψ1)∨Eq(ψ2) in L. If the pushout exists, so the equivalence relations are compatible, then L0 is a legitimate object of Sys. Generally, lattices and semi-lattices (such as subset unions) do not have the cancellation property so we cannot use the standard Grothendieck construction. Nevertheless, Red(L) has a rich set of properties. (Indeed, monoids play a signiﬁcant role in representing classes of automata Perrin [1990]). Let Mon denote the category of monoids.

Proposition 2.8.1. Red : Sysepi → Mon is a contravariant functor from relational landscapes with epimorphic relational translations to (Abelian) monoids and injective homomorphisms.

Proof. Any left representable functor from Sysepi to sets is contravariant. Furthermore Sysepi(L, _) changes epimorphisms to injective functions so, given ϕ : L1 L2 the induced set map Red(ϕ) Red(L2) −−−−→ Red(L1)

57 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

ϕ ψ 0 is one-to-one. We need to check the algebra. If ϕ : L1 −→L2 then every ψ : L2 −→L in Sysepi 0 composes with ϕ to give an epimorphism from L1 to L . The pushout of ψ1 and ψ2 gives a pushout of (ψ1 u ψ2) ◦ ϕ which is (ψ1 ◦ ϕ) u (ψ2 ◦ ϕ) and so respects the u addition and is a homomorphism of Red(L2) to Red(L1).

Example: Suppose the following diagram is the category of epimorphisms of L in Sysepi.

L ? 1 ψ1

ψ ψ 1u 2 / 00 L L> ( pushout )

ψ2 ψ21 % L2 / L21 9/ ω ( pushout) ψ22 ! L22

Red(L) is the set

{ψ1, ψ2, ψ1 u ψ2, ψ12 u ψ22, ψ2k ◦ ψ2, (ψ1 u ψ2) u (ψ2k ◦ ψ2) = ω} where k = 1, 2.

The results of Sections 2.7.2 and 2.7.4 enable us to interpret the above diagram in terms of the structure of L. We have:

1. ψij u ψij = ψij.

2. ψ12 ◦ ψ2 u ψ22 ◦ ψ2 = ω indicating no legitimate pushout.

3. Likewise (ψ1 u ψ2) u ψ2k ◦ ψ2 = ω.

[n] Label the unlabeled paths φ1 and φ2 so φ1 ◦ ψ1 = φ2 ◦ ψ2 = ψ1 u ψ2. Given R (x1, x2, ..., xn) in L suppose:

˜ 1. ψ1 : R(x1, x2, ..., xn) 7→ ψ1(R)(x1, x2, ..., xn) = R(y1, y2, ..., yp, xi1 , xi2 ...xik ). Here yi ∈ cospt(ψ1) hence 2p + k < n.

˜ 0 0 0 2. ψ2 : R(x1, x2, ..., xn) 7→ ψ2(R)(x1, x2, ..., xn) = R(y1, y2, ..., yq, xj1 , xj2 ...xjl ) with 2q + l < n ˜ ˘ 3. φ1 : R(y1, y2, ..., yp, xi1 , xi2 ...xk) 7→ R(z1, z2, ..., zr, xµ1 , xµ2 ...xµs ) where 2r + s < 2p + k < n

58 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

˜ 0 0 0 ˘ 0 0 0 4. φ2 : R(y1, y2, ..., yq, xj1 , xj2 ...xjl ) 7→ R(z1, z2, ..., zu, xµ1 , xµ2 ...xµs ). In this case t + u = r + s

0 0 0 0 0 0 Here we assume cospt(ψ2) = {y1, y2, ..., yq}, and cospt(φ2) = {z1, z2, ..., zu}. The equation φ1 ◦ ˘[t+u] 0 0 0 ψ1 = φ2 ◦ ψ2 gives two ways of obtaining the relation R (z1, z2, ..., zu, xµ1 , xµ2 ...xµs ) which, in 00 ˘[r+s] L , is logically equivalent to R (z1, z2, ..., zr, xµ1 , xµ2 ...xµs ). There are, therefore, two ways to construct these relations as reductions of R(x1, x2, ..., xn). Suppose that ψ1 u ψ2 is such that ψ1 u ψ2 u ψ3 = ω for an arbitrary reduction ψ3. In this case spt(ψ1 uψ2) is nearly the full potential of L for further abstraction or reduction “along those lines”. For example, suppose x1, x2, x3 are different sets of courses and u and v are different qualiﬁcations.

In u pairs of courses from x1 and x2 count towards the qualification. For the qualification v only pairs of courses from x2 and x3 count. Suppose ψ1 identifies x1 and x2, ψ2 identifies x2 and x3 and ψ3 identifies x1 and x3 but in all case u and v remain distinct. The combined pushout ψ1 u ψ2 u ψ3 results in identifying x1, x2 and x3 while u and v remain separate. ψ3 u ψ1 u ψ2 destroys this logic and therefore ψ3 u ψ1 u ψ2 is set to ω. A consequence of this is that cospt(ψ1) = cospt(ψ2) does not imply ψ1 = ψ2 or even that they are compatible: the cosupport can have many incompatible “cases”.

Sub-monoids of Red(L)

The phrase above “along those lines” suggests that there are classes of reductions that can be added together to produce a non-trivial, aggregated effect. A maximal sub-monoid H of Red(L) is a set

ψi, i = 1, 2, ...r such that

ψ1 u ψ2 u ... u ψr 6= ω but for all other φ ∈ Red(L), φ∈ / H,

ψ1 u ψ2 u ... u ψr u φ = ω.

Red(L) and hence L is characterized by the set {H1,H2, ...Hk} of maximal sub-monoids. Hmax =

H1 ⊕ H2 ⊕ ... ⊕ Hk = Red(L). This follows as anything left out, say ψ ∈ Red(L) \ Hmax, generates a sub-monoid with no elements from Hmax. Thus ψ would be a generator for its own sub-monoid to be added to Hmax. 0 0 0 0 If φ : L L and H is a maximal submonoid of Red(L ), the elements of H are of the form 0 0 0 0 0 ψ ◦ φ and H is formed by a maximal set ψ1 ◦ φ, ψ2 ◦ φ,..., ψs ◦ φ which will be a submonoid of a maximal submonoid of Red(L). A single maximal H ⊂ Red(L) can be a composite of a number of maximal submonoids from a number of reductions of L.

59 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

2 As with rings, the sub-monoids of monoids form a lattice with intersection and direct sum . 1L is in every sub-monoid so intersections are never empty. A reduction ψ that takes an innocuous or isolated partition of a data-type to single data-type can easily be part of all maximal Hi. This allows at least some of the sub-monoids to intersect with more than one element.

Hi1 ∩ Hi2 ∩ ... ∩ Hir , ir ≤ k always contains 1L but can also contain other “innocuous” reductions. The minima among such sub-monoids forms a stable set of reductions that do not result in incompatible reductions. The distribution of the non-empty elements of descending series

Hi1 ∩ Hi2 ∩ ... ∩ Hir , ir ≤ k over all subset {i1, i2, ..., ik} ⊂ {1, 2, ...k} is an indication of the distribution of compatible reductions in Red(L) and, as a characteristic of Red(L), is an invariant of relational landscapes.

Red(L) and factoring relational translations

In Section 2.7, reductions were decomposed into local reductions to study their compatibility. This can be generalized to “covers” of L, sets of sub-relational landscapes the union of which is all of 0 L. Let V = {Li ( L | i = 1, 2, . . . , m} be a cover L and each Li has a map ψi : Li → L for which spt(ψi) is non-empty. Hence

0 φi = 1L\Li t ψi : L L is deﬁned and is a reduction. Hence an element of Red(L). The covering set is compatible if φ = φ1 u φ2 ··· u φm 6= ω. In which case the reductions 0 φi, i = 1, 2, . . . , m, are in at least one maximal sub-monoid H ⊂ Red(L). If L has no further 00 reductions, so that for any other reduction of θ : L L , θ u φ = ω and φ is in the intersection of the maximal sub-monoids. 0 The compatible covers can be characterized as decompositions φ : D(L) D(L ) and the maximal submonoids H correspond to “saturated” decompositions. The study of these decompositions links circumstances when the equivalence relation (deﬁned in Section 2.7.3) Eq(ψi) can compose with Eq(ψj) and when it cannot, in which case compatibility fails. These comments gives another perspective on the monoid Red(L) by linking it to coverings of L by sub-relational landscapes.

2.8.3 Calculating Red(L) versus Calculating C(L)

ψ Every reduction L −→L0 is built from the following ingredients

2If S is a semigroup, then the intersection of any collection of sub-semigroups of S is also a sub- semigroup of S. Similarly, the direct sum of two sub-semigroups is generated by all their products. Conse- quently, the sub-semigroups of S form a complete lattice

60 CHAPTER 2. RELATIONAL LANDSCAPES AND THE COUPLING RING 2.8. THE REDUCTION MONOID

1. An observation that x1 and x2 can play the same role so ψ(x1) = y = ψ(x2) which can only

happen if there is a relation R(x1, x2, ....) 7→ ψ(R)(y, ...)

0 2. The reduction of a relation to a single data-type: R(x1, x2, ...xn) 7→ ψ(R)(y) = P(y) ∈ L . This

reduces all the relations logically implied by R(x1, x2, ...xn) to P(y).

From these two types of reductions we can use pushouts to combine them to obtain Red(L). Given L, calculating C(L) is a decidable problem achieved by listing relations and their domains. Listing ideals and their intersections follows from knowing C(L) and then one can work out direct sums of ideals. From this data, verification of ideal equations can be checked. On the other hand, Red(L), being sensitive to the implications of logical partitions, probes much more deeply into the system. Seeing the possibility of some of the obscure, small equivalence classes among two or three data-types in thousands seems to be difficult to achieve algorithmically. Also a reduction need not start from the intention to reduce a system but to find abstractions or equivalence classes that can be useful in certain circumstances. A typical example being substitution of parts when maintaining machinery. The reductions of a reporting structure are not something that one designs into the structure but are features that come out of the entire logic. This is what makes the reduction functor a summary of connective properties but hard to calculate.

61 Chapter 3

CORRELATION HOMOLOGY

3.1 Introduction

One of the most significant achievements of corporations in the first half of the twentieth century was to develop management structures that coordinated business processes Chandler [1995]. In- formation systems have formalized this and taken it to a much more detailed level. Enterprises are replete with screens and portable devices accessing data-bases that deliver information that helps sequence the flow of work. Every schedule is an allocation of tasks linking a staff member, a type of work, when it is due and where it might be located. Such schedules take into account the flow of orders, the work required to complete them. The schedules coordinate specialist workers such as welders and electricians, auditors and insolvency experts, logistic companies and so on. They also draw on information about the priorities of current and future commitments to resources and their replenishment. In our terms, schedules are the higher relations in a reporting structure and appear in a relational landscape as nested relations. We now introduce new structures that can link relations that are not actually coupled or, if so, only through long tenuous chains of coupling. This new level of correspondence we shall call “adjoint relations” and they arise from the logic contained in the nesting of relations.

We start with the observation that R(R1,R2) relates the queries dependent on aspects of R1, so J(R1), and those in J(R2). Even if R1 and R2 are not coupled this is still an indication of a well formed relation. Queries on the subject of R2 are also associated with queries on R1. When this two way association is well behaved we obtain an adjoint relation (Section 3.2.4). An adjoint relation ensures that there is a reasonable degree of matching between the queries in J(R1) and those in J(R2).

Higher-relations, relations with the form R(R1,R2) or

62 CHAPTER 3. CORRELATION HOMOLOGY 3.1. INTRODUCTION

0 00 R(R(R (R1,R2,R3)),R (R3,R4,R5))R6,R7, ~z) and beyond, are difficult to translate in terms of ideals. In practical terms they require a great deal of testing, building up and checking all the components. However, every large construction or assembly takes a form even more complicated than R. Thus there are good reasons to develop tools to be able to distinguish among the higher-level relations, the form of which is not a good indication of the role they play in L. We shall develop concepts that see higher-relations themselves as being finite categories of finite categories that have distinct structures that are important in the coordination of activities in the enterprise. These structures form a hierarchy. Each level of the hierarchy is a high-level relation that takes relations from the previous level as its domain relations (Section 3.2.4). Consequently, the number of ordinary, non-nested relations required to supply data to high-level structures in the hierarchy grows quickly. Each non-nested relation adding to a growing number of data-types. This produces a new problem. We can end up with a level of coordination among twenty or thirty ordinary relations but the whole construction is rather “floppy”. Although each local area of the whole high-level adjoint relation works well, the propagation of coordination degrades over the aggregate relations. To avoid this, in Section 3.3 we adapt the techniques of homology from topology [Spanier, 1966]. This is used to define an algebraic criterion on high-level adjoint relations that have high levels of coordination throughout their entire domains. Those hierarchies of adjoint relations that are detected by the homology are called correlations. The homology lists the correlations for each level of the adjoint relation hierarchy. The approach here uses adjoint functors to select relations that are “tractable” in terms of testing them. We recall some details about adjoint functors.

Adjoint Functors

Given two categories C and D, with functors F : C → D and G : D → C and for every object A of C and B of D

A → G(B) in C corresponds to a map F (A) → B in D and vice versa, then F is called the left adjoint of G and G is the right adjoint of F and we write F a G. This is equivalent to the sets C(A, GB) and D(F A, B) being in one to one correspondence. Adjoint functors play a large role in category theory (and indeed in much of modern mathematics, for an example in topology see [Adams, 1978]). They are important for a number of reasons, the ﬁrst being the important (and old) result [Freyd, 1964, ch. 3].

Theorem 3.1.1. Let F : C → D, G : D → C with F a G then G preserves limits and F preserves colimits.

63 CHAPTER 3. CORRELATION HOMOLOGY 3.1. INTRODUCTION

Indeed, once you have limits and colimits in the appropriate categories you can construct adjoint pairs. Adjoints can considered as "nearly inverses" or “weak equivalences”. In ordinary sets, a func- −1 tion f : X → Y has an inverse g : Y → X if g ◦ f = 1X and f ◦ g = 1Y (usually we write g = f ). In categories and other mathematical structures we can use the ”relatedness” of objects to slightly relax the idea of a strict inverse. An adjoint pair on partially ordered sets allows g ◦ f ≤ 1X and f ◦ g ≥ 1Y (so things don’t shrink to nothing or grow to a maximum).

Nested relations as adjoint functors

What does it mean to have a relation R(R1(~x),R2(~y))?

As with R(x, y) we can have a “graph” of a relation R, R1 ← R → R2. R relates changes in the values of the domain data-types of R1 to changes in the values of the domain variables in R2.

We reinterpret R in terms of a relation between the couplings (hence queries) of R1(~x) and R2(~y). α In this way R(J(R1),J(R2)) is a pair of order preserving (or reversing) maps J(R1) −→ J(R2) β and J(R2) −→ J(R1). The order on J(Ri) corresponds to classes of queries and a morphism of queries, r0 → r, corresponds to ordering r0 ≤ r when r0 is a specialization of r so r0 = r#R¯, hence 0 J(r ) ⊆ J(r). This is the way we shall understand the nature of R(R1(~x),R2(~y)). The higher-order relations R of interest here are the ones for which the maps α and β act as adjoint functors so that β ◦ α ≤ 1 and 1 ≤ α ◦ β (alternatively the ≤ are reversed). If ri ∈ J(Ri), i = 1, 2 we get β ◦α(r1) ≤ r1 and r2 ≤ α◦β(r2). If R1 relates each value of ~x to many values of ~y and

R2 relates each ~y to many values of ~x, then sub-relations of R1(~x), such as r1 = R1#R¯, are mapped 0 0 to sub-relations of R2(~y), such as r2 = R2#R¯ where R¯ and R¯ can be independent. R becomes an adjoint relation if it guarantees the existence of functions α and β that satisfy the constraints: if

α : r1(~x,~w) 7→ r2(~y, ~z) then

0 β : r2(~y, ~z) 7→ r1(~x,~u), and ~u will depend on ~z and ~y which, in turn, depend on r1(~x,~w).

The query β(α(r1)) is then likely to be a coupling of r1 and so we can think of it as a specializa- 0 tion of r1 with an added variable so r1(~x,~u) ≤ r1(~x,~w).

64 CHAPTER 3. CORRELATION HOMOLOGY 3.1. INTRODUCTION

Example of an adjoint pair

I shall denote an adjoint relation with the pair of functors α and β as α : J(R1) J(R2): β. The functors do not depend on R1 and R2 being coupled. A simple example is as follows.

α - J(R1) m β J(R2) (3.1)

r21 β α t , r11 l β r22

α , r12 l β r23 Gives α ◦ β ≤ 1 and β ◦ α = 1. We need to check

J(R1)(r1k, β(r2k)) = J(R2)(α(r1k), r2k).

The only problematic case is r11:

1. r11 ≤ β(r21) matches with α(r11) = r22 ≤ r21

2. r11 ≤ β(r22) matches with α(r11) = r22 ≤ r22

This is about the smallest example of adjoints and their natural transformations, in this case the ≤ relations. L is ultimately the source of adjoints. In (1) and (2) above, we can always have examples where α and β fail to be adjoints because the sets of possibilities in each relation is not constrained unless

R(R1,R2) has a structure that provides them. The following examples provide some reporting structure examples.

1. The relation R =“Orderline” links a product with options chosen from a multitude of option-

sets: R1(ordered product, option) with the set of components needed to make the options:

R2(option , component set). Many products are determined by their component set. 1 ≤ β◦α as β picks up all the option-sets with common components. Thus an adjoint pair expresses the correspondence of the ordered product with the necessary components (in this case the α picks out components while β gives the products with those components).

2. R =“TreatmentCapability”. R1(disease classification, treatment options) lists disease symp- toms and classiﬁcation and treatment protocols.

65 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS

R2(treatment equipment/resources, staff competency) list the available resources for diag- nosing and treating patients and whom among the staff are trained to use them.

The importance of adjoints is that they restrict higher-order relations of relations by making clear exactly what a relation of relations is doing. Functions of functions do not need this. We now generalize all this.

3.2 Adjoint Relations

Not all relations of relations can be represented in terms of adjoint functors on the principal ideals of their domain relations. For example statistical reports which take variables from many different reports to calculate sets of related statistics. In this section I deﬁne a hierarchy of nested relations that have a special form that can always be cast in terms of adjoint functors. This is the hierarchy of “adjoint relations” and gives a new structure in L. Adjoint relations are intended to form the domain for the homology functors of L and eventually this will justify their form.

3.2.1 Level 1 Adjoint Relations

Whenever R(R1,R2) can be represented in terms of adjoints α : J(R1) J(R2): β, R(R1,R2) is a member of the ﬁrst level of adjoint relations.

3.2.2 Level 2 Adjoint Relations

From now on relations that can be expressed in terms of adjoint functors between the queries of !p their domains will be signiﬁed by writing R with p indicating the depth of nesting. So R(R1,R2) ! !2 !! can be written as R (R1,R2). The next step is the R “canonical” form (also written as R ). !2 ! ! ! The second level of adjoint relations of relations have the form R (R1,R2,R3). We shall also !! ! ! ! !2 write these as R (R1,R2,R3). R corresponds to a category (of categories):

! R (R1,1,R1,2) (3.2) 1 = a

R˜ R˜ 12 R!! 13

} ˜ ! ! R23 ! R2(R2,1,R2,2) o / R3(R3,1,R3,2),

66 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS where R!! in the center can be thought of as a mapping between all the queries that can be made ! among the domains of Ri(Ri,1,Ri,2), i = 1, 2, 3. This is a subcategory of L that contains those ! queries that can be formulated in J(Ri,j), i = 1, 2, 3, j = 1, 2 subject to constraints of Ri and then R!!. Diagram 3.2 deﬁnes a hierarchy of categories:

! 1. The J(Ri) are categories of categories J(Ri,j) with adjoint functors αi : J(Ri,1) J(Ri,2): ! βi. This makes J(Ri) 2-categories [Street and Kelly, 1974] with objects sets of queries.

!! ! ! ! 2. R has objects J(Ri) and stipulates adjoint relation pairs of J(Ri) and J(Rj). This makes it a 3-category.

˜ ! ! ˜ Rij defines the adjoint relation between the two categories J(Ri) and J(Rj). The three Rij !! ! ! ! ˜ define the total class of queries defined by R ⊂ J(R1) ⊕ J(R2) ⊕ J(R3). Rij is defined in terms of ! ! ! ! grouping pairs of queries in J(Ri) ⊕ J(Rj). J(Ri) ⊕ J(Rj) is the collection of tuples

((ri,1, ri,2), (rj,1, rj,2)), with ri,j ∈ J(Ri,j)), i, j = 1, 2.

Each R˜ij can be thought of as a subset of (J(Ri,1) ⊕ J(Ri,2)) ⊕ (J(Rj,1) ⊕ J(Rj,2)) or as a relation

R˜ij

(J(Ri,1) ⊕ J(Ri,2)) ⊕ (J(Rj,1) ⊕ J(Rj,2)). which we shall call the “graph” form of

˜! Rij(J(Ri1 ∨ Ri2),J(Rj1 ∨ Rj2)).

! The reformulation of R˜ij in the form of R˜ will be called “aggregating the faces” R˜ij to make 2 R! an adjoint relation (in the “canonical” form). Although the diagram suggests one can compose around the triangle this can vary in many ! ways. Suppose each J(Ri(Ri,1,Ri,2)) has many aspects or dimensions of queries which we put into classes Qi,ri . These will correspond to different classes of couplings affecting the Ri,j, j = 1, 2.

R˜ij represents the graph of these adjoint mappings so we can write (as a shorthand)

˜ Rij : Qi,ri Qj,rj . ˜ Suppose Q1,ri has 3 classes Q1,1, Q1,2, Q1,3 and Q2,ri has 4 classes Q2,1, Q2,2, Q2,3, Q2,4 and R1,2 treats them as Q1,1,Q1,2 ∪ Q1,3 and Q2,1 ∪ Q2,2, Q2,3, Q2,4, so R˜1,2 mixes the dimensions with Q1,2 ∪

Q1,3 and Q2,1 ∪ Q2,2 but Q1,1 and Q2,4 map to either a maxima of one and the minima of the other.

67 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS

An example of a level 2 adjoint relation from scheduling is as follows. Factories have ongoing schedules in which the factory workers are assigned a sequence of task that have well deﬁned times or work-slots. This requires a combination of equipment, skills and materials. As new orders come in from customers they have an agreed data of delivery and need to be scheduled. This starts with the current orders and their status (status= in progress, planned for dates, priority1). To add new activities to the ongoing schedules the planners have to check what activities, equipment and personnel are required to make each new order. This is matched against planned inventory: available (unreserved) components onhand and the expected replenishment of material stock. (In the following the topic of a relation is given as a function to show how the listing is sorted, so RequiredActivities(order)). This gives

! R1(R11(CurrentOrders, Status),R12(ScheduledActivities(order)) Here we assume tha individual activities for a particular time slot are all about a single order.

! 1. R2(R21(UnscheduledOrders, priority)),R22(RequiredActivities(order), RequiredInventory(order)).

! 2. R3(R31(RequiredEquipment(activity)), R32(AvailableDates(RequiredEquipment), AvailableDates(SkillRequired))).

!! ! ! ! !! to produce R (R1,R2,R3) R that uses this information to populate future vacant work-slots with activities. A schedule is usually a rolling display of what is happening currently and the future already has the future unassigned work slots that R!! populates. The existence of adjoint relations such as R!! is never guaranteed, it is a mark of well disciplined planning.

Adjoint functors can be quite coarse. Given two partially ordered sets, P1 and P2 which we can treat as categories, suppose P1 which has a maximum and P2 has a minimum with order preserving maps f : P1 → P2 and g : P2 → P1 then

P1(a, g(b)) = P2(f(a), b) is satisﬁed if g(b) is always the maximum in P1 and f(a) is always the minimum of P2. The weak equivalence is no more than the posets have either a single maximum or minimum. Without some constraints on adjoint functors the composition of adjoints can lead to these rather uninformative compositions. This can become more of a problem when we seek adjoints between large classes of queries such as J(R1 ∨ R2 ∨ ...Rm) = J(R1) ⊕ J(R2) ⊕ · · · ⊕ J(Rm) and 0 0 0 J(R1 ∨ R2 ∨ ...Rk). Keeping these adjoints “informative” leads to homology (Section 3.3).

1Orders always have an agreed date of delivery but some customers have overriding priority, for example for warranty service agreements or work for emergency services.

68 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS

!3 ˜ ! 1 1 o R / ! 2 2 o R2 / ! 2 2 R1,1(R1,1,R1,2) R2,1(R2,1,R2,2) R2,2(R3,1,R3,2) 9 O e O e 9 O R˜1 R˜2

R˜1 R˜2 y % 3 % y R˜1 R! ! o !3 / ! o !3 / ! 2 2 !3 R1,2(R2,1,R2,2) R R1,3(R3,1,R3,2) R R2,3(R1,1,R1,2) R O O O

3 R! R˜4 !3 ! 3 3 o !3 / ! 4 4 o !3 / ! 4 4 R R3,1(R1,1,R1,2) R R4,1(R2,1,R2,2) R R4,2(R3,1,R3,2) 9 e e 9 R˜3 R˜4

R˜3 R˜4 y ˜ % !3 % y ! 3 3 o R3 / ! 3 3 o R / ! 4 4 R3,2(R2,1,R2,2) R3,3(R3,1,R3,2) R4,3(R1,1,R1,2)

Figure 3.1: A level 3 adjoint relation expanded.

3.2.3 Level 3 Adjoint Relations

!3 !! !! !! !! !2 The next level has the form R (R1 ,R2 ,R3 ,R4 ) and is a category of R adjoint relations with the overall diagram:

!! ! ! ! R1 (R1,1,R1,2,R1,3) (3.3) = O a

R˜13 R˜14 . R˜12

} ˜ ! !! ! ! ! R34 !! ! ! ! R3 (R3,1,R3,2,R3,3) o . / R4 (R4,1,R4,2,R4,3) i 5 R˜24 ˜ R23 ) u !! ! ! ! R2 (R2,1,R2,2,R2,3)

!! This is easier to see this as a tetrahedron with R2 the closest vertex. Expanding the four vertexes and reforming the diagram so it fits on the page we get Figure 3.1: 3 In Figure 3.1 all the dotted lines are defined by R! and some are implied. For example the ! 3 3 ! 2 2 !3 passage from R3,2(R2,1,R2,2) to R2,2(R3,1,R3,2) could be composition of maps defined in R that include lower level functors and natural transformations. To include all such arrows would create a pointlessly cluttered diagram. The arrangement of the diagram has been chosen to fit the width

69 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS on the page. Going back to diagram 3.3 we “aggregate” the representation of a “face” of the tetrahedron. !! Each face is deﬁned by three of the Ri , i = 1, 2, 3, 4

˜!! ! ! ! Ri (Ri,1,Ri,2,Ri,3) 4 f

t & ˜!! ! ! ! . ˜!! ! ! ! Rj (Rj,1,Rj,2,Rj,3) n Rk(Rk,1,Rk,2,Rk,3)

˜!! ! ! ! Each Ri deﬁnes a subset of J(Ri,1) ⊕ J(Ri,2) ⊕ J(Ri,3). Each subset can be identiﬁed a sequence of ! queries zi = (ξi,1, ξi,2, ξi3), each ξi,m being a query in J(Ri,m) hence associated with a sub-ideal of ! ! J(Ri,m). Ri,m has its own adjoint pairs aligning these queries but now we are gathering them up ˜!! ˜!! into larger sets which will be related by adjoints to those of Rj and Rk. The ijk triangle ensures there are functions:

zi = (ξi,1, ξi,2, ξi,3) 4 g

t . ' zj = (ξj,1, ξj,2, ξj,3) n zk = (ξk,1, ξk,2, ξk,3) which can be collected in:

˜!! ! ! ! ! ! ! ! ! ! Rijk((Ri,1 ∨ Ri,2 ∨ Ri,3), (Rj,1 ∨ Rj,2 ∨ Rj,3), (Rk,1 ∨ Rk,2 ∨ Rk,3)).

In terms of the underlying category this is part of the coherence rules [Eilenberg and Kelly, 1965] for

˜!! ! ! ! ! ! ! ! ! ! J(Rijk) ⊂ J(Ri,1 ∨ Ri,2 ∨ Ri,3) ⊕ J(Rj,1 ∨ Rj,2 ∨ Rj,3) ⊕ J(Rk,1 ∨ Rk,2 ∨ Rk,3), or ˜!! ! ! ! ! ! ! ! ! ! J(Rijk) ⊂ J(Ri,1) ⊕ J(Ri,2) ⊕ J(Ri,3) ⊕ J(Rj,1) ⊕ J(Rj,2) ⊕ J(Rj,3) ⊕ J(Rk,1) ⊕ J(Rk,2) ⊕ J(Rk,3).

p This will be the standard pattern for faces for arbitrary R! .

Finding ascending levels of adjoint relations

Here we have outlined ascending levels of adjoint relations; high levels of abstract structure in L beyond the coupling ring and certainly beyond the purview of information systems engineering techniques. Where might we ﬁnd such relations? Large-scale engineering has components and assemblies which are then matched with spares and equipment for manufacture, repair and maintenance. Assemblies are part of a hierarchy that builds toward high value products. High value products are those used in infrastructure construction projects or in the manufacturing of large

70 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS engineering products that have to be serviced and maintained over years. For example mining equipment, aircraft, shipping and dockside equipment. No part of the entire cycle including quality assurance of supplied components can be ignored [Andersen, 1994, Knolmayer, Mertens, and Zeier, 2002].

3.2.4 The Hierarchy of Level p Adjoint Relations

Deﬁnition 15. The Hierarchy of Adjoint Relations Lk of L is the sequence of sets:

1. L−1 is the set of ﬁnite conjunctions of data-types from D(L),

2. L0 is the set of relations of L that are relations of an element of L−1. As L−1 contains conjunctions

P(x1) ∧ P(x2) ∧ · · · ∧ P(xn) this covers relations R(x1, x2, . . . , xn).

3. For k > 0. Lk is the set of relations of k +1 elements of Lk−1 and between any pair there is an adjoint relation.

! Hence L1 has elements of the form R (R1,R2)) and for k > 1,

!k !k−1 !k−1 !k−1 !k−1 Lk has R (R1 ,R2 ,...,Rk ,Rk+1 ).

We shall refer to relations in Lk when k ≥ 1 as (level k) adjoint relations. !p !k !p The span of R is the entire set of L0 relations that are contained in all the R , k ≤ p, in R .

k Proposition 3.2.1. The span of R! contains at most (k + 1)! relations.

! Proof. 1. R (R1,R2) spans 2 relations

!! ! ! ! ! 2. R (R1,R2,R3) spans 3× the span of R objects

!!! !! !! !! !! !! 3. R (R1 ,R2 ,R3 ,R4 ) spans 4× the span of R objects and so on by induction.

p Thus the higher the p in the superscript of R! the greater the span and the signiﬁcance for L.

3.2.5 Example of a Level Three Adjoint Relation

The following is an example of the data that have to be coordinated before negotiations for a construction projects can be ﬁnalized. (Components of this example are from [Magad and Amos, 1989, Groover, 2007, Sarmiento and Nagi, 1999, Macfarlane, 2014, Knolmayer, Mertens, and Zeier, 2002].) A project, such as housing development of an area of land in another country can go on for

71 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS years and require large-scale building components to be delivered by specialist logistic companies. These companies need to be booked well ahead of the actual transportation. In many cases negotiations can go through various iterations before the ﬁnal contract is signed. Each iteration has to coordinate the new information, rescheduling people, supplies and delivery logistics. In the following the relations link business concepts that ﬁnd expression in relations, such as schedules that list who is doing what at a particular time and for which project. This is a sketch of such a project and outlines connections among large-scale objects. Inevitably, there are many other details.

!! ! ! ! R1 (R1,1,R1,2,R1,3)(Requirements) 8 O f

R˜13 R˜14

R˜12 x ˜ & !! ! ! ! R34 !! ! ! ! R3 (R3,1,R3,2,R3,3)(Expertise) o / R4 (R4,1,R4,2,R4,3)(Capacity) j 4 R˜24 R˜ 23 * t !! ! ! ! R2 (R2,1,R2,2,R2,3)(Costing)

Each domain relation expands:

!! ! ! ! Ri (Ri,1(R(i,1)1 ,R(i,1)2 ),Ri,2(R(i,2)1 ,R(i,2)2 ,Ri,3(R(i,3)1 ,R(i,3)2 ))

!! ! ! ! 1. R1 (R1,1,R1,2,R1,3) (Requirements. What is wanted, where and when?)

! (a) R1,1(R(1,1)1 ,R(1,1)2 )

i. R(1,1)1 (Customer, InitialRequirements, T oBeDeliveredBefore)

ii. R(1,1)2 (Requirements, Drawings) ! (b) R1,2(R(1,2)1 ,R(1,2)2 )

i. R(1,2)1 (Drawings, Assemblies, Components)

ii. R(1,2)2 (Components, Suppliers, Leadtimes) ! (c) R1,3(R(1,3)1 ,R(1,3)2 )

i. R(1,3)1 (Assemblies, DeliveryRequirements, SpecialLogisticsRequirements)

ii. R(1,3)2 (DeliveryRequirements, LogisticsSchedule)

!! ! ! ! 2. R2 (R2,1,R2,2,R2,3) (Costings, can we make a proﬁt?)

72 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS

! (a) R2,1(R(2,1)1 ,R(2,1)2 )

i. R(2,1)1 (Requirements, StandardBOM, StandardOperatingOverhead)

ii. R(2,1)2 (StandardCostings, SupplierV olumeDiscounts) ! (b) R2,2(R(1,2)1 ,R(1,2)2 )

i. R(2,2)1 (SpecialOptions, Machining, Components)

ii. R(2,2)2 (ComponentLicensing, P rocessCostings) ! (c) R2,3(R(1,3)1 ,R(1,3)2 )

i. R(2,3)1 (AreaLogisticsRequirements, Quotes)

ii. R(2,3)2 (LastdateAreaDeliveries, LogisticsSchedule, AdditionalCosting) (d) Note: Standard Bill Of Materials is linked to standard costings. Logistics requirements can include special containers or packaging and temperature control requirements. De- livery might depend on special access to be in place and avoiding dangerous weather.

!! ! ! ! 3. R3 (R3,1,R3,2,R3,3) (Expertise: Do we know how to do it?)

! (a) R3,1(R(3,1)1 ,R(3,1)2 )

i. R(3,1)1 (SpecialistOnSiteInstallations, P ossibleT askDates, RequiredExpertise)

ii. R(3,1)2 (Expertise, CurrentCommitments, StandardCostings) ! (b) R3,2(R(1,2)1 ,R(1,2)2 )

i. R(3,2)1 (SpecialOptions, MachiningComponents, ExpertRequirements)

ii. R(3,2)2 (ExpertMachinist, CurrentCommitments, P rocessCostings) ! (c) R2,3(R(1,3)1 ,R(1,3)2 )

i. R(3,3)1 (SpecialOptions, OutsourcedCosts)

ii. R(3,3)2 (ScheduleCompletionOfOutsourcedW ork, P enaltiesF orUnschedldDeliveries)

!! ! ! ! 4. R4 (R4,1,R4,2,R4,3) (Capacity: Do we have the factory capacity in the required time? Can we fund start up work before progress payments start?)

! (a) R4,1(R(3,1)1 ,R(3,1)2 )

i. R(4,1)1 (CurrentW orkOnhand, CurrentReservedP roductionInventory, CurrentLogisticsSchedule)

ii. R(4,1)2 (CurrentInventoryRequirements, CurrentScheduleOfSupplierDelveries)

73 CHAPTER 3. CORRELATION HOMOLOGY 3.2. ADJOINT RELATIONS

! (b) R4,2(R(1,2)1 ,R(1,2)2 )

i. R(4,2)1 (SpecialistSchedule, ScheduleOfOutsourcedW ork, SpecialistMachiningComponents)

ii. R(4,2)2 (F irstdateScheduleOfDelivery, RiskAssessment) ! (c) R4,3(R(1,3)1 ,R(1,3)2 )

i. R(4,3)1 (CurrentW orkOnhand, P rojectedCashflowOfP ayments)

ii. R(4,3)2 (F undingNewW ork, CreditRequirementsforNewW ork)

L is the setting that welds this collection of relations into a hierarchy of adjoint relations.

3.2.6 Transforming R[n] to an Adjoint Relation

[n] [n−1] [n2] [nr] Arbitrary relations of the form R (R1 ,R2 , ....Rr , ~z) where ni ≤ n − 1 do not conform to this hierarchy. Nevertheless they might conceal sub-programs or sub-relations, that can be ﬁtted into the adjoint relations hierarchy. Also there might be equivalent expressions or reformulations some of which can be in the form of an adjoint relation.

[3] ! [2] 1. A relation such as R (R1(R11,R12),R2 (R21,R22),R3,R4,R5)) has a principal ideal

[3] ! ! J(R ) = Re ⊂ [(Rf1 ∩ (J(R11) ⊕ J(R21)) ⊕ (Rf2 ∩ (J(R21) ⊕ J(R22)) ⊕ J(R3) ⊕ J(R4 ∨ R5)))]

Note that J(R4 ∨ R5) can be replace by J(R4) ⊕ J(R5). As we are working in the lattice J(C(L)) we can reform the last part of this as

[2] Rg ∩ (J(R3) ⊕ J(R4 ∨ R5))

! which is a relation potentially of the form R˜ (R3,R4 ∨ R5).

[n] [n] 2. R (R1,R2, ...Rm) has J(R (R1,R2, ...Rm)) ( J(R1) ⊕ J(R1) ⊕ ... ⊕ J(Rm) where the ( is what deﬁnes R so we can write this as

[n] [n] J(R (R1,R2, ...Rk)) = Rg ⊂ (J(R1) ⊕ J(R1) ⊕ ... ⊕ J(Rm).

This puts a relation in its graph form. If {R1,R2, ...Rkt} can be collected or partitioned into

sets Ri1 , Ri2 ,...,Rik , Rj1 , Rj2 ,...,Rjm so that J(Ri1 ,Ri2 ,...,Rik ) and J(Rj1 ,Rj2 ,...,Rjm ) are categories related by adjoint functors then R[n] might be able to be represented as R˘!. [n] Similar if {R1,R2, ...Rk} can be broken into n+1 subsets aggregating might reformulate R n as R˘! .

74 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

Not all relations R[n] can be reformed like this. Functions are usually the high n valued relations R[n]. They are not always good candidates for adjoint relations as mapping back from an output ﬁle to all possible inputs would require selecting a generic representative of what is in the output. Although functions create the connections upon which an adjoint relation is based, those connections need not come from a single function. They frequently come from the union of the ranges of many functions. Functions deﬁned on data from a number of relations to present statis- [n] [n−1] [n1] [nr] tics also have the form R (R1 ,R2 , ....R , ~z) where ~z is a set of selection parameters. An adjoint pair would require a map from each value of the range of the statistics to a representative value from the relations. Each representative value would be from an equivalence class of values giving the same value for the particular statistic. But this might vary from sample to sample and so has no natural or permanent form.

3.3 Correlation Homology of L

This section introduces a homology functor that imitates the construction of the classical simplicial complex approach to homology in algebraic topology [Greenberg, 1967]. The definition of such functors is rather lengthy. The payoff has been well proven since they were introduced by Poincaré at the beginning of the twentieth century [Dieudonne, 1985, ch. VI, section 36 ff]. The new functors are constructed using the hierarchy of adjoint relations. From here on the development is mainly algebra. The primary function of the homology functor here is to define classes of adjoint relations that are “correlated” and thereby define an invariant of relational landscapes, hence a structure that characterizes isomorphism classes of relational landscapes. The definition of “correlated” is designed to get around the problem that an adjoint relation with large p can be rather uninteresting because of “coarse” adjoint functors. The homology of a relational landscape, L, can be considered as the spectrum of specially correlated adjoint relations defined on the adjoint relation hierarchy

Lp(L), p > 0.

p Categories Cat[R! ]

!! ! ! ! ! R (R1,R2,R3) is a category of 3 categories Ri each of which is a pair of categories J(Ri,1), J(Ri,2) which give the relational structures in terms of sets of queries that are linked with other sets of queries. Adjoint relations are represented as towers of categories with the morphisms between categories being pairs of adjoint functors. To emphasis the category of category structure we use 2 the notation Cat[R! ]. In terms of queries, this category can be thought of as a collection of queries:

75 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

!2 ! ! ! Cat[R ] ⊆ J(R1(R1,1,R1,2)) ⊕ J(R1(R1,1,R1,2)) ⊕ J(R1(R1,1,R1,2)) but with the set of sets of queries being related. It is precisely at this level of sets of sets that we get the higher structures. It is useful to formalize the lattice of ideals as a category of categories.

Definition 16. Define the category associated with J(C(L)), denoted J(C(L)), as having objects the categories J(R) and J(P(x)), R ∈ L or x ∈ D(L). These are defined as categories through their partial orders. In the case of R, each coupling R¯#R becomes an object of J(R) and J(R¯#R) a subcategory of J(R). The morphisms of J(C(L)) are the embeddings J(R1) → J(R2) ⇔ J(R1) ⊂ J(R2).

In all these categories, coherence rules [Eilenberg and Kelly, 1965, Street and Kelly, 1974] allow us to work out equations of morphisms, usually commuting diagrams.

3.3.1 The Correlation Complex: Faces and Boundary Maps

Homology functors are a sequence of modules over a ring of coefficients. Here the ring is modulus 2 arithmetic Z/2.Z that was used in the definition of the coupling ring in Definition 6. Notation: We shall use a common notation from algebraic geometry and algebraic topology, see Hartshorne [1977, Ch. III ], which indicates an omitted variable in a list of variables by a ˆ above the omitted variable. For example

{R1,R2,..., Rck,...,Rn} denotes {R1,R2,...,Rk−1,Rk+1,...,Rn}.

Deﬁnition 17. The p-th chain complex module, Cp(L), is the Z/2Z module generated by the elements of Lp(L) and is equipped with a “boundary map” δ : Cp → Cp−1 deﬁned by the linear extension of

!p !p−1 !p−1 !p−1 p+1 k δ(R (R1 ,R2 ,...,Rp+1 )) = Σk=1(−1) Fk

!p−1 !p−1 !p−1 !p−1 where the “k-face”, Fk, is the category J(R¯ (R ,R ,..., R[ ,...,R )) which is i1,i2,...,ibk,...,ip+1 1 2 k p+1 !p−1 !p−1 \!p−1 !p−1 !p the intersection of the categories J(R1 ) ⊕ J(R2 ) ⊕ ... J(Rk ) · · · ⊕ J(Rp+1 ) with J(R ).

Consequences of this deﬁnition.

!p !p p+1 k 1. R is a cycle if δ(Cat[R ]) = 0 so if Σk=1(−1) Fk = 0 where this is interpreted as

!p !p−1 !p−1 \!p−1 !p−1 ∼ × Cat[J(R ] ∩ {J(R1 ) ⊕ J(R2 ) ⊕ ... J(Rk ) · · · ⊕ J(Rp+1 )}] = k odd !p !p−1 !p−1 \!p−1 !p−1 × Cat[J(R ] ∩ {J((R1 ) ⊕ J(R2 ) ⊕ ..J(Rk ) · · · ⊕ J(Rp+1 )}] (3.4) k even

76 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

for all cases for p > 0. In the case p = 0 the categories are J(R1) and J(R2), we do not

demand that these are isomorphic. J(R1) and J(R2) and are already linked by the existence of adjoints so that the categories are weakly equivalent. The higher cycles impose constraints

on the p = 0 case. Matching up (ξ11 , ξi2 ,..., ξ\k(odd). . . . , ξp+1) on the left hand side of equation

3.4 and (ζ11 , ζi2 ,..., ζ\k(even) . . . , ζp+1) on the right hand side with an isomorphism imposes strong constraints on these collections of adjoints.

2. We require δ2 = 0. This is illustrated in the case

!! ! ! ! ¯ ! ! ¯ ! ! ¯ ! ! δ(R (R1(~x),R2(~y),R3(~z)) = R23(R2,R3) − R13(R1,R3) + R12(R1,R2).

2 !! ¯ ! ! ¯ ! ! ¯ ! ! ¯ ¯ ¯ Then δ (R ) = δ(R23(R2,R3) − R13(R1,R3) + R12(R1,R2)) = δ(R23) − δ(R13) + δ(R12). As δ will always be the alternating sum of sets of n − 1 domain variables from the n domain variables we get:

! ! ! ! ! ! (R2 − R3) − (R1 − R3) + (R1 − R2) = 0

p By using the R! notation, but interpreting the formal equations in the categories associated with p the R! , we can proceed to evaluate these equations. This makes the usual homological algebra a !p ¯ two step process of formal algebraic manipulation for R and Ri1,i2,...,ip+1 expressions followed by interpretation in the category of models, which, in this case, is a category of categories. We now interpret all these as categories as in section 3.2 so

!p 0p−1 0p−1 0p−1 δ(R (R1 ,R2 ,...,Rp+1 )) = 0 means

¯ ∼ ¯ × J(R1,2,...,k,...,pˆ +1) = × J(R1,2,...,k,...,pˆ +1). k odd k even where the product is the Cartesian product of categories and =∼ signiﬁes an isomorphism linking the two sides of the equation2.

We have the standard homological algebra deﬁnitions of

δp 1. The subgroup of cycles: Zp = Zp(L) =defn ker(Cp(L) −→ Cp−1(L)).

δp+1 2. The subgroup of boundaries: Bp = Bp(L) =defn im(Cp+1(L) −−−→ Cp(L)).

2If this seems to be at a very high level of abstraction, this is because the objects of study are among our most complex mathematical constructions.

77 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

3. The quotient group of p homology classes: Hp =defn Hp(L) = Zp(L)/Bp(L). [Lang, 1969, Greenberg, 1967]. These will be illustrated in the next subsection. ! The deﬁnition of δ requires the idea of a face. R (R1,R2) ∈ L1(L) has faces R1 and R2. C0 is generated by relations R(x1, x2, . . . xm) ∈ L0(L) and has the face P(x1) ∧ P(x2) ∧ · · · ∧ P(xm) of L−1(L). If we deﬁne δ0 as:

δ0 : C0 → Z/2.Z, δ0(R(x1, x2, . . . xm)) = 0, then C0 = Z0 and H0 = Z0/B0. Z0 is therefore the non-nested relations of L0. ! B0 is the span of the R ∈ L1(L) making H0 classes of relations R1 ∼ R2 if the difference R1 −R2 0 0 0 0 is of the form R1 − R2 ∈ B0. That is R1 + R2 = R2 + R1. This becomes

0 0 J(R1) ⊕ J(R2) = J(R2) ⊕ J(R1). (3.5)

! R1 and R2 make up the difference between the domains of some R . Hence query classes (ξ1, ζ2) ! 0 0 0 0 (ξ2, ζ1) whenever ζ1 and ζ2 are related by R (R1,R2) so α : J(R1) J(R2): β. Calling the equivalence relation given by equation 3.5 “quasi adjoint,” H0 becomes the quasi-adjoint equivalent ! 0 classes. For relations not in the span of some R , say R and R neither of which are in B0, they are their own quasi-adjoint equivalence class unless R − R0 is a boundary, in which case they are one quasi-adjoint pair.

3.3.2 The Meaning of δ: Low Dimension Examples

The concept of a cycle is best seen from the diagrams in Section 3.2 and identifying the various levels of adjoint relations. These are simplexes of categories. Diagram 3.2 is a triangle of categories and the cycle tells us that it has the property of “strong factorization” which means any two sides give a composition that gives the third side but also, the third side factors into the other two sides. Diagram 3.3 is a tetrahedron of categories; a cycle means any two of the triangular faces determines the other two. In the following we also interpret the meaning of “degenerate” cases where some of the components of the chain map are the same.

1 cycle

! δ1(R (R1,R2)) = R1 − R2 = 0 if and only if J(R1) ≈ J(R2). This means each sub-relation of Ri, 3 which we might think of as a SQL query , can be matched with one of the sub-relations of R2. For

3Structured Query Language is a query language that lists data from a relation with additional conditions effectively the same as a coupling

78 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

example, course tutorial time and place (relation R1) matches with a tutor assignment to a course

(relation R2). Another example: each student who is enrolled in a course (relation R1) will have a record of payment (R2).

2 cycle.

!! ! ! ! ˜ ! ! ˜ ! ! ˜ ! ! δ2(R (R1,R2,R3)) = R23(R2,R3) − R13(R1,R3) + R12(R1,R2). (3.6)

˜ !! ! ! ! Here we have faces Rij that are in L1. The 2 cycle deﬁnition δ2(R (R1,R2,R3)) = 0 is

˜ ! ! ˜ ! ! ˜ ! ! R23(R2,R3) − R13(R1,R3) + R12(R1,R2) = 0 so

˜ ! ! ˜ ! ! ˜ ! ! R23(R2,R3) + R12(R1,R2) = R13(R1,R3).

This to be interpreted in categories as

˜ ! ! ˜ ! ! ∼ ˜ ! ! Cat[R23(R2,R3)] × Cat[R12(R1,R2)] = Cat[R13(R1,R3)]. (3.7)

! ! ! Maps between ideals J(R1),J(R2),J(R3) denoted in an obvious way on the left hand side of equation 3.7 by ((ξ2, ξ3), (ξ1, ξ2)), are always matched with a map from the composed (ξ1, ξ3) on the right 0 0 0 0 hand side to related ((ξ2, ξ3), (ξ1, ξ2)) back on the left hand side. This means not only can we compose ((ξ2, ξ3), (ξ1, ξ2)) to get (ξ1, ξ3) but, whenever we are given (ξ1, ξ3), we can factorize it, maybe in more than one way, to get a ξ2. !! We can see this as the adjoints deﬁned by the R˜ij match up so that R is a coherent set of conditions on the entire set of queries. The boundary case !! ! ! ! What happens when δ2(R (R1(~x),R2(~y),R3(~z))) 6= 0? Suppose have

!! ! ! ! ˜ ! ! ˜ ! ! ˜ ! ! ¯! δ2(R (R1,R2,R3) = R23(R2,R3) − R13(R1,R3) + R12(R1,R2) = R ∈ C1(L).

In this case (ξ2, ξ3), (ξ1, ξ2)) ∈ R¯13 ×R¯12 is not always matched with some (ξ1, ξ3), or vice versa. !! ! ! ! The relation R (R1,R2,R3) gives us a criteria for matching triples ((ξ2, ξ3), (ξ1, ξ2), (ξ1, ξ3)) with the ﬁrst two giving the third and vice versa. While composition seems obvious, factorization is not. Factorization implies:

α R˜13(R1,R3) −→ R˜12(R1,R2) ∨ R˜23(R2,R3).

A minimum expectation is that there is a relation of the form

79 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

! ! R˜ ⊂ J(R1) ⊕ (J(R2) ⊕ J(R3)) = J(R¯ (R1, (R2 ∨ R3))) which gives

!! ! ! ! ˜ ! ! ˜ ! ! ˜ ! ! ¯! δ(R (R1,R2,R3)) = R23(R2,R3) − R13(R1,R3) + R12(R1,R2) = R (R1, (R2 ∨ R3)) so that

˜ ! ! ˜ ! ! ˜ ! ! ¯! R23(R2,R3) + R12(R1,R2) = R13(R1,R3) + R (R1, (R2 ∨ R3)).

Or, in the categorical representation,

˜ ! ! ˜ ! ! ∼ ˜ ! ! ¯! Cat[R23(R2,R3)] × Cat[R12(R1,R2)] = Cat[R13(R1,R3)] × Cat[R (R1, (R2 ∨ R3))].

! Cat[R¯ (R1, (R2 ∨ R3))] provides information that maps queries in J(R1) to queries in J(R2) or 0 J(R3). This allows (ξ1, ξ3) to “pick” the appropriate ξ2 or another ξ3 related to ξ3 by the adjoint relations. This is a general principle: if δ(R!n) 6= 0, it defines a relation that will allow us to fill in the conditions on the data to balance the category equations and adjoints. The additional data is a n measure of the failure of R! to be a sufficiently precise set of adjoints required to give the cycle condition.

The degenerate case

What if all the R˜ij are the same? This is the “degenerate” case. All the relations in the faces and their lower level adjoint relations are also all the same “all the way down”. These collapse to a set of relations Rij all the same and the corresponding faces are ﬂattened. The Diagram 3.2 becomes

ﬂattened. In terms of homology this is not an element of C2(L) but a sum of adjoint relations in

C1(L).

3 cycle.

The Diagram 3.3 is a tetrahedron with categories at each vertex. All the categories are weakly equivalent.

!3 !! !! !! !! δ3(R (R1 ,R2 ,R3 ,R4 )) = ˜ !! !! !! ˜ !! !! !! ˜ !! !! !! ˜ !! !! !! R234(R2 ,R3 ,R4 ) − R134(R1 ,R3 ,R4 ) + R124(R1 ,R2 ,R4 ) − R123(R1 ,R2 ,R3 ) = 0 (3.8) if and only if ∼ Cat[R˜234] × Cat[R˜124] = Cat[R˜134] × Cat[R˜123] (3.9)

80 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

where the R˜ijk belong to C2(L). The R˜ijk can be represented as

˜ !! !! !! Rijk ⊂ J(Ri ) ⊕ J(Rj ) ⊕ J(Rk)

˜ !! !! !! ! Thinking of Rijk(Ri ,Rj ,Rk) as a category, it has objects ξijk = (ζi, ζj, ζk) where ζi : J(Ri1) ! ! J(Ri2) J(Ri3) and similarly ζj and ζk. Equation 3.9 implies a one-to-one onto matching

(ξ234, ξ124) (ξ134, ξ123)

This, in turn, implies that three of the four ξijk determine the fourth triple say ξpqr. Geometrically !! !! !! !! we have a picture of the R1 , R2 , R3 , R4 occupying the four vertexes of a tetrahedron in which any three faces determine the fourth. The boundary case !3 !! !! !! !! What happens when δ3(R (R1 ,R2 ,R3 ,R3 , )) 6= 0? The required matching

(ξ234, ξ124) (ξ134, ξ123)

!3 !! !! !! !! fails. This means δ(R (R1 ,R2 ,R3 ,R4 , )), which is an element in C2(L), is a function (as it will 0 0 be) that gives (ξ234, ξ124) (ξ134, ξ123) and we need another relation that has the recipe to create 0 0 a mapping to take (ξ134, ξ123) 7→ (ξ134, ξ123) (which might be parametrized). Such a relation is a 3 measure of how far R! deviates from being a cycle. Suppose the information, R, that is added to either side, say the right hand side of equation 3.9, says what values should be provided to make the isomorphism in equation 3.9. We assume that R can be represented as a category so that

R × Cat[R˜134] × Cat[R˜123] → Cat[R˜234] × Cat[R˜124].

By “Currying” we also have

Cat[R˜134] × Cat[R˜123] → Cat(R, (Cat[R˜234] × Cat[R˜124])).

Starting with the (ξ134, ξ123) in Cat[R˜134] × Cat[R˜123] this picks out an item of information

π ∈ R. The π parameterize the functions into subsets of Cat[R˜124] and many of these sets of functions π(ξ134, ξ123) can correspond to a straight forward mapping of (ξ134, ξ123) to a single possible

(ξ234, ξ124). When this doesn’t happen, which is the reason for the failure of the cycle condition, we get a number of different (ξ234, ξ124) related back to (ξ134, ξ123) and the information in R. This is the 3 case when the adjoints in R! are “coarse” or “ﬂoppy” with R being a measure of that coarseness.

81 CHAPTER 3. CORRELATION HOMOLOGY 3.3. CORRELATION HOMOLOGY OF L

4 cycle.

4 By 3.2.1 the span of R! is 120.

!4 !3 !3 !3 !3 !3 δ4(R (R1 (~x1),R2 (~x2),R3 (~x3),R4 (~x4),R5 (~x5))) ˜ !3 !3 !3 !3 ˜ !3 !3 !3 !3 ˜ !3 !3 !3 !3 = R2345(R2 ,R3 ,R4 ,R5 ) − R1345(R1 ,R3 ,R4 ,R5 ) + R1245(R1 ,R2 ,R4 ,R5 ) ˜ !3 !3 !3 !3 ˜ !3 !3 !3 !3 − R1235(R1 ,R2 ,R3 ,R5 ) + R1234(R1 ,R2 ,R3 ,R4 ) = 0. (3.10) if and only if

∼ Cat[R˜2345] × Cat[R˜1245] × Cat[R˜1234] = Cat[R˜1345] × Cat[R˜1235] (3.11) which means any collection of sub-relations in the left hand side of the equation, hence having the form (ξ1, ξ3, ξ5) (subscripted by the omitted number), has a matching (ξ2, ξ4) on the right hand side. p cycles for p an even number have unbalanced cycle equations as in the case of equations 3.7 and 3.11. Equation 3.11 appears to have different degrees of freedom. In terms of the mappings of (ξ1, ξ3, ξ5) and (ξ2, ξ4) there is a degree of “wriggle room”. This is seen from the 2 cycle case. This implies that (ξ2, ξ4) “dominate” the order in the system; the choice of (ξ2, ξ4) forces the !4 choice of objects in R˜2345, R˜1245, R˜1234. This might occur when parts of R set the scene for other things to happen. For example, budget allocations deﬁne what projects can be undertaken or work schedules constrain what follows from them in determining schedules of supplies, and logistics. !4 !3 The boundary case when δ4(R ) is not zero has been discussed for δ3(R ) and follows the same pattern.

The pattern of the low dimensional cases

The pattern that has emerged is that for p an odd number, the p-th homology group counts the number of adjoint relations that have a level of symmetry across the high-level relations. For p an even number one side of the cycle equation “dominates” by having more terms. This is balanced by the side with fewer terms creating order by such mechanisms as factoring the terms in the dominant side. For all p, the p-th homology group gives a strong measure of order in those parts p of the system that are covered by the higher relations R! that are cycles.

82 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

3.3.3 Sums of Relations

The way the adjoint relations hierarchy as been defined allows us to make up general collections ( ( ( !p ! p−1) ! p−1) ! p−1) of adjoint relations that overlap to form expressions such as R (R1 ,R2 ,...,Rm ), m > p + 1. These collections of adjoint relations can intersect below the level p but still contribute to the correlation homology as follows. !p !p !p The chain complex module Cp(L) contains formal sums R1 + R2 + ... + Rm (with Z/2.Z coefficients this is just the set). Such expressions are useful in defining more arbitrary correlation ! ! forms. The simplest case is R1(R1,R2) + R2(R2,R3). If this forms a cycle then it contains the in- !! ! ! ! formation that R1 and R3 are in an adjoint relation. Similarly a homology class R1 (R1,R2,R3) + !! ! ! ! !! ! ! ! ! ! ! R2 (R3,R4,R5) + R3 (R4,R5,R6) gives us information that R1, R2, ...R6 are all linked by adjoint relations and might be part of a higher level adjoint relation. !p !p !p !p !p !p p !p !p δ is linear so δp(R1 +R2 +···+Rm) = δp(R1 )+δp(R2 )+···+δp(Rm). If δp(R1 +R1 +···+Rm) = !p 0 and not all of the δp(Ri ) = 0, i = 1, 2, . . . m, these adjoint relations must overlap in a way that produces cancellations in the overall boundary. This reinforces the information used to match up !p sequences of queries; each Ri adding to the context of other adjoint relations in the collection. R!p + R!p + ··· + R!p R!p+1 (R!p ,R!p ,...,R!p ) Alternatively 1 2 m is itself the boundary of a sum of i i1 i2 ip+2 with !p+1 !p !p !p δp+1(ΣiRi ) = R1 + R2 + ··· + Rm.

3.4 Calculation of the Homology for a Simple Relational Landscape

p p In this section we calculate the correlation homology of L =∼ R! when R! is a cycle. It will be shown that even if L is a top level adjoint relation this does not determine everything about it. !p !p−r If R is a cycle, to what extent does it determine the Ri , 1 ≤ r < p in its overall domain? !! ! ! ! Start with the simplest case of R (R1,R2,R3). This is Diagram 3.2

! R (R1,1,R1,2) 1 = a

R˜ R˜ 12 R!! 13

} ˜ ! ! R23 ! R2(R2,1,R2,2) o / R3(R3,1,R3,2)

! 0 Suppose R2 only relates two classes of queries Q1, Q2 in J(R2,1) to three classes of queries Q1, 0 0 ! Q2,, Q3 in J(R2,2). These are the objects in R2 and can be taken to be the example as in diagram

83 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

! ! 3.1, Section 3.1. We shall assume R1 and R3 have rich structures, with many classes of queries being mapped to similarly many classes of queries on both sides of the adjoint relation.

The sides R˜12, R˜23 and R˜13 are the faces deﬁned in Section 3.2.2. These are relations R˜ij ⊂

J(Ri,1) ⊕ J(Ri,2)) ⊕ (J(Rj,1) ⊕ J(Rj,2), i 6= j = 1, 2, 3. But R˜12 ⊂ J(R1,1) ⊕ J(R1,2)) and R˜23 ⊂ ! J(R2,1) ⊕ J(R2,2)) are the graph of the adjoint in R2 so have only 5 maps. Suppose R˜1,3 ⊂ J(R1,1) ⊕ J(R1,2)) ⊕ (J(R3,1) ⊕ J(R3,2) has a large number of query classes, much greater than 5. The cycle condition that

∼ Cat[R˜12] × Cat[R˜12] = Cat[R˜13]

!! ! will fail. That is, if R is a cycle there is very little room for the Ri(Ri,1,Ri,2) to fail to be a cycles. !3 !! !! !! !! !! !! !3 Consider now if R (R1 ,R2 ,R3 ,R4 ) has the R1 = R above, that fails as a cycle, can R still be a cycle? We have the commuting diagram.

!! ! ! ! R1 (R1,1,R1,2,R1,3) = O a

R˜13 R˜14 . R˜12

} ˜ ! !! ! ! ! R34 !! ! ! ! R3 (R3,1,R3,2,R3,3) o . / R4 (R4,1,R4,2,R4,3) i 5 R˜24 ˜ R23 ) u !! ! ! ! R2 (R2,1,R2,2,R2,3)

From the 3 cycle equation

∼ Cat[R˜234] × Cat[R˜124] = Cat[R˜134] × Cat[R˜123] where the R˜ijk can be represented as

˜ !! !! !! Rijk ⊂ J(Ri ) ⊕ J(Rj ) ⊕ J(Rk)

!! ˜ ˜ with R1 appearing in Cat[R124], Cat[R134]. The left hand side of the cycle equation in category form is

˜ ˜ !! !! !! !! !! !! R234 × R124 ⊂ J(R2 ) ⊕ J(R3 ) ⊕ J(R4 ) ⊕ J(R1 ) ⊕ J(R2 ) ⊕ J(R4 ).

Collecting terms

84 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

˜ ˜ !! !! 2 !! !! 2 R234 × R124 ⊂ J(R1 ) ⊕ J(R2 ) ⊕ J(R3 ) ⊕ J(R4 ) and ˜ ˜ !! 2 !! !! 2 !! R134 × R123 ⊂ J(R1 ) ⊕ J(R2 ) ⊕ J(R3 ) ⊕ J(R4 ).

2 2 2 2 On a straight count of adjoint relations in the R˜ijkl this is variety xy zw = x yz w or, canceling, yw = xz which has many solutions in N. The possibility remains that if R!! fails the cycle condition sufﬁciently badly it remains an ob- 3 p struction to R! being a cycle. This becomes less and less likely as the dimension p in R! increases as there is more and more “wriggle room” with p + 1 faces between one dimension and the next. Nevertheless, we introduce a very strict criterion on a class of adjoint relations.

!p !p−r Deﬁnition 18. An adjoint relation R that is a cycle is homogeneous if for any adjoint relation Ri , !p !p−r 1 ≤ r < p, contained in R , Ri will also be a cycle.

!p 3.4.1 Hk(L) for L = Homogeneous R

We have, as usual,

Hp(L) = ker(δp : Cp → Cp−1)/im(δp+1 : Cp+1 → Cp).

We calculate this for p ≤ 3.

! 1. L is a single cycle R (R1,R2). ! δ1(R (R1,R2)) = R1 − R2 gives H1(L) with one dimension (which is Z/2.Z). L0(L) = {R1,R2} and R1 ∼ R2 as R1 − R2 is in B0. But this is also C0 so that H0 = {0}.

!! ! ! ! !! 2. L is a cycle R (R1,R2,R3). We have dim(H2) = 1 being generated by R . !! ! ! ! ˜ ! ! ˜ ! ! ˜ ! ! ˜ ! ! δ2(R (R1,R2,R3)) = R23(R2,R3) − R13(R1,R3) + R12(R1,R2). We require R23(R2,R3) − ˜ ! ! ˜ ! ! ˜! ˜! ˜! ˜ R13(R1,R3) + R12(R1,R2) = 0 which would give R13 = R12 + R23. The faces Rij are level 1 ! adjoint relations Rij representing the adjoint relations J(Ri,1 ∨ Ri,2) J(Rj,1 ∨ Rj,2) and are cycles by the homogeneity assumption. Individually they are not boundaries and provide ! at most two dimensions to H1. The Ri, i = 1, 2, 3, by the deﬁnition of homogeneity, are also cycles so contribute up to 3 dimensions to H1 giving a maximum of 5 dimensions. If their domains interact they might not be independent or be related by a boundary. ! ! C0, so Z0, is generated by all the domains of relations Ri, (6 of the them) and the domains Rij: Ri,1 ∨ Ri,2, Rj,1 ∨ Rj,2. To calculate H0 we have R13,1 ≈ R13,2, R12,1 ≈ R12,2 and R23,1 ≈ R23,2 ! ! and the last two imply the ﬁrst one. The homogeneity assumption for Ri = Ri(Ri,1,Ri,2) implies the domains satisfy Ri,1 ≈ Ri,2. Depending on whether any Ri,k are in adjoint relations !! with Rij,l the maximum dimension for H0 is 5. Generally, span(R ) is 6. Even at this level

85 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

the homology detects whether L is a well separated hierarchy of adjoint relations or whether it intersects itself in some way.

!3 !! !! !! !! 3. L is a cycle R (R1 ,R2 ,R3 ,R4 ). As before, this generates a single top level cycle giving dim(H3) = 1. !3 !! !! !! !! ˜ !! !! !! ˜ !! !! !! ˜ !! !! !! ˜ !! !! !! δ3(R (R1 ,R2 ,R3 ,R4 ) = R234(R2 ,R3 ,R4 )−R134(R1 ,R3 ,R4 )+R124(R1 ,R2 ,R4 )−R123(R1 ,R2 ,R3 ) each of the R˜ijk has the form

!! ! ! ! !! ! ! ! Rijk(R1,ijk,R2,ijk,R3,ijk), for example R234(R1,234,R2,234,R3,234).

3 R! is a cycle so that

!! !! !! !! R234 + R124 = R134 + R123

!3 This contributes 3 dimensions to dim(H2(R )). These homology classes are generated by !! !! any 3 cycles of any selection of four of the Rijk. We also have the cycles Ri , i = 1, 2, 3, 4, !3 !3 adding 4 classes. Thus, for R , dim(H2(R )) ≤ 7. !2 We can then use the same arguments for the homology of H1(R ) above, to show each of !! !! !3 R124 and R123 contributes 2 dimensions to H1(R ) giving, in total, 6 possible dimensions to !3 !! ! ! ! dim(H1(R )). Add to this the domains of Ri (Ri,1,Ri,2,Ri,3) which, by homogeneity, must also be cycles that give 4 × 3 = 12 more potential homology classes. !3 H0(R ) can have a maximum of 24 dimensions which is the largest span of a non-homogeneous !3 ! ! R . In the homogeneous case the Ri,1, Ri,2 are in B0; the only new question is whether any of the underlying L0 relations are the same: each Ri ≈ Rj deletes a dimension from the !3 maximum of 24 dimensions of H0(R ). This differentiates or classiﬁes the different cases of r R! into homology classes for r ≥ 1.

!p !p The homology classes of homogeneous R will always have dim(Hp(L)) = 1. If L = R is not homogeneous each failure of a face takes away one dimension of dim(Hp−1(L)). This does not stop p − 2 level adjoint relations, including those faces of p − 1 relations that are not cycles, from !p being cycles and so contributing to dim(Hp−2(L)). Thus, for non-homogeneous R , the dimension of the Hp−r(L) has to be dealt with case by case, but it always bounded by the homogeneous case. !p The greatest use of Hp−1(L), L = R homogeneous, will be in the general case L when adjoint relations do not intersect. In particular, this shows that the homology is calculable. In the case of intersecting adjoint relations the long exact sequence and excision theorem described below, can add to the ability to calculate the Hp(L), p ∈ N.

86 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

3.4.2 Long Exact Sequence and Excision

The introduction of relative homology in topological spaces required going from a space X to (X,A) with A ⊂ X. We follow the standard topological approach [Greenberg, 1967, Part II sections 13 and 15]. 0 0 0 0 0 Let S ⊂ S with L(S ) = L L = L(S). Deﬁne Cp(L, L ) =defn Cp(L)/Cp(L ).

0 0 δ¯ : Cp(L, L ) → Cp−1(L, L ) is to be deﬁned so the following diagram commutes

π 0 Cp(L) / Cp(L)/Cp(L )

δp δ¯p π 0 Cp−1(L) / Cp−1(L)/Cp−1(L ),

!p 0 0 !p where π is the homomorphism from a module to a quotient. If R ∈ L then δp(L )(R ), the δ operator for L, must also be in L0. This means all the faces

R¯ (R!(p−1),R!(p−1), ...R\!(p−1)...R!(p−1)) i1,i2,..ibk,...ip+1 1 2 k p+1 (after reformulation) have to be in L0. This will be the case if L0 is closed under the adjoint functor relation. The appropriate notion here is that of “adjoint closure” or “correlation closure”. Recall that the span of a nested relation R¯ is all the L0 relations that have to be deﬁned in order to deﬁne the domain relations of R¯ (see Section 3.2.1). p The correlation closure of R! is

!p 0 !p 0 CCl(R ) =defn {R ∈ L0 | ∃R¯ ∈ span(R ) ∧ (R ≈ R¯)},

0 !p where L0 = L0(L ). That is, CCl(R ) is any relation, R, for which J(R) is linked by an adjoint p relation to something in the span of R! . p The closure of the closure is the closure. It corresponds the adjoint functor component of R! and takes in all the intersecting adjoint relations and their intersections. By demanding that L0 is adjoint closed in L, so that

CCl(L0) = L0

0 0 0 then all the constructions for the homology of L are kept in L . In particular, H0(L ) is given by the relations that are adjoint to the domains of adjoint relations R! in L0 and therefore contained in L0. 0 0 Deﬁne the category Sys2 of pairs of objects (L, L ), each from Sys, with L ⊂ L being an adjoint closed embedding, and morphisms

87 CHAPTER 3. CORRELATION HOMOLOGY 3.4. CALCULATION OF THE HOMOLOGY FOR A SIMPLE RELATIONAL LANDSCAPE

(ϕ,ϕ| 0 ) 0 L1 0 (L1, L1)) −−−−−→ (L2, L2).

Lemma 3.4.1. The following are easily seen to follow from the deﬁnition of δ¯

0 π 0 1. The image of in Hp(L) in Hp(L, L ) comes from the maps Cp(L) −→ Cp(L)/Cp(L ).

2. H(L0) is a submodule of H(L).

0 0 3. ker(Hp(L) → Hp(L, L )) = Hp(L ).

0 0 4. im(Hp(L ) → Hp(L, L )) = {0}.

Proof of 1. This is a matter of deﬁnition. 0 0 Proof of 2. δ : Cp(L) → Cp−1(L ) so L is self contained as a submodule of Cp(L). 0 Proof of 3. If ζ maps to Hp−1(L ) as ζ = δp−1(ξ), ξ in Zp(L) it will be zero as δp−1 ◦ δp(ξ) and 0 0 therefore be in the kernel of Hp(L, L ) → Hp−1(L ). 0!p 0!p 0 Proof of 4. Let [R ] be the homology class of R in Cp(L ). By the deﬁnition of the relative chains this is contained in the zero class.

0 0 0 Corollary 3.4.2. · · · → Hp(L ) → Hp(L) → Hp(L, L ) → Hp−1(L ) → ... is exact.

The proof is standard in books of homology and algebraic topology. 2 and 3 in 3.4.1 give 0 exactness at Hp(L). The exactness at Hp(L, L ) follows from 1 and 4.

3.4.3 Excision Theorem

The relational landscape version of the excision theorem is as follows. If L2 ⊂ L1 ⊂ L with CCl(L2) ( L1 and CCl(L1) ( L then

∼ Hp(L\L2, L1 \L2) = Hp(L, L1). (3.12)

!p Conceptually this is straight forward. If R has a boundary in L2 its span and then everything else is in the closure of L2 which is L1. This means the only non-zero classes of the left hand side of !p equation 3.12 are those disjoint from L2. This leaves those R that can have a boundary in L1 but the boundary is disjoint from L2. Such adjoint relations are the non-zero elements of in Hp(L, L1).

For γ in Cp(L) we denote its homology class by [γ] (when deﬁned).

Starting with γ in Cp(L), if span(γ) ⊂ L2 it is in L1 so in δ(γ) ∈ L1 and the homology class [γ] is in the homology classes of both sides of equation 3.12. Let ζ be in the right hand side of 3.12 which we want to map g : ζ 7→ g(ζ) with span(g(ζ)) ∩

Obj(L2) = ∅. This follows from the next lemma.

88 CHAPTER 3. CORRELATION HOMOLOGY 3.5. NOTE ON THE FORMAL DEVELOPMENT OF CORRELATION COMPLEX

Lemma 3.4.3. If γ is in Cp(L) with [γ] = ζ and span(ζ) ∩ Obj(L2) 6= ∅ then γ ∈ Cp(L2).

!p !p !p !p Proof. Suppose ζ is represented by R1 +Bp(L), [γ] by R2 +Bp(L). [γ] = [ζ] means R1 −R2 ∈ Bp(L) !p !p !p+1 !p !p !p+1 so that R1 − R2 = δ(R ). So R1 = R2 + δ(R ). These are all collections of adjoint relations. If 0 0 0 R ∈ span(ζ) and R ∈ L2, there is a chain of adjoint relations linking R to any relation in ζ at least one of which will be in γ, therefore γ is in the closure of L2 which is L2. Hence all the components of γ are in Cp(L2).

If ζ is disjoint from Cp(L2) then there is no γ overlapping Cp(L) that can map it to Cp(L2) and make it zero in Hp(L\L2, L1 \L2). That is, the identity map on this set of classes is one- to-one. Furthermore the proof of the lemma implies that any ζ that overlaps Cp(L2) will be zero in Hp(L\L2, L1 \L2), therefore all homology classes will come from those in Hp(L, L1) giving a surjection:

Hp(L, L1) Hp(L\L2, L1 \L2).

I have previously used the adjective “large-scale” for reporting structures without deﬁning it. We can use correlations to give a measure of large.

Deﬁnition 19. A reporting structure is large if dim(H5(L(S))) ≥ 0 and large of order of n, n > 5, if dim(Hn(L(S))) ≥ 0

Whilst the starting point of the term “large” is arbitrary, dimension 5 implies a 120 relations involved in levels of adjoint relations. This presents a diagram that is a 5 simplex and so difﬁcult to represent as a coherent diagram in two dimensions.

3.5 Note on the Formal Development of Correlation Com- plex

We now give a more formal development of this homology that frames it as an analogy with simplicial homology [Greenberg, 1967]. To do this we need to reformulate the Lp hierarchy as its own category.

Deﬁnition 20. Deﬁne a category of Level templates Cat(Lk)(L) for k = −1, 0, 2, 3... as follows.

1. Cat(L−1) is a ﬁnite set of discrete categories with no morphism other than identity morphisms.

2. Cat(L0) is subsets of objects of Cat(L−1) A morphism X Y in this category takes the form

89 CHAPTER 3. CORRELATION HOMOLOGY 3.5. NOTE ON THE FORMAL DEVELOPMENT OF CORRELATION COMPLEX

X = {xi1 , xi2 , . . . , xim } 7→ {xi1 , xi2 , . . . , xim , xim+1 , . . . , xm+r}.

3. An object in Cat(L1) is the category of pairs (X1, X2) of objects in Cat(L0)). The objects (X1, X2) inherit the sub-objects from Cat(L0). Morphisms within this category are adjoint pairs Sub(X1) Sub(X2).

4. If Cat(Lk) is deﬁned, the objects of Cat(Lk+1)) are k + 2-tuples of objects of Cat(Lk) with their

inherited sub-objects and morphisms follow the pattern of Cat(L1): : if Y = (Yi1 ,Yi2 ,...Yik+2 )

there will be adjoint functors Sub(Yij ) Sub(Yik ) for all j, k.

sptk+1 These form a tower of categories Cat(Lk+1) −−−−→ ℘(Cat(Lk)). Within each part of the tower we can define sub-objects to get a tower of sub-objects. An object has height n if it is not defined beyond Cat(Ln). Recall definition 16 of J(C(L))). The objects of this category are the principal ideals of data- types or relations with their objects being sub-ideals that correspond to classes of queries. The order of sub-ideals provides the morphisms. In particular J(P(x)) is a maximal ideal so for every

J(R) with J(R) ∩ J(P(x)) 6= ∅, there is a (unique) J(R) → J(P(x)). The existence of the map

J(R1) → J(R2) embeds J in L as R1 → R2 typically implies R1 has more logical constraints on the domain variables than R2. J(P(x)) is maximal in J so J(P(x)) is a non-unique terminal object.

Deﬁnition 21. Deﬁne a correlation complex as an indexed set of functors S F : Cat(Lk) J(C(L)) k

sptk+1 commuting with Cat(Lk+1) −−−−→ ℘(Cat(Lk)), k = −1, 0, 1, 2....

Fk Cat(Lk) / J(C(L))

sptk domain objects F k−1 k ℘(Cat(Lk−1)) / J(C(L))

The images are all in the category of categories J(C(L)). The components of a correlation complex are illustrated as follows

1. F−1 : Cat(L−1) = {1, 2, . . . , n} J(D(L)), F−1 : k 7→ J(P(xk)) so the image of F−1 is a set of

categories {J(P(xi)) | i indexed by {1, 2, . . . , n} ∈ Cat(L−1)} ,

2. F0 : Cat(L0) J(C(L)) as F0 : {1, 2, ...n} 7→ J(R(F0)(F−1(1),F−1(2),...,F−1(n))) so is a principal ideal J(R(x1, x2, . . . , xn)). Note that {1, 2, . . . , m} ⊂ {1, 2, . . . , m, m + 1, . . . , m + r} then, by the convention in Cat(L0), {1, 2, . . . , m, m + 1, . . . , m + r} {1, 2, . . . , m}.

90 CHAPTER 3. CORRELATION HOMOLOGY 3.5. NOTE ON THE FORMAL DEVELOPMENT OF CORRELATION COMPLEX

F ({1, 2, . . . , m + r}) = J(R2(xi1 , xi2 , . . . , xim , xim+1 , . . . , xim+r ))

J(R1(xi1 , xi2 , . . . , xim )) = F ({1, 2, . . . , m}).

and, as a special case every J(R(x1, x2, . . . , xm)) J(P(xi)). So, by the constraint on the deﬁnition of Cat(L0), the correlation map F0 is covariant.

3. F1 : Cat(L1) J sends

(X1,X2) 7→ F1[(F0({i1, i2, . . . , ir}, (F0({j1, j2, . . . , js})]

or more explicitly

F1(J(R1(xi1 , xi2 , ...xir ),J(R2(xj1 , xj2 , . . . , xjs )) ∈ Cat({J(R1),J(R2)}).

The morphisms of Cat({J(R1),J(R2)}) are pairs J(R1) J(R2) that are order (or sub- ! object) preserving. Relations R(R1,R2), especially adjoint relations R (R1,R2), can be formulated this way.

4. F2 : Cat(L2) J. The objects of Cat(L2) are tuples (X1,X2,X3) from Cat(L1) and each pair form a Cat(Xi,Xj) these must be mapped to J:

F2(X1,X2,X3) = F2(F1(X1),F1(X2),F1(X3)) therefore ! ! ! ! ! ! ! ! ! F2(J(R1),J(R2),J(R3)) 7→ (J(R1) J(R2),J(R1) J(R3),J(R2) J(R3)) in J.

¯!2 ! ! ! This then deﬁnes a relation R (R1,R2,R3).

5. From these examples the pattern is set and, in the case of F3, we have F3 : Cat(L3) J for (Y1,Y2,Y3,Y4) with each Yi in Cat(L2) and each triple (Yi,Yj,Yk) forming a category in which each morphism from Yi to Yj is a pair Yi Yj. So the functor F3

(Y1,Y2,Y3,Y4) 7→ F3(F2(Y1),F2(Y2),F2(Y3),F2(Y4)) !2 !2 !2 !2 Giving J(R123), J(R234),J(R134),J(R124),

for each triple indicated in the subscripts. The morphisms must also be deﬁned so giving the usual way to deﬁne

!3 !2 !2 !2 !2 F :(Y1,Y2,Y3,Y4) 7→ R (R123,R234,R134,R124)

all of which illustrates the pattern with each level Fk, k ≥ 1 deﬁning a k form.

91 CHAPTER 3. CORRELATION HOMOLOGY 3.5. NOTE ON THE FORMAL DEVELOPMENT OF CORRELATION COMPLEX

Such functors can be seen as corresponding to the formats of the diagrams 3.2 and 3.3 which are various categories of categories. Homology asks whether these also have a specific desirable set of maps that indicate large-scale structure. It picks out high-level relations that have a specific criterion of large-scale structure. This structure is more abstract than concepts available by using only C(L). We now define the Z/2.Z chain complex C(L) as the free group on

{F | F is a correlation complex }.

The image of F is always an adjoint relation and we can identify each F with such a relation. We now deﬁne the correlation homology for a given L.

1. Cp = free group on the set {Fp | F is a correlation complex with Fp its p-th component}

2. δ : Cp → Cp−1 deﬁned by the linear extension of

!p !p−1 !p−1 !p−1 k δ(Cat[R (R1 ,R2 ,...,Rp+1 ))] =defn Σ (−1) Fk k=1,2,...,p+1

where Fk is the “face opposite the k vertex”. The origin of this term is seen in the triangle 3 of relations [D 3.2 R!!] and the tetrahedron of relations [D 3.2 R! ]. In particular each face !p Fk(R ) is a category

!p !p !p−1 !p−1 \!p−1 !p−1 Fk(R ) = Cat[J(R ) ∩ {J((R1 ) ⊕ J(R2 ) ⊕ ..J(Rk ),..., ⊕J(Rp+1 )}] ⊆ !p−1 !p−1 \!p−1 !p−1 Cat[R1 ] × Cat[R2 ],..., Cat[Rk ],..., ×Cat[Rp+1 ].

!p−1 ¯!p−1 It is important that the categories Cat[R2 ] can be represented as some Rk as illustrated in Section 3.2 for the cases p = 2 and p = 3. This ensures that δ : Cp → Cp−1 is well deﬁned.

0 3. δ1 : C → C−1 is not deﬁned on C0. The augmentation proposed in Section 3.4 is a possible solution.

This gives the foundation of the correlation complex and the rest of the development follows in that standard way for complexes, see [Lang, 1969, Greenberg, 1967, Spanier, 1966].

92 Chapter 4

THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS

The previous chapters have examined the concordant properties of a single relational landscape representing a reporting structure. In this chapter we introduce the dynamics of relational landscapes. In an article “More is different” the physicist Phillip Anderson [1972] succinctly described what happens as we scale up concepts:

“ . . . at each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other.” Anderson [1972, p. 393].

4.1 Elaborations of L

An obvious criticism of the representation of a reporting structure as a relational category is that the category is a static representation. But organizations respond to changes in markets, products, suppliers, and equipment as well as legislative and socio-cultural pressures and adapt their reporting structures to address these changes. The evidence derives from the process of software maintenance of their underlying information systems: large-scale systems are constantly being adapted in process of software maintenance. Indeed, Robert Glass [2003, p. 115 (Fact 41)] notes that

“Maintenance typically consumes 40 to 80 percent (average, 60 percent) of software costs. Therefore, it is probably the most important life cycle phase of software”.

93 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

This suggests that we generalize L = L(S) to a family of systems “sprouting” from a starting stage L0. Every member, L¯, of the family will come from the ancestral L0 as a common reduction: ¯ φ : L L0. These families will be called elaborations of L0 and are defined below. We shall use the notation L¯ to indicate an elaboration of L0. The idea of a mathematical study of the growth of systems might seem quixotic as there are so many random factors. However, if we confine ourselves to certain types of growth, the overall dynamics comes into sight. There are many examples of mathematical families where growth of systems can be studied; examples that come to mind are Galois Theory [Cassels and Frohlich, 1967], Expander graphs [Hoory et al., 2006], Moore Postnikov sequences of fibrations [Spanier, 1966, Ch. 8], Projective and Injective Resolutions [Eilenberg and Cartan, 1956] and, no doubt, there are many more.

Elaborations provide tractable models for relational landscape evolution. L0 evolves by adding the capability to make distinctions; by splitting data-types. This is described in Section 2.7 and concepts of the support and cosupport of relational translations are pertinent here. Splitting data- types arbitrarily does indeed provide an elaboration as just described; but this is controlled by insisting that L¯ is at least as correlated as L0. The correlation homology (Section 3.3) is the measure of correlation so this chapter builds on all the concepts that have already been introduced. The topics addressed in this chapter concern the properties we might expect from sets of elaborations. If the theory of reporting structures is abstract then the theory of their possible evolutions is likely to be even more abstract. A number of new concepts are required to deal with classes of possible futures of a given system. Topology will play a key role. Sheaf theory and a rather ex- otic re-interpretation of Etal´ e´ space are used to give families of elaborations a topology. With this topology we have another set of concepts to describe a system. System can vary by having more or less clustered possible evolutions. As much as possible I have avoided the technical aspects of presheaves and sheaves and introduced them in the relevant context. The chapter ends with an section on Stacks (also known as 2-Sheaves) which deﬁne a new higher level of concordance structures. I give conditions for two of the functors introduced in this chapter to be stacks.

4.1.1 The Category E(L0) and consistent families

In the following the initial (ancestral, original) relational landscape is L0.

Deﬁnition 22. An elaboration of a relational landscape L0 is a reduction

¯ φ : L L0 such that elaborations conserve correlation:

94 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

∀p dim(Hp(L¯)) ≥ dim(Hp(L0)).

¯ ¯ φ1 Elaborations form a category with objects elaborations φ : L L0 and a map from L1 −→L0 ¯ φ2 ¯ ¯ to L2 −→L0 being a reduction ψ : L1 L2 that is an elaboration as well, such that

ψ L¯1 / L¯2 φ1 ~ φ2 L0 commutes.

E(L0) denotes the category of elaborations of L0. As a set it is a subfunctor of the right representable functor Sysepi(_ , L0) and so is a (covariant) functor of Sysepi. A slightly weaker form of preserving correlation can be given. An elaboration conserves cor- !p !p+r relations if R gives a homology class in Hp(L0) then, for any elaboration L¯, there is a R¯ , r ≥ 0, !p !p+r corresponding to a homology class in Hp+r(L¯) and R is part of R¯ . This can change the in- p p+r equality in the definition 22 if R! becomes a boundary of some R¯! . This would make a difference to the category of elaborations. But at this stage in the development of concepts it would complicate matters. Consistency ¯ Two elaborations can be in contradiction. Let φi : Li L0 for i = 1, 2. Let P(y) be the data- ¯ type of student in L0 which in L1 becomes P(¯y1) ⇔ tP(¯y1i). This divides the class of students ¯ into subclasses defined by discounted and concessional fees. P(y) becomes Py¯2 ⇔ tP(y2j ) in L2 and these are punitive categories according to religion which require higher fees for people not in the dominant religion. Minorities are frequently represented outside the dominant religion and therefore would be subject to contrary fee requirements. In both cases φi(yi) = y producing an overlap, hence a coupling, on the domains of relations charging for course fees. Here the expansion of distinctions causes conflicts which prevent there being a common elaboration of L¯1 and L¯2. Section 2.7.4 discusses the problems of incompatible reductions which have the same cause. ¯ Given two elaborations L¯1 and L¯2 we write L¯1 ] L¯2 for the pullback in E(L0). That is, for L¯ an elaboration of L¯1 and L¯2 we have:

95 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

L¯

∃ψ

0 ¯ ¯ ψ2 0L1 ] L2 ψ1 ψ2 ψ Ô { 1 # L¯1 L¯2

φ1 # { φ2 L¯0. ¯ ¯ ] makes E(L0) a monoid, the unit being 1L0 . L1 ] L2 need not be the pullback of the categories

L¯1 and L¯2 over L0 in Sysepi. E(L0) is a subcategory of Sysepi and subcategories need not have the same limits or colimits as the larger category. This is particularly so here as elaborations have to satisfy homology requirements that the Sysepi pullbacks need not satisfy. These comments prompt us to deﬁne consistent families of elaborations.

Deﬁnition 23. A consistent family of E(L0) is a maximal set of elaborations S ⊂ E(L0) in which every pair of elaborations has a pullback in S.

It follows that

Proposition 4.1.1. If L1 and L2 belong to consistent family S of elaborations of L0, L1 ]L2 is non-empty.

Consistency plays a subtle role in the monoid of elaborations. It is another illustration of “more is different” Anderson [1972]. With E(L0) a single system becomes just the ﬁrst stage of an ensemble of other related systems. This “more,” this expanded vision, requires us to invent new concepts to deal with relations among the ensemble. Two further examples illustrate the interplay between reporting structures, the capability to make distinctions, consistent families and the conservation of correlations. In order to meet standard demand a manufacturer (perhaps of bread, so a baker) purchases a raw material from many suppliers at different prices. In L0 this is regarded as one material. One elaboration is to separate the raw material into quality categories for accounting purposes but continue to produce undifferentiated batches of a single product. Another possible elaboration is to differentiate the raw material as above, keep them separate for making differently priced products. The ﬁrst elaboration (no change in product) will not add to a level 1 adjoint relation as prices data and product price become unrelated while in the second case the price of supplies and the price of the products stays in relation (and an adjoint relation) with one another. These cannot co-exist in a pullback as in one case all supplies go to the one product while in the other case this is not true. Hence they must be consigned to different consistent families.

96 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

Suppose R1 = Enrol(s, c, x, q) = student s enrolled in course c in semester x for qualification q. This is related to R2 = Asgn(t, c, r, h) = tutor t is assigned to course c to be held in room r at hour h. R1 R2 divides the c students into equivalence classes defined by r and h. Suppose additional statistical information is required for each student such as domestic / international, refugee status, language abilities and so on. This gives many more ways for c to support queries and ways in which students might be serviced such as streams for different groups so, perhaps, refugees have exam scripts customized in their primary language, and these produce dimensions of queries on students that must be matched in J(R2). To conserve the adjoint relation, R2 will need to differentiate at least between rooms and teachers. Without this, the conservation of adjoint relations can be broken. At this stage we do not investigate the possible number of different consistent families. It is possible that they share many characteristics. They all start with a common set of variables they can expand. What frequently separates them is what is conserved. In the first case the single product from multiple suppliers. In the second case, breaking correlations at a lower level can break the correlations at a higher level. The intention to special students for various reasons needs to have supporting structures in many parts of the system.

4.1.2 Topologies on D(L0) and the Sheaf Z(L0)

We now introduce a (pre)sheaf that contains much of the information about E(L0) by describing how each elaboration is deﬁned by creating subclasses of various y ∈ D(L0). φ1 Given L¯1 −→L0 and y ∈ D(L0), y is expanded into n sub-classes in D(L¯1). Suppose a reduction ¯ ψ21 ¯ −1 L2 −−→ L1 leaves p of these variables as they are but expands the rest of φ1 ({y}) into m subclasses in D(L¯2). For the composition we get.

−1 #(φ1 ◦ ψ21) ({y}) = p + (n − p).m. (4.1)

We want to keep track of this type of data as it gives detailed information information about the category E(L0). #φ−1({y}) is always an integer. If y does not expand, #φ−1({y}) is 1 so #φ−1({y}) ≥ 1. ¯ ˜ Given φ : L L0 ∈ E(L0) deﬁne a function φ : D(L0) → N as:

˜ −1 Deﬁnition 24. φ(y) =defn #φ {y} − 1

From the equation 4.1, φ˜1(y) is smaller than φ^1 ◦ ψ21(y) = φ˜2. The cosupport of a relational translation, cospt(φ) (Section 2.7.3), can be rewritten in terms of φ˜:

97 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

cospt(φ) = {y ∈ D(L0) | φ˜(y) > 0}.

If U ⊂ D(L0) and φ˜(y) = 0 for all y ∈ U we say φ is “flat” over U. φ is supported over U if ∀y ∈ U with φ˜(y) > 0. If φ is supported over U so that φ−1(y) has a non-trivial fiber: at least two data-types map to y. The knowledge of all of these sets U that have non-trivial fibering over all elaborations gives a picture of the data-types that can be be expanded or split.

This allows us to deﬁne a topology on D(L0).

Deﬁnition 25. The ﬁbering topology on D(L0) is given by the basic open sets:

U = {U ⊂ D(L0) | ∃φ ∈ E(L0), ∀y ∈ U φ˜(y) > 0}.

The intersection of basic open sets U1 and U2 is clearly open. The topology, the collection of open sets, is the obtained from all unions of the basic open sets.

Continuity among elaborations

φ Pulling back the ﬁbering topology on D(L0) along L¯ −→L0 induces a topology on D(L¯) with basic −1 ¯ open sets φ (U), U open in D(L0). This induced topology makes φi : D(Li) D(L0) continuous −1 ¯ ψij ¯ as φi (U) is always open for each open set U ⊆ D(L0). Given Li −−→ Lj in E(L0), if V is open in ¯ −1 D(Lj) it is so because there is an open U set of D(L0) and V = φj (U).

−1 −1 −1 −1 −1 ψij (V ) = ψij (φj (U)) = (φj ◦ ψij) (U) = φi (U) which is open. Thus ψij is continuous. We conclude:

¯ ¯ Proposition 4.1.2. For an arbitrary φi : Li L0 deﬁne the topology on D(Li) by the open sets {W ⊂ ¯ 0 ˜0 −1 D(Li) | ∃φ ∈ E(L0), ∃U ⊆ D(L0), φ (U) > 0 and W = φi (U)}. Then all the φ in E(L0) are ψij continuous on D(L¯) with respect to the ﬁbering topology as are all the maps L¯i −−→ L¯j in E(L0).

The sheaf Z(L0)

Deﬁne a collection of modules, Z(L0)(U), indexed by open sets U of D(L0), to be the Z/2.Z mod- ˜ ˜ ules generated by the set of “sections” {φ |U | φ | U > 0} over U. Thus Z(L0)(U) is the set:

˜ ξ(U) = Σkak.(φk | U), ak ∈ Z/2.Z

The “global sections” are sections deﬁned for every y in D(L0) so that every data-type is split.

The module of global sections, Z(L0)(D(L0)) is usually empty. The “local sections” are the elements of Z(L0)(U) for some open U, U ( D(L0).

98 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

The collection of modules Z(L0)(U), U ∈ U form a presheaf ([Macdonald, 1968] or almost any book on category theory). The collection Z(L0) is a presheaf if, for any pair V , U in U, with V ⊂ U, there is a restriction map Z(L0)(U) → Z(L0)(V ). As every φ˜ in Z(L0)(U), hence every generator, must have φ˜(y) > 0 for every y ∈ U and so for every y ∈ V , we have a natural restriction map

Z(L0)(U) → Z(L0)(V ). restriction Z(L0)(U) −−−−−−−→Z(L0)(V ) need not be surjective. If the restriction is always surjective

Z(L0)(V ) is merely the restriction of everything that contains it, hence all the information is in the ˜ global sections coming from φ : D(L0) → N. If the restriction Z(L0)(U) → Z(L0)(V ) is not always surjective, it starts to differentiate the elaborations by their effects on areas of D(L0).

A presheaf is a sheaf if given a set of open sets Ui, i ∈ I some indexing set, and ξi are deﬁned on Ui and for each pair i, j, ξi | Ui ∩ Uj = ξj | Ui ∩ Uj, there is a ξ¯ deﬁned on ∪Ui that restricts to i each of the ξi. That is, ξ¯ extends the ξi.

Proposition 4.1.3. Each consistent family S gives rise to a sheaf ZS(L0) in which every section ξ ∈ ˜ ZS(L0(U) corresponds to some combination of φ restricted to U.

Proof. ZS(L0) is the presheaf Z(L0) for a consistent class of elaborations. To obtain the sheaf we use the associated sheaf that can always be constructed from a presheaf. (The construction of the associated sheaf is described in Hartshorne [1977, p. 64], Macdonald [1968, ch. 4], and in Spanier [1966, p. 324] wherein the associated sheaf is referred to as the completion of a presheaf.)

To construct the associate sheaf we do the following

1. Deﬁne the stalk over y ∈ D(L0) as the total set of values at y of all the possible sections ˜ ˜ ξ = Σkai,k.φi,k so ξ(y) = Σkai,k.φi,k(y). This set of values is the stalk at y. The entire set of

stalks over all y ∈ D(L0) form a set E with a map p : E → D(L0) taking any value in the stalk of y to y.

2. Deﬁne the germs of each stalk. A germ over y starts with a possible value, n, in the stalk over y. Take sections ξ that coincide on some open set containing y such that ξ(y) = n. The ξ(y) converge to y on smaller and smaller neighborhoods of y. If y ∈ W and W is the smallest open set about y then a germ at y is the set of sections that are the same on all of W so deﬁned ˜ by a value ξ(y) = Σkak.φk = n. This is an equivalence class of sections on W .

3. Deﬁne E˜ = Z^S(L0) as E˜(U) is the set of sections of germs over U and deﬁne p : E˜ → D(L0)

as p(ξ(y)) = y. E˜ is given the topology that makes each section over any open U of D(L0) continuous. The collection E˜(U) made up from sections of germs (necessarily continuous) is again a presheaf but now with the extension properties required for a sheaf.

99 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

E˜ = Z^S(L0) is the associated sheaf of the presheaf ZS(L0). If ZS(L0) is already a sheaf then the sections deﬁned on E˜ for an open set U are just the set = ZS(L0)(U).

Dropping the subscript S from now on and writing Z(L0) for the sheaf we have:

0 Lemma 4.1.1. A reduction ϕ : L0 L is an open map of the ﬁbering topology U of D(L0): ϕ(U) ⊂ V the open cover of D(L0)

φ Proof. Let L¯ −→L0 be an elaboration in E(L0). If φ makes U ⊂ D(L0) open so φ˜ > 0 on U then, for 0 y ∈ ϕ(U), ϕ]◦ φ(y) ≥ φ˜ > 0 so ϕ(U) is open in the topology of D(L ). Thus, if U is open in D(L0), ϕ(U) is open in D(L0) so ϕ is an open map.

0 Z(ϕ) 0 Proposition 4.1.4. A reduction ϕ : L0 L induces a homomorphism Z(L0) −−−→Z(L ).

Proof. By lemma 4.1.1 ϕ takes opens sets to open sets. If φ˜ is a generator of Z(L0)(U) with φ˜(x) > 0 for all x of U then ϕ]◦ φ(ϕ(x) > 0 for all y = ϕ(x) ∈ V = ϕ(U). This deﬁnes Z(ϕ)(φ˜) in Z(L0).

A sheaf gives us the property that sections that are equal where they overlap can be patched together to extend to a bigger section. ˜ ˜ ¯ φi Suppose each ξi = φi. Each generating section φi : Ui → N comes from some Li −→ L0. If φ˜i = φ˜j on Ui ∩ Uj then we can certainly extend these functions to the union U. Does this ¯ ¯ φi correspond to a section over some F : L L0? We take the limit of the Li −→L0 which exists by the deﬁnition of a consistent family. This gives:

L¯

pullback

× ! L¯1 L¯2 ... L¯n)

v v L0 where

F : Lim(L¯i → L0)

¯ is the reduction that commutes with all the individual φi : L2 L0. Because the construction of Lim(L¯i) as a set of pullbacks of a set U it has to accumulate the splittings of each y in the intersections of the Ui and so F is more like the sum of the φi but not an

100 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

extension. The existence of F shows that each L¯i can sustain further elaboration that propagates to other elaborations.

The open sets U can have isolated open sets {y} for a data type y ∈ D(L0). Such sets make −1 D(L0) into a disconnected topology [Hu, 1964, p. 77]. For example φi (P(y)) provides various −1 ¯ F independently parametrized versions of y say φ (P(y)) = tP(y ) and so the joint pullback L −→ i j ij φi L0 has all the different variations of the expanded y from L¯i −→L0.

The properties of relational landscapes require that the distinctions yij exist in each L¯i because they take part in relations in L¯i. The effect of these relations must be restricted otherwise they will be mapped to relations of y in L0 and possibly break the isolation of y. To avoid this, the relations −1 remain in each ﬁber φi (P(y)). Such elaborations will typically add more and more logic over P(y) but it is constrained to remain in the ﬁber. This will give rise to the open set {y} which make it possible to divide D(L0) into two open sets which is a mark of a disconnected topological space.

The isolated open sets such as {y} means an enhancement from L0 to L¯ can only create new −1 relations in the ﬁber φi (P(y)). Much more common are elaborations of data-types with many relations in which many domain variables need to be split into classes. (This is the bane of maintenance programmers when the logic of the system is poorly understood; one apparently innocent change leads to a another and then to another in a struggle to maintain consistency. In this case

P(yi), i = 1, 2...n in D(L0) all become part of a single elaboration that cannot be decomposed into stages.)

4.1.3 The Cohomology of Z(L0)

When the context is clear we drop the L0 from Z(L0). Starting with sheaf Z deﬁned on U and following Hartshorne [1977, III section 4] deﬁne the p cochain modules

p C (U, Z) =defn Π Z(Ui1 ∩ Ui2 ∩ · · · ∩ Up+1). i1

The superscript on the Cp counts the number of intersections of open sets. U includes all ﬁnite 0 intersections so that the intersections Uij ∩ Uik ∩ · · · ∩ Uim are also part of the C . p ˜ p p p+1 If Φ ∈ C it is a sequence of functions φi1,i2,...,ip+1 ∈ Z(Ui1 ∩ Ui2 ∩ · · · ∩ Uip+1 ), ∂ : C → C is deﬁned as

p p k ˜ ∂ (Φ) =< ∂ (φi1,i2,...,ip+1 ) >=< Σ(−1) φ ˆ | Ui1 ∩ Ui2 ∩ ... Uci · · · ∩ Uip+2 ) >. k i1,i2,...,ik,...,ip+2 k where the angle brackets remind us that Φ is vector of expressions indexed by p + 1 intersections of open sets. Note that φ˜ is well deﬁned on the restriction Ui ∩ Ui ∩ ... Ui · · · ∩ Ui i1,i2,...,iˆk,...,ip+2 1 2 ck p+2

101 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L which has p + 1 intersections and so Φ is an element of Cp yet ∂p(Φ) is a well deﬁned component of Cp+1. We can deﬁne the usual cocycles, coboundaries and cohomology classes.

1. The p th cocycles are the group Zp = ker(∂p) = {Φ ∈ Cp | ∂p(Φ) = 0}.

2. The p th coboundary is the group Bp = im(∂p−1) = {Φ ∈ Cp | ∃Ψ ∈ Cp−1, ∂p−1(Ψ) = Φ}. This is a subgroup of the cocyles because ∂p ◦ ∂p−1 = 0.

3. The p th cohomology classes, Hp are the elements of the quotient group Zp/Bp.

The deﬁnition of the cohomology is also applicable to presheaves.

Coboundaries and the meaning of cohomology ˜ A p coboundary is a section ξi = Σjai,j.φi,j deﬁned on an intersection of p + 1 open sets Ui1 ∩ Ui2 ∩ p p−1 · · · ∩ Uip+1 (so p ∩ symbols) in C and there is a section Ψ with ∂ (Ψ) = ξ. Ψ will be a sum of sets ˜ ˜ of φi,j,k deﬁned on subsets of p − 1 subsets of Ui1 ∩ Ui2 ∩ · · · ∩ Uip+1 . The k in φi,j,k telling us which particular subset of the i-th intersection Ui1 ∩ Ui2 ∩ · · · ∩ Uip+1 has been missed out. Rearranging p−1 the resulting ∂ (Ψ) gives a set of expressions for φ˜i,j(y) in terms of the restrictions of some of the

φi,j,k to Ui1 ∩ Ui2 ∩ · · · ∩ Uip+1 . p p p p What is the effect of this on H =defn Z /B ? H is to be seen as the set of equivalence classes ξ1 ∼ ξ2 whenever the difference between ξ1 and ξ2 is a coboundary. This will happen when there is a common subset of the φ˜ in the expressions of ξ1 and ξ2 that are from a coboundary; the p−1 difference ξ1 − ξ2 is in the image of ∂ . This, in turn, means that cohomology classes are new phenomena that appear at the p level, sets of overlapping φ˜ that cannot be expressed in terms of fewer intersections.

Interpretation of cocycles in low dimensions

The key to understanding what the cohomology tells us is in the nature of the cocycles. Here we work through what cocycles tell us in the lowest dimension.

1. p = 0. 0 C = Z(Ui1 ) ⊕ Z(Ui2 ) ⊕ · · · ⊕ Z(Uim+1 ).

0 ˜ ˜ ∂ : Φ = (φi1 ∈ Z(Ui1 ), φi2 ∈ Z(Ui2 )), ....) 7→ φ˜ (U ∩ U ) − φ˜ (U ∩ U ) ... φ˜ (U ∩ U ) − φ˜ (U ∩ U ) ... ) i2 i1 i2 i1 i1 i2 ij ij ij+1 ij+1 ij ij+1

102 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.1. ELABORATIONS OF L

where the ... cover all pairs of intersections. 0 ˜ ˜ If ∂ (Φ) = 0 each φij ∈ Z(Uij ) matches up with all other φik on the non empty intersections

Uij ∩ Uik . As Z is a sheaf these must result in an extension to the union of the open sets.

This implies ∪ cospt(φij ) = ∪U. The collection Φ deﬁnes a section in which every ij |j=1,2,...n data-type contained in an open set is included. In other words, the largest possible open set

of D(L0). Here we exclude D(L0) if it is not covered by unions of sets in U. This demonstrates:

¯ Proposition 4.1.5. If there exist reductions L L0 with every ﬁber for each data-type in each open 0 set having at least two elements then H (Z(L0)) 6= 0.

2. p=1 1 C = Z(U ∩ U ) ⊕ Z(U ∩ U ) ⊕ ...Z(U ∩ U 0 ).... i1 i2 i2 i3 ij ij

1 ˜ ˜ ˜ ∂ :(φ3 ∈ Z(Ui1 ∩ Ui2 ), φ1 ∈ Z(Ui2 ∩ Ui3 ), φ2 ∈ Z(Ui1 ∩ Ui3 ) 7→ ˜ ˜ ˜ −φ3(Ui1 ∩ Ui2 ∩ Ui3 ) + φ2(Ui1 ∩ Ui2 ∩ Ui3 ) − φ2(Ui1 ∩ Ui2 ∩ Ui3 )

is a cocyle if the local sections of Φ have the cocyle relation φ˜3 + φ˜1 = φ˜2. This must be interpreted as

−1 −1 −1 #(φ2 ({y})) = #(φ3 ({y})) + #(φ1 ({y})) for all y ∈ Ui1 ∩ Ui2 ∩ Ui3

One of the ways this can occur is when φ2 is restricted to Ui1 ∩ Ui2 ∩ Ui3 , it is the pullback of

φ3 and φ1 combining their effects over Ui1 ∩ Ui2 ∩ Ui3 .

3. p=2 2 2 C is the product of factors Z(Ui1 ∩Ui2 ∩Ui3 ) with i1 < i2 < i3. Φ ∈ C is therefore a sequence

of φjkl ∈ Z(Uij ∩ Uik ∩ Uil ). The calculation is on intersections of four open sets indexed by i1 < i2 < i3 < i4. Writing φk for the reduction with the missing index ik we have

2 ˜ ˜ ˜ ∂ (Φ)i1,i2,i3,i4 = (φ1 ∈ Z(Ui2 ∩ Ui3 ∩ Ui4 ), φ2 ∈ Z(Ui1 ∩ Ui3 ∩ Ui4 ), φ3 ∈ ˜ Z(Ui1 ∩ Ui2 ∩ Ui4 ), φ4 ∈ Z(Ui1 ∩ Ui2 ∩ Ui3 ) 7→ ˜ ˜ ˜ ˜ −φ1(Ui1 ∩Ui2 ∩Ui3 ∩Ui4 )+φ2(Ui1 ∩Ui2 ∩Ui3 ∩Ui4 )−φ3(Ui1 ∩Ui2 ∩Ui3 ∩Ui4 )+φ4(Ui1 ∩Ui2 ∩Ui3 ∩Ui4 )

which is a cocycle if φ˜1 + φ˜3 = φ˜2 + φ˜4. In terms of the reductions this is

−1 −1 −1 −1 #(φ1 ({y})) + #(φ3 ({y})) = #(φ2 ({y})) + #(φ4 ({y}))

103 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

for each y ∈ Ui1 ∩Ui2 ∩Ui3 ∩Ui4 . Clearly this can be satisﬁed in a variety of ways. If cospt(φi)

is concentrated around the sets Uij , j = 1, 2, 3, 4 the cocycle equation might be a consequence

φ˜1 and φ˜3 having a common pullback that has the same local structure as the pullback of φ˜2

and φ˜4. These become two possible routes to achieve an enhancement of L0 for a particular set of data-types.

The cohomology of Z(L0) contains much of the information about E(L0) especially which reductions compose, if only in small areas, and where reductions expand the data-types of D(L0). These examples set the scene for interpreting the deeper algebraic structure of E(L(S)).

4.2 The Homology Tableau for Z(L0)

4.2.1 Distinguishing Elaborations ¯ ¯ Let φ1 : L1 L0 and φ2 : L2 L0 and W open in D(L0). ¯ −1 Over W , L1 has additional logic generated by relations among split data-types, φ1 ({y}), y ∈ W , and functions involving these additional distinctions. In the Figure 4.1 below, the ∗, ∗,..., ∗ represent “layers” of logic in L¯1 and are the relations that are constrained in and among the −1 ¯ layers. φ1 (W ) relates to other parts of L1 only through relations expressed in terms of the ~xi,

—– : − − − ∗, ∗, ∗, ∗, ∗, ∗ φ1 D(L¯ ): − − − ∗, ∗, ∗, ∗, ∗, ∗ 1 O φ 1 D(L¯ ): − − − ∗, ∗, ∗, ∗, ∗, ∗ 1 O φ 1 D(L¯ ): − − − ∗, ∗, ∗, ∗, ∗, ∗ 1 O φ1 ¯ D(L1): − − − ∗, ∗, ∗, ∗, ∗, ∗ φ1 —– : − − − o [~x1, ~x2, ..., ~xn] / etc φ1 D(L0): − − − o W = [y1, y2, ...yn] / etc

Figure 4.1: An elaboration with few implications from split data-types

−1 −1 i = 1, 2.3...m ≥ n. The only elements of φ1 (W ) that relate to anything outside φ (W ) are in −1 φ (W ) ∩ {~x1, ~x2, ..., ~xn}.

104 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

¯ −1 The elaboration φ2 : L2 L0 has φ2 (W ) with equally many vertical layers but each layer is well connected to other parts of L¯2 through implications. This is represented in Figure 4.2. L¯1 might be a preliminary elaboration before integration with adjoint relations that is completed in

L¯2

o / —– : − − −− ∗, ∗, ∗, ∗, ∗, ∗ 49 etc φ s ¯ * D(L2): − − − o ∗, ∗, ∗, ∗, ∗, ∗ 4 etc k O : φ2 s ¯ *% D(L2): − − − ∗, ∗, ∗, ∗, ∗, ∗ 4 etc k OO 9 φ2 s ¯ * D(L2 : − − − o ∗, ∗, ∗, ∗, ∗, ∗ 4 etc k O φ2 s * —– : − − −− o [~x1, ~x2, ..., ~xn] / etc φ2 D(L0): − − − o W = [y1, y2, ...yn] / etc

Figure 4.2: An elaboration with many implications ﬂowing from split data-types.

˜ ˜ While it is possible that W ( U and φ1 is ﬂat on D(L0) \ U it is unlikely that this is so for φ2 −1 especially if φ2 (W ) is part of an adjoint relation. We would like to have information where the concentrated logic of each elaboration is placed relative to the information in Z(L0).

4.2.2 The Homology Tableau ¯ ¯ ¯ Each φ : L L0 has the associated homology H∗(L) (denoting the sequence Hp(L) for p = 0, 1, 2,... , not, in this case, the direct sum, which would complicate the definition of the concepts to follow). This tells us of the existence of correlations, hence the enhancement of coordination capabilities, but not which data-types they connect. Z gives an indication of how the various ¯ ˜ φ : L L(S) are locally related but not what is giving them the values of φ. In this section the two sets of information are linked. Recall the definition of the span of an adjoint relation (Definition 3.2.1). Define the domain of p R! as the domains of its span:

!p 0 dom.span(R ) =defn ∪ dom(R ). R0∈span(R!p )

p If R! is in L¯ this is a subset of D(L¯). ¯ !p !p Let ζ ∈ Hp(L) be the class of a sum of Ri , i = 1, 2, ..., n. Although ζ is represented by R , it 0 !p can also be represented by another (set of) (R ) if the difference is in the boundary Bp(L¯), hence

105 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

!p 0 !p in the image δp+1(Cp+1(L¯)). This set of boundary relations will be part of span of R and (R ) . We make the deﬁnition:

!p !p ¯ Deﬁnition 26. dom.span(ζ) =defn ∪i{(dom.span(Ri ) | Ri + Bp((L) = ζ}

This can be extended when the homology class is a sum of adjoint relations. In most cases this p simply dom.span(R! ).

Definition 27. Define the “Homology tableau” as follows. ¯ For open U ⊂ D(L0) and φ : L L0 define

hp(φU ): Hp(L¯) → Z(L0)(U) given by

˜ ˜ !p hp(φU )(ζ) = φ(dom.span(ζ) ∩ U) = φ(dom.span(R ) ∩ U) with the convention φ˜(∅) = 0.

From now on we denote the set of functions hp(φU ) by h(φU ) acting on H∗(L¯).

Lemma 4.2.1. h(φ) is sub-additive: h(φ)(ζ1 + ζ2) ≤ h(φ)(ζ1) + h(φ)(ζ2).

Proof. For each open U, ˜ h(φU )(ζ1 + ζ2) = φ(dom.span(ζ1 + ζ2) ∩ U

!p !p −1 Assume ζi, i = 1, 2 are represented by two different Ri . If y is in φ(Ri ) then φ ({y}) counts the !p −1 !p new subclasses in both of the Ri . If these are disjoint then φ ({y}) gives the sum from R1 and !p R2 so h(φU )(ζ1) + h(φU )(ζ2). !p !p If dom.span(R1 ) ∩ dom.span(R2 ) 6= ∅ it is possible that !p !p φ(dom.span(R1 ) ∩ dom.span(R2 ) ∩ U !p !p −1 is non-empty. For y ∈ φ(dom.span(R1 ) ∩ dom.span(R2 ) ∩ U), φ ({y}) is less than the sum of h(φU )(ζ1) and h(φU )(ζ2).

A Homology Tableau is intended to make explicitly just where the correlated areas that generate homology classes are localized relative to D(L0). This gives a picture in the manner of Figures 4.1 and 4.2 where L0 has been expanded. It also identiﬁes which part of each elaboration is the origin of the sections in Z(L0)(U).

106 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

! Expanding adjoint relations R (R1,R2)

Homology classes play a role in expanding variables and act as "propagators" of expansion. The homology tableau is defined in order to describe how the higher homology classes are located relative to D(L0). We illustrate this with simple examples. The Figures 4.1 and 4.2 illustrate that over an open set W there can be a large number of relations in the fiber but also relations in L¯ intersecting the fiber. For example, suppose we have the following cycle

!! ! ! 0 0 0 0 ! 000 000 R (R1(R1(x11, x12),R2(x21, x22)),R2(R1(x11, x12),R2(x21, x22)),R3(R1 (y11, y12),R2 (y21, y22))) representing ζ ∈ H2(L¯). Given the assignments

0 1. φ(x11) = φ(x11) = y1,

0 2. φ(x12) = φ(x12) = y2,

0 0 3. φ(x21) = φ(x21) = y1,

0 0 4. φ(x22) = φ(x22) = y2, and

0 5. φ(zij) = yij. we can represent this “over” L0 as follows.

! o !! .RO 1 R

| " R1(x11, x12) R2(x21, x22)

Ð ¯ ! L R2

t } 0 0 0 0 0 0 ! R1(x11, x12) R2(x21, x22) R3 φ & 00 00 .R1(y11, y12) R2(y21, y22)

˜ ˜0 0 0 00 00 L0 : R(y1, y2) R (y1, y2) R1(y11, y12) R2(y21, y22)

! 0 ! 0 } ! D(L0): y1 y2 y1 y2 y11 y12 y21 y22

107 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

Here, the adjoint relations joining their domain components by adjoint functors are represented by dotted arrows. We have

˜ ˜ 0 1. φ(yi) = 1 and φ(yi) = 1 for i = 1, 2 so W = {y1, y2, y3, y4} is open

2. φ˜(yij) = 0 so for this reduction V = {y11, y12, y21, y22} is not “seen” to be open.

In this case, if ζ represents the cycle associated with R!! then ζ can be split into ζ(↔) + ζ(⊥), “horizontal” and “vertical” parts of ζ relative to L0. Neither ζ(↔) nor ζ(⊥) are in the same homology ! ! ! dimension (the p in Hp) as ζ. Here R1 and R2 are in ζ(⊥) and R3 in ζ(↔). ˜ 0 0 The example has many interpretations: y1 and y2 are data-types related by R, y1 and y2 and R˜0 similarly, perhaps students and outcomes in courses and tutorial groups and results of coop- ! ! erative projects. Splitting the set of students into two classes gives a clear relation R1 and R2 but with a third set of variables, perhaps age and the existence of partners, the z variables, gives the most comprehensive adjoint relation. But unless the set of people is divided into classes as in the elaboration, no two way functional relation is able to be deﬁned. ! An adjoint relation R (R1,R2) in L1(L0) requires adjoint functors α : J(R1) J(R2): β. Sup- pose R1 = R1(y1, y2) and R2 = R2(z1, z2) and these need not be coupled (although they probably are). Recalling the adjoint functor data in diagram 3.1 each query type r1i on R1 gives

1. α : r1i 7→ r2j which implies that, if

2. β : r2j 7→ r1k ≤ r1i then, if

0 3. β : r 2l 7→ r1m then

0 0 4. α : r1m 7→ r 2r ≥ r 2l.

φ ! ! Given L¯ −→L0 in E(L0), R (R1,R2) is the image of a similar R¯ (R¯1, R¯2) in L1(L¯). Without loss of generality we can assume R1(y1, y2) expands by replacing y1 with subclasses {y11, y12, ...y1p}. 0 ¯ J(R1) becomes ⊕ J(R1j) = J(R1). j=1,2,...p The list of the effects of α and β above expands as follows.

0 1. α : {r1rs | r, s ≤ p, deﬁned by 2j} 7→ r2j which implies that if

2. β : r2j 7→ r1r0s0 ≤ r1i then if

0 3. β : r 2l 7→ r1r0s0 then

0 0 0 4. α : r1r0,s0 7→ r 2r ≥ r 2l

108 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

The domain of the “expanded” β0 in L¯ can only have the same number of different queries as 0 0! β ◦ α ≤ 1 0 R before and these define classes for which J(R1). These classes group the queries (in ) ! in terms of the clustering around the original queries defined in R (R1,R2). This “stratifies” the 0 queries of the expanded ⊕ J(R1j) via the β ordered function. j=1,2,...p ! If y2 similarly expands, R¯ is a category of queries

¯ 0 0 ¯ J(R1) = ⊕ J(R1j) ⊕ J(R2k) = J(R2). j=1,2,...p k=1,2,...q

0 ¯! 0 and a new β reﬁnes the classes of queries in R to more closely match those in the various R2k.

A R!! example.

!! ! ! 0 0 ! 00 00 If R (R1(R1,R2),R2(R1,R2),R3(R1,R2)) is in L2(L0), it is a triangle of adjoint functors among φ !! the domain categories. This must lift to L2(L¯) for all elaborations L¯ −→L0. A fully expanded R¯ ! might replace R (R1,R2) expressed as α : J(R1) J(R2): β with the domain or range ¯ ¯ αjk ⊕ J(R1j) ⊕ J(R2k): βkj j=1,2,...p k=1,2,...q

! 0 0 ! 00 00 and similarly for the lifting of R2(R1,R2) and R3(R1,R2). These expanded adjoint relations are !! further interrelated by the lifting of correlations to R that must apply to the αjk and βkj. Here we can see the effect of the propagation of the growth of distinctions and how it increases !p the requirements of lifting R from L0 to an elaboration L¯. This lifting can be a signiﬁcant part in realizing the potential of the elaboration L¯. Indeed it can take months or years to be fully realize in an corporate reporting structure. φ L¯ −→L0 ∈ E(L0) gives the homomorphism Hp(φ): Hp(L¯) → Hp(L0). Suppose ζ ∈ H2(L¯) ! but all the relations in ζ map to R (R1(~y1),R2(~y2)) in L0. H2(φ): ζ 7→ H2(φ(ζ)) but φ(ζ) = ! R (R1(~y1),R2(~y2)) which is not in L2(L0) so not in H2(L0). This makes H2(φ)(ζ) = 0. Thus, in !! Y = dom.span(φ(ζ)) which contains dom.span(φ(R¯ )), there are components yij of ~yi, i = 1, 2, for which φ˜(yij) > 0 and so Y contains an open set. This generalizes to

Lemma 4.2.2. For each ζ ∈ ker(Hp(φ)), if Y = dom.span(φ(ζ)) then Y contains an open set.

The condition ζ ∈ ker(φ) simple alerts one that this is a new adjoint relation not found in L0. p This identiﬁes elaborations that create adjoint relations R! “in the ﬁber”.

109 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

4.2.3 The Homology Tableau as a Functor

Deﬁne the category of the homology tableau over a relational landscape L0, denoted ht(L0), as h having the objects the sequences of maps Hp(L¯) −→Z(L0) for p = 0, 1, 2, ... where L¯ ∈ E(S0). This is a set of maps doubly indexed by open sets and the dimension of the homology. The morphisms of ht(L0) are commuting triangles:

H∗(ψij ) H∗(L¯i) / H∗(L¯j) (4.2)

h(φi) h(φj ) $ y Z(L0).

This is a collection of diagrams one for each p and U:

Hp(ψij ) Hp(L¯i) / Hp(L¯j) h(φi,U ) h(φj,U ) & x Z(L0)(U).

These morphisms are very strict. If we follow a homology class ζ ∈ Hp(L¯i), h(φi,U )(ζ) is part of the function φ˜ | U. For the diagram to commute ψij(ζ) ∈ Hp(L¯j) cannot be zero. For each U, h(φi,U )(ζ) is to be the same function as h(φj,U )(ψij(ζ)). This does not mean L¯i is isomorphic to L¯j, ψij only that the reduction L¯i −−→ L¯j does not change any part that intersects the span of homology classes. We can summarize this as:

Proposition 4.2.1. The morphisms of ht(L0) deﬁne isomorphic homology classes of elaborations of L0 though the elaborations need not be isomorphic away from the span of homology classes.

ϕ 0 Proposition 4.2.2. ht is a category valued functor on Sysepi. A reduction L0 −→L induces a functor ht(ϕ) 0 ht(L0) −−−→ ht(L ).

h(φ) Proof. Let H∗(L¯) −−→ Z(L0) be an object in ht(L0). By proposition 4.1.4 ϕ induces a homomor- Z(ϕ) 0 0 phism Z(L0) −−−→Z(L ) producing a homology tableau in ht(L ):

h(φ) Z(ϕ) 0 H∗(L¯) −−→Z(L0) −−−→Z(L ).

This deﬁnes the functor ht(ϕ) on objects. Let Hp(ψij) be as in the diagram 4.2. This becomes a morphisms in ht(L0) by appending the map Z(ϕ)

110 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

Hp(ψij ) Hp(L¯i) / Hp(L¯j) h(φi,U ) h(φj,U ) & x Z(L0)(U)

Z(ϕ) Z(L0)

As h(φi,U ) = h(φj,U ) ◦ Hp(ψij) so also

Z(ϕ) ◦ h(φi,U ) = Z(ϕ) ◦ h(φj,U ) ◦ Hp(ψij) which deﬁnes ht(ϕ) on morphisms.

0 The category ht(L0) maps to a subcategory of ht(L ). The mapping is onto if every elaboration 0 of L factors through L0.

We now deﬁne the homology tableaux functor, H, on Sysepi to the category of sets by taking the colimit of all the homology tableau over a given L0, Colim(ht(L0)), giving a new functor H(L0). ¯ ¯ This colimit is over all the reductions ψi,j : Li Lj in E(L0). As a standard exercise in category theory we have

0 Proposition 4.2.3. If ϕ : L0 L is a reduction then the homology tableaux is a covariant functor H Sysepi −→ Sets.

Proof. The proof is just a matter of showing the internal maps that deﬁne the colimit to obtain h H(L0) = Colim (H∗(L¯) −→Z(L0)) L∈E¯ (L0)

ϕ 0 Z(ϕ) 0 compose with L0 −→L and Z(L0) −−−→Z(L ) to give a map to

0 0 h 0 H(L ) = Colim (H∗(L¯ ) −→Z(L )) L¯0∈E(L0) but this is assured by the propositions 4.2.2.

ψ1 ψ2 ψj−1 The elements of the set H(L0) are sequences ζ1 −→ ζ2 −→ ... −−−→ ζj that give the same h value for an open set U ⊂ D(L0). H is a functor that captures the global details of L0. H(L0) is a very comprehensive functor of L0 giving all the homology classes of elaborations but distinguishing them by the classes of data-types they divide into sub-types. If we think of how a relational landscape L0 might evolve, at least per consistent family of elaborations, H(L0) gives the details.

This functor H : Sysepi → Sets carries with it a great deal of structure about L0 and suggests that the image of the functor needs to be investigated for what it can tells us. This will be taken up in the Section 4.5.

111 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

4.2.4 Homology Tableau and Sheaf Cohomology

Suppose we have

h(φi,U): H∗(L¯i) → Z(L0)(U),

¯ ˜ then, for any ζi ∈ Hp(Li), i = 1, 2 . . . m, we have φi : dom.span(ζi) ∩ U → N. We have the following cases:

1. dom.span(ζi) do not overlap on U.

2. dom.span(ζi) = Vi open sets, and ∩iVi 6= ∅. If, also, there is some algebraic relation among

the φ˜i this could be a cocycle in the m-th cohomology of Z.

˜ What are the conditions for φi : dom.span(ζi) ∩ U → N to give a cocycle ? ¯ ¯ ψi ¯ φ ¯ Suppose L¯ −→ L¯i, L¯ −→L0, φ¯ = φi ◦ ψi and ζ¯ ∈ Hp(L¯) maps to ζi ∈ Hp(L¯i) where U is the ˜ ˜ largest subset in D(L0) for which φ¯ is greater than zero in dom.span(ζ¯), If φ¯(y) = ξ at y then φ¯ ¯ maps ξ different data-types in D(L¯) to y. Then, also, φ¯ = φi ◦ψi maps these variables to y. Splitting data-types, especially around a correlation class, will split in ways that carries to other, related data-types. These will then have positive φ˜i values. That is, we expect there are open sets Vi all including y and for which the φ˜i(Vi) > 0. For variables linked to y by various relations, equations such as φ˜1 + φ˜3 = φ˜2 + φ˜4 might hold. In such a case we can conclude that φ˜1 + φ˜3 = φ˜2 + φ˜4 is 3 true for V1 ∩ V2 ∩ V3 ∩ V4 so is a cocycle in Z (Z). ¯ Define G(U) to be the equivalent class of the values of {h(φ, U) | φ : L L0 ∈ E(L0)}. This equivalence class is the “localization” of ht(L0) over U. G defines a presheaf with values in the category ht. ψ ψ2 ψj−1 As each element of H is a sequence ζ1 −→ ζ2 −→ ... −−−→ ζj defining the same h φ˜ function for some U. These are the non-zero values in some G(Ui) where dom.span(ζ) ∩ Ui is non-empty. This defines a single function [ζ1](Ui) = φ˜1(ζi) on Ui. Define

Ξ: H(L0) → Π G(Ui) as ζ 7→ [ζ]: U → N. Ui open

Of course, on Ui, we have [ζ1](Ui) = [ζ2](Ui) ... [ζj](Ui). We also have

0 Π G(Ui) → C (Z) Ui open as every class in G(Ui) is a single function in Z.

Each class in G(Ui) is identiﬁed by the value of some h(U, φ). Deﬁne

112 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.2. THE HOMOLOGY TABLEAU FOR Z(L0)

∆1 .−. 1 Π G(Ui) −−→ Π G(Ui) × G(Uj) −−→ C (Z) Ui open Ui∩Uj 6=∅

∆1 : ([ζi1 ], [ζi1 ],... [ζiN ]) 7→ the sequence of ([ζik ], [ζij ] | (Ui ∩ Uj)) ... ) and

. − . : ([ζik ], [ζij ]) | (Ui ∩ Uj) 7→ (h(φik ,Ui ∩ Ui)(ζik ) − h(φik ,Ui ∩ Ui)(ζij ) ∈ Z(Ui ∩ Uj)

We now have

Ξ / ∆1 / ∆2 / H(L0) Π G(Ui) Π G(Ui1 ) × G(Ui2 ) Π G(Ui1 ) × G(Ui2 ) × G(Ui3 ) Ui open Ui1 ,Ui2 open Ui1 ,Ui2 ,Ui3 open .−. .−.+.

0 1 Z0(Z) / ker / C0(Z) ∂ / C1(Z) ∂ / C2(Z)

The induced map denoted by the dashed line comes from the commuting of the ﬁrst square and so induces a map to the kernel of ∂0 where we treat the kernel as a set deﬁned as the equalizer of 0 0 1 ∂ and the constant map from C (Z) to 0 ∈ C (Z) This can be repeated to get maps from H(L0) so we get:

Ξ / ∆1 / ∆2 / H(L0) Π G(Ui) Π G(Ui1 ) × G(Ui2 ) Π G(Ui1 ) × G(Ui2 ) × G(Ui3 ) Ui open Ui1 ,Ui2 open Ui1 ,Ui2 ,Ui3 open .−. .−.+.

% 1 Z1(Z) / ker / C1(Z) ∂ / C2(Z) and so on to obtain induced maps to any Zp(Z), p = 0, 1, 2 ... and then by projection to the cohomology classes Hp(Z).

This constructs a map from H(L0) to the cohomology Z(L0) which is the “tableau” of sets of correlation classes and their combined contribution to the cohomology of the topology on D(L0). Over consistent classes of the evolution of the reporting structure represented by L, this is an encyclopedia of what is possible.

We end this section with a question. To what extent are L¯i “cohomologically independent” when they do not share any Z cohomology classes?

113 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

4.3 The “Global Theory” of Elaborations

4.3.1 Topologies on E(L0)

Let EN (L0) denote all possible elaborations up to homology dimension N for L0. As N increases, adjoint relations that represent homology classes span factorial N relations and so homology classes run out of relations in L0. Consequently, as N increases, the high N adjoint relations exist vertically (as in ζ(⊥) towards the end of Section 4.2.2) over open sets in L0.

Each L¯ in EN (L0) is associated with sets of reductions:

¯ ¯ ¯ L Li1 Li2 ··· L0.

0 Likewise there might be a variety of embeddings from L0 to L¯ and if L¯ is similar to L¯ then we would like to a have a clear idea on this similarity. The homology tableaux captures much of the algebraic level of information in the functor E(L0) but much of the intuition of similarity lies outside the homology classes. To capture a sense of the way elaborations overlap or move easily into one another, or not, we put a topology on EN (L0). Topology gives an overall concept of nearness or “being in the neighborhood of”. (A reference for general topology is Hu [1964] but any modern reference will do). The advantage is that we can use a well deﬁned sense of nearness to formulate new concepts about different enhancements of the reporting structure of L0 (and their supporting information system). This provides a foundation for investigating the space of phases or stages (as in theater) where enhancements “play out”.

4.3.2 The Category E¯N (L0)

Deﬁnition 28. The category E¯N (L0).

1. The objects of E¯N (L0) are the objects of the categories L¯ of E(L0) with the “cutoff” N being the highest correlation homology dimension of L¯.

ψi,j 2. A morphism in E¯N (L0) will be any composition of the form ψj,k ◦ f ◦ ψi,j where L¯i −−→ L¯j and ψj,k L¯j −−→ L¯k are reductions and f is a morphism, hence a dependency, in L¯k.

3. Deﬁne

¯ ¯ Φ: EN (L0) L0

¯ ¯ such that Φ¯ restricted to L¯i is φi and Φ(¯ ψj,k ◦ f ◦ ψi,j) is evaluated as

114 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

φj (f) 0 0 φi(R¯) = φj(ψi,j(R¯)) −−−→ φj(R¯ ) = φk(ψj,k(R¯ )).

¯ φi Φ¯ is the limit of the diagrams L¯i −→L0. ¯ ¯ 4. A section s is a functor from L0 to EN (L0) such that Φ ◦ s = 1L0 . As a diagram:

E¯ (L ) N? 0

s Φ¯

0 P L0

¯ where for R¯ ∈ L¯i, R¯ 7→ φi(R¯) = Φ(¯ R¯).

Note that these sections are “global” sections. The domain of s is all of L0. (In this way they correspond more to sections of a ﬁber space over a base space rather than the local sections of a sheaf.)

Sections exhibit “descending behavior”: s(R1) is in L¯i, s(R0) in L¯j, s(R2) in L¯k, and we have the following commuting diagram of reductions.

L¯i ψi,j L¯j ψj,k φi ¯ φj Lk

× φk × w w L0 which explains the adjective “descending”.

Everything in E¯N maps to L0 by one reduction or another. Recalling Figure 2.4 each L¯i is a inter- ¯ weaving of layers each one of which can be a section s : L0 Li, each section being a relational 00 00 translation. For example, each relation, say R3(~x3) in the Figure 2.4 will be a component that 00 00 projects down to a relation R3(φ(~x3)) ∈ L0. Thus we have s(R3(φ(~x3))) = R3(~x3). Another section 0 ¯ 0 0 0 0 0 s might pick the Li object R3(~x3) so s (R3(φ(~x3))) = R3(~x3) which might be in a reduction ψij. This gives a local description of s, one deﬁned on one connected level or “slice” of an elaboration

L¯i. The more high-level adjoint relations L¯i has, the more slices sections can have in L¯i that will have the same projection to L0. In this way E¯N is a ﬁbration over L0.

115 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

0 0 0 If R0 → R0 ∈ L0 a section, as a functor, must also map R0 → R0 to s(R0) → s(R0) where the arrows are in one category at a time. Sections of E¯N (L0) are more flexible; if s(R0) is in the L¯i ¯ ¯ 0 ¯ ¯ component of the fiber over R0 and if ψij : Li Lj ∈ E(L0) then s(R0) can be in Lj. If, in Li, 0 s(R0) and s(R0) are both part of a cycle and all variables in all relations of the cycle are similarly 0 ¯ expanded then s(R0) is also expanded in Li. The definition of sections is less constraining, so that 0 ¯ ¯ 0 ¯ s(R0) can be in Lj. s(R0) ∈ Li and s(R0) ∈ Lj might never appear in the same elaborations but they can appear in the same section. How many maximum elements can we have in any one section? This brings to light a new property of L0. L0, as with any relational landscape, will have relations Rmax for which there is 0 0 no R with R → Rmax. These relations minimal principal ideals. {R | Rmax → R} is usually large or else Rmax is isolated. Sections will take the Rmax ∈ L0 to an object in some L¯i and then descend from s(Rmax). The image is of sections “draped over” the s(Rmax) and descending from them. The Rmax play special role in that they are accumulation or saturation relations where data is summarized. These often display significant co-ordinating roles such as large-scale allocation of resources. The possible number of maximal relations is a significant measure of the logic of L0.

4.3.3 Topologies on L0 and E(L0)

The aim of this sub-section is to construct a topological space

EcN = EcN (L0) that is a ﬁber space over the base space L0. This requires a topology on L0, not just D(L0), that is compatible with sections. As each section is a functor the topology will reﬂect the category structure.

We start with choosing the largest open neighborhood about an arbitrary R0 of L0. This is the set of R1 with R1 → R0 in L0. It includes couplings of R0.

Deﬁnition 29. The conditional topology on L0.

The conditional topology of L0 is the topology with basic open sets of the form:

N(R) =defn {R1 6= R | R1 → R ∈ L0}. where R can be any object in L0 including properties P(x).

N(R) is all logical entailments or business rules requiring R. If R = P(x) this is everything that can map to P(x). N(R) or N(P(x)) is deﬁned in terms of L0 not in terms of couplings. The ideals of the coupling algebra can furnish us with larger open sets J(R) and J(x) especially if L0 is not

116 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS a complete category. Suppose R = R(x, y) is a “must have” relation such as every qualiﬁcation must be conferred by a faculty, then N(R) is all the rules that depend on the existence of R(x, y). If N(R) does not include R, the open neighborhoods containing R are the N(R0) R → R0. The

“must have” relation R(x, y) is in N(P(x)) and N(P(y)) which are the large open sets. Open sets are closed under intersection so in this case N(R) is everything dependent on R and anything else in

N(P(x)) ∩ N(P(y)). This is the topology we work with below. But ﬁrst we note some of its features.

1. Open sets include all ﬁnite intersections of the basic open neighborhoods and all unions.

Intersections of basic open sets are already open. If R → R1 and R → R2 are both in L0 and

the conjunction of R1 and R2 exists (so the relations do not operate at different times.) then,

as R1 ∧ R2 is the pullback, R → R1 ∧ R2:

R1 ∧ R2

Ô z $ R1 R2

which generalizes to N(R1) ∩ N(R2) = N(R1 ∧ R2).

0 0 2. An object R0 of L0 is contained in all the open sets N(R ) for every R’ with R0 → R . Conse-

quently the set of open neighborhoods containing R1 is the same as the open neighborhoods

containing R2, which implies all the variables of R1 and R2 are the same but does not make

R1 = R2. R1(x1, x2, . . . , xn) and R2(x1, x2, . . . , xn) can be different relations using the same data to extract different information as in the case of Enrol(s, c, q, d, f) being the basis for many types of statistical reports, each of which is a different program hence different rela-

tions. Thus, with this topology, L0 will seldom be Hausdorff.

3. Recalling the discussion on minimal ideals in Section 2.5.2, these are ideals

∩ J(xi) where X is set of variables for which there is a relation R in L0 with dom(R) = xi∈X⊂X(S) X but there is no relation R with dom(R) ) X. The smallest neighborhoods of L0 are N(R)

with very few or no maps to R. R can be a relation that uses a seldom used property P(x) that determines the way R relates to other relations. R can be both a summation of data and

the initiation of other actions. Such relations might be thought of as the boundary of L0 with

117 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

this topology1. Because N(R0) contains R for every R0 with R → R0, R, itself, can have many open neighborhoods converging to it.

4. There are other more reﬁned topologies (so more open sets). Deﬁne the “unfactored” topology as having sub-basic open sets.

0 0 N˜(R) =defn {R 6= R | R → R ∈ L0 and cannot be factorized }

The unfactored topology has more, smaller open sets than the conditional topology but every basic open set of the conditional topology is a union of open neighborhoods of the unfactored

topology. This makes the identity morphism of L0 continuous from the unfactored topology

to the conditional topology. An implication that cannot be factored in L0 need not remain that way in an elaboration, so, unless there are good reasons for adopting the unfactored topology, we shall work with the conditional topology as the default. The sets N¯(R) are

sub-basic open neighborhoods as N¯(R1) ∩ N¯(R2) is seldom N¯(R1 ∧ R2). Frequently there is no best topology. Algebraic geometry developed with 3 topologies: Zariski, Flat (making ﬂat morphism continuous) and Étale [Hartshorne, 1977, p. 2, III. 9 and Appendix C]. In our case, the coarser topology of the conditional topology allows us to construct a topological

space EcN (L0) that has a declining gradient in the direction of entailment.

¯ We construct the topological space EcN (L0) by taking all relations R and properties in P(¯x) of the elaborations in EN (L0) as the underlying set of the space EcN (L0) and give it the weakest topology so that each section is continuous. ¯ ¯ −1 ¯ The topology we seek for EcN is the family of sets U ∈ U such that, for each section s, s (U) is ¯ open in L0. For any section s, Φ ◦ s = 1L0 . In particular (by normal set theory)

−1 ¯ s (U¯) = Φ(¯ U¯ ∩ s(L0)). (4.3)

The topology with the least number of open sets for continuity of sections, otherwise called the weak topology, gives the most information. It becomes a baseline topology for all other topologies that add more properties on sections or morphism in E¯N .

Deﬁnition 30. The elaboration topological space EcN (L0). ¯ The sub-basic open sets of EcN (L0) are given by the family U having members

0 0 N(R¯) = {R¯ ∈ E¯N (L0) | R¯ → R¯ ∈ E¯N (L0)}.

1Perhaps the term “horizon” is more appropriate as boundary is usually used for a subspace rather than a full space. As with the boundary, the horizon of a horizon is zero.

118 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

0 0 f◦ψij Here R¯ → R¯ can be a composition R¯ −−−→ R¯ so N(R¯) has all the E¯N (L0) maps that end in R¯ as given in Deﬁnition 28. ¯ Any R¯ in an elaboration in E¯N (L0) has to have an underlying object R0 = Φ(¯ R¯) in L0. Similarly

(R¯1 → R¯2) ∈ E¯N (L0) must have an underlying ¯ Φ(¯ R¯1 → R¯2) = R1 → R2 ∈ L0.

Any set of the form N(R¯0) in E¯N (L0) can contain all the images of sections s that give R0 7→ R¯0 ¯ and for R¯ → R¯0, Φ(¯ R¯ → R¯0) = (R → R0) ∈ N(R0) there can be an section that takes R to R¯. All −1 such a sections map the open set N(R0) ⊂ L0 to the N(R¯0) and, in each case, s (N(R¯0)) = N(R0) as given by equation 4.3. Thus we get:

Proposition 4.3.1. The open sets of the topology of EcN (L0) are generated by the images of open subsets of ¯ L0 “lifted” by sections s : L0 EN (L0).

EcN is a functor

We ﬁnish the introduction to the topological space EcN by showing it is indeed a structure or property of relational landscapes (so reporting structures) and has the all important property of being a functor.

Proposition 4.3.2. EcN is a functor from Sysepi to topological spaces.

ϕ 0 We shall prove that if L0 −→L ∈ Sysepi is continuous then the induced

EcN (ϕ 0 EcN (L0) −−−→ EcN (L ) is continuous. ¯ 0 Given V open in EcN (L ) that is also in the image of E(ϕ) (which need not be onto), we want to −1 ¯ show E(ϕ) (V ) can be constructed as an open set in EcN (L0).

¯ 0 ¯ EN (ϕ) ¯ 0 ¯ Lemma 4.3.1. An epimorphism ϕ : L0 L induces a morphism EN (L0) −−−−→ EN (L ) so EN is a functor from Sysepi to 2-categories.

Proof. The set of objects of E¯N can be embedded in E(L0) which, as a subfunctor of the left representable functor Sysepi(_, L0), is a covariant functor. This gives the following commutative diagram.

119 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

E(ϕ) 0 EN (L0) / EN (L )

Φ¯ Φ¯ 0 ϕ 0 L0 / L

It remains to prove that the additional morphisms of the form ψjk ◦ f ◦ ψij : L¯i → L¯k transform correctly. 0 In fact L¯i, L¯j, L¯k are elaborations of L via the reductions ϕ ◦ φi, ϕ ◦ φj and ϕ ◦ φk respectively.

Given R¯1 ∈ L¯i, the f is an implication ψij(R¯1) → R¯2 ∈ L¯j. Thus we must have φj(ψijR¯1 → R¯2) = 0 0 (say) R1 → R2 in L0 which gives ϕ(R1) → ϕ(R2) in L . Thus E¯N (L0) is a subcategory of E¯N (L ) ¯ ¯ EN (ϕ) 0 via the mapping ϕ ◦ Φ¯. This gives the mapping E¯N (L0) −−−−→ E¯N (L ) between the 2-categories as required.

0 To complete the proof of the proposition we have to show that if V is an open set EcN (L ) ¯ ¯ −1 intersecting the image of EN (ϕ), then EN (ϕ) (V ) is open in EcN (L0).

0 0 0 0 0 0 Proof. Suppose V = N(R¯ ) where R¯ ∈ L¯ ∈ E¯N (L ). For N(R¯ ) to be in the image of E¯N (ϕ), L¯ 0 ¯ must be the image of an elaboration over L0 but represent an elaboration over L via ϕ ◦ Φ¯. Taking ¯0 ¯ ¯0 ¯ 0 0 U = N(R¯ ) in E¯N (L0) where ϕ(Φ(¯ R¯ )) = Φ¯ (R¯ ) then E¯N (ϕ)(U) = V and is the pullback of V along

E¯N (ϕ).

R¯0 itself might be a composite of relations that all map to R¯0 in which case U might be many- sheeted over V .

4.3.4 Convergence in Eˆ

Topology makes precise the way a sequence of elements of the topological space can be taken as converging to another element, the limit of the sequence. This can be done by deﬁning a metric or by looking at the way convergence is deﬁned by open neighborhoods.

Deﬁne the norm of a section s as follows. Given s(R(~y)) = R¯(~x) ∈ L¯i then, if dim(~y) gives the number degrees of freedom in ~y, (or the number of components in the vector) deﬁne

d(s(R)) =defn dim(~x)/dim(~y).

For example, if ~y = (y1, y2, y3) and ~x is obtained by splitting y1 three ways we get dim(~x) becomes three times that of dim(~y). If ~x is obtaining by splitting y1 three ways and y2 four ways then the dimension or degrees of freedom of ~x is 12 times the degrees of freedom of ~y. Deﬁne the norm of a section as:

120 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.3. THE “GLOBAL THEORY” OF ELABORATIONS

µ(s) =defn maxd(s(R)). R∈L0 If µ(s(R)) > 1, there must be some y in the domain of R which splits into sub-classes. If Φ(¯ s(R)) = φ(s(R)) then φ˜(y) > 0 from which we have:

Proposition 4.3.3. For every relation of L¯i that is in the image of s, µ(s(R)) > 1 implies dom(R) contains an open set in D(L0).

This gives a link with Z(L0). We also have a normed based metric, hence a metric space of sections:

d(s1, s2) = max(|d(s1(R)) − d(s2(R))|. R∈L0 The link with continuity

Sections are not only continuous they are also open, taking open sets to open sets. Suppose sλ,

λ ∈ Λ a directed set, converges in the elaboration topology EcN (L0) to s. Open sets around s(R0) are

0 0 00 0 00 00 s({R | R → R0 and ∃R with R → R and R0 → R })

00 00 00 with R 7→ sλ(R ) lifting N(R ) which is an open neighborhood of R0. For point-wise convergence sλ(R0), given any neighborhood of R0, a conﬁnal set of sλ must eventually be in the neighborhood of s(R0) in EN . Consequently sλ(R0) is either the same as s(R0) or there is a reduction ψij in E¯N that carries sλ(R0) to s(R0). It follows that

dsλ(R0) ≥ ds(R0) as the increase in the degrees of freedom of ds(R0) cannot be greater than that of dsλ(R0).

The picture we get of convergence sλ → s is that it proceeds with sλ(R0) descending through each ﬁber over R0. It gives, as it were, no trace of the convergence in L0. The smallest open set containing R0 might be N(R1) which can be very large and with a descending s point-wise convergence around large N(R1) conveys little. We can say convergence is a movement that is

“perpendicular” to the base space L0. Convergence ﬂows down the gradient towards L0 often decreasing correlation. The metric based on the norm is indifferent to this gradient, allowing the convergence to be independent of reductions and the increase or decreased of correlation. The norm based topology allows more sequences to converge, giving a more natural sense where one section is close to another. Nevertheless we take the elaboration topology as the standard. It incorporates more of the categorical structure of the elaborations and has an intrinsic gradient which we shall shortly exploit.

121 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.4. SECTIONS AND SPECIFICATIONS; THE DYNAMICS OF CHANGE

The business implications of continuity

If we identify E¯N (L0) with the future possibilities of the existing reporting structure S0 with L(S0) =

L0, the topology of EcN (L0) tells us how closely packed are the development possibilities of L0. Of course this depends on what is to be conserved; the constraints on the possibilities.

Suppose we have U open in L0 and a large set of sections si, i ∈ I, all close to one another over U, either in the metric or in the conditional topology. Then, over U, there are few constraints to move from one section to another; elaborations as specifications can be freely planned without fear of constraints. Although this is hardly a big step in practical terms it shows that the topology has significance in business. If Hp(L0) is zero beyond p = 1 or at most p = 2 and low dimensional in both cases there are few correlations and they involve few relations. Thus, to begin with, away from the correlation classes, sections will be malleable, easily changed to another. This means, ini- tially, lots of small elaborations are possible, typically small enhancements to the reporting structure that elaborate the system in increments. Thus the topology of EcN (L0) is reflected in the ease of making incremental changes. The existence of large correlations that separate sections into sets that create correlations and those that do not, separates EcN (L0) into topological regions defined by the correlations. Speci- fications, so sections, that require the creation of adjoint relations to produce a homology class

ζ ∈ Hp(L¯) p > 3, require a high degree of coordination themselves. Such sections are unlikely to reach their goal through a set of incremental changes. This bears on a sense of inertia that the logic imposes. This inertia becomes a force to be overcome in the management of a system when not enough details are known or sufﬁciently documented to change within a speciﬁed time. Cases of such systems threatening to be too big, or too unwieldy, to change are documented in Hanseth and Ciborra [2007]. In these case studies the authors see integration the problem, which can now be seen in terms of correlations. The counter to “too big, too unwieldy to change” requires a serious study of how such “high inertia” or immutable systems might occur. Something is unchangeable if what needs to be conserved blocks all possible or needed changes in a particular area. The study of elaborations is just the beginning of this wider study.

4.4 Sections and Speciﬁcations; the Dynamics of Change

In this section of the chapter, sections s : L0 EcN (L0) are identified with specifications for enhancement. With this identification the elaboration topology can be seen as the topology of change requests.

122 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.4. SECTIONS AND SPECIFICATIONS; THE DYNAMICS OF CHANGE

As mentioned in Section 4.3.4 the topological space EcN (L0) has an in-build gradient that comes ψi,j from the reductions L¯i −−→ L¯j. This gradient involves a loss of the ability to make distinctions. The evolution of a system is against this gradient. We shall use this to deﬁne a new class of operators on this space.

4.4.1 Operators of Elaboration Dynamics

The space EcN (L0) can be very large but it is bounded as relational landscapes are defined as finite ¯ and that the subject matter of each L in E(L0) is a refinement of the data in some of the P(x) ∈

D(L0). Nevertheless, the variety of possibilities can be classified into trajectories of evolution. This classification can be defined so that we have a taxonomy of evolutions giving a cladistic study of E(L0). In fact this is what is done on a small scale. Many projects to enhance reporting structures are broken into phases especially in “agile” projects [Cockburn, 2006], each phase offering added functionality in small helpings while on the path to a major development or implementation. The whole process goes to a plan defined by a series of criteria. We might suppose a project is then a series of elaborations (where time goes from left to right):

φ1 φ2 φn Θ: L0 ←− L¯1 ←− ... ←− L¯n.

Each φi is “shaped” by some business or project requirements Ii(σi1 , σi2 , . . . σm(i)) where each σ is a specification. I is the integration of the separate specifications. This makes I more than the sum of the parts {σi1 , σi2 , . . . σm(i)} just as a car is more than the sum of its parts. The achievement of a set of specifications {σi1 , σi2 , . . . σm(i)} can be identified with a section sij having “lifted” L0 to some elaboration.

Suppose Ii(σi1 , σi2 , . . . σm(i)) is a language that can apply to any elaboration. We can then develop of language of “reduction differentials” of “reduction operators”. For the sequence Θ above we deﬁne the operator

δ(φi) = Ii(σi1 , σi2 , . . . σm(i)).

The intention here is that all the σik are deﬁned mathematically, possibly as functors on Sys. The correlation homology is an example. Our speciﬁcations might be that the homological dimension, the largest p for which Hp(L¯i−1)) is not zero, is to be increased.

dim(H(Li)) − dim(H(Li−1)) =notation δ(dim(H)/δ(φ) > 0.

If the adjoint relations can be separated by different “subjects”, for example human resources and payroll or factory operations, we can split the homology into the direct sum of the subject homologies and have different dynamical criteria in each case. In other cases, we might want to extend and

123 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS elaborate a humanitarian emergency reporting structure but each elaboration needs to have multiple reductions that can be applied in areas where supplies and expertise might be compromised, we might have the dynamic speciﬁcation:

δ(φ) = F[(H(subject1)(Li−1),H(subject2)(Li−1), ....H(subjectn)(Li−1)), Red(Li−1)]. (4.4) where F is a multifunctor, covariant in some domain functors and contravariant in others. Equation

4.4 defines the change from Li−1 to Li as a function of the correlations in various subject areas as well as certain simplifications that allow for substitutes and can be identified for Li−1. The operator acts as abstract specification saying what needs to be done in terms of the structures of Li−1. Beyond this is the study of elaborations for which

δ(dim(H)/δ(φ) = dim(H(Li)) − dim(H(Li−1)) > 0, is asymptotically true or “generally” true which allows growth of order but with some lapses. As further functors on Sys are formulated, the range of operators will expand giving us new ways to express structural dynamics.

4.5 On Stacks

This section has provided examples of 2-categories over topological spaces. The ﬁrst, the data- types with the elaboration topology, the second with the conditional topology over the full L0. This suggests that the stacks (also known as 2-sheaves) might have some bearing on the subject. I show that this is the case and so connect reporting structures with a current area of research which has deep roots in the Grothendieck approach to algebraic geometry and later his interest in homotopy theory [Brown, 2005].

4.5.1 What is a Stack?

The following is a brief review of stacks (also known as 2-sheaves). I have followed the Wikipedia article on Algebraic Stacks and [Romagny, 2003]. Other introductions include [Canonaco, 2004, Edidin, 2003, Gomez, 1999, Olsson, 2016]. This is a subject that has been developed for algebraic geometry and homotopy theory and so the examples in the references are concerned with the schemes and Grothendieck topologies of various kinds. In this section I show that the concept of a stack applies in the study of elaborations and the elaboration topology and an extension of the homology tableau category. Such stacks give a criterion for claiming very large reporting structures

124 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS can be built from “local knowledge”. The usual treatment of stacks requires familiarity with Étale coverings and deﬁnes algebraic spaces then stacks. This is route taken by Olsson and Gomez. [Ols- son, 2016, p. 98] gives our starting deﬁnition as a consequence of lemma 24.7 on the equivalence of the existence of a sheaf and the properties of a covering. Gomez [1999] starts with moduli spaces over a scheme so is slightly closer to the point of view here of a category bundle over a topological space.

Fibered Categories

Let C and D be categories with objects A, B in C, U, V in D. T : C → D. T (B) = Y and X → Y ∈ D then there is a “pullback”:

AO 0 h

f A )/ B

T T ↓

F V / U( in D) O 5 H W

∗ A is called the pullback of B along F and denoted F (B). In our case, D is the open sets of L0 with the conditional topology; V → U means V ⊆ U then the object A in category C comes from a restriction of U to V (and the largest such object).

Prestack

C is ﬁbered over D. In this case D is a category with a Grothendieck topology so coverings with pullbacks replacing intersections. For any object U ∈ D we have the ﬁber over U which is a set of objects in C for which T (B) = U. We consider all the covering maps V → U (in our case the open cover of U) and take the set Hom(F ∗A, F ∗B). In a prestack this is a sheaf. Thus all the various maps V → U have a value Hom(F ∗A, F ∗B)(V ) being the restrictions to V and these can be “patched” according the sheaf rules.

125 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

Descent Data

In a Grothendieck topology the topology is in terms of coverings of an object, in our case an open set by subsets V ⊂ U (corresponding to a covering map Vi → U). We want the pullbacks of Vi and

Vj over U (so Vi ∩ Vj, Vi ∩ Vj ∩ Vk) to satisfy what comes down to associativity of ∩. More generally, descent datum consists of a covering of an object U of D by a family Vi for which the objects, Ai, ∗ in the ﬁber over Vi have restriction morphisms fji from Ai to F (Vij) over Vij = Vi ×U Vj these 2 restrictions satisfy the compatibility condition fki = fkj ◦fji. The descent datum is called effective if the objects Ai are pullbacks from an object in the ﬁber of U.

Stacks

A prestack is a stack if the descent data for the Grothendieck topology is effective. In ordinary set theoretic topology with D the open sets of the topology τ, this stipulates that the fiber over U is F defined for every open U and for every V −→ U (so V ⊆ U) the pullback F ∗ (the restriction) is well defined in the C category.

4.5.2 The Stack of EcN (L0) over the Conditional Topology of L0

In our case the category D above, the base space, is L0 with the conditional topology. This is a set based topology and the descent data is always effective. This will be the context of what follows so it is omitted. U is an open set in the conditional topology.

EcN (U) can be deﬁned as all possible sections s : U → EcN (L0) or, more categorically, the ﬁbered pullback.

EcN (V ) / EcN (U)

V / U.

¯ Given the category nature of EcN deriving from the underlying EN with morphisms

2Knutson [1971, p. 31] deﬁnes a stable class of maps D (which we shall assume) in a category C as satisfying effective descent if, given a sheaf F and a covering Ui of U and a sheaf map F →= C(_,U) such that for each i, C(_,Ui) ×U F = C(_,Wi) for some Wi in C and Wi → U in D, then F is representable. If C is a topology, D inclusion of open sets (a stable set), C(_,U)(V ) is either (V ⊂ U) or empty. C(_,Ui) ×U F =

F |Ui = C(_,Wi), Wi an open set). From this we have F is indeed representable as C(_,W ) = C(_, ∪iWi). This follows as ∪ and ∩ distribute in open sets (∩ = pullbacks of open sets) and open sets have arbitrary unions. That is, effective descent is automatically satisﬁed in the category of sets and set based topology.

126 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

ψ L¯1 / L¯2 φ1 ~ φ1 L0, and sections defined on open sets, restriction is straightforward and EcN (U) a presheaf. If we adopt the first, section oriented, definition of EcN (U), we can show fibering in this way gives a category-valued sheaf.

Given si deﬁned on Ui such that, for every pair Ui1 , Ui2 , the restrictions of si1 and si2 coincide on Ui1 ∩ Ui2 and, similarly, with si3 coincide on Ui1 ∩ Ui2 ∩ Ui3 . The objects and morphisms in these 0 φ sections contain the dependencies R → R that are lifted by the sections to some L¯ −→L0. As these match up on the subsets Ui of ∪iUi we can deﬁne s that restricts to the individual si.

Lemma 4.5.1. EcN is a sheaf

0 ¯ For each U, EcN (U) is a category. Morphisms s → s are deﬁned using the maps in EN so that ψ s(R) in L¯ maps to s0(R) either by an internal map in L¯ or, if s0(R) ∈ L¯0 by a reduction L¯ −→ L¯0. 0 To get a prestack, the set of maps Hom(s, s ) in each category EcN (U), is to be a sheaf: partial maps can be glued appropriately. For Ui ⊂ U open in L0 and we have a well deﬁned set 0 0 Hom(s, s )(Ui) and when they can be glued appropriately we get back Hom(s, s )(∪iUi).

Proposition 4.5.1. EcN is a prestack.

0 0 0 Proof. If fi ∈ Hom(si, si)(Ui) then, for an arbitrary R → R in Ui, we have fi defined on si(R → 0 0 R)). The restriction Hom(si, si)(Ui) → Hom(sij, sij)(Ui ∩ Uj) is well defined on these sets of maps by forgetting all but the section values on the Ui∩Uj relations, and this means fi restricts to fi |Ui∩Uj 0 0 or fij. For any R → R in ∪iUi we can define s(R → R) unambiguously giving its value on each 0 of the Ui so the value si(R → R). si | (Ui ∩ Uj) = sj | (Ui ∩ Uj) so there is no problem of multiple 0 0 values. f is then defined in terms of the fi by fi(si(R → R)) and this will be part of a section si(Ui) f 0 0 which defines the extension to the fi, giving s −→ s ∈ Hom(s, s )(∪iUi).

Corollary 4.5.2. EcN is a stack.

Proof. As noted above, the criterion of being “effective” in ordinary set based topology is equivalent to the associativity of intersection.

4.5.3 The Category of Sections to htb

As deﬁne in Section 4.2.3 the category ht(L0) is already deﬁned locally over the elaboration topology of D(L0). Is there a similar localization over the conditional topology on L0? As in the case of

127 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

EcN we approach this by using sections. If U is open in the conditional topology we can certainly ask, for a given R in L0, whether an adjoint relation depends on R? (And, of course, vice versa in a “co-conditional theory” base on open sets deﬁned by neighborhoods N op(R) = {R0 | R → R0 ∈

L0}.) An adjoint relation that gives a homology class ζ will be a possible value of a section deﬁned of N(R). 0 Let U be an open set in the conditional topology of L0. U is given by a set of maps R → R,

R ∈ U, and is a full subcategory of L0. We want to deﬁne local sections, s, as functors from open ¯ h(φ,W ) 0 sets in L0 to ht(L0) with s(R) = ζ ∈ H∗(L¯) −−−−→ Z(W ). Suppose R → R ∈ U, for s to be a 0 0 0 ? functor, s(R ) → s(R) in ht(L0) corresponds to s(R ) = ζ −→ ζ = s(R). We need the unknown 0 map to take correlation class ζ to the correlation class ζ in Hp(L¯i). The only maps available are isomorphisms. There is nothing in ht that tells us that ζ0 and ζ are connected in any way. We need to add maps to create a new category: htb .

The leads to a new category with objects the homology classes of the H∗(L¯), L¯ ∈ E(L0), and the maps derived by ﬁlling in the top dotted line in the diagram:

ζ0 / ζ (4.5)

span(ζ) span(ζ0)

R0 / R.

That is, this new category, htb (L0) (or htb when the context is clear), requires defining extra maps expressing “weak dependencies” in ht. The category htb is clearly fibered over the conditional 0 0 topology as each open set U is a union of open neighbors N(Rk) = {R | R → Rk} in the finite category L0. 0 0 0 0 Note that, if (R → R) ∈ L0, R¯ and R¯ are in L¯ with φ(R¯ ) = R and φ(R¯) = R, then, as φ is onto, 0 0 there must be a morphism R¯ → R¯ that maps to (R → R) ∈ L0. This provides the link between the homology classes: some part of ζ0 depends on some part of ζ. As homology classes are strongly connected by the definition of adjoint relations that satisfy the correlation criterion, the new maps do indeed imply ζ0 must have some level of dependence on ζ. Furthermore this extends to the ht ψ1 ψ2 ψj−1 classes ζ1 −→ ζ2 −→ ... −−−→ ζj.

Sections from L0 are functors to htb

U is open in the conditional topology and (R0 → R) ∈ U.

128 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

Deﬁnition 31. A homology tableau section is a functor s : U → htb (L0) that assigns values to R ∈ U as h(φ,W¯ ) s(R) = ζ ∈ H∗(L¯) −−−−→Z(W ) if R ∈ span(ζ) and W ⊂ U.

There can be many sections for a given R as long as it belongs to a ζ that satisfy the conditions. ψ1 ψ2 ψj−1 The nature of objects in H(L0) (the end of Section 4.2.3) as sequences, ζ1 −→ ζ2 −→ ... −−−→ ζj that give the same h value for an open set W ⊂ D(L0) imply that the germs of these sections (deﬁned by a given R) correspond to the number of homology classes with W ∩ dom(R) 6= ∅. 0 00 000 00 000 Example. In the diagram below put U = {R1,R2,R ,R ,R } and V = {R2,R ,R }. Suppose 0 0 0 0 0 s(Ri) = ζi, i = 1, 2 and R ∈ span(ζ ) so we want to put s(R ) = ζ . This is consistent as R is dependent on R1.

ζ0 = s(R0) F s(10)

00 * 00 s(1 ) s(R ) / s(R1) = ζ1 000 9 F 00 G s( s ) s(2 )

s(3) , s(R000) / s(R ) = ζ H 2D 2 s s

s 1 s

00 00 1 /* ;R R1 000 200

000 3 , R / R2

00 00 R is dependent on R1 and R2 so s(R ) has to be dependent on ζ1 ∈ Hp(L¯1) and ζ2 ∈ Hq(L¯2).

If L¯1 and L¯2 are consistent we can form the pullback L¯1 ] L¯2 that will have the adjunct relations that form ζ1 and ζ2. This combined adjunct relation has weak dependency of relations in ζ1 and ζ2. Denote this combination by ζ1 Z ζ2. In the category htb it is the product of ζ1 and ζ2 and, by our 00 00 00 construction, exists when the pullback L¯1 ] L¯2 exists in E(L0). If s(R ) = ζ then, for s(R ) to be 00 00 well deﬁned, we expect ζ has a weak dependency ζ / ζ1 Z ζ2 . Finally suppose s(R000) = ζ000. For s to be a functor ζ000 has a weak dependency on ζ00 hence on 000 ζ1 Z ζ2: ζ / ζ1 Z ζ2 . 00 000 The restriction of s to V is just the diagram of s values on {R2,R ,R }.

Lemma 4.5.3. The presheaf of sections is a sheaf.

129 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

As usual given a set of si deﬁned on Ui that are equal on intersections we need to ensure they can be patched together to form s on ∪iUi, each patch over Ui is consistent in terms of dependency.

Proof. We need to show that even though si and sj are equal on Ui ∩ Uj there is no possibility that 0 0 0 there is an R in Ui and an R in Uj such that R → R but si(R ) fails to be weakly dependent on sj(R). Uj is an open set so is the union of basic open neighborhoods N(Rk). In particular if R is in

Uj its open neighborhood N(R) is in Uj. (If R is one of the Rk this is obvious. If R is not an Rk it is in N(Rk) for at least one case and then N(R) ⊂ N(Rk).) However, N(R) does intersect Ui ∩ Uj as 0 0 0 R ∈ N(R). Thus R is in the domain of sj and therefore sj(R ) is weakly dependent on sj(R).

Therefore, as we deﬁne a section s over ∪iUi by giving it the values of si on the respective Ui, we do not get a situation where R0 → R breaks the weak dependency that would obstruct the creation of s.

Maps between sections

Deﬁnition 32. A map η between sections s and s0 deﬁned on U is a given by a set of commutative diagrams

s(R0) / s(R)

η(R0) η(R)

s0(R0) / s0(R) for every pair R0 → R ∈ U.

p This can be thought of as an intersection diagram. If s(R) = ζ is the homology class of R¯! in 0 0 0 0!q !p 0!q L¯1 and ζ = s (R) is the homology class of R¯ in L¯2 then R ∈ φ1(R¯ ) ∩ φ2(R¯ ). This allows us to define the set of maps between sections in htb so that we get Hom(s, s0) to be all possible morphisms, hence natural transformations, between a functor s and all other s0 on which s is weakly dependent. We can also define Hom(s, s0)(U) as being Hom(s | U, s0 | U). Definition 32 describes maps between sections in local terms, essentially per entailment. So, by exactly the same arguments given in Lemma 4.5.3, section maps of Ui that are equal on the intersections Ui ∩ Uj can be glued together to form a single map on ∪iUi giving

Proposition 4.5.2. Hom(s, s0) is a sheaf.

And

Corollary 4.5.4. The category of sections from the conditional topology of L0 to the category htb is a prestack.

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As before (Corollary 4.5.2), the topology is ordinary set topology and so satisﬁes the Grothendieck effective descent requirements.

Corollary 4.5.5. The category of sections from the conditional topology of L0 to the category htb is a stack.

The stack is an additional property of a reporting structure that adds to the information in the category of the homology tableau. Each section s selects an adjunct relation that includes the domain relations and the topology keeps control of dependency. L0 is the “ancestral legacy system” for all the L¯ and each set of correlations has a part that traces back to the R in s(R), a relation, so a listing, in L0. The class (in this case category) of those sections provides the record of what traces of the ancestral dependencies remain in all the futures of L0. The sections to htb can be seen as the projects that create correlations and so require adjoint relations. These are the substantial projects that required signiﬁcant planning and possible staging (so a sequence of maps 0 from the ﬁnal stage, sfinal, to intermediate stages s on which sfinal depends. In some cases stages will create correlations that include sets of relations that are unchanged. A poorly documented function (perhaps a complicated program) that is to become part of a homology class that has many dependencies can cause problems for each stage of the project (for a case study see [Blechar and Hanseth, 2007]). More generally, the more we know how coordinating structures are related to all possibilities, the deeper is our knowledge of the possibilities for L0.

4.5.4 A Note on Abstraction

This section has taken the study reporting structures, as represented by relational landscapes, into highly abstract realms. The categories E(L0) and EcN (L0) are examples of stacks, as indeed is the category htb constructed from homology tableau. (See [Canonaco, 2004, Edidin, 2003, Gomez, 1999]). This connects them with a burgeoning area of mathematical research and indicates that reporting structures have deep connections with important areas of mathematics. This is an new application of stacks as they are generally concerned with aspects of algebraic geometry. The idea of studying classes of potential futures is not common in technology and certainly not common in systems engineering. Sometimes though, the exploration into abstraction opens up a subject to new perspectives. Here we have seen that there are signiﬁcant mathematical structures that arise from considering futures of a reporting structure; futures that increase the resolution of a reporting structure by increasing the information in critical areas and, also, at the same time, ensure that coordination structures are maintained. This simply mirrors much of the intention to exploit the large repositories of data now available. That exploitation needs to feed into actions that can be focused to improve enterprise outcomes. These are only achieved by using the data for a more detailed level of coordination. It takes “big-logic” to exploit “big data”.

131 CHAPTER 4. THE DYNAMICS OF REPORTING STRUCTURES: ELABORATIONS 4.5. ON STACKS

There are many reasons to expect that the number of global enterprises will increase, driving competition. Part of the response to competition will be elaborating and integrating reporting structures and their supporting systems. The options, and their interactions, live in the elaborations of what we have now. The concepts developed in this chapter open up this future of reporting structures. I have seen no better justiﬁcation for exploring this area than the following quotation.

“. . . from a methodological point of view, what we have here is a technology for replacing inﬁnitely many hypotheses (about disparate small objects) with a single hypothesis (about one large object): the large object – the p-adic integers in this case – can be studied by methods of algebra or arithmetic. This process of mathematical reiﬁcation is of course quite traditional. . . but is a particular characteristic feature of 20th century mathematics, (Hilbert space, representable functors. . . )” Segal [2008, p. 1]

In this case we can substitute “category of elaborations” for “p-adic integers” and “topology” for “arithmetic”.

132 Chapter 5

HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS

5.1 Introduction: Adopting Different Scales of Understand- ing

In other areas of mathematics we seek phenomena above the details of the dynamics by deﬁning classes of states that are linked by recurrence such as orbits or stability over long time frames (see for example the discussion on “scaling” orbits of dynamical systems Bowen [1978, p.17], and generally studying orbits of group actions on a set or on a topological space [Lang, 1969, p. 22]). In these cases a state is deﬁned by its time evolved relation with another state or set of states. This is a type of reduction: for a class of dynamical systems there is a class of local reductions by collapsing a closed subsystem to a single point in a scaled up system. In relational landscapes the dynamics is that of the morphisms and so entailment. Long sequences of dependency such as

R0 ← R1 ← R2 ← · · · ← Rn or “processes” can be characterized by an outcome such as the making a product or a history a client, student or patient’s interaction with the organization. This can be further expanded as

R0/L ← R1/L ← R2/L ← · · · ← Rn/L, (5.1) which gives, say, a history or a sequence of relational states to be achieved and all their attendant dependencies. Processes and subcategories R/L were studied in Macfarlane [2017] where tracing

133 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.1. INTRODUCTION: ADOPTING DIFFERENT SCALES OF UNDERSTANDING the processing of data through a system is broken down into scenarios as described above in Sec- tion 2.6.1. In that publication scenarios formed a functor Scn : Sys → Sets while selected sets of subsystems (subcategories R/L) formed the variables of a subsystem functor Sub : Sys → Set. In enterprise reporting structures, a sequence of processes can be accessed through various menus of work options or data-base actions. Listing menus and their relations is also a common way of getting impressions of what a system can do while avoiding the details of the data-types and the definitions of relations. Menus can have sub-menus as options, so the structure of menus has its own class of relations. It is another high-level view of a reporting structure but not one we concentrate on here as, in a relational landscape, menus are simply relations of relations and can be included in subcategories R/L. Interpreting the functor Scn(L) (introduced in [Macfarlane, 2017]) as test scenarios gives a scale functor at a lower level than subsystem or process testing: many scenarios might be required to test R1 → R2 especially when R1 and R2 are functions or adjoint relations. Each scenario can involve many data-types so an individual scenario usually requires all the dependent relations and scenarios to be part of the test. Scn and Sub are two scale functors that are easy to define. We shall define a general class of scale functors for which Scn and Sub are examples.

Deﬁnition 33. Let T be a type theory with terms σ1, σ2,... and relations R(σi1 , σi2 , . . . σim ),(m ∈ N) and T be the category of sub type-theories of T . A scaling functor is a functor Λ: Sys → T such that 0 if Λ (L) is the set of objects of L that become terms in Λ(L) = T ∈ T and if η :Λ1 → Λ2 is a natural transformation with

0 0 0 η |L:Λ1(L) Λ2(L), the following diagram commutes: 0 η |L Λ0(L) / / Λ0(L) (5.2) 1 _ 2 _

η|L Λ1(L) / / Λ2(L).

In consequence, the type theory grows with the number of variables.

5.1.1 Relations among the Variables of Λ(L)

0 Let Λ be a scaling functor for which Λ (L) are sets of processes or subcategories σi = Ri/L or a mixture of both as in the sequence 5.1. For Λ(L) to be a type theory there must be relations among their variables. These relations will written in Fraktur script to distinguish them from relations in

134 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.1. INTRODUCTION: ADOPTING DIFFERENT SCALES OF UNDERSTANDING

L. The variables in Λ(L) will be denoted by σ and we study relations R(R1/L,R2/L,...,Rn/L) = [n] R (σ1, σ2, . . . , σn). We cannot simply identify R/L with R as there can be many business rules or logical entailments R0 → R00 with R → R00 in R/L so R is dependent on R00 but not necessarily on 0 R . The possible relations between R1/L and R2/L is more varied than those between R1 and R2. Macfarlane [2017, Section 7.1] lists a number of standard relations that arise from the processes or category structure of L, especially dependency on prior conditions. These include:

[n] 1. Rinit(σ1, σ2, ..., σn) if all the σi have the same initial condition (dependency).

[n] 2. Rterm(σ1, σ2, ..., σn) if all the σi have the same ﬁnal state (sufﬁcient conditions).

[n] 3. R∩ (σ1, σ2, ..., σn) if all the σi coincide at some point. Rinit ∨ Rterm ⇒ R∩.

4. Operational relations: branching and clustering relations. Let σ] denote the condition "on termination of σ".

[n+1] (a) Rcond (σ], σ1, σ2, ..., σn) if all the σi are conditional on σ having just ﬁnished.

(b) If Y {x1, x2, ...xn} means one of the set {x1, x2, ...xn} then

Rbr(σ, σ1, σ2, ....σn) ⇔ σ] Y {σ1, σ2, ..., σn}

is a branching relation. (c) Time constraints:

Rt(σ, σ1, σ2, ....σn) ⇔ (σ] {σ1, σ2, ..., σn} must ﬁnish within t (units of time)).

(d) Rexcl(σ1, σ2) if an only if σ1 and σ2 cannot occur together. There can be many variations on this theme. They reﬂect capacity restraints or restraints arising from incompatible conditions.

5. Introducing properties of subcategories of the reporting structure L(S) provides further relations:

(a) R∩,P (σ, σ1, σ2, ....σn) is true if {σ1, σ2, ....σn } have a non-empty intersection containing σ which has a property P . (This can indicate that if P is an information exchange then

all the σi have a subsystem for sharing data among the rest.)

(b) RP (σ, σ1, σ2, ....σn) is true if {σ1, σ2, ....σn } are subcategories of σ which has a property

P . (This can indicate that if P is a secure system then all the σi take part in that level of security.)

135 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.1. INTRODUCTION: ADOPTING DIFFERENT SCALES OF UNDERSTANDING

Among the set of properties of the σ are “extremal properties”, the combinatorial property of being a maximal or a minimal set to satisfy some relation. For example the greatest number of subsystems of different classiﬁcations that overlap or share some data-types. Extremal relations are properties of the operations of an enterprise. They are the answers the questions:

1. Capacity problems. How many projects can we carry out at one stage? This depends on how they are related. For example, what classes of projects classiﬁed by requirements for expertise can we do at any one time?

2. When we reach certain states in some processes we are committed to a sequence of other states. Does our reporting structure automatically make these connections and ensure all the resources are available for the committed states or are we still reliant on experienced

people? This a problem of coordination. Given σ1, σ2, . . . , σn is there a σ¯ and a relation

R(¯σ, σ1, σ2, . . . , σn) that says σ¯ allocates resources to the set of subsystems σ1, σ2, . . . , σn.

3. How many subsystems with certain desired classiﬁcation does it take to "cover" the entire system? The answer to this question can be important in the distribution of expertise in the enterprise systems.

4. How modular or intertwined ("entangled") are subsystems of certain classes? This includes questions of security and vulnerability. This defines classes of couplings that are classified by external criteria. Another example of this is the extension of the idea of algebraic integration (defined is Section 2.6) by requiring that we want our ideals to have certain properties such as the required expertise or a specified level of security.

These are questions about the capability of a reporting structure and so about the reporting structure; we need not expect the questions to be formulated or answered within the structure. To verify relations deﬁned about subsystems of a reporting structure requires considerable work with the details of the associated relational landscape. The relations above will be taken as a generic class of relations that are included in the type system T . The category of sub-type theories of T , T, completes our deﬁnition of scaling functors of interest here.

As defined, there is no restriction on what Λ creates as the terms. The relations R allow σ1 ⊂ σ2 so Λ is not restricted to taking a class of subcategories (for example) as Λ0 and all below that set disappears from sight. Λ can be defined in a way so that subcategories are separated by properties that allow σ1 ⊂ σ2 to be well defined. The idea is that sub-systems (processes, scenarios) have associated classifications P1, P2, ... , PM . We then consider those relations R that can be defined among the objects satisfying the classifications. These relations are not defined in L, they belong

136 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.2. VIEWPOINTS

to the “afﬁliated logic” with P1, P2, ... , PM . Such a logic is called a viewpoint of L. Relations [n] [n] R = R (σi1 , σi2 , . . . , σin ) are “placed” in a class of relations “over” the set of criteria Pi1 , Pi2 ,

... , Pin for k = 1, 2, . . . , n. The scale functor Λ represents the “high-level” in a high-level specialist description. The viewpoint adds the “specialist” and the two together give the description. This will be given a precise expression in Section 5.5 in Deﬁnition 42 of the category of viewpointed systems.

5.2 Viewpoints

Viewpoints and viewpoint homology were introduced in Macfarlane [2017] as a way to analyze relations in Λ(L) relative to an interpretation of the variables of Λ(L). Λ is always a functor so ϕ Λ(ϕ) L1 −→L2 will give Λ(L1)) −−−→ Λ(L2). Here we give the essential deﬁnitions.

Deﬁnition 34. A viewpoint, V is a set of classiﬁcations, P1, P2, ... , PN for which there exists a semigroup operation Pi ∗ Pj and for all ∗ products there is a “face map” or projection map:

projk Pi1 ∗ Pi2 ∗ · · · ∗ Pim −−−→ Pi1 ∗ Pi2 ∗ · · · ∗ Pik−1 ∗ Pk+1 · · · ∗ Pim (5.3)

It is also convenient to have a “null” element ∅ when faces are not deﬁned. The face maps make V into a category for which ∅ is a terminal object.

0 m m Let V = {P1,P2,...,PN } and V the set of m ∗ products. Elements of V will be called (m) W 0 “places”. We can define a universal place U = P1 ∨ P2 ∨ · · · ∨ PN (or V ) for which Pi ∗ U = Pi which in practice is a very minor constraint on ∗. This makes V into a monoid and a category. A viewpoint is also a type-theory that has variables which are terms for the types, the classifications, P1, P2, ... , PM . These classifications are terms for ∗ expressions which are the types of the type system. An interpretation of this type theory for a specific relational landscape gives us the monoid.

Deﬁnition 35. Interpretation in a viewpoint. A class of objects Λ(L) is interpreted in a viewpoint V if there 0 0 0 is a mapping ι :Λ (L) → V and relations R(σi1 , σi2 , . . . , σim ) of objects of Λ (L) are interpreted relative m to the place Pi1 ∗ Pi2 ∗ · · · ∗ Pim in V where ι(σik ) = Pik .

ι maps variables of Λ(L) to the classifications of V0 and so they become terms of the type theory of V. As ι(σ) can be ∅ it acts as a filter throwing out variables and relations of Λ(L) that cannot be classified. The ∗ operation conditions the placing of relations by imposing additional logic. For example there is a big difference in the placing of relation R(σ1, σ2, σ3) in two viewpoints wherein,

137 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.2. VIEWPOINTS

in the first case, ∗ = ∩ and in the second case by ∗ =(. In the first case R(σ1, σ2, σ3) is placed over P1 ∗ P2 ∗ P3 if σ1 ∩ σ2 ∩ σ3 is non-empty and in the second case only if σ1 ( σ2 ( σ3. An important non-commutative case is when the viewpoint is used to define time development and

∗ =≺ (precedes). If Pi ≺ Pj and ι(σi) = Pi, ι(σj) = Pj then σi precedes σj and nothing comes between them. Notation.

0 • If ι(σ) = Pi ∈ V we write P (σ). For example, in the above paragraph σi precedes σj

becomes Pi(σi) ≺ Pj(σj).

• We shall write Pi1 ∗ Pi2 ∗ · · · ∗ Pim (R(σi1 , σi2 , . . . , σim )) to signify that R(σi1 , σi2 , . . . , σim ) is

placed over Pi1 ∗ Pi2 ∗ · · · ∗ Pim so, for k = 1, 2, . . . m, Pik (ι(σik )) is true.

• If ∗ is a particular operation such as ∩, ∪ or ≺ we write the viewpoint with the value of ∗ as

a subscript, so, for example, if ∗ =≺, we write V≺.

A viewpoint interpretation can be considered as a hypothesis that the rescaled system Λ(L) can be usefully described in the terms of V. This is a conceptual (logical or epistemological) hypothesis not a statistical one and we need to develop ways to test this hypothesis.

How do we interpret a relation such as R(σ1, σ2, . . . , σn) that asserts these σi are the most productive work areas of a factory? The only place these symbols have models is in L. The choice of σ of Λ(L) as being of the form R/L gives us an immediate model or interpretation of the symbol σ as a subcategory in L. It also has another significance. A minimal model of L is a set of data that satisfies all examples of all cases of each data-type; satisfies (is an example of) all relations and is evaluated by all relevant functions to give legitimate values of results. This might require long time-sequences of data. A minimal model of σ, µ(σ), is the restriction of a minimal model of L to σ as a subcategory. From this we define the minimum [n] [n] model of R (σ1, σ2, . . . , σn), µ(R ), as follows.

[n] 1. µ(R ) exists if, in the place Pi1 ∗ Pi2 ∗ ...Pin , there are σk, with Pik (σk), k = 1, 2, . . . n and µ(σk) 6= ∅.

2. µ(R[n]) has meaning as a subset of

0 0 0 ^ {(s1, s2, . . . sn) | ∃σi ∧ (ι(σi) = ι(σi)), (µ(σi) |= si) ∧ (L |= si)}. (5.4) σ0 i i

0 0 Here si is typically an instance of a relation R with Rσ → R and Rσ/L = σ.

138 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.3. VIEWPOINT OPERATIONS: FACES, BOUNDARIES AND CYCLES.

3. This allows us to deﬁne the µ(R[n]) by what would falsify it: what restricts its meaning from

being vacuous and applying to all σi relevant to its place.

In the next sections we require a formal sum of relations. What is the model of R1 +R2? Again, we look to a minimal model of L (as a set) that gives us minimal models of each relation and deletes contradicting statements leaving us with a set of statements true for both R1 and R2. What if

R1 + R2 are placed over separate intervals in V≺? R1 and R2 can be relations about suppliers in a long chain of processes, the earlier ones affecting the later ones. The latter ones can be working on work that came through from the earlier supplies and there is no overlap of µ(R1) and µ(R2). They are related over time. We can only conclude

µ(R1 ∩ R2) ⊆ µ(R1 + R2) ⊆ µ(R1 ∪ R2).

That is R1 + R2 is “assessed” by the set of statements that can always be true. This will become important when we consider the homology of a viewpoint. We shall need the concept of an equivalence class of models later on to deﬁne viewpoint homology.

Deﬁnition 36. R1 and R2 are logical equivalent, written R1 ≈ R2 if and only if V ∀p ∈ µ(R2)∃{s1, s2, . . . sm} ⊂ µ(R1) such that si =⇒ p and vice versa with R1 and R2. i

A simple example: R1 relates a sequence of relations reporting the result of inspecting sets of goods for shipment, packing them and labeling the packages and, on receipt, R2 records the labeling and unpacking and inspecting the goods for storage. (This can be put in terms of adjoint functors α : µ(R1) µ(R2): β. Statements can be ordered 0 0 by implication so implies S = {s1, s2, . . . sm} and S ⊂ S =⇒ α(S) =⇒ α(S ) and similarly for β).

5.3 Viewpoint Operations: Faces, Boundaries and Cycles.

The faces of a relation Equation 5.3 deﬁnes for viewpoints the role of the face in topology where it is a side of the 0 simplest polyhedron for each dimension. Instead of point vertexes we have the properties Pi ∈ V . The boundary of the polyhedron is then taken as an alternating sum of faces. We shall keep the topological vocabulary here. For relations such as those in Section 5.1.1 or many others that might apply to Λ(L) we deﬁne the following partial order.

139 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.3. VIEWPOINT OPERATIONS: FACES, BOUNDARIES AND CYCLES.

R1(σ1, σ2, . . . σn) < R2(σ1, σ2, . . . , σn, σn+1, σn+2, . . . , σn+r) if, for all values of σ1, σ2,... σn satisfying R1(σ1, σ2, . . . σn), there is at least one set of values of an+1, an+2, ... , an+r of the variables σn+1, σn+2, ... , σn+r for which R2(σ1, σ2, . . . , σn, an+1, an+2

..., an+r) is true.

Deﬁnition 37. The k-face of a relation R(σ1, σ2, . . . σn) ∈ Λ(L) is the largest relation F Rk in Λ(L), such that

F Rk((σ1, σ2,... σck, . . . , σn) < R(σ1, σ2, . . . , σn).

In particular F Rk has one less variable than R.

It is always the case that the k-face can be logically deﬁned as

∃σk,Pk(σk) ∧F Rk(σ1, σ2, σk−1, σk, σk+1 . . . , σn) in which case the intersection of the k face and R itself will satisfy the definition. In fact neither computer systems nor enterprise operations want definitions that require some variable(s) to be existentially quantified. It is not efficient when a concept means searching for something especially if that something can be as complicated as a sub-system or a process of some class. Thus a face must already be defined in Λ(L). Definition 37 has an implied existential quantifier in the definition of the partial order. This definition of order is not itself a relation in Λ(L).

We can deﬁne the k-face of R(σ1, σ2, . . . σn) ∈ Λ(L) in terms of the data in the relation R.

The k-face F Rk(σ1, σ2, . . . , σk−1, σk+1, . . . , σn) ∈ Λ(L) will be a largest relation F Rk such that

Data(F Rk) ⊂ Data(R(σ1, σ2, . . . σn)) such that for every

F Rk(a1, a2, . . . , ak−1, ak+1, . . . , an) there is a ak such that

R(a1, a2, . . . , ak−1, ak, ak+1, . . . , an) is true.

Each value of ak provides a section of

a˘k :F Rk(σ1, σ2, . . . , σk−1, σk+1, . . . , σn) R(σ1, σ2, . . . , σk−1, ak, σk+1, . . . , σn).

That is, each section of F Rk is an image in R and has to have the “shape” of R in such a way that

F Rk is a series of slices corresponding to

[ a˘k(F Rk). (5.5)

a˘k

140 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

Suppose R(σ1, σ2, σ3, σ4, σ5) asserts the dependence of σ5 on the set {(σ1, σ2, σ3, σ4} as a minimal set of suppliers. For many σ5 classiﬁed by P5 there can be a number of minimal supplier sets due to the ability to substitute materials. Assume σ1 and σ3 relate to the supply of important catalysts, σ2 and σ4 relate to the supply of important material supplies.

1. R1(σ2, σ3, σ4, σ5) and R3(σ1, σ2, σ4, σ5) can be deﬁned in Λ(L) as emergency cases but will

not match anything in R(σ1, σ2, σ3, σ4, σ5) and so are not faces.

2. Likewise R2(σ1, σ3, σ4, σ5) and R4(σ1, σ2, σ3, σ5) make little sense as there can be no delivery

to σ5 and so are not faces.

3. F R5(σ1, σ2, σ3, σ4) is likely to be another relation altogether so is not face.

The existence of faces of a relation multiplies the possible relations in Λ(L) and it is of interest when they cannot be deﬁned or when they are all the same in one way or another. Viewpoints themselves encode logic in the classiﬁcation and in the ∗ operation. For example,

∗ =≺. Suppose we have σ1 ≺ σ2 · · · ≺ σn is R(σ1, . . . , σn) with ≺ = “precedes” (alternatively, share a vertex, share a sub-system, are disjoint,...) then R adds information on top of the classiﬁcation and the relation σi ≺ σi+1. The sequence of ≺ is encoded in P1 ∗ P2 ∗ · · · ∗ Pn and so contains some measure of dynamics with R being additional information. This suggests the more logic (external to that of L) that can be put into the viewpoint classes and the ∗ operation the less that R needs to express. In this way we can design viewpoints to carry “scale” properties combining the scales of connectivity and coordination with purpose or legal and other business characterizations.

5.4 Deﬁnition of Viewpoint Homology

A viewpoint is a rather minimal logic of classifications. Viewpoints are independent of L but linked by an interpretation of processes or subcategories of L. As viewpoints are modeled on the category of simplexes in algebraic topology [Gabriel and Zisman, 1967] they lead naturally to a novel homology [Macfarlane, 2017]. This homology identifies a class of relations that are significant properties that can be ascribed to Λ(L). In this way viewpoint homology can be used to distinguish reporting structures.

141 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

5.4.1 The Boundary of a Relation

If R is deﬁned in Λ(L) deﬁne the “boundary” of R by the following expression.

δ(Pi1 ∗ Pi2 ∗ · · · ∗ Pim (R(σi1 , σi2 , . . . , σim )) = k Σ (−1) Pi1 ∗ Pi2 ∗ · · · ∗ Pi −1 ∗ Pk+1 · · · ∗ Pim (F Rk(σi1 , σi2 ,..., σi , . . . , σim )) (5.6) k=1,2,...,m k ck

The term “boundary” is standard in homology so we keep it here. The viewpoint boundary is a formal or symbolic sum of expressions that need to be “interpreted” just as we interpreted formal sums of adjoint relations in terms of categories in Chapter 3. As in simplicial homology in topology, the important relations are the cycles, when the boundary is zero: so that

δ(Pi1 ∗ Pi2 ∗ · · · ∗ Pim )(R(σi1 , σi2 , . . . , σim )) = 0 if and only if

Σ Pi1 ∗ Pi2 ∗ · · · ∗ Pik−1 ∗ Pk+1 · · · ∗ Pim (F Rk(σi1 , σi2 ,..., σik , . . . , σim )) ≈ k is even c Σ Pi1 ∗ Pi2 ∗ · · · ∗ Pik−1 ∗ Pk+1 · · · ∗ Pim (F Rk(σi1 , σi2 ,..., σik , . . . , σim )). k is odd c This is interpreted in terms of logical equivalence as given in Deﬁnition 36. 0 0 0 (In Macfarlane [2017] the expression R1(σ1, σ2, . . . σn) + R2(σ1, σ2, . . . σn) was deﬁned in terms of the greatest lower bound in the lattice of relations. This interprets + in terms of the lattice of relations in Λ(L) that have a place (are interpretable) in the viewpoint.

R1 + R2 = glb(R1, R2).

If such a greatest lower bound exists, then it has a place in the viewpoint. In terms of models in L it is the largest model that is accommodated in the smallest models containing R1 and R2. Thus if ι : Λ(L) → V is lattice complete, the two interpretations of + are the same. ) When a face is not deﬁned in Λ(L) it is set to zero. A zero is just a vacuous “true” when it comes to models. This allows us to interpret a sum: ΣRk + 0 ≈ ΣRk. k k Starting with Λ(L), a viewpoint V and an interpretation ι : Λ(L) → V0 we can deﬁne the modules Q(ι)m as follows

Deﬁnition 38. The “qualiﬁer” Q(ι)

Deﬁne Q(i)m to be the free Z/2.Z module generated by the expressions Pi1 ∗Pi2 ∗· · ·∗Pim (R(σi1 , σi2 , . . . , σim )) factored by the relation R1 + R2 ≈ R3 when the smallest subcategory of L containing R1 + R2 is the same as the smallest subcategory containing R3. Furthermore, we deﬁne the boundary homomorphism

142 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

δm : Q(i)m → Q(i)m−1 that is the linear extension to the boundary operator δ deﬁned in equation 5.6.

Using < x1, x2, . . . xm > to mean the Z/2.Z module generated by subsets of the set of symbols {x1, x2, . . . xm} we deﬁne:

Q(ι)1 = ⊕ < Pi(σ) >. 0 Pi∈V Using this convention we can write

Q(ι)2 = ⊕ < Pi ∗ Pj(R(σ1, σ2)) > and 1 Pi∗Pj ∈V Q(ι)3 = ⊕ < Pi ∗ Pj ∗ Pk(R(σ1, σ2, σ3)) > where Pi(σ1), Pj(σ2), Pk(σ2). 2 Pi∗Pj ∗Pk∈V We obtain a sequence of modules, called the Q(ι) chain complex:

δ1 δ2 δ3 δ4 δN Q(ι)0 ←−Q(ι)1 ←−Q(ι)2 ←−Q(ι)3 ←− ... ←−−Q(ι)N

0 where N is the total number of classiﬁcations in V . If there are no relations R(σi1 , σi1 , . . . , σim ), m ≤ N, then Q(ι)m will be zero. (The background to this is Macfarlane [2017] but much of the algebra is outlined in books of Algebraic Topology such as [Greenberg, 1967, J.R.Strooker, 2009].)

The way δ has been deﬁned in the modules Q(ι) we have, at the algebraic level, δn−1 ◦ δn = 0. From this we have the standard deﬁnitions from homological algebra.

1. The n-th module of cycles Zn = Zn(Λ(L), V) =defn ker(δn).

2. The n-th module of boundaries Bn = Bn(Λ(L), V) =defn im(δn+1).

3. The n-homology class is Hn = Hn(Λ(L), V) =defn Zn/Bn.

The chain complex Q(ι) is “augmented” by deﬁning Q(ι)0 to be Z/2.Z and δ1(Pi) = 0 in all cases. This makes Q(ι)1 = Z1(Λ(L)).

The homology Hn(Λ(L), V) is a way of counting relations that are “seen” in the viewpoint, and are signiﬁcant in Λ(L). 0 0 The homology functor Hn(Λ(L), V) can be extended to pairs (Λ(L)), Λ(L ) with L ⊂ L, with the same V, to create the relative viewpoint homology that yields the standard homology long exact sequence and a version of the excision theorem as in Algebraic Topology [Greenberg, 1967]. This is proved in Macfarlane [2017]. We illustrate how the homology works in the lowest dimensions and show the interplay of the viewpoint and the boundary equation for a cycle.

143 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

R(σi, σj).

δ(Pi ∗ Pj(R(σi, σj)) = 0 ⇔F Ri(σj) ≈F Rj(σi).

The relations F Ri, F Rj only have meaning and are true if they are equivalent to Pj and Pi respectively. So for δ(Pi ∗ Pj)(R(σi, σj)) = 0, Pj ≈ Pi (for them to be logically equivalent). This means R(σi, σj) is a relation within the class of a single Pi. Hence Z2 gives the relations among variables with the same classiﬁcation.

R(σi, σj, σk).

δ(Pi ∗ Pj ∗ Pk(R(σi, σj, σk)) = 0. In this case we get

F Ri(σj, σk) +F Rk(σi, σj) ≈F Rj(σi, σk).

1. The faces exist. Therefore the least subsystem (processes or subcategories) in L for which

F Ri(σj, σk) and F Rk(σi, σj)) can be true is the same as that for F Rj(σi, σk).

This means that there is a relation between F Rj(σi, σk) that implies the relations F Ri(σj, σk)

and F Rk(σi, σj) in the context of R(σi, σj, σk)). One way this can be interpreted is that of

a connection or composition. If F Rj(σi, σk) exists then there is a σj that connects σi to σk

which gives us F Ri(σj, σk) and F Rk(σi, σj) which, in turn gives F Rj(σi, σk)). This gives us a level of connectivity in L in terms of subsystems of particular classiﬁcations.

proj 2. The faces F Rk are not deﬁned in Λ(L). This is the case that Pi ∗Pj ∗Pk −−→ 0, the projections

R(σi, σj, σk) have no interpretation, and there is no relation among pairs of subsystems σi,

σj, σk that intersects R. As will be examined below this is suggestive that R is an extremal property.

R(σi, σj, σk, σl).

δ(Pi ∗ Pj ∗ Pk ∗ Pl(R(σi, σj, σk, σl)) = 0.

1. All the faces of R exist. In this case we get

F Ri(σj, σk, σl) +F Rk(σi, σj, σl) ≈F Rj(σi, σk, σl) +F Rl(σi, σj, σk).

Or, as it must be interpreted, the minimal model of F Ri(σj, σk, σl) and F Rk(σi, σj, σl) to-

gether is the same as for F Rj(σi, σk, σl) and F Rl(σi, σj, σk). As an example, suppose Pi

classiﬁes the relations Ri,r and so the subcategories σi,r = Ri,r/L(S) related to components

of a product. Similarly Pj and Pk classiﬁes the sub-systems Rj,s/L(S) and Rk,t/L(S) which

144 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

are other sets of components. These components are only shipped as assemblies which is

done in σl which is includes the registering of shippable products (or work in progress into

inventory) implying all of σi, σj, σj but not being the same as them. The faces are the following:

(a) F Ri(σj, σk, σl) and F Rk(σi, σj, σl) are subsequences that produce shippable components though, quite likely, they will be different sets.

(b) F Rj(σi, σk, σl) and F Rl(σi, σj, σk) with F Rj(σi, σk, σl) another possible sequence to producing shippable goods probably different from the other i, j and j, k sequences.

Finally F Rj(σi, σk, σl) exists as a sequence producing yet a another set of goods. These

of course might be precisely what ﬁlls in for the σj. In all this the minimal models means that each side of the equation ﬁlls in information that might be missing on the other side.

This can occur if there is a σ¯ for which Pl(¯σ) =⇒ Pi(σi) ∗ Pj(σj) ∨ Pi(σj) ∗ Pk(σj) .

We note the contrary case of the example after Equation 5.5. R(σ1, σ2, . . . , σn) states the contingency contacts when only n−1 supply systems σi are operational. Thus F Ri(σ1, σ2,..., σbi . . . , σn) is the contingency plan when σi resources are cut off. In this case R(σ1, σ2, . . . , σn) is logically equivalent the disjunction of the faces. When does this imply a cycle?

(a) n = 3: F R1 +F R3 ≈F R2 is not automatically the case here.

(b) n = 4: F R1 +F R3 ≈F R2 +F R4 is not automatically the case here. Again, the example after Equation 5.5 illustrates this. In terms of models each of these sides of the equation is quite different and so do not form a cycle. For n sufﬁciently large, this can be a cycle if each side of the cycle equations is covered by a subset of plans that are logically equivalent. In other words running through the faces is redundant.

2. The faces of R are not deﬁned in Λ(L). This is the case where Pi ∗ Pj ∗ Pk ∗ Pl → ∅ which

is typically the case when R is extremal or when Pi ∗ Pj imply a transitive relation such

as precedence (≺); for example, Pi(σi) and Pi+1(σi+1) imply Pi ∗ Pi+1(R(σi, σi+1)) means

σi ≺ σi+1) as well as being related by R. In the general case each face would interrupt the precedence and so would not be deﬁned.

In general, if R[n] has zero boundary then it has faces with some degree of symmetry with the various σ playing similar roles among the faces or else, when R[n] has no faces, R[n] is a "concept"

145 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY referring to particular sets of subsystems for which R[n] applies only for sets of n variables and is not deﬁnable for smaller sets.

5.4.2 Characterizing Cycles and Acyclic Viewpoints

The nature of cycles

Given R, δ(R) = 0 implies Σ F Rk ≈ Σ F Rk so: k odd k even

µ(Σ F Rk) ≈ µ(Σ F Rk) k odd k even By (Deﬁntion 36) every statement on each side has its “counterparts” on the other.

Furthermore, at least some of the values produced in the a˘i mappings have to be in both sides otherwise there can statements about the omitted a˘i embeddings that cannot be matched. Conse- quently, both sides of the model equation correspond to a consistent set of statements that are the evidence for a model of R.

We can deﬁne Cyc(R) to be a non-empty “logical enclosure” of µ(F R1,F R2,... ) or equally

µ(F R2,F R4,... ). The logical closure being all the statements q true in µ(R) that are a consequence of some p in µ(F R1,F R3,... ) (hence also µ(F R2,F R4,... ). As defined Cyc(R) need not be all of R. There might be a number of self-consistent sets of statements associated with sets of subcategories {σ1, σ2, . . . σn} which satisfy R(σ1, σ2, . . . σn). (For example, R could be a listing of maximally co-operative university schools σi, perhaps as mea- sured by the ability to cross teach in at least n − 1 departments.). This is evidenced by statements sij ∈ µ(σi) ∩ µ(σj). Only are few show as many as n − 1 strong cooperativeness giving a small region of R where the faces are defined.) [n] [n] [n] Suppose R splits into Cyc(R ) plus a residue R¯ . Suppose also there exists F R¯ i then it [n] would be over the same place as F Ri and be part of the definition of the i-face of R as the largest n − 1 relation in Λ(L) to be in R. This would mean Cyc(R) = Cyc(R) ∪ µ(R¯ [n]) We get.

Theorem 5.4.1. If R is a cycle with a full set of faces that splits into Cyc(R) t R¯ then R¯ does not have any faces and so is also a cycle.

Thus a relation with a full set of faces has Cyc(R) = µ(R) or it splits into a pair of cycles Cyc(R) and R¯ .

Corollary 5.4.2. If Cyc(R) = µ(R), the set of even, or the set of odd faces contains sufﬁcient information to determine R.

146 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

Proposition 5.4.1. Given R in Λ(L) and

1. The order of the variables in R(σ1, σ2, . . . , σn) does not matter. (This implies that these subcategories are interpreted over commutative places).

2. If σi = Ri/L and for each i and j ∃αij : Ri/L → Rj/L and αij is bijective. (This is usually the case when L has parameterized levels over the same set of data-types as illustrated in Section 2.7.5 ).

Then δ(R(σ1, σ2, . . . , σn)) = 0

Proof. 1. The relation R(σ1, σ2, . . . , σn) is still true after any permutation of the σi but the place

is the same. This also means Data(F Ri) can be embedded into Data(R) with some value

ai in the i-th place. The embedding will also be true with swapping with ai with the j-th

place if σi and σj are swapped. Thus F Ri and F Rj are not distinguished when σi and σj are swapped. Hence the models are all the same so logically equivalent and the minimal models are all the same for any mixture of faces and so the boundary is zero.

2. We can use the bijective αij to substitute the σi in each F Ri so the µ(F Ri) are all the same

and so µ(F Ri +F Rj) ≈ µ(F Ri) in all cases, so that the boundary is zero

Corollary 5.4.3. An extremal property is a cycle.

Proof. Suppose R(σ1, σ2, . . . , σn) asserts that {σ1, σ2, . . . , σn} is optimal for some property such as throughput or least cost. Being a property of a set it is indifferent to order. Faces of R require existential quantiﬁcation for the omitted variable and so are not deﬁned in Λ(L).

Consequences of ∗

Definition 39. A viewpoint is a flow if the viewpoint properties are partially ordered, that is V0 and ∗ defines the partial order: Pk ∗ Pk+1 only if Pk(σ1) and Pk+1(σ2) imply σ2 follows directly after σ1.

Deﬁnition 40. If V is progressing if it is a ﬂow and ∀σ Pi ∗ Pj =⇒ ¬(Pi((σ) ∧ Pj(σ)).

That is, σ cannot have the same properties in succeeding classiﬁcations.

Proposition 5.4.2. If V is progressing it is acyclic.

Lemma 5.4.4. If V is a ﬂow and places relations on intervals in V0 then R[n] can only be a cycle if n is even and greater than 3.

147 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.4. DEFINITION OF VIEWPOINT HOMOLOGY

[n] [n] [n] Proof. A relation R placed on a chain of n ≥ 3 has two faces: F R1 and F Rn none of the others can be deﬁned. To do so would be to leave gaps at the places of the omitted σk which violates the progression. In the case of n > 3 and even, the faces will be on different sides of the boundary equation for a cycle. If these are the same the R[n] will be a cycle. If n is odd they are on the same side of the equation and therefore the cycle equation fails.

The proposition follows by ruling out an unchanging interval in which the only two possible [n] [n] faces: F R1 and F Rn , n even are equal. Thus the case n even and ≥ 3 fails as the only two faces of different sides of the cycle equation are unequal and for n odd the equation is never fulﬁlled.

5.4.3 What is the Interpretation of Bn?

Suppose δ(R[m+1]) = R0[m] 6= 0. For example in the case m = 3

[4] 0[3] δ(R (σi, σj, σk, σl) = Ri(σj, σk, σl) − Rj(σi, σk, σl) + Rk(σi, σj, σl) − Rl(σi, σj, σk) ≈ R .

So:

0[3] R + (F Rj(σi, σj, σl) +F Rl(σi, σj, σk)) ≈F Ri(σj, σk, σl) +F Rk(σi, σj, σl)).

Or, as it must interpreted,

0[3] µ({R ,F Rj(σi, σj, σl),F Rl(σi, σj, σk)}) ≈ µ({F Ri(σj, σk, σl),F Rk(σi, σj, σl)})

0[3] Hence µ(R ) restricts µ({F Rj(σi, σj, σl),F Rl(σi, σj, σk)}) to equal µ({F Ri(σj, σk, σl),F Rk(σi, σj, σl)}). This restriction is likely to be equations among the domain variables of R[m+1] in a way that restricts the variables to a subset of possibilities that exist in R[m+1].

This gives us an interpretation of the boundary group Bn and, in particular, Hn = Zn/Bn. If

R1 and R2 are two cycles they belong to the same class if R1 − R2 = r ∈ Bn so R1 ≈ R2 + r. r ≈ δn+1(R) provides a context for R2 to have the same minimal model of R1. This means both

µ(R2) ( µ(R1). and

µ(R1) ≈ µ(r, R2).

Which means the number of non-zero boundaries r ∈ Bn are a way of “locating” common places of sets of cycles.

148 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

5.5 The Category of Scaled Viewpoints as A Functor

In this section we deﬁne the category of scaled viewpoints which expands the category of viewpointed systems introduced in Section 4 of Macfarlane [2017]. Viewpointed systems were shown to be the domain of the homology H (this was proved for a cohomology in that reference). We shall prove that the homology is a functor on the category of scaled viewpoints by building on the established result noted below. The objects of viewpointed systems V are pairs (Λ(L), V) where the elements of Λ(L) are interpreted in V. This will be the same for scaled viewpoints and so we keep the symbol V which, after the deﬁnition of the morphisms of scaled viewpoints, will be the meaning of V.

Deﬁnition 41. A viewpoint morphism υ : V1 → V2 is

0 0 1. a function υ : V1 → V2 in which υ : Pi 7→ υ(Pi),

2. υ is an ∗ homomorphism so that υ(Pi1 ∗ Pi2 ∗ ...Pin ) = υ(Pi1 ) ∗ υ(Pi2 ) ∗ ... ∗ υ(Pin ).

3. υ is a functor from V1 to V2 so projections and their composition are preserved.

This defines the morphisms for the category of viewpoints V, the objects being viewpoints. A viewpoint morphism υ does not necessarily conserve the index of V so that we do not have n υ n n υ n−r V1 −→V2 or even V1 −→V2 with r dependent on n. It does preserve the ∗ product but the change in the number of variables is dictated by υ | V0. υ 0 An onto homomorphism V −→V such as V = V1 × V2 and υ :(P1,P2) 7→ P1 ∧ P2, can merge independent sets of properties. (P1,P2) in V means ι(σ) is a combination of properties in the Carte- sian product that can be merged to the conjunction of products in V0. In general, the distribution of ∧ over ∗ (or vice versa) has to be checked for each case. The functors Λ are often defined with a viewpoint in mind. What subcategories are to be the variables of Λ(L) for a class of relational landscapes is influenced (possibly determined) by the classifications in the viewpoint. Viewpoint homomorphisms then arise from adding or omitting properties defining the classification. For example, Pi(σi) in one viewpoint might correspond to σ having properties pi,1, pi,2, . . . pi,m while in another viewpoint, a more inclusive classification, is 0 0 0 define by pi,1, pi,2,... pi,m or pj,1, pj,2,... pi,k so maps between viewpoints and Λ will arise through possible Boolean combinations of properties of subcategories of relational landscapes.

Deﬁnition 42. The category of viewpointed systems and translations, V, has as objects the assignments ι Λ(L) −→V0 which we write as pairs (Λ(L), V) where L is an object of Sys. Morphisms of this category will Λ(ϕ) be called translations and are pairs (ϕ, υ) : (Λ(L1), V1) → (Λ(L2), V2) with Λ(L1) −−−→ Λ(L2) induced

149 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

ϕ υ by the relational translations L1 −→L2 and the viewpoint homomorphism V1 −→V2 so that the following diagram commutes: Λ(ϕ) Λ(L1) / Λ(L2) (5.7)

ι1 ι2 0 / 0 V1 υ V2 .

Given σ with P (σ) then Λ(ϕ)(σ) is interpreted as υ(P ) so that υ(P )(Λ(ϕ)(σ)) is true:

P (ι(σ)) ⇒ υ(P )(ι(Λ(ϕ)(σ))).

Λ(_),_) A scaling functor Λ creates the functor Sys×V −−−−→ V. A scaling functor Λ with a viewpoint Λ(_),V) V is then a functor of relational landscapes: Sys −−−−→ V: L 7→ (Λ(L), V).

Translations (ϕ, υ) : (Λ(L1), V1) → (Λ(L2), V2) give rise to homomorphisms

H∗(ϕ,υ) H∗(Λ(L1), V1) −−−−−→H∗(Λ(L2), V2). (5.8)

This is proved in Macfarlane [2017, Section 5].

η Extending Homology to include maps Λ1 −→ Λ2

Equation 5.8 can be extended to compare two scalings. Given L and two functors Λ1 and Λ2, each η satisfying the commuting diagram 5.7, and a natural transformation Λ1 −→ Λ2, what additional properties does the natural transformation have to have to give a homomorphism

H∗(η,υ) H∗(Λ1(L), V1) −−−−−→H∗(Λ2(L), V2)? (5.9)

This is answered by adding to viewpointed systems the following morphisms.

η Deﬁnition 43. A general viewpoint scaling translation is a natural transformation of functors Λ1 −→ Λ2 that commutes with viewpoint homomophisms giving the following commuting diagram.

ηL Λ1(L) / Λ2(L) (5.10)

ι1 ι2 0 / 0 V1 υ V2 . and η is algebraic:

R1 + R2 ≈ R3 ∈ Λ1(L) =⇒ ηL(R1 + R2) ≈ ηL(R3). (5.11)

150 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

Equation 5.11 is equivalent to:

(µ(R1 + R2) ≈ µ(R2)) =⇒ µ(ηL(R1) + ηL(R2)) ≈ µ(ηL(R3)). (5.12)

An example of such a natural transformation is where the viewpoint closely mirrors the functor

Λ. If Λ1 selects subsystems distinguished by various types of expertise, so are interpreted in a viewpoint V1, and also by location (in the case of a company with many locations) to be interpreted υ in a viewpoint V2, we can deﬁne Λ1 × V2 −→V1 as the projection. Λ2 is not interested in location. The substantial relations of interest among subsystems are determined by expertise and all the ι2 expertise is shared among locations. This gives Λ2 −→V2. What is true for relations among σi is remains true without consideration of location. The subsystems being aggregates of all the local subsystems means Λ2 has fewer, larger variables σ¯j. We can therefore deﬁne a general viewpoint η scaling translation Λ1 −→ Λ2.

Deﬁnition 44. The category of scaled viewpoints V is the category with objects (Λ(L), V) with an inter- ι pretation Λ(L) −→V) (as in viewpointed systems) and the morphisms are viewpoint translations (Diagram 5.7) and general viewpoint scaling translations (Diagram 5.10).

(ηL,υ) Theorem 5.5.1. A general viewpoint scaling translation (Λ1(L), V1) −−−−→ (Λ2(L), V1) induces a homomorphism

H∗(η,υ) H∗(Λ1(L), V1) −−−−−→H∗(Λ2(L), V2).

We sketch the proof. We start by deﬁning the homomorphism

Q(ηL,υ) Q(ι1 :Λ1(L) → V1) −−−−−→Q(ι2 :Λ2(L) → V2) so that

Q(ηL,υ) Q(ι1)m / Q(ι2)m

δm δm

Q(ηL,υ) Q(ι1)m−1 / Q(ι2)m−1 commutes. 1 Given σi and with ι1(σi) = Pi and R(σ1, σ2, . . . , σm) in Λ1(L) then R(σ1, σ2, . . . , σm) is repre- 1 1 1 sented by the expression P1 ∗ P2 ∗ ...Pm(R(σ1, σ2, . . . , σm)) in Q(ι)m. (ηL, υ) maps

σi 7→ η(σi),

R(σ1, σ2, . . . , σm) 7→ η(R)(η(σ1), η(σ2), . . . , η(σm)) ∈ Λ2(L),

151 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

1 1 2 and, if Pi ∈ V1 then υ : Pi (σi)) 7→ Pj (η(σi)). 1 0 Evaluating diagram 5.10 with ι1 : σi 7→ Pi ∈ V1 , the top path through Λ2(L) gives σi 7→ 1 1 2 ηL(σi) 7→ ι2(ηL(σi)). The lower path through V1 gives σi 7→ Pi 7→ υ(Pi ) = Pj . By commutativity

1 2 υ(Pi )(σ1) = ι2(ηL(σi)) = Pj (ηL(σi)).

Therefore we can deﬁne Q(ηL, υ) as mapping

1 1 1 1 1 1 P1 ∗ P2 ∗ ...Pm(R(σ1, σ2, . . . , σm)) 7→ υ(P1 ) ∗ υ(P2 ) ∗ . . . υ(Pm)(η(R)(η(σ1), η(σ2), . . . , η(σm))) ∈ Q(ι2).

This is the mapping on generators. Q(ι) is the quotient of the free module by equations of the type:

R1 + R2 ≈ R3.

ηL Λ1(L) −→ Λ2(L) preserves this algebraic relation.

Now δ : Q(ι)m → Q(ι)m−1 is deﬁned for each place and this is sufﬁcient to map from places in

V1 to those in V2 and faces in Λ1(L) to faces in Λ2(L). Where V1 and V2 are isomorphic as monoids the homology of relations R is preserved. Where υ identiﬁes classiﬁcations and m places becomes k places, k < m, a m placed relation that is a cycle in Λ1(L), so in Zm(Λ1(L), V1), is mapped by Q(ηL,υ) Qk(ι1) −−−−−→Qk(ι2) to zero even though it might appear as a cycle in Zk(Λ2(L), V2).

ϕ Corollary 5.5.2. A relational landscape translation L1 −→L2 and a general viewpoint scaling translation η Λ1 −→ Λ2 induce a homomorphism

H∗(ϕ◦η,υ) H∗(Λ1(L1), V1) −−−−−−→H∗(Λ2(L2), V2).

Proof. The natural transformation gives the commuting diagram:

ηL1 Λ1(L1) / Λ2(L1)

ϕ ϕ ηL2 Λ1(L2) / Λ2(L2).

Λ1(ϕ) Λ1(L1) −−−→ Λ1(L2) induces the homology homomorphism

H∗(ϕ,υ) H∗(Λ1(L1), V1) −−−−−→H∗(Λ1(L2), V2).

The horizontal maps give:

H∗(η,υ) H∗(Λ1(L), V1) −−−−−→H∗(Λ2(L), V2).

Composition of these homomorphisms provides the required homomorphism.

152 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

ϕ ϕ1 ϕ2 When we can factor L1 −→L2 as L1 −→ L and L −→ L2 the commuting diagrams can be combined to give:

Λ1(ϕ1) ηL Λ2(ϕ2) Λ1(L1) / Λ1(L) / Λ2(L) / Λ1(L2)

ι1 ι1 ι2 ι2 $ z 0 / 0 V1 υ V2 .

In this diagram ιi acts as a type of natural transformation from the functor Λi to Vi.

5.5.1 The Category V(L) of Scaled Viewpoints of L

Let V(L) be the full subcategory of V for L so the entire set of scalings and viewpoints for L. For a given Λ the morphisms in V(L) are commuting diagrams:

Λ(L)

ι1 ι2 | " / V1 υ V2.

If υ is a surjective homomorphism then (Λ(L), V1) is a refinement of (Λ(L), V2) (as V2 has fewer classifications (bigger classes of classifications) than V1. This gives a “second-order” of scaling within the scale defined by Λ(L). We also have morphism of the form of Diagram 5.10. A relational landscape can now be associated with a category of all scaled viewpoints as a class of descriptions of L, the objects are the objects of V(L) and the morphisms are

(ηL,υ) 0 0 (Λ1(L), V) −−−−→ (Λ2(L ), V ) that factorize as

(ηL,1V ) (Λ1(L), V) −−−−−→ (Λ2(L), V) (Λ(ϕ),υ) 0 0 (Λ2(L), V) −−−−−→ (Λ2(L ), V )

ϕ υ where L −→L0 is a reduction and V −→V0 is a surjection. This category is an epistemological object. We can have many descriptions of reporting structure as can be illustrated by the following story. I go to a hotel that has a convention center with the plan of attending a one day conference of business analysts and stay overnight before leaving. I interact with and see many types of staff and infer the existences of others: the reception staff, waiters, cleaners, chefs, and maintenance staff. One can see or guess the existence of conference planners, caterers, purchasing, accounting and the part of the reporting structure that ensure the

153 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR plans and assets and catering is all available when and where they need be. This is a scaled object and my guesses might be true of a reduction of the actual reporting structure. I might have missed out security functions, gardeners and cleaners of the car-park which I lump in with maintenance. My viewpoint is that of a customer, but there are other viewpoints, perhaps for architects, investors, insurers or staff trainers. These are specialist descriptions and the category is class of partial knowledge that “surrounds” the phenomena of the reporting components of a hotel reporting structure.

The category of scaled viewpoints is a functor on Sysepi by Deﬁnitions 42, 43 and Corollary 5.5.2. As with the homology tableaux it contains a very large number of new details of the properties of L, especially extremal properties, that cannot be extracted directly from L. It also gives new details of the way two relational landscapes can differ. In the case of information systems that claim to provide a reporting structure, a lot can ride on the different details among systems. The wrong choice can be costly. ξ A scaled viewpoint (Λ(L), V) is not a relational landscape for which these is a map L −→ Λ(L).

We only need σi ∩ σj 6= ∅ for there to be no obvious map ξ (in either direction).

5.5.2 Scaling Relations from L to Λ(L)

Each functor Λ is deﬁned by selecting the level of relations for which σ = R/L. If the cutoff is relatively low, corresponding to low level sub-systems in the reporting structure, we expect many of the relations deﬁnable in Λ(L) to come from relations in L. If σi = Ri/L, then, for an adjoint ! [2] [2] relation R (Ri,Rj) to have a representation as R (σi, σj), µ(R ) ⊂ µ(σi) ∩ µ(σj). This means ! sets of statements Si ⊂ µ(σi) map to, or are cross-referenced with, sets Sj ⊂ µ(σj). R (Ri,Rj) is a correlation cycle if α : J(R1) J(R)2 : β is an adjoint. On its own this does not imply logical equivalence. Implicit in the development of correlation homology is that the adjoints and isomorphisms of the categories interpreting the cycle equations are always true. They hold for any minimal model of L. Assumption: L has no fortuitous equations: We make the assumption that any homological equation is true for ANY minimal model of L. In such a case the set of statements matched by the correlation must arise from the logic L imposes on any minimal model.

Proposition 5.5.1. Given “L has no fortuitous equations,” then, for sufﬁciently large n, if an correlation cycle that “lifts to” (can be deﬁned in) Λ(L), and is interpretable in V it becomes a cycle in the viewpoint homology.

154 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

!! ! ! ! ! Proof. Consider a correlation cycle R (R1,R2,R3). It is a level 2 adjoint relation. The Ri(Ri1,Ri2), i = 1, 2, 3 are adjoint relations. Let αi : J(Ri1) J(Ri2): βi be the adjoints. αi and βi map queries which are satisﬁed by data of coupled relations irrespective of the model of L. Each element of data corresponds to a proposition modeled by Rij, j = 1, 2. These are independent of the minimal models so must have a basis in logic. This implies there is a logical equivalence among sets of propositions. ¯ ! ! This argument can be extended to logical equivalences to the category of queries Cat[R13(R1,R3)] ¯ ! ! ¯ ! ! which is isomorphic to Cat[R12(R1,R2)] × Cat[R23(R2,R3)] (using the notations of Section 3.3.2). The isomorphism imposes a strict constraint on the queries that follow from R¯13, R¯12 and R¯23. By the “L has no fortuitous equations” principal, these must arise from logical connections and so statements satisfying µ(R¯13), must imply statements in µ(R¯12, R¯23) and vice versa so proving logical equivalence. This argument provides the induction argument for any correlation starting above the level of the σ are deﬁned.

The viewpoint homology is a more general homology than the correlation homology. It applies to all relations in Λ(L) while the correlation homology only applies to adjoint relation in L.

Deﬁnition 45. 1. The relevance of V for L is the union of the domains of the set of σ classiﬁed by the properties in V0:

S dom(σ) ⊆ D(L). 0 σ∈Λ(L)∧∃Pi∈V ∧Pi(σi)

(dom(σ) = ∪{dom(R) | R ∈ σ}).

2. The “homological cover” of the viewpoint homology is the set of σ that are included in the relations that generate homology classes.

Relevance and homological cover give the extent to which we can transfer relations in L to Λ(L) and so transfer correlation homology classes to viewpoint homology classes.

5.5.3 Product Viewpoints and Increasing Homological Cover

Many of the relations R in Section 5.1.1 apply to systems which flow and systems with reporting structures that monitor those flows. This gives little information about capacity or what can be happening across the flows and particularly what happens when flows intersect. To obtain this information we need a more general viewpoint that includes branching and activities in and across

155 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

[n+1] the ﬂow. The relations Rinit and Rterm, Rcond (σ], ~σ) are all branching or cross ﬂow relations. We have to choose a viewpoint that places these relations. The following process diagram of subsystems will demonstrate that using a product viewpoint can increase relevance and homological cover.

σ / σ / σ / σ < 11 12 13 < 14

" " σ σ / σ σ / σ 21 < 22 < 23 < 24 < B 25

" " σ31 / σ32 / σ33 / σ34

" σ41 / σ42 σ43 // σ44

" " σ51 / σ52 / σ53

Here the σij are each classiﬁed by Pi ∧ Pj. The Pj,j = 1, 2,..., 5 are progressing. The Pi,i = 1, 2,..., 5 are commutative. Left to right subsystems are progressing so there are no cycles but for every i = 1, 2,..., 5 each set of vertical σij classiﬁed by Pi ∧ Pj can be in a relation

[k] Pi ∧ Pj ∗ Pi+1 ∧ Pj ∗ · · · ∗ Pi+k ∧ Pj(R (σij, σi+1,j . . . σi+k,j)) which is possibly a cycle. Indeed, each set {σi,j, σi+1,j, . . . σi+k,j} can be the maximal Pj antichain for the appropriate i and k.

This means cycles, and indeed homology classes, include all of Λ(L) except σ21, σ24, and σ25. If dom(µ(σ)) is a large subset of D(L), this also implies that much of L is covered. This should be taken as a measure of the relations that can be placed in V and the way they connect the subsystems. In this case it shows how the cycles and, in this case, the homology classes, apply to a majority of the subsystems defined by Λ and classified by V so this gives quite general information about the structure of L. The fact that the more comprehensive viewpoint is a product suggests that there is a sense of dimension of viewpoints that summarizes highest level of non-zero homology of each of the product factors. In the case of the flow this number is always zero; we need to include it by adding one to the highest number. This area needs much more research in order to disentangle the contribution of “non-orthogonal” viewpoints.

156 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

5.5.4 Integration Relative to a Viewpoint

We now have the tools to continue the discussion in Section 2.6 on general concepts of integration. Algebraic integration was deﬁned in Section 2.6 as a set of prime ideals of C(L) the direct sum of which has no annihilator. A different “integrated” could be in terms of classes of expert users (being the viewpoint) who can transfer information (data, answers to queries, data in arbitrary multimedia formats) to each other. That is, “integrated” in terms of data exchange.

Suppose the Pi classify the expert users. Putting ∗ = ∧, assume there is largest set σi, i = [m] 1, 2, . . . , m satisfying a cycle Rintg(σ1, σ2, . . . , σm) that says anyone working with the subsystem σi can exchange information with anyone using σj, i, j ≤ m. If the largest place is P1 ∧ P2 ∧ · · · ∧

PM then that is the most we can divide up the types of expertise that can be interpreted by the viewpoint. The two cases are:

[M] [m] 1. Rintg(σ1, σ2, . . . , σM ) is true then similar statements hold for Rintg(σ1, σ2, . . . , σm), m < M [M] so Rintg has faces. In this case the minimal models are all of the type that a message sij can be sent from σi to σj so each side of the cycle equations will have all the same messages and [M] will be logically equivalent. We assume Rintg is a cycle and that the L is integrated according to the new viewpoint deﬁnition.

[M] [M] [M] 2. Rintg(σ1, σ2, . . . , σM ) has no faces; This means R is defined in terms that preclude R from applying to a smaller set. That is R[M] is defined as a relation for a set of subsystems that has representatives from every expert group that has been classified. In this case R[M] has

no sub-relations in Λ(L(S)). This is a property of the system and of the set {σ1, σ2, . . . , σM } which is has no hierarchy deﬁned in Λ(L) that intersects R[M].

In both cases the minimal model will be in terms of information being exchanged between the experts concerned with the working of subsystems σi. This gives us a new way to define integration via a viewpoint. Given a viewpoint that identifies subsystems we define integration as a relation holding among a particular maximum set of subsystems of different types. If this is defined as an extremal relation that gives us a special homology class that defines a “relativist” concept of integration. This type of easily interpreted structure can become a desired enhancement of a reporting structure L. This adds to the type of project operators as given in Section 4.4. It also illustrates that the existence of viewpoint homology classes can be used to define properties of L.

5.5.5 Consistency Classes of Viewpoints

Viewpoints should be a seen as analytic techniques to shape the use of homology to highlight aspects of the structure of Λ(L) and hence of L.

157 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.5. THE CATEGORY OF SCALED VIEWPOINTS AS A FUNCTOR

There is a sense in which two viewpoints of Λ(L) are inconsistent. Given V1 and V2 with 0 0 ι1 0 ι2 0 V1 ∩ V2 6= ∅ and the pullback of Λ(L) −→V1 and Λ(L) −→V2 is a set E ( Λ(L). We can ask whether this can be extended to an enlarged viewpoint accommodating both V1 and V2? This would the case if there was a pushout W = V1 + V2 (ﬁbered coproduct) V1∩V2 V ∩ V / / V 1 2 1

V2 / W. If there is no such pushout W there must be a logical obstruction to forming it hence some inconsistency between the two viewpoints.

Deﬁne a mutually independent set of classiﬁcations as a set P˜ = {P1,P2, ...., Pn } such that for any σ the truth of Pi(σ) says nothing about the truth of Pj(σ) and vice versa. It is a simple exercise to create a viewpoint with such a set. Two or more independent sets can be used to create independent viewpoints that are consistent as long as all the properties are independent. These can then be multiplied to create a new viewpoint

0 Deﬁnition 46. The viewpoints V1 , V2 on Λ(L) are consistent if there exist V0 and V0 so that the following pullback and the pushout diagrams exist as viewpoints on Λ(L). The pullback V1 ×V0 V2 exists:

V1 ×V0 V2 −−−−→ V1    u y y 1 u2 V2 −−−−→V0. The pushout W of two viewpoints exists:

0 υ1 V0 −−−−→ V1   υ   2y y

V2 −−−−→W. In the ﬁrst place a mutual reﬁnement exists and in the second a mutual generalization can be created. This creates a symmetric relation among viewpoints and, as pullbacks of pullbacks are pullbacks and similarly for pushouts, the consistency is transitive and thus we can talk about consistency equivalence classes or consistent families. Consistent families of viewpoints represent the many ways in which we make judgments about enterprise reporting structures. Most public companies will have to conform to the accounting rules and the share-market regulations where they are registered. It is an open question whether consistent classes of viewpoints along with their homologies can characterize an enterprise reporting structures. Indeed there are many questions along theses lines.

158 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.6. THE STACK OF SCALED VIEWPOINTS OVER L

5.6 The Stack of Scaled Viewpoints Over L

In this section the connection between the relations of L and the type theory of an arbitrary (Λ(L), V) is investigated. This leads to sections that can span a number of scaled viewpoints. This in turn raises questions of how we can do this in a logically consistent way. The techniques developed in the Sections 4.5.2 and 4.5.3 lead to developing a new stack of V(L over L equipped with the conditional topology. The stack gives us local views of the logic of (Λ(L), V) relevant to open sets of L. Interpreting (Λ(L), V) as a high-level description of L, the stack gives us the catalog of local descriptions in a way that can be a patched together. The structure of the stack ensures that η,υ viewpointed translation (Λ1(L), V1) −−→ (Λ2(L), V2) can be localized and also these local maps can be patched together as with sections of sheaves.

(Λ(L), V) as a sheaf

Let τ be the conditional topology on on L (Section 4.3.3). In this section we treat τ as a category where for V and U in τ, if V ⊂ U, then the map V → U is in category τ. Usually, sheaves are developed over τ op so that when V ⊂ U we have restriction maps going from U to V . We want to gather the information in the viewpointed objects (Λ(L), V)(L) into a stack-like construction over L with the conditional topology. ι Given (Λ(L), V), and so Λ(L) −→V, deﬁne:

1. The relevance function ρ : (Λ(L), V0) → τ by

ρ(σ) = S N(R). R∈σ

ρ(σ) is everything that depends on σ.

[m] 2. For R (σ1, σ2, . . . , σm), m ≥ 2, σi, i = 1, 2, . . . , m deﬁne ρ(R) as the intersection of the ρ(σi). As this is a ﬁnite set of open sets it is open.

[m] Although the minimal model µ(R ) can be smaller than the intersection of the ρ(σi) it is whether it is deﬁnable in U that is important. For U open in the conditional topology, deﬁne

[m] (Λ(L), V)(U) =defn {R (σ1, σ2, . . . , σm) | m ≥ 2, σi, i = 1, 2, . . . , m, and ρ(σi) ⊂ U}.

By the deﬁnition of ρ above, ρ(R[m]) ⊂ U.

159 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.6. THE STACK OF SCALED VIEWPOINTS OVER L

If V ⊂ U define (Λ(L), V)(V ) → (Λ(L), V)(U) as a embedding; if ρ(σ) is a subset of V it is a subset of U. (If we make (Λ(L), V)(U) a pointed set with the symbol ω as the point in the category of pointed sets, we can define a restriction (Λ(L), V)(U) → (Λ(L), V)(V ) as identity for those σ ∈ (Λ(L), V)(V ) and map everything else to ω). (Λ(L), V) gives a presheaf over τ. (Equally, we have a presheaf over τ op by adding a point, ω, to each V˜(U) and then defining a restriction map.)

Lemma 5.6.1. The following diagram commutes

(Λ(L), V)(Ui ∩ Uj ∩ Uk)

t * (Λ(L), V)(Ui ∩ Uj) (Λ(L), V)(Ui ∩ Uk) (Λ(L), V)(Uj ∩ Uk)

(Λ(L), V)(Ui) (Λ(L), V)(Uk) (Λ(L), V)(Uj)

* t (Λ(L), V)(Ui ∪ Uj ∪ Uk)

Proof. This is simply a matter of set theory.

The lemma gives the descent datum for a prestack over the conditional topology. (In this case it is more like “ascent datum”.)

The sheaf of (Λ(L), V)(U)) sections

Definition of sections A section is defined to be a segment of logic in (Λ(L), V)(U)) in a way that keeps dependencies together. (In Definition 28 sections are functors from one L0 but here their range (Λ(L), V)(U)) is not a category.)

Deﬁnition 47. A section s : U → ℘((Λ(L), V)(U)) is given by the following assignments

1. s(R) = {σ1, σ2, . . . , σr} ⊂ (Λ(L), V)(U) where R ∈ ρ(σi) i = 1, 2, . . . , r. This implies {σ1, σ2, . . . , σr} are all in (Λ(L), V)(U) and all contain R.

2. s : R¯(R1,R2,...Rm) 7→ R(σ1, σ2, . . . , σr) ∈ (Λ(L), V)(U) then,

(a) ρ(σi) ⊂ U, i = 1, 2, . . . , r and (b) µ(R) ⊇ J(R¯) ∩ U.

160 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.6. THE STACK OF SCALED VIEWPOINTS OVER L

Denote the set of U sections as Γ(U, (Λ(L), V)).

The reason for 2(b) in Deﬁnition 47 is that µ(R) is a set of statements formulated in L. Such statements are relational statements S asserting that couplings of relations are true. We can view

R¯ as a collection of statements true for a subset of the couplings of the domain relations Ri, (see

Theorem 2.5.4). These statements must be able to be formulated in U which follows from Ri being in the sets ρ(σj) of R. The definition of a section gives a subset of (Λ(L), V)(U) that has connections with each relation in the domain of the section and which is also in U. Each section defines a “fragment of logic” in Λ(L). It might not be possible to define a section to (Λ(L), V)(U); U might contain a number of nested relations for which there is no appropriate R. There might be only one possible section or there might be many sections over U. Sections to viewpointed objects, including local sections (Λ(L), V)(U), are contained entirely in a single (Λ(L), V)(U). If s(R¯) = R, half of which is in 0 (Λ1(L), V1)(U) and the other half in (Λ2(L), V2)(U), this forces R → R¯ to be in the intersection of

(Λ1(L), V1)(U) ∩ (Λ2(L), V2)(U). The sections in Γ(U, (Λ(L), V)) are a collection of connected items in (Λ(L), V)(U). Thus, given si ∈ Γ(Ui, (Λ(L), V)) with si | Ui∩Uj = sj | Ui∩Uj for each pair i, j these are all consistent fragments having common components. The aggregate set deﬁned over ∪iUi is given by s(R) = si(R) for

R ∈ Ui. The condition si | Ui ∩ Uj = sj | Ui ∩ Uj ensures there are no ambiguous assignments (an application of the Lemma (5.6.1). This gives a sheaf Γ((Λ(L), V)). Generally:

Proposition 5.6.1. Any object of (Λ(L), V) of V(L) gives rise to sheaf over the category of conditional open sets τ

Generalizing the category V(L)

V(L) is the category of objects (Λ(L), V) for fixed L. Definition 41 in Section 5.5 defines viewpoint morphisms and viewpointed systems and trans- (η,υ) lation. Here we require a general viewpoint scaling translation (Definition 43) (Λ1(L), V1) −−−→ η (Λ2(L), V2) which is a natural transformation Λ1 −→ Λ2 satisfying the diagram 5.10:

ηL Λ1(L) / Λ2(L)

ι1 ι2 0 / 0 V1 υ V2 .

161 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.6. THE STACK OF SCALED VIEWPOINTS OVER L and an algebraic condition, Equation 5.11, that does not concern us here. ι ι Given Λ1(L) −→V deﬁne the local interpretation V(U) as the image im(Λ1(L)(U) −→V).

Lemma 5.6.2. The following diagram commutes.

V (U ) 8 1 i

* V1(Ui ∩ Uj) V1(Ui ∪ Uj) 8 υ * V1(Uj)

υ υ V (U ) 8 2 i υ * V2(Ui ∩ Uj) V2(Ui ∪ Uj) 8

* V2(Uj)

ι Proof. Starting with Λ1(L) −→V the top and bottom are deﬁnitions corresponding to ι im(Λ1(L)(Ui ∩ Uj) −→V1).

The diagram corresponds to the same diagram for Λ1(L), V)(W ) with W ranging through V =

Ui ∩Uj, Ui, Uj, Ui ∪Uj. The image of η, η(Λ1(L)) ⊂ Λ2(L), gives η(Λ1(L))(U) ⊂ η(Λ2(L))(U) which η|U ι2 then is the start of Λ1(L)(U) −−→ Λ2(L)(U) −→V2. The proof is then an application of Lemma 5.6.1 and the commuting diagram 5.10.

ι A similar diagram checking exercise can be done following Pi1 ∗Pi2 ∗· · ·∗Pim in im(Λ1(L)(U) −→

V1) by using Lemma 5.6.2 and the deﬁnition of relevance in the deﬁnition of Λ1(L)(U). If σi is in

Λ1(L)(U) then ι(σ) = Pi will be in V(U). If R(σ1, σ2, . . . , σm) is in Λ1(L)(U) so also P1 ∗P2 ∗· · ·∗Pm will be in V(U) and R(σ1, σ2, . . . , σm) must be placed over P1 ∗ P2 ∗ · · · ∗ Pm. Example.

Λ1(L) allows towers of subsystems σ1 ⊂ σ2 ⊂ · · · ⊂ σm and can differentiate among stages of a process. V1 has two dimensions with ∗ corresponding to the ordering of the tower together with 0 1 2 classes of expertise. V1 = {(Pi ,Pj ) | i = 1, 2, . . . r, j = 1, 2, . . . s} and

1 2 1 2 0 0 0 (Pi ,Pj )(σ) ∗ (Pk ,Pl )(σ ) =⇒ σ ⊂ σ and σ is a sub-specialty of the expertise characterizing σ .

162 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.6. THE STACK OF SCALED VIEWPOINTS OVER L

η Λ1(L) −→ Λ2(L) maps the subsystems containing segments of processes to the top of the largest subsystem containing any subsystem dealing with part of the process. (We assume η(σ) can be deﬁned uniquely in Λ2(L)). υ : V1 → V2 is a projection of the ﬁrst component of the properties of

V1 and V2 is the image of the projection.

The stack of 2-sections

We now deﬁne the 2-sheaf Γ(V[(L)) which is the V(L)-valued sheaf over τ. The objects in Γ(V[(L))(U) are 2-sections from U with values in any (Λ(L), V) such that, given 0 0 (η,υ) s(R ) in (Λ1(L), V1) and s(R) in a different (Λ2(L), V2) then, if R → R, there must be a (Λ1(L), V1) −−−→ 0 (Λ2(L), V2) such that µ(η(s(R ))) will depend on some state in µ(s(R)). The prestack sheaf requires

Γ(V[(L)((Λ1(L), V1), (Λ2(L), V2)), which we abbreviated to

Hom((Λ1(L), V1), (Λ2(L), V2)), to be a sheaf.

Start with Ui with ∪iUi = U. Let si : Ui → (Λ1(L), V1)(Ui) and (η, υ)(si): Ui → (Λ2(L), V2)(Ui).

This is clearly well deﬁned. If si(R) is placed over Pi1 ∗ Pi2 ∗ · · · ∗ Pim in V1 then η(si)(R) is placed over υ(Pi1 ∗ Pi2 ∗ · · · ∗ Pim ) in V2.

As si | Ui ∩ Uj = sj | Ui ∩ Uj so η(si) | Ui ∩ Uj = η(sj) | Ui ∩ Uj. These standard sheaf conditions give us a extension s deﬁned on Λ1(L), V1)(∪iUi) as (Λ(L), V) is a sheaf (Proposition

5.6.1). This is also true of the image η(si) in (Λ2(L), V2)) which was created by patching the natural transformations η | Ui ∈ Hom((Λ1(L), V1), (Λ2(L), V2))(Ui) together. This proves

Proposition 5.6.2. Hom((Λ1(L), V1), (Λ2(L), V2)) is a sheaf.

Corollary 5.6.3. Γ(V[(L)) is a stack.

Proof. The descent datum is given by Lemma 5.6.2.

Γ(V[(L)) relates open sets U, hence dependent areas of L, with classes of scaled viewpoints in V(U), U a conditional topology open cover of L. Here we have the class of high-level descriptions, already parametrized by viewpoints now localized so we can consistently patch the localized descriptions together. The application here is the class of “ultra-large-systems” [Cliff and Northrop, 2010, Northrop and Pollak, 2006] or systems of enterprise systems and complex systems as discussed by Gorod et al. [2015, Ch 1] and Bernus et al. [2016]. Gorod et al. [2015] present case studies

163 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.7. CONCLUSION: SCALED VIEWPOINTS AS LIMITED EPISTEMOLOGY of very large information systems (supporting very large reporting structures). In such cases, the reporting structures can only be understood or speciﬁed locally with the underlying assumption that they can be patched together. Beyond the level of details, there is the necessity to communicate the contents of these local reporting structures by “tailored,” high-level, specialized descriptions (so scaled viewpoints) for various audiences and, perhaps, the most accurate technique is by scaled viewpoints. The existence of stacks presents us with a criterion of epistemological tractability: that the representation of very large reporting structures in terms of relational landscapes and scaled viewpoints can be approximated locally and these local representations can be merged and aggregated consistently.

5.7 Conclusion: Scaled Viewpoints as Limited Epistemol- ogy

The development of viewpoints in this thesis has provided a conceptual mechanism for relating high level properties, properties that are above the level of data-types but maintain a connection with the relational landscape L. Developing ways in which to integrate scales of description for systems as diverse as enterprises, [Bernus et al., 2016], biochemistry of behavior and emotions Goranson and Cardier [2013], and the properties of molecules from the atomic configurations [An- derson, 1972], and, in our case, relations of data-types to relations of subcategories (viewpoints in our case) is very much a work in progress for a range of scientific endeavors. Accomplishing this integration so that it works smoothly with each vocabulary informing the other has given rise to different approaches. Goranson and Cardier [2013, p. 157] have developed a two-sorted situational logic to describe some of the considerable subtleties of our sense of smell and how it interacts (both ways) with our emotions. The approach adopts two classes of “infons” (relations describing specific facts, very similar to the lower level relations in a relational landscape). The first level of infons are factual and typically cause and effect. The second level “captures the structure of causality.” The objects of this level are categories that “comprise a universe of causal agent structure”, Goranson and Cardier [2013, p. 157]. A relational landscape is a related collection of infons (or infon place holders, in the sense that a relation can be in L but not always have data). Nested relations give context to the domain relations and coordination and correlation add high level structures. The language can surely be couched in terms of the two-sorted situation logic of Goranson and Cardier. In our case causality is not explicit but viewpoints can be defined to investigate conformance to legal, safety, and quality certification requirements. In such cases subsystems are classified as conforming to tax laws, stock

164 CHAPTER 5. HIGH-LEVEL DESCRIPTIONS: SCALING AND VIEWPOINTS 5.7. CONCLUSION: SCALED VIEWPOINTS AS LIMITED EPISTEMOLOGY exchange regulations, audit reporting and so on. In this way the viewpoint gives the reason (the cause) for the existence of certain subsystems. Viewpoints are a way in which the social agendas of the enterprise can be related to the general “outline” of the reporting structure. Again, there is much research to be done in this area which is likely to sustain many approaches. I end with the way scaled viewpoints resemble “approximations” of a reporting structure. How we know it without knowing any details. Suppose for a moment that L does not exists. A new enterprise has grown rapidly and needs a integrated reporting structure. This is frequently the case when a new technology is being scaled- up to produce new products. There are many things that need to be controlled which come to light as materials are produced in bulk as opposed to a laboratory. We might produce a series of “ﬁrst cuts” that start with subsystems that will address the way operations and administration need to interact. This suggests classes of relations that would be expected in these “ﬁrst cut” approximations. The result is a cluster of (Λ(L), V) objects in the category V(L) (Section 5.5.1). The largest consistent (and locally consistent) cases form the stack Γ(V[(L)), the “library of approximations” that provides the fullest “high-level” picture of L.

165 Chapter 6

CONCLUSION

6.1 Functors Resulting from the Research Topics

This thesis has given organizational reporting structures a mathematical expression in order to investigate a number of their properties. The definition of reporting structures is that of a type theory but for the reasons discussed in Section 1.2.1 I have taken the approach that reporting structures are readily represented by a certain class of categories, herein called relational landscapes. Rela- tional landscapes were originally defined in Macfarlane [2017] and are the models of a type theory that would include our Definition 2 of a reporting structure. Relational landscapes have made it easy to address the Research Topics in terms of functors. Four Research Topics were posed in Section 1.1.1. The first two questions are on the theme of connective and coordinating structures in reporting structures; the third question asks for a framework for change and evolution and the fourth question asks how we can obtain high-level descriptions of a reporting structure for specialist audiences. Here I review the novel mathematical aspects that have arisen in answering these questions. Aside from the individual questions discussed below, it is surprising that the idea of a reporting structure, a quintessential corporate and bureaucratic concept going back centuries to at least the origin of banking, should be so simply defined and modeled by a category. That this can be done so readily, and so productively, shows that this is a very natural application for category theory.

166 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS

Research Topic 1: Connective structure.

The coupling ring

The coupling ring (Section 2.5) is a simple idea but it gives the algebra of connections in its fullest extent as well as leading to a number of new functors and interpretations that play important roles in this work. The following are especially novel aspects that arise from the coupling ring.

• The lattice of ideals (Section 2.5.2) provides a new way to see a widening series of couplings as possible futures paths, or “narratives” of fundamental entities such as students, orders or patients in various organizations. The re-interpretation of queries as couplings was unfore- seen. This makes ideals the tool for analyzing the query capability of a reporting structure including queries about functions such as all functions that contain particular data-types.

• The study of relational translations (Section 2.7) is an application of the coupling ring and ideals that demonstrates that these are truly tools for studying the way a reporting structure is connected. The study of relational translations also gives meaning to the monoid structure of the left representable functor of a relational landscape in Sys. While the calculation of reduction monoid described in Section 2.8.2 will require further research, it is the algebra of all simpliﬁed representations of a given reporting structure.

• Higher or nested relations play an important part in the overall structure of the ideals. They have complicated expressions in the lattice of ideals that express how the nested, or “higher”, relations use parts of their domain relations. Nested relations play an important part in deﬁning the Krull dimensions via the initial calculation of new data-types (Section 2.5.3). This makes the Krull dimension a measure of the sophistication of a reporting structure.

The discussion after the statement of this research question 1 mentions a number of reasons for studying the way the reporting structure is connected. These are Testing the system, specifically test planning, planning to migrate data and relations from an old system to a new one and the subsequent planning of testing the conversion, debugging errors and anomalies Beyond the technical aspects the ability to explain the propagation of data changes is valuable. In this thesis we have concentrated on the mathematical aspects. The future of the coupling ring lies in using it to quickly establish the limits of data propagation without resorting to database diagrams and program code. This requires solving equations: whether R1#R2 is zero or not in specific cases. Any software that contains the full definition of C(L), hence its multiplication table, would have the answer.

167 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS

Aside from information technology applications, the coupling ring should be seen as a tool for the enterprises’ own needs to understand its own structures, how it controls the multitude of activities and events that make it what it is. The Krull dimension, perhaps given a local version over open sets, is new measure of intermediate processes in the management of the enterprise. This indicates the number of functional measures or concepts (whether physical assemblies, intermediate compounds or service components) used in controlling the enterprise.

Research Topic 2: Coordination and the correlation homology.

Identifying relations with their principal ideal of couplings suggests treating nested relations themselves in terms of categories. This gives rise to “adjoint relations” and the hierarchy of adjoint relations. The hierarchy itself is a new concept and could not be defined without its relational landscape setting (or a similar setting in a similar class of categories). The hierarchy of adjoint relations, an abstraction of coordination, turns out to be a surprisingly fecund approach to define the classes of relations that match what is to be provided with resources to provide it. Adjoint relations can model the allocation of resources for projects, cash flow for investments, the management of annual budgets for overall operations, wages and material for research projects. These coordinating structures, higher level concordant properties, are the signa- tures of the size and capability of enterprises. However, adjoint relations are not so tightly defined that they are always precise enough to capture the larger-scale interwoven and effective coordination in the larger organizations. The approach taken in this thesis is to select the well-defined adjoint relations by applying a homology theory. This new homology theory, the “Correlation Ho- mology” is loosely modeled on classical simplex homology but applied to the hierarchy of adjoint relations. The discussion in Section 1.1 listed some of the mathematical achievements that relied on (co)homology, so their development and use in new subjects without obvious topological or geometric aspects, is always of interest. Certainly, the subject of reporting structures is a new area distinct from the historical topological or the algebraic homological applications [Eilenberg and Cartan, 1956, Ch. 9]. What is particularly nice is that the new correlation homology is central to the description of coordination and demonstrates its importance in the third research topic. The mathematics of coupling is foundational for the exposition of correlations and correlation homology which provide a new and important class of knowledge for an enterprise. This is especially so as the size of modern enterprises is unprecedented and a recent phenomena in history. The special knowledge or corporations is the how to scale up the production of goods or services. Knowing how to make a car is one type of knowledge. To make 400 cars per day1, day after day, to

1This was the ﬁgure for Ford Australia as reported in The Australian newspaper, 18 November 1999.

168 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS exactly the same quality is a different class of knowledge, knowledge that includes knowing how to make one car. This knowledge requires a high level of coordination in all resources, something that is not easily seen by operations staff doing the jobs on a day-to-day basis. Knowledge of correlations is valuable for decision making as equipment can fail and priorities can change. Knowing the correlations, how they are interdependent and intersecting structures gives management operational control. This is obviously valuable knowledge and represents a high level of enterprise self-knowledge: the design of the “operating system” of the the enterprise itself.

Research Topic 3: Dynamics; a framework of change.

The third Research Topic asks for a framework of change. This is answered by deﬁning a special class of extensions for any given relational landscape L0, namely the elaborations. L0 is a reduction of these extensions and the extensions have at least as many correlations as L0. This deﬁnes the objects of the category of elaborations of L0, the morphisms being compositions of reductions. Elaborations of L0 are a subfunctor of Sysepi(_, L0), the right representable functor in

Sys of reductions to L0. This “category way of thinking” encompasses all extensions and mixtures of extensions and reductions to throw a wide net of changes starting at L0. The concept of an elaboration is new and opens up the exploration of the evolution of reporting structures. Corporations seek mastery of their domain all the time; they and their reporting structures are constantly evolving. There is nothing comparable in the software engineer’s toolbox that addresses this impetus for evolution. The category of elaborations, itself a functor of relational landscapes, is the most novel and most surprising aspect of this thesis. Its definition involves all the preceding concepts and brings forth a range of new tools for studying a special dynamics of reporting structures. Most surprising of all is the depth of analysis that can be achieved. The degree of analysis is demonstrated particularly with the homology tableau (Section 4.2.2, Definition 27. This collection of functors combine the correlation homology and a sheaf that defines the extent that an elaboration expands the original reporting structure (using concepts from Section 2.7). A homology tableau is a functor of a single elaboration and the set of all tableau functors form a category. Taking colimits over the entire category of elaborations for L0, we get a new functor defined on the category of relational landscapes: the homology tableaux. The homology tableaux can be considered as parameterizing classes of correlations by the new subclasses of data-types they require. Finally, the stack associated with the homology tableaux functor includes all the information contained in the conditional topology which encodes dependency of relations. The stack is an encyclopedia of the evolution of concordance for each reporting structure; a level of abstraction that would be hardly expected in a study of reporting structures.

The topology on the category of elaborations of L0 provides a new topology derived from

169 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS

the category structure of L0. The topology makes all local embeddings (i.e. sections) continuous and so is a type of Étale space (Section 4.3.2). This functor is further developed as a stack over the conditional topology. This emphasizes the intrinsic gradient of this new topological space. Surprisingly, the topology provides a way of studying which projects can be factorized into many small projects while others are isolated and so need more planning. This interpretation turns the topology functor into a type of phase space which supports “operators” that classify trajectories of enhancement phases. The phase space becomes the arena of the dynamics for operators that guide structural changes that bring new properties into being. For an enterprise, correlations represent a major tool in controlling the near-future activities of enterprise. A higher level of enterprise development is strategic planning. This requires mapping out what is missing in the development of products or services for the market and what is needed to develop and defend markets. Furthermore, what will be needed to produce the reporting structure that enables these plans to work properly? Competition produces the need to refine operations and the cost-effective fulfillment of orders. Refinement in customer tastes, processing materials and a changing legal and auditing environment can require further development of the reporting structure. The concept of elaborations is always a first step. The category of elaborations clarifies the class of possible evolutions. We have identified local sections of Eb(L0) with enhancement projects giving a new, precise vocabulary when discussing portfolios of projects. The projects in these portfolios are defined for overlapping open sets, so conditioned by relational states, and 2 must be merged to give an overall enhancement to a global system of systems . The stack Eb(L0) is the primary object for this information with the stacks htb giving information on new or enhanced correlations and Vb (L) merging the high-level descriptions.

Research Topic 4: Develop a framework for high-level descriptions

The framework of high-level descriptions combining both “scaling up” and interpretation of the high-scale relations, is built on the idea of viewpoints and viewpoint homology first published by this author for the scale functor defined by “scenarios” [Macfarlane, 2017]. That publication developed the homology and cohomology of viewpoints. To produce a theory of high-level descriptions I have broadened the entire approach. The category of viewpointed systems introduced in the first publication is now a subcategory of the category of scaled viewpoints with the same objects but more morphisms. The concept of creating, what might be called, an “approximation” of a system or its category theoretic representation by scaling up seems to be new. This is not the same as approximating

2The description of an enterprise system in Gorod et al. [2015] and includes Humanitarian Relief logistics and Whole of Nation Health Capabilities

170 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS via morphisms within a category as we have used when an image of a relational translation is a retract of the domain. A scaling Λ(L) has no natural map back to L. Instead we get an entirely new category V(L) that contains, what is obvious in retrospect, the many ways to scale-up and interpret the relational landscape. The more complex the object to be described, in our case a reporting structure, the more ways we can look at it. These are the various descriptions of “the intricately interlocked software elephant.” [Brooks, 1986, p. 4] as quoted in 2.5.3. The advantage of the category of scaled viewpoints, so V(L), is that it captures all the high-level descriptions and the maps between them, the general viewpoint scaling translations (Definition 43). We are also in a position to characterize scaled viewpoints by their homology. All of this is put into the stack of scaled viewpoints over a relational landscape: Γ(V[(L)) (Section 5.6). As mentioned at the end of Chapter 5, Γ(V[(L)) is a library of all the different ways we can describe L by scaling and interpreting. It is the ways of knowing about L without knowing all the data- types. It applies when L is a large information system. Programmers do not get the complete story to create L, they get scaled viewpoints and an act of imagination creates the full details of L. Thus Γ(V[(L)) is what we can know about L without the details. This “epistemological functor” identifies L completely if there is a fully faithful functor that is the inverse of Γ(V[(_)). The novelty of this approach is that it turns the common place idea of a high-level description for a specific audience into a deep mathematical subject overlapping many areas of current research such as category-valued functors over topological spaces. The chapter on Viewpoints (chapter 5) discusses applications of viewpoints to organizations. !p Here we make an additional observation. We can replace scenarios σi with σi = N(Ri) or σi = Ri and can use viewpoints to define high-level relations R(σ1.σ2, . . . σn) placed according to the definition of ∗. R(σ1.σ2, . . . σn) then describes how the σi interact or intersect according to the definition of ∗. Taking the product of this viewpoint with one defined to give reasons in terms of governance (as in section 5.5.3) provides a mathematical description for the need for very high level structures. This is something that is difficult to define and study with diagrammatic techniques.

Beyond the Research Topics

As mentioned above, the various stacks summarize category level properties arising from reporting structures. Because of the social importance of reporting structures, these stacks amount to a signiﬁcant application of category theory (indeed n-category theory) in an entirely new area. They are intrinsic to the problem of modeling large-scale social and engineering structures. To claim a very large reporting structure of a global enterprise or a national organization, sometimes described as a system of enterprise systems [Gorod et al., 2015, Ch 1], is a stack implies a highly

171 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS consistent and coherent structure that can be assembled from smaller sub-reporting structures. A large reporting structure needs to give a consistent picture across all its activities across all its divisions. Inconsistencies in reporting such as incorrect conversions of standard measures used in different countries, or different testing regimes for materials, cause misunderstandings and acci- dents. This is especially so with control panels3. The ability to take reports from many areas that at least agree where they overlap and merge them is a measure of consistency and integration that is meant by the phrase “ the single source of truth” (discussed in Section 1.2.2). The stack is a mathematical expression of this.

Beyond organizational reporting structures

The definition of a relational landscape goes well beyond the logic of reporting structures. Any conceptual realm that seeks relations among a set of starting concepts or primary phenomena that can be interpreted as a data-type can be a candidate for a relational landscape. This can include the definitions in a technical subject, the relations being the theoretical concepts linking the definitions. Examples might include those listed in Table 6.1 such as aspects of ecosystems, particularly niche theory and trophic levels as discussed in such books by May [1976], Sugiharha [2017]. The categories and functors of this work can be developed in any category that allows the definition of relational hierarchies for which correlation-like homology functors can be defined. The maps between the analogues of relational landscapes also have to have mono and epimorphisms. Table 6.1, is a speculative extension of relational landscapes to other realms. If we drop the condition of finiteness in the definition of a relational landscape, a range of applications come into view. For example, if the data-types are the set of algebraic numbers and level zero relations are polynomials with integer coefficients, so have roots in the set of algebraic numbers, higher relations become classes of maps between sets of roots and so morphisms of varieties Fulton [1969]. As long as each object of the category of relational landscapes, Sys, has pullbacks, the coupling ring can be defined. If, also, Sys has pullbacks and pushouts, elaborations and reductions can be defined. In the examples in the table 6.1, evolution starts with elaboration. In the case of exchanges, elaboration corresponds to specialization of goods and trades. In the case of ecosystems, the division of species into subspecies and cellular system evolves by specialization of metabolic processes which often underline adaptation of species..

3The spacecraft Mars Climate Orbiter failed due to software on the ground generating commands in pound-force (lbf), while the orbiter expected newtons (N). Source: https://en.wikipedia.org/wiki/List_of_software_bugs#Military.

172 CHAPTER 6. CONCLUSION 6.1. FUNCTORS RESULTING FROM THE RESEARCH TOPICS

Relational Economic Exchanges Ecosystems Cellular systems Landscapes D(L) = Data- Commodities Species Baseline biochemical types environment Relations of First level commodity Predator-Prey rela- Cell functions convert- data-types trading (no added tions ing inputs to metabo- value) lites Relations of First level of added Trophic levels Dependence relations relations value. Sets of com- (cells that rely on other modities become com- cells metabolites) ponents for trading in more complex items Adjoint re- Second level of added Interdependent Interdependent cell col- lations of value. Component trophic levels and lections (organelles) adjoint rela- trading to manufacture food webs tions of domestic products For an arbi- As n increases, this is As n increases, this As n increases net- trary n > 2, the set of high level, models the effect works of interdepen- relations of n well connected and on trophic levels dent cell types that are levels of nest- coordinated trading to ecosystems con- coordinated at every ing including connections producing nected by migrating level. Such networks the set of n high-value products animals. Migrating are likely in the con- level adjoint species elicit eco- trol of physiological relations logical responses responses to environ- such as predator mental conditions. reproductive cycles R1 → R2 R1 cannot be traded R1 becomes estab- R1 is a cellular func- until R2 trade occurs lished only after R2 tion that is sustained by is established (for products or metabolites example, a predator from the interactions in prey relations can R2 only happen after the prey species is established

Table 6.1: Other systems with possible relational hierarchies.

173 CHAPTER 6. CONCLUSION 6.2. FURTHER RESEARCH

The diagram of functors

The Dramatis personæ of this work are the functors. They are the structure and reason for the thesis. They stand as evidence that reporting structures, through their representation as relational landscapes, are richly endowed with mathematical properties. They are the evidence that this thesis is a new application of category theory, with new concepts in a new area of social importance. In Figure 6.1 they are mapped together with their conceptual links.

6.2 Further Research

Much remains to be explored with these functors and beyond them lie different classes of relational translations preserving more or less structure. The following are a set of problems that arise from this analysis.

1. Knowledge of the ideals is necessary for assessing whether nested relations form an adjoint relation. This needs to be formulated in terms of properties of J(C(L)). What types of coupling rings C(L) or lattices of ideals J(C(L)) put a limit on the level of correlations in adjoint hierarchy? Very ﬂat lattices of ideals or coupling rings that are direct sums of many small rings will have low Krull dimensions that will limit adjoint relations. To what extent does

this limit elaborations as well? How does it affect the topology of EcN (L)?

2. The coupling ring C(L) contains information about the way the adjoint relations intersect, but nothing on whether the intersections can be enclosed in adjoint relations that are correlations. The correlation information is in the correlation homology modules. Is there a “signature” of correlation that can be formulated in the coupling ring? What are the links between the coupling ring and the correlation homology modules?

3. Techniques for calculating Red(L). This seems to be a more mysterious functor than C(L) and contains different information. Each reduction creates equivalences classes. These have additional algebraic structure as groupoids. Analogies with algebraic geometry [Hartshorne, 1977] suggest using the scheme associated with the C(L) as a basis for topological concepts (rather than topologies on L). Section 2.7.4 can be considered as the ﬁrst steps towards the possibilities of sheaves of reductions and their obstructions that are the basis of calculating the reduction monoid Red(L). Red(L) gives algebraic structures at a very local level but also in a way that is expressed “on the factory ﬂoor” in terms of “this can substitute for that as long as ...”.

174 CHAPTER 6. CONCLUSION 6.2. FURTHER RESEARCH

V(L) localization / Γ(V[(L))(Stack) 9 O

∈ ∈

V iewpoint interp v.point homology Λ(L) / V / Hp(Λ(L), V) O

2−ary logic

logic couplings Ideals Abstract relns.s S / L / C(L) / J(C(L)) / Lp(L) 8

reductions reduction<−>ideal % x extensions Red(L) homology

E(L) o H (L) correlations p

homology tableau topology an object of % E (L) H(L) o ht x cN colimit

localization localization

EcN (L)(Stack) htb (Stack)

Figure 6.1: Functors of a relational landscape.

175 CHAPTER 6. CONCLUSION 6.2. FURTHER RESEARCH

4. Elaborations are only one class of extensions to reporting structures and their underlying systems. Another class of extensions are relational translations that maintain or improve algebraic integration as given in Definition 11 in Section 2.6. I have concentrated on correlations as that is what enterprises want to achieve and having integrated systems are a path to that goal. A more general theory should include reductions to reflect simplifications and generalizations of obsolete subclasses. These more general extensions can be modeled by a mixture of elaborations in one part of the system and reductions in other parts all the while maintaining some level of algebraic integration. Composition of these extensions might lead to systems that have little in common with the original reporting structure. This suggests that new concepts will be needed to study the way disparate structures can evolve to converge while similar structures can diverge.

5. The investigation of the category of elaborations EN (L0) and its associated topological space

EcN (L0) leads to a number of analytic or operator approaches. These have been presented only in a sketch. These operators deﬁne “strategic projects”: intentions to extend a reporting structure over many phases. These strategic projects might be driven by conformance to government regulations, for example a legal change in the European banking environment [Wolf, 2014, Ch. 7]. Viewpoints can be used to interpret whether a subsystem conforms to regulations or not. In a such a case the viewpoints make an entry into the expression of the operators.

6. What is the effect of localizing relational translations? We do not always want to compare reporting structures in terms of relational translations between their representing relational landscapes. In an analogy with algebraic geometry we can start with a finite “affine” covering of L, in this case L/R (so an open set in the conditional topology), and define a “locally f C(f) proper relational translation” L1 −→L2 as a functor for which C(L1/Ri) −−→ C(L2/f(Ri)) is

a homomorphism for each Ri of the cover. A locally proper relational translation preserves the coupling structure in different ways in certain regions of the reporting structure. Such a functor will be characterized by the cohomology of the resulting sheaf. Generally, classes of functors, either weaker or stronger than relational translations, can be deﬁned in terms of preserving natural transformations between locally deﬁned functors on L. The advantage here is that the evolution of reporting structures does not always proceed uniformly any more than any area of technology. Certain structures (hence functors) are to be maintained in some areas of the reporting structure while others are less important. This research will be relevant to different classes of elaborations mentioned above.

7. Can EcN (L0) be simulated in some way? A possible approach is to use Monte Carlo methods

176 CHAPTER 6. CONCLUSION 6.2. FURTHER RESEARCH

to generate EcN (L0) by random specifications that preserve correlations in L0. Each generated specification becomes a part of an existing iteration for further specification generation.

8. Consistency classes of elaborations and viewpoints have not been studied. They indicated a level of connection. Consistency classes of elaborations are the largest scale of future developments that can be made without introducing contradictions. Inconsistency in viewpoints means there are ways of looking at parts of the system that can give rise to dissonant descriptions.

9. The category of scaled viewpoints (Definition 44) is a work in progress. Viewpoints and their resulting homology are a property of L and the choice of Λ(L). Nothing has been said about the class of relational landscapes for which a scaling Λ(L) can be defined. The implications of a viewpoint’s homology needs to be elucidated. What does zero homology tells us? For example, “progressing” viewpoints (directed graphs) which model process flows give zero ι homology. This suggests that the assignments Λ(L) −→V have to be well chosen to yield much information. (The chapter on viewpointed systems contains many pointers to future research and they are not repeated here.)

10. Much remains to be done to clarify the connections between adjoint relations and relations in viewpointed systems. In Section 5.5.2 the connection between adjoint relations in L and the homology of relations in (Λ(L), V) is mentioned. The role of dependency in L appears in the stack of scaled viewpoints but how this affects the relations among the high-level relations needs to be fully investigated. These are all likely to be part of the “epistemological problem” of ﬁnding conditions for the existence of an inverse, or even an adjoint, of the functor L 7→ Γ(V[(L)).

11. Nothing has been said about modules and the way these open up the use of concepts from K Theory [Silvester, 1981], or the older Hochschild cohomology of algebras where one might calculate the cohomology of C(L) relative to the coupling ring of a reduction of L, C(ϕ(L)), [Eilenberg and Cartan, 1956].

These are mathematical avenues to be explored. The most immediate priority for further research will involve converting the mathematics into systems that can assess classes of information systems documentation and produce reports on the reporting structure that give concordance information about the structure that can be widely understood. This mapping and the associated user testing will provide further impetus for reﬁning the mathematics in directions that are most likely to bear fruit.

177 6.3. FINAL OBSERVATION

6.3 Final Observation

Reporting structures, especially those with thousands of data-types are the big-logic required to handle the big data that is processed daily in corporations throughout the world. According to Ge- offrey Sharman [2016, p. 60], in 2016 the number of on-line transactions corresponded to 14 transactions for every person alive today; two or three orders of magnitude greater than the number of Google searches per day. Those on-line transactions are captured in reporting structures design to enable the corporations to use these torrents of data to operate more effectively and to maintain or grow their social standing, however they measure it. This thesis has shown that these reporting structures have easily deﬁned mathematical representations that seem to have been overlooked. Important properties of these representations exist in what I have called concordant structures. These have required substantial mathematical effort to make them explicit. The result is a new area of mathematics which has surprising depth. Its importance lies in the widest perspective of what we need to know to manage the ﬂows of information that are the invisible infrastructure of our society. The ultimate reason is best expressed by Alfred Dupont Chandler.

“In all industrial, urban and technologically advanced societies where the large enterprise, either private or public, has acquired an essential role in planning, coordinating and appraising economic activities, a lack of systematic structure within these organizations can lead to wasteful and inefﬁcient use of resources. Further studies of the way in which the great enterprise has grown and become administered have, then, more than mere scholarly value.” Chandler [1995, p. 396].

178 Bibliography

J.F. Adams. Inﬁnite Loop Spaces. Princeton Universitiy Press, New Jersey, 1978.

E.J. Andersen. The Management of Manufacturing, Models and Analysis. Addison Wesley, Reading, Massachusetts, 1994.

P.W. Anderson. More is different. Science (New Series), 177(4047 (Aug.4,1972)):394 – 397, 1972.

E. Artin and J. Tate. Class Field Theory. American Mathematical Society Chelsea Publishing, Provi- dence, Rhode Island, 1967.

M. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces. American Mathematical Society, Providence, Rhode island, 1961.

M. Atiyah and I. Macdonald. Introduction to Commutative Algebra. Addison Wesley, Reading, Mas- sachusetts, 1969.

M. Atiyah and I. Singer. The Index of Elliptic Operators III. Annals of Mathematics, 87:546–604, 1968.

H.P. Barendregt. Functional programming and lambda calculus. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B Formal Models and Semantics, pages 321–363. Elsevier, Amsterdam, 1990.

B.M. Beamon. Supply chain design and analysis: Models and methods. International Journal of Production Economics, 55:281–294, 1998.

P. Bernus, L. Nemes, and T.J. Williams (Eds). Architectures for Enterprise Integration. Chapman and Hall, London, Glasgow, 1996.

P. Bernus, T. Goranson, J. Gotze, A. Jensen-Waud, H. Kandjani, A. Molina, O. Noran, R. J. Rabelo, D. Romero, and P. Saha. Enterprise engineering and managment at the crossroads. Computers in Industry, (79):87–102, 2016.

179 BIBLIOGRAPHY

G. Birkhoff. Lattice Theory. American Mathematical Society, 1967.

J. Blechar and O. Hanseth. From risk management to ’organized irresponsibility’? Risks and risk managment in the mobile telecom sector. In Ole Hanseth and Claudio Ciborra, editors, Risk, Complexity, and ICT, pages 136–153. Edward Elgar Publishing Limited, Cheltenham, U.K., 2007.

R. Bowen. On Axiom A diffeomorphisms. American Mathematical Society, 1978.

C.B. Boyer. A History of Mathematics Second Edition (Revised by Uta C. Merzbach). John Wiley and Sons, New York, Chichester, Brisbane, 1991.

F.P. Brooks. No silver bullet - essence and accident in software engineering. In Proceedings of the IFIP Tenth World Computing Conference, pages 1069 – 1076. Elsevier Science B.V., Amsterdam, 1986.

D. Brookshier. BPMN 2.0 tutorial, 2011. URL http://www.omg.org/news/meetings/workshops/ SOA-HC/presentations-2011/14_MT-2_Brookshier.pdf. Retrieved April 2015.

R. Brown. The Origins of Alexander Grothendieck’s "Pursuing Stacks". 2005. URL http://groupoids.org.uk/pstacks.htm. Retrieved February 2018.

C.S. Calude and G. Longo. The deluge of spurious correlations in big data. Foundations of Science, 21(1), 2016.

P.J. Cameron. Combinatorics, topics, techniques, algorithms. Cambridge University Press, Cambridge, England, 1998.

A. Canonaco. Introduction to algebraic stacks. 2004. URL http://www.mat.uniroma1.it/ncerca/seminari/geo-alg/dispense/stacks.pdf. Retrieved, June 2011.

J.W.S Cassels and A. Frohlich, editors. Algebraic Number Theory. Academic Press, London, 1967.

A.D. Chandler. Strategy and Structure. Chapters in the History of the American Industrial Enterprise. The MIT Press, Cambridge, Massachusetts, 1995.

Donald H. Chew and Stuart L. Gillan. Global Corporate Governance. Columbia University Press, New York, 2009.

D. Cliff and L. Northrop. The global ﬁnancial markets: an ultra-large-scale systems perspective, 2010.

180 BIBLIOGRAPHY

A. Cockburn. Agile Software Development: The Cooperative Game, 2nd edition. Addison-Wesley Pro- fessional, New York, 2006.

P.J. Cohen. Set Theory and the Continuum Hypothesis. W.A. Benjamin, New York, 1966.

J. Cooper. The integration of a lean manufacturing competency-based training course into university curriculum, 2009.

J.W. Cortada. The Digital Hand, Volume 2: How Computers Changed the Work of American Financial, Telecommunications, Media, and Entertainment Industries. Oxford University Press, Oxford, New York, 2006.

C.N.G Dampney and Michael Johnson. Enterprise information systems: Specifying the links among project data models using category theory. In J. Filipe, B. Sharp, and P. Miranda, editors, Enterprise Information Systems III. Kluwer, Norwell, Massachusetts, 2002.

B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, Cambridge, 2002.

K.Y. Degtiarev. Systems analysis: mathematical modeling and approach to structural complexity measure using polyhedral dynamics approach. Complexity International, 7, 2000.

K. Devlin. Situation theory and situation semantics. Handbook of the History of Logic, 7:601–664, 2006.

J. Dieudonne. A Panorama of Pure Mathematics. Academic press, London, 1982.

J. Dieudonne. History of Algebraic Geometry: An Outline of the History and Development of Algebraic Geometry. Wadsworth Mathematical Series, Monterey, California, 1985.

M. Drack and D. Pouvreau. On the History of Ludwig von Bertalanffy’s General Systemology, and on its Relationship to Cybernetics - Part III: Convergences and Divergences. International Journal of General Systems, 44(5):523–571, 2015. doi: http://doi.org/10.1080/03081079.2014.1000642.

D. Edidin. What is a stack. Notices of the American Mathematical Society, 50(4):458–459, 2003.

S. Eilenberg and H. Cartan. Homological Algebra. Princeton University Press, Princeton, New Jersey, 1956.

181 BIBLIOGRAPHY

S. Eilenberg and G.M. Kelly. Closed categories. In Samuel Eilenberg, D. K. Harrison, and et al., editors, Proceedings of the Conference on Categorical Algebra (La Jolla 1965), pages 422 –562. Springer- Verlag, Berlin Heidelberg New York, 1965.

S. Eilenberg and S. MacLane. General theory of natural equivalences. Transactions of the American Mathematical Society, 58:231 – 294, 1945.

S. Eilenberg and N. Steenrod. Foundations of algebraic topology. Princeton University Press, Prince- ton, New Jersey, 1952.

M. Felderer, C. Haisjackl, V. Pekar, and R. Breu. An exploratory study on risk-based testing approaches (short paper). In D. Winkler, S. Bifﬂ, and J. Bergsmann, editors, Software Quality. Software and Systems Quality in Distributed and Mobile Environments. 7th International Conference, SWQD, Vienna, Austria January 20-23, 2015 Proceedings , pages 32–43. Springer Cham, Heildel- berg, 2015.

A. Ferreira and D. Otley. The design and use of management control systems: An extended framework for analysis. 20, 07 2009.

L. Fischman. Evolving function points. Crosstalk: The Journal of Defense Software Engineering, 14(2): 24 – 27, 2001.

D.W. Fogarty, J.H. Blackstone, and T.R. Hoffmann. Production and inventory management. South- Western Publishing Company, Cincinnati, Ohio, 1994.

C. Forzaa and F. Salvadora. Product conﬁguration and inter-ﬁrm co-ordination: an innovative solution from a small manufacturing enterprise. Computers in Industry, 49(1):37–46, 2002.

P.J. Freyd. Abelian Categories, an Introduction to the Theory of Functors. Harper and Row (now available online), 1964.

B. Friedland. Control System Design : An Introduction to State-Space Methods. Dover Publications Inc., Publication City/Country New York, United States, 2005.

William Fulton. Algebraic Curves. W.A. Benjamin, New York, Amsterdam, 1969.

P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory. Springer Verlag, Berlin Hei- delberg New York, 1967.

R. Ganeshan, E. Jack, M. J. Magazine, and P. Stephens. A Taxonomic Review of Supply Chain Manage- ment Research. Springer, 2000.

182 BIBLIOGRAPHY

R. Garner. Two dimensional models of type theory. Mathematical Structures in Computer Science, 19: 687 – 736, 2009.

P. Gerrard and N. Thompson. Risk Based E-Business Testing. Artech House, Inc, Boston, London, 2002.

R.E. Giachetti. The Design of Enterprise Systems. Theory Architecture and Methods. CRC Press of Taylor and Francis, Baton Roca, Florida, USA, 2010.

R.L. Glass. Facts and Fallacies of Software Engineering. Addison-Wesley, Boston, 2003.

T.L. Gomez. Algebraic Stacks. arXiv:math/9911199v1 [math.AG] 25 Nov 1999, 1999.

T. Goranson and B. Cardier. A two-sorted logic for structurally modeling systems. Progress in Biophysics and Molecular Biology, (113):141–178, 2013.

A. Gorod, B.E. White, V. Ireland, J. Gandhi, and B. Sauser. Case Studies in Systems of Systems, Enterprise Systems, and Complex Systems Engineering. CRC Press, Taylor and Francis Group, Boca Raton, Florida, 2015.

M. Greenberg. Lectures on Algebraic Topology. W.A Benjamin, New York, 1967.

M.L. Gromov. In a search for a structure, part 1: On entropy. In Jacek Juzwiszyn, editor, 6th European Congress of Mathematics, Krakow, July 2012, pages 51–78, Zurich, 2013. European Math- ematical Society.

M.P. Groover. Automation, Productions Systems, and Computer-Integrated Manufacturing (3rd Edition). Prentice Hall, London, 2007.

P.W. Gross and P.R. Kotiuga. Data structures for geometric and topological aspects of ﬁnite element algorithms. Progress In Electromagnetics Research (PIER), 32:151 – 169, 2001.

T. Haigh. Annual Review of Information Science and Technology, 45:431–487, 2011. doi: 10.1002/aris.2011.1440450116.

O. Hanseth and C. Ciborra. Risk, Complexity, and ICT. Edward Elgar Publishing Limited, Chel- tenham, U.K., 2007.

J. Harrison, J.Horrocks, R.L. Newman, and R.W. Jenkin. Fundamentals of accounting, 2nd. ed. Long- man Paul, Auckland, N.Z, 1987.

183 BIBLIOGRAPHY

R. Hartshorne. Algebraic Geometry. Springer Verlag, Berlin Heidelberg New York, 1977.

Anders Haug, Lars Hvam, and Niels-Henrik Mortensen. Deﬁnition and evaluation of product conﬁgurator development strategies. Computers in Industry, 63(5):471–481, 2012. ISSN 0166- 3615.

S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439 –561, 2006.

W.J. Hopp. Supply Chain Science. Irwin/McGraw Hill, 2007.

S.T. Hu. Elements of general topology. Feffer and Simons, Inc., New York, 1964.

N. Immerman and P.G. Kolaitis, editors. Descriptive Complexity and Finite Models, Proceedings of a DIMACS Workshop January 14 - 17 1996, Princeton University, volume 31 of DIMACS: Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1997.

S. Iyanaga and Y. Kawada, editors. Encyclopedic Dictionary of Mathematics. MIT press, Cambridge, Massachusetts, 1977.

Editor J. van Leeuwen. Handbook of Theoretical Computer Science, Volume B Formal Models and Seman- tics. Elsevier, Amsterdam, New York, Oxford, 1990.

J.R.Strooker. Introduction to Categories, Homological Algebra and Sheaf Cohomology. Cambridge Uni- versity Press, Cambridge, 2009.

H. Kandjani, A. Mohtarami, A. E. Andargoli, and R. Shokoohmand. A conceptual framework to classify strategic information systems planning methodologies. 2014.

C. Kaner. Cem Kaner on Scenario Testing: The Power of "What If..." and Nine Ways to Fuel Your Imagination. Testing and Quality Engineering magazine, October 2003, 2003.

G. Knolmayer, P. Mertens, and A. Zeier. Supply Chain Management Based on the SAP Systems R/3 4.6 APO 3.0 Order Management in Manufacturing Companies. Springer, Berlin Heidleberg New York, 2002.

D. Knutson. Algebraic Spaces. Number 203 in Lecture Notes in Mathematics. Springer, Berlin Heidleberg New York, 1971.

J. Kosur. CFOs need to deﬁne a ’single source of truth’ for all their business need. Business Insider Australia, 2015-10, 2015.

184 BIBLIOGRAPHY

S. Lang. Algebra. Addison Wesley, Reading, Massachusetts, 1969.

F.W. Lawvere. Introduction. In F.W Lawvere, editor, Toposes, Algebraic Geometry and Logic, number 274 in Lecture Notes in Mathematics, page 189. Springer verlag, Berlin Heidelberg New York, 1972.

A. L. Lederer and V. Sethi. The implementation of strategic information systems planning methodologies. MIS Quarterly, 12(3):445–461, 1988.

J. Lee. Machine performance monitoring and proactive maintenance in computer- integrated manufacturing: review and perspective. International Journal of Computer Integrated Manufacturing, 8(5):370–380, 1995. doi: 10.1080/09511929508944664. URL https://doi.org/10.1080/09511929508944664.

M. Lewis and N. Slack. Operations management. Routledge, London, 2003.

I. Macdonald. Algebraic Geometry, Introduction to Schemes. W.A. Benjamin, New York, 1968.

A. Macfarlane. Structures of systems 1. cohomology of manufacturing and supply network-like systems. International Journal of General Systems, 43(5):470 – 507, 2014.

A. Macfarlane. Structures in systems 2: viewpoint cohomology of general systems. Interna- tional Journal of General Systems, 46(2):158–199, 2017. doi: 10.1080/03081079.2017.1293667. URL http://dx.doi.org/10.1080/03081079.2017.1293667.

E.J. Magad and J. Amos. Total Material Management. Van Nostrand, 1989.

Tim A. Majchrzak. Improving Software Testing: Technical and Organizational Developments. Springer, Heidelberg, New York, Dordrecht, London, 2012.

V. Mandrekar and P. R. Masani. Proceedings of the Norbert Wiener Centenary Congress, 1994. Num- ber 52 in Proceedings of Symposia in Applied Mathematics. American Mathematical Society, Rhode Island, USA, 1994.

R.M. May. Models for two interacting populations. In R.M. May, editor, Theoretical Ecology, Princi- ples and Applications, pages 49 – 70. Blackwell Scientiﬁc, Oxford, 1976.

J. Milnor. Differential topology 46 years later. Notices of the American Mathematical Society, 58(6): 804–809, 2011.

185 BIBLIOGRAPHY

J.C. Mitchell. Type systems for programming languages. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B Formal Models and Semantics, pages 365–458. Elsevier, Amsterdam, 1990.

B. Nordstrom, K. Petersson, and J.M. Smith. Martin-löf’s type theory. In S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 5 Algebraic and Logical Structures. Oxford University Press, Oxford, 2001.

L. Northrop and W. Pollak, editors. Ultra-Large-Scale Systems. The Software Challenge of the Future. Software Engineering Institute, Carnegie Mellon, 2006.

M. Olsson. Algebraic Spaces and Stacks, volume 62 of Colloquium Publications. American Mathemat- ical Society, Providence, Rhode Island, 2016.

R. Overy. Why The Allies Won. PMLICO, London, 1996.

G. Peirson, R. Bird, R. Brown, and P. Howard. Business Finance, Sixth Edition. McGraw-Hill Book Company, Sydney, New York, 1997.

D. Perrin. Finite automata. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B Formal Models and Semantics, pages 3–57. Elsevier, Amsterdam, 1990.

W. Perry. Effective Methods for Software Testing. John Wiley, 1995.

A. R. Ravindran. Operations Research and Management Science. CRC Press, 2007.

F. Redmill. Exploring risk-based testing and its implications. Software testing, veriﬁcation and relia- bility, 14(1):3 – 15, 2004.

M. Romagny. A straight way to algebraic stacks. 2003. URL https://perso.univ-rennes1.fr/matthieu.romagny/notes/stacks.pdf. Pre and post doc. seminar notes from Kunglia Tekniska Hogskolan (KTH) Stockholm. Retrieved Dec 2017.

J.B. Rosser. Simpliﬁed Independence Proofs. Academic Press, New York and London, 1969.

J. Rumbaugh, I. Jacobson, and G. Booch. The uniﬁed modeling language reference manual. Addison Wesley Longman, Inc., Reading, Massachusetts, 1999.

B. Russell. The Principles of Mathematics. Cambridge University Press, Cambridge, 1903.

186 BIBLIOGRAPHY

A.M. Sarmiento and R. Nagi. Recent directions in integrated analysis of manufacturing- distribution systems. IIE Transactions on Scheduling and Logistics, special issue on Manufacturing Logistics, 31(11):1061–1074, 1999.

D. Segal. Some aspects of proﬁnite group theory. 2008. URL arXiv:math/0703885v1 [math.GR]. Accessed October 2017.

G. Sharman. The evolution of enterprise computing. ITNOW (British Computer Society), 2016(June): 60–61, 2016.

S.S. Shatz. Proﬁnite Groups, Arithmetic, and Geometry. Princeton University Press, New Jersey, 1972.

J.R. Silvester. Introduction to Algebraic K-Theory. Chapman and Hall, London, 1981.

P.K. Singh. Database Managment System Concepts. V.K. (India) Enterprises, New Delhi, 2009.

E.H. Spanier. Algebraic Topology. McGraw Hill, New York, 1966.

D.I. Spivak. Databases are categories - MIT Mathematics. math.mit.edu/ dspivak/informatics/talks/galois.pdf, 2010a. [Online; accessed 20-April-2015].

D.I. Spivak. Databases are categories II- MIT Mathematics. math.mit.edu/ dspivak/informatics/talks/refinements_and_extensions.pdf, 2010b. [Online; accessed 20-April-2015].

J.M Spivey. The Z Notation: A Reference Manual. Second Edition. J.M. Spivey, Oxford, England, 2001.

R. Statman. Herbrand’s Theorem and Gentzen’s Notion of a Direct Proof. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 897 – 912. North Holland, 1977.

R. Street and G.M. Kelly. Review of the elements of 2-categories. In Proceedings of the Category Seminar Sydney 1972,1973, number 420 in Lecture Notes in Mathematics, pages 75 – 103. Springer Verlag, New York, 1974.

George Sugiharha. Niche Hierarchy, Structure, Organization, and Assembly in Natural Systems. J. Ross Publishing, Plantation, Florida, USA, 2017.

R. A. Teubner. Strategic informamtion system planning: A case study form the ﬁnancial services industry. Journal of Strategic Information Systems, 16:105 – 125, 2007.

187 BIBLIOGRAPHY

Alessio Trentin, Elisa Perin, and Cipriano Forza. Product conﬁgurator impact on product quality. International Journal of Production Economics, 135(2):850–859, 2012. doi: doi:10.1016/j.ijpe.2011.10.023.

B. van den Berg and R. Garner. Topological and simplicial models of identity types. ACM Transac- tions on Computational Logic, 13(1, Article 3):3.1–3.44, 2012.

F.B. Vernadat. Enterprise Modelling and Integration; Principles and Applications. Chapman and Hall, London, 1996.

S.A. White. Process modelling notations and workﬂows. 2004. URL http://www.bptrends.com/Archives / White Papers and Technical Briefs / Process Modeling Notations and Workflow Patterns. Accessed August 2017.

M. Wolf. The Shifts and the Shocks, what we have learned, and have still to learn, from the ﬁnancial crisis. Allen Lane, London, 2014.

E. Yourdon. Modern Structured Analysis. Prentice Hall, Upper Saddle River, New Jersey, 1988.

M.B. Zaremba, editor. Information Control Problems in Manufacturing Technology 1992. Select papers from the 7th IFAC/IPIP/IFORS/IMACS/ISPE Symposium, Toronto, Ontario, Canada 25-28 May 1992. Pergamon Press, Oxford, New York, Seoul, Tokyo, 1993.

188 Appendix A

EXAMPLES AND BACKGROUND

A.1 Example Reporting Structures

In this Appendix I give two contrasting examples of reporting structures. The ﬁrst is a reporting structure for student enrollments which is the basis of many of the simple examples. The second example, is derived from a mathematical concept, a combinatorial system, that was used to model supply networks and factory processes. This is used to illustrate some of the more complicated concepts.

A.1.1 Student Enrollments

The data-types of this system are students, courses, qualiﬁcations, faculties, departments, fees, instructors, classroom, time, semester (x), and timetables. We have the following relations

1. enrol(student, course, semester)

2. Confers(faculty, qualification)

3. CreditsIn(course, qualification, no.ofcredits)

4. F ee(course, fee)

5. isP rerequisite(Course1, Course2)

6. T eaches(department, course)

7. Instructor(instructor, course) One course can have many instructors.

189 A.1. EXAMPLE REPORTING STRUCTURES

8. InstrDept(instructor, department)

9. AssignedT o(enrol(student, course, semester), classroom, time) (a nested relation)

We can think of all these in terms of lists used in the administration of the institution. Functions Most functions will be statistics: how many students currently enrolled in what courses, for what qualiﬁcation? The statistical functions are deﬁned on the “coupling,” or logical composition, of relations. For example,

enrol(student, course, semester) ◦ CreditsIn(course, qualification, no.ofcredits) ◦ Confers(faculty, qualification) ◦ T eaches(department, course).

(The component relations will appear as various sub-relations of Enrol(s, c, x, q, d, f)). Indeed there will be statistical functions for almost any pair and triple of data-types.

Qualification is a function that defines a qualification. This is a formal sum of the type Σ(ni.Ci) where Ci is a set of courses and ni is the least number required from this set in order to get the qualification. Timetabling is a major function but we do not use it here.

A.1.2 Manufacturing and Supply Networks

(i) Combinatorial Systems

A combinatorial system A is described in Macfarlane [2014] as a ﬁnite, partially ordered set of non-commutative rings acting on modules over integers (in the simplest case of discrete items). Usually A has a maximum element that represents a ﬁnal state of production. A has the following components.

1. The partially ordered set, A, represents suppliers or specialist work centers and, for any two

elements a1, a2, there is an element a¯ ≥ a1, a2. a¯ need not be unique.

2. Each a ∈ A is associated with a non-commutative ring of actions W (a), and an integer module, Inv(a). Inv(a) is generated by inventory items which contains supplies and work

in progress (partly made goods). A single item in inventory with be denoted ei while a

set of inventory items in Inv(a) (or, for U = {a1, a2, . . . am}, Inv(U)) will be a vector x = q1.e1 + q2.e2 + ··· + qm.em in Inv(a) (or Inv(U)) with qi ∈ Z. (Negative inventory allows us to calculate deﬁcits that might occur some time in the future given a rate of use).

190 A.1. EXAMPLE REPORTING STRUCTURES

3. w ∈ W (a) acts on the expressions of Inv(a), mainly by doing nothing which is an idempotent operation : Inv(a) → Inv(a). Each action w ∈ W (a) does nothing unless acting on multiples of a vector of Inv(a) that deﬁnes its starting condition to make one thing from its minimal supplies. Thus, if w is “make a chocolate cake” it does nothing unless it has all the ingredients as represented by a formal sum of the sufﬁcient quantities of each individual ingredient. The change in Inv(a) is that one chocolate cake is added to the inventory and the corresponding quantities of ingredients are subtracted.

4. In W (a), w1 + w2 is the independent action of w1 and w2, w1 ◦ w2 is the action of w1 acting on

the state of Inv(a) immediately after w2. w1 ◦ w2 deﬁnes the ring multiplication. This is an uncapacitated system in that no bounds on capacity are given; n.w is a legitimate expression for any n and any w. This can be changed by generating W (a) as a Z/2.Z module over the actions or adding various constraints.

5. A lead time, L(w), is deﬁned for each action w. This is the time w needs to go from start to

ﬁnish. The units of time are the same for all a in A. L(w1 + w2) = max(L(w1),L(w2)) and

L(w1 ◦ w2) = L(w1) + L(w2).

6. b ≺ a (b precedes a) in the order of A if and only if Inv(a) ∩ Inv(b) 6= ∅ and for all x in Inv(a) ∩ Inv(b), w ∈ W (a), v ∈ W (b), w(x) = x or w(x) ∈ Inv(a) \ Inv(b) and v(x) = x. In manufacturing terms, Inv(a) ∩ Inv(b) is the set of items produced by b that are needed for some actions in a. If w(x) = y 6= x then x is in the “bill of materials” for y.

In addition to these properties, Macfarlane [2014] characterizes a supply network as a combinatorial system that has a replenishment logic. A simple replenishment logic has the following properties. If w(Σnkek) = y then nk units of ek are taken out of Inv(a) and one y is added. Then, for w to operate to produce one more y, there must be a further nk (or more) units of ek in Inv(a) or else there is a wait until there are at least nk units of each ek resupplied to Inv(a). Schedules are defined for subsets U ⊆ A as pairs ξ = (~w, s) with ~w defined in W (U) = ⊕ W (a) a∈U and s maps the w ∈ ~w to a sequence of starting times in a calendar common for all A. There are the usual rules: w1 ◦w2 means s(w1) ≥ s(w2)+L(w2) so an action must finish before the next one in that sequence starts. ~w is usually the sum sequences of actions wi = wi,n(i)◦wi,n(i)−1◦wi,n(i)−2◦· · ·◦wi,1), i = 1, 2,...,N where N is as many actions that are required to fulfill the schedule in U and i keeps track of the independent sequences. Schedules are therefore the non-zero polynomial expressions of W (U). The actions must be allocated enough time for their lead time per unit of action to complete. The schedule has to take into account when there is a limit on how many actions of any one type can be operating at the same time.

191 A.1. EXAMPLE REPORTING STRUCTURES

Combinatorial systems contain many of the ingredients common in the operations research studies of supply chains [Beamon, 1998, Ganeshan, Jack, Magazine, and Stephens, 2000, Hopp, 2007]. What could be a reporting structure for such a system? A factory full of machines does not give an impression of what it takes to run the factory effectively any more than a restaurant kitchen full of gadgets tells us how to run a restaurant expeditiously. Much of what follows derives from the recommendations and practices of American Production Inventory and Control Society - Supply Chain Operations Reference model APICS SCOR. https://www.apics.org/apics- for-business/products-and-services/apics-scc-frameworks/scor. A standard reference is [Fogarty et al., 1994]. Here we shall see how a reporting structure requires knowledge of the reported structure, in this case a combinatorial system. The point of studying combinatorial systems, and indeed supply networks or factory operations is to coordinate schedules so that time is not lost waiting for supplies while, at the same time, not overstocking with supplies that will never be used.

(ii) The reporting structure

Data and Relations

1. The set of data-types D starts with the set A of suppliers or production areas of A and their data; so S W (a) ∪ S Inv(a). Each a ∈ A will have data-types to describe who they are, a∈A a∈A who to contact, their legal business description and marketing and engineering information.

2. The state of all inventories can be represented by a vector < q1, e1, q2.e2, . . . , qM .eM >∈

⊕ Inv(a) where Inv(a) is treated as an integer module and qi is the quantity of ei in its a∈A respective units. Time needs to be incorporated in D, perhaps as a variable T with values in a calendar in days or a sequence in hours.

3. The lowest level of relations will be the partial order on A, b ≺ a. Set membership is expressed as a relation: W (a, w) ⇔ w ∈ W (a), Inv(a, e) ⇔ e ∈ Inv(a), dom(w, ~x) ⇔ w(~x) 6= ~x.

4. Each item of inventory accrues a set of properties that become data-types. For each e we want to know what it is called or known by for various groups of users, how much we have on-hand, how much is on-order and with whom, when it is it due, what is the price per unit, what are the units, what is the rate of use, how many units of e can be stored, is it perishable and, if so its expiry date. This set of data becomes the state of the inventory; it is a list that is the main relation linking the inventory data-types.

192 A.1. EXAMPLE REPORTING STRUCTURES

5. Likewise each w ∈ W (a) has its associated reporting data that become data-types. What is the overall description of w? What inputs are required? What is its lead-time? For each w what is the required equipment? Is each item of the equipment in good working order or does it currently require maintenance or repairs? What is the probability of equipment failure for any length of time (mean time between failures and mean time for repair), whether specialists are required to operate the equipment, and whether they are to be certified as operators with legally required qualifications and possible renewal times. The reporting structure will keep maintenance records, warranty documentation, records of similar failures, contacts for supplier technical help and contacts for expertise for quick fixes, whether via a manual or an on-screen expert systems.

6. The reporting structure requires all the costs of running equipment, the labor costs of maintaining the equipment, and all the skilled people to do the ﬁnishing and checking the products before dispatch to a customer.

7. Sales and Sales Orders. These are the “potential” that drives the entire system. Orders arrive at the maximal a of A and drive supplies of components and assemblies to the maximal a. In our case we have put all the maximal elements into one maximum (rather like a large aggregation of production units in the one area). An order will be a relation among the following data-types.

(a) The customer (or an identiﬁer for the customer) and their address, contact details, perhaps for a number of people involved in the order and its possible after sales services, details of other relations giving credit history, partnership contracts, and so on. (b) What is wanted. This can be as simple as recording a product code and a quantity (for example, one jar of coffee for immediate sale) or as complicated as a deep ocean oil drilling platform, or the development of an ofﬁce tower. (c) Quoted cost (which might have taken months, see Section 3.2.5). (d) Delivery date (or sequences of deliveries for parts of the order). (e) Reference to payment contracts and part payments. (f) Details of individuals who are the contact people in this order.

There are many other details such as guarantees, warranties, insurance and so on.

8. Purchase orders to suppliers, often in the form of contracts for supply, have a similar structure to sales orders so have similar relations.

193 A.1. EXAMPLE REPORTING STRUCTURES

9. Logistics. In some cases there are a ∈ A that are logistic companies or a specialist group in a single company. They receive orders to ship goods or materials, often as large components to be sent overseas in specialist containers or with associated installation equipment. Logistic specialists have their own schedules and they are also the ones that have to keep records for customs and quarantine regulations. A purchase order to a logistics supplier can be as complicated as any other purchase order.

10. Production incidents: recording interruptions to production caused by failures in actions. This builds a relation recording the action w ∈ W (a), operators at the time, incident type or description, start of the interruption, recommencement time, how fixed. Incident type can be machine breakdowns, substandard inventory, fights on factory floor etc. Documentation of incidents might be needed as part of compliance to legally mandated safety requirements.

11. Quality assurance and compliance documentation for contracted speciﬁcations. This can involve a number of tests for each batch of items or each item of an assembly shipped to a customer. The failure of tests can involve reworking or scrapping which in turn needs to be recorded to calculate the cost of production.

Functions

1. Lead time for a ∈ A to receive a set of items {x1, x2, . . . xn} ∈ Inv(a) ∩ Inv(b) from b. At best,

this will be the maximum time it takes b to make any one of the xi assuming they can be

made “in parallel”. Otherwise b will supply a schedule of times the xi (or part thereof) will be available. The function is part of the data for overall production scheduling. This function can be extended to lead time from b to a where b < a but not necessarily a predecessor.

2. Inventory functions.

(a) The state of each inventory might be divided into inventory available for proposed schedules and reserved stock for current schedules. (This splitting of the general class of stock items into items for special purposes drives the elaboration of systems.) (b) Inventory is always bounded, the total number of items always has a limit. Finding the optimal numbers for maximum holdings of fast and slow moving supplies or stock, especially perishables, requires operations research techniques which need to be re- sponsive to changes in the overall market. (c) The current expected rate of use for each item e ∈ Inv(A). This can be a single number or a function based on statistics over the last n calendar months.

194 A.1. EXAMPLE REPORTING STRUCTURES

(d) The reorder function for each input item e in Inv(a) uses the expected rate of usage and the lead time from order to receipt to calculate the amount of e to be ordered to avoid both stock-out and overstock. In simple systems, this is triggered by the on-hand quantity of x falling below a re-order point and re-ordering the standard re-order quantity. More sophisticated systems use a ﬂuctuating lead time function and the current rate of usage, rate of spoil, seasonal demand and known scheduled demand.

3. The calculation of the available capacity of of a supplier a ∈ A. This is the capacity that is not already committed to a current schedule (and there might be a number of parallel schedules running in a.

4. The calculation of the total number of units that the actions of W (a) can produce per interval of time is a function that depends on the lead times of the combinations of actions that can be accomplished at the same time in a.

5. Schedules are relations assigning resources, people, equipment and inventory items to ac- complish a set of actions to produce a quantity of goods for inventory (to be used by others). To create an effective schedule you need to know the capacity of a, for each a in the schedule, and what is to be achieved in terms of output by what time. Calculating the capacity must take into account whether there must be built-in idle time because of machine change-overs, expectation of length of delays, reworks, and stock-outs.

6. The integration of current schedules. This function looks at all the existing schedules across A. Subsets of A, especially those who sell outside the set of elements of A, negotiate their own schedules. The integration of current schedules tries to correlate them and, in terms of Macfarlane [2014], ﬁnd a global schedule.

7. The robustness of schedules: what changes can they withstand? Given a change in a such as a reduction in capacity (affecting some dimension), how are other upstream (for a0 > a) units affected?

8. Maintenance Scheduling. This function uses data from machine use plus information in the production incident report to create a report that schedules necessary maintenance for the equipment in a.

I have listed about 120 items of data here. I have not given the level of detail that is required to produce a mature reporting structure. Many details of the factory maintenance, quality assurance and compliance documentation are mentioned with barely a sketch of details. Between this sketch of a reporting structure and what eventuates is subject to the “Glass multiplier”. Robert Glass

195 A.1. EXAMPLE REPORTING STRUCTURES

[2003, Fact 26] noted that between high-level requirements as have been sketched here, and actual design, there is an “explosion” of lower level details that can result in design requirements being be up to ten times longer than the business description. (This is also discussed in Section 1.2.2). The result is that the set of data-types is easily over one thousand. I have not included all the financial, asset management, human resources, legal or other reporting. There are many functions associated with integrating operational schedules with cash-flow estimates and overall budgets. Budgets and cash-flow can be seen as the scheduling of financial resources to various activities, including asset management (replacement of plant). The number of data-types can easily be in the thousands. The number of relations perhaps a tenth of the number of data-types. Functions can be more numerous than relations. Many relations, for example sales figures, give rise to many levels of analysis so many functions providing a variety of statistics about a single relation. This description of a reporting structure shows that it derives not only from the structure of the reported entity but also from the context of business exigencies and the evolution of good operating practices. It is interesting to note the number of suppliers (the a ∈ A). Richard Overy [1996, p.195] notes that General Motors used nineteen thousand specialist subcontractors during the Second World War to manufacture weapons for the US forces. The reporting structures were accomplished without computers and with clerks following instructions from managers and su- pervisors. Computers have enabled an enormous scaling up of reporting structures. The logic that deals with combinations of cases scales non-linearly. Tracing the links and the connections is the mathematical challenge.

196 A.2. HISTORICAL ANTECEDENTS

A.2 Historical Antecedents

Definition 2 of a reporting structure is disarming and perhaps disingenuous when it is claimed to be a reporting structure of a corporation. All this falls away when we consider the number of data- types can run to thousands and relations numbering in hundreds and be described by complicated algorithms (see Appendix A.1). This section gives a historical background on how the parts of the reporting structures arose from the desire of corporate officers responsible for accountancy, finance, or operations and engineering to administer their areas with better and better knowledge of the events and actions for which they were responsible. What was created gained its own importance and, in many ways, distinguished enterprises as well run or not. I also comment on why there is so little academic literature. It is likely that the growth of banking in the 1400s on necessitated the first modern reporting structures based on double-entry bookkeeping. The fact that Luca Pacioli’s Summa complete in 1487 contained a section on double-entry bookkeeping indicates that the Italian bankers were experimenting with reporting structures at that time (see [Boyer, 1991, p. 279]). The growth of the modern manufacturing corporation has been, in part, an evolution towards refining reporting structures and their standards. Alfred D. Chandler [1995, pp. 303-311] outlines how organizational innovators had to articulate the role of “administration” and argue its benefits in entrepreneurial corporations. This started in the 1920s and 1930s, sometimes earlier, and gave rise to the modern multifunction divisional structure [Chandler, 1995, p. 311]. Much of this was driven by a desire to have a reliable method for assessing the state of the corporation; that state, in itself, is a concept that has many dimensions and a range of interpretations. This led to a corporate hierarchy to gather and report results in order to control the company’s activities. The basic information was the lowest level of activity: who has paid, or is yet to pay, for their goods; who is working on what shift; what supplies have been ordered or are on-hand. At this level the reporting structure is about relations that concern people and activities such as work assigned, work done, purchases and payments. People, orders, payments, services, products, and costly mishaps and failures in the management in operations, become the data-types and statements that link them correspond to the relations in the reporting structure. Each layer of the corporate hierarchy was dealing with reports of data summarized by the preceding level of corporate officers. Such reports are functions on selected rows in the listings or reports: the relations that can be defined in the reporting structure. Each set of functions extended the span of activities summarized in the previous level, creating new relations between previously summarized data; “this group achieved these goals, for this cost, during this period of time” is the basis for comparison between groups.

197 A.2. HISTORICAL ANTECEDENTS

Extensive, detailed reporting structures must have been in place for the efforts of all parties in the Second Wold War to produce the enormous output of armaments. Above all, the US effort testiﬁes to a management capability that can only come from detailed record keeping and careful scheduling. Richard Overy [1996, Ch. 6] in the chapter “Economies at War” gives the statistics of how the US out-produced all other nations in munitions. Liberty ships that took 355 days to build at the beginning of war took 8 days by the end [Overy, 1996, p. 194]. At the end of the war, bombers were being produced by the Ford company at the rate of one every 63 minutes. Each B24 bomber involved thirty thousand different types of parts and a total 1,550,000 parts in all [Overy, 1996, p. 196]. Yet it is rare to ﬁnd detailed studies of reporting structures or their supporting mechanisms, the corporate information systems, in either histories of information technology or in academic studies of reporting in various industries. Even in Overy’s informative chapter on “Economies of War” there is no mention of the reporting structures that kept the factories working. The dearth of historical studies is discussed by Thomas Haigh [2011]. The following quotes give an impression of the state of historical studies of information systems applications, hence reporting structures. First the historical emphasis:

“The computer industry features prominently in recent overviews of the history of computing, and has received a focused summary (J. Yost, 2005). Before the Computer (Cortada, 1993) puts the industry into its historical context, showing the extent to which the mainframe computer industry grew out of the earlier ofﬁce machine industry. Likewise (Arthur L Norberg, 1990) showed the importance of punched card machine companies to scientiﬁc computing practice even before the development of electronic computers.” p. 24

And what it leaves out.

“But studying information technology itself cannot tell us how or why it has been used to change the world, as Tom Misa pointed out in an argumentative survey of the existing literature (Misa, 2007). Neither will studying the hardware or software industries, however fascinating the stories we uncover. These constitute only a tiny part of the economy. We can only understand the importance of information technology to society by studying it in use.” p. 44

In the section “Applications in Administrative Work and Business” Haigh [2011, p. 50] concludes with: “To call this coverage of administrative computing patchy would be to greatly exaggerate its comprehensiveness.” The only exceptions are the work of James Cortada.

198 A.2. HISTORICAL ANTECEDENTS

“His three volume opus The Digital Hand chronicles the use of computers in manufacturing, transport and retail industries (Cortada,2003), service and communication industries (Cortada, 2006) and the public sector (Cortada, 2007). Each section examines the introduction of successive waves of computer technology with a focus on the applications and technologies most distinctively associated with the industry in question. These chapters will provide valuable starting points for future historians and Cortada’s voluminous footnotes are an impressive resource.” (p. 50)

Cortada [2006] gives a useful outline (for example) of what banks do but the book does not give details of reporting structures. In Table 1.2 Cortada lists the main reporting areas of a bank but conveys little beyond what might be considered specialist job titles; no details of the data are given. Nevertheless the book underlies the continuity of business rationales and the reason for overall reporting structures. The point of mentioning historical studies is that they underly the socially perceived importance of an area of science and technology. They also document how new strands of studies emerge and affect other disciplines. Some areas of study, even though they are recognized, are ignored because of the huge number of abstract details. I argue below that this is what bedevils the study of reporting structures.

Why the gap in the literature?

What lies behind the dearth of studies mentioned above by Thomas Haigh? What are the features of reporting structures that lead to the disinterest? Certainly the mathematical definition I have given does not sit in any specialist area in mathematics. But what of this more general disinterest as indicated by the quotations above. As noted before, the impressive research done in [Overy, 1996] does not state anything about, what must have been, an army of clerks and mangers that keep the material flowing to the correct factories that made a wide range of armaments during the Second World War. Modern industrial enterprises require reporting structures to make them work effectively. Reporting structures, and pre-computer reporting systems, seem to be subjects that nobody wants to own. For a start, reporting structures are neither new nor obvious; they have been a feature of all bureaucratic societies for centuries. Their very associations with governments and bureaucracies limits their interest. Previously, they were not associated with any technology of interest, nor any easily identified social change. Yet I would claim that reporting structures are how we model what we reasonably claim as important to know about large organizations. Not only large organizations but also large-scale engineering constructions such as oil refineries, deep ocean oil rigs and state

199 A.2. HISTORICAL ANTECEDENTS or national power grids. There, reporting structures are the many displays in control rooms (see Zaremba [1993] for detailed case studies, though light on speciﬁcations). The similarity between information systems design and control panel design is noted in Bernus et al. [1996, p. 13]

“Developers of control systems (both hardware and software) will hopefully ﬁnd that this book can give them guidance in a way similar to information technology experts. The control engineers situation differs in our view only in they cover a different sub- part of the entire enterprise domain from the information technologist. That is they concentrate particularly on control system description techniques, methods for control system design and/or tools to support these various methods rather than on data handling techniques, data base design, etc.”

Modern control systems have had considerable attention in specific areas such as optimization and automatic control of physical and chemical processes but, as with administrative systems, their very size is forbidding. The size of corporate reporting requirements which can incorporate data from factory devices means they can only be described in very general terms. Even Appendix A.1 of this work makes no attempt to go beyond outlining a fraction of the data-types, relations and functions. Full reporting structures cover the specialist data of accountants and financial experts; operations, including factory management, supply management and logistics - all huge areas; risk management, legal issues, human resources, sales and marketing are also interwoven with no clear boundaries and many levels of coordination [Harrison, J.Horrocks, Newman, and Jenkin, 1987, Peirson, Bird, Brown, and Howard, 1997, Lewis and Slack, 2003, Knolmayer, Mertens, and Zeier, 2002]. Studying these structures requires access across departments and divisions of the enterprise. It also requires access to people who know the reasons for the way things are. The original moti- vation for the structure can be lost in time as the original specialists leave the enterprise. The first thing that new staff do on starting in a company is to “learn (their part of) the system”. This means the history of the system is eroded over time. All of these reasons mean that these reporting structures, except in their briefest outline, are too unwieldy for the confines of academic papers or even books. This is especially so if one is studying the coordination structures, so correlations, in a system that is “large” by the definition 19. A book of, say, the reporting structure of a bank is unlikely to find a publisher. The only published materials are proprietary user manuals that are bought with the software.

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