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Reporting Structures: Category Theory, Algebraic and Topological Properties

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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 in- formation systems and control panels for factories or large engineering constructions. Re- porting structures can be represented by a class of finite categories, herein called “rela- tional landscapes”. Relational landscapes are categories of nested relations connecting a large number of data-types; as categories they have attracted little attention except, per- haps, 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 underly- ing information systems but are not properties expressed in current software engineering techniques. Reporting structures have a long history, predating current corporate information sys- tems. 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 co- ordination 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 proper- ties 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 report- ing structure (or information system) but there is no mathematical study of how these descriptions can be rigorously derived from the original reporting structure. A mathe- matical 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 ex- presses properties of the entire relational landscape not just components. Calling these “concordant properties," they include the propagation of changes in data, how coordina- tion 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 func- tors 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 val- ued functors. Exploiting natural (categorical) topologies of relational landscapes leads to both stacks and phase-space structures. The high-level description of reporting struc- tures proceeds with the introduction of “scaled viewpoints” combining a “scaling” func- tor 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 (con- nective and coordinating) properties of the entire reporting structure. These are all func- tors that can be used to characterize and compare the specifications of information sys- tems. By studying reporting structures we obtain a new application of category theory that reveals properties of reporting structures that lie beyond normal software engineer- ing 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 Copyright in Relation to This Thesis Copyright 2019 by Andrew Ian Macfarlane. All rights reserved. vi 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 ad- ditional 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 .................................
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