DOCSLIB.ORG

Research.Pdf (1003.Kb)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Research.Pdf (1003.Kb) PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS A Dissertation presented to the Faculty of the Graduate School, University of Missouri, Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by KILLIAN MEEHAN Dr. Calin Chindris, Dissertation Supervisor Dr. Jan Segert, Dissertation Supervisor MAY 2018 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS presented by Killian Meehan, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Associate Professor Calin Chindris Associate Professor Jan Segert Assistant Professor David C. Meyer Associate Professor Mihail Popescu ACKNOWLEDGEMENTS To my thesis advisors, Calin Chindris and Jan Segert, for your guidance, humor, and candid conversations. David Meyer, for our emphatic sharing of ideas, as well as the savage question- ing of even the most minute assumptions. Working together has been an absolute blast. Jacob Clark, Brett Collins, Melissa Emory, and Andrei Pavlichenko for conver- sations both professional and ridiculous. My time here was all the more fun with your friendships and our collective absurdity. Adam Koszela and Stephen Herman for proving that the best balm for a tired mind is to spend hours discussing science, fiction, and intersection of the two. My brothers, for always reminding me that imagination and creativity are the loci of a fulfilling life. My parents, for teaching me that the best ideas are never found in an intellec- tual vacuum. My grandparents, for asking me the questions that led me to where I am. ii Contents Acknowledgements ii Abstract vi 1 Introduction1 2 Preliminaries3 2.1 Category Theory..............................3 2.1.1 Universal Properties: Kernels and Cokernels..........5 2.1.2 Additive and Krull-Schmidt Categories.............9 2.2 Quiver Theory................................ 11 2.2.1 The Path Algebra.......................... 13 2.2.2 Quivers with Relations....................... 14 2.2.3 Representation Type........................ 15 2.2.4 Equivalence of Categories..................... 17 2.2.5 Auslander-Reiten Theory..................... 17 2.3 Persistent Homology............................ 18 2.3.1 Simplicial Complexes....................... 18 2.3.2 Simplicial Homology........................ 19 2.3.3 GPMs................................. 22 iii 2.3.4 Process................................ 23 2.4 Stability.................................... 25 2.5 Interleaving Metric............................. 25 2.5.1 Fixed Points............................. 27 2.5.2 Submodules and Quotient Modules............... 28 2.5.3 Weights and Suspension at Infinity................ 31 2.6 Bottleneck Metric.............................. 33 2.7 The Category Generated by Convex Modules.............. 35 3 Categorical Framework 37 3.1 Motivation.................................. 37 3.2 Manifestations of the Structural Theorem................ 37 3.2.1 Introduction............................. 37 3.2.2 Matrix Structural Theorem.................... 45 3.2.3 Categorical Structural Theorem and Structural Equivalence. 50 3.3 Proving the Categorical Structural Theorem............... 51 3.3.1 Persistence Objects and Filtered Objects............. 51 3.3.2 Chain Complexes and Filtered Chain Complexes....... 54 3.4 Categorical Frameworks for Persistent Homology........... 57 3.4.1 Standard Framework using Persistence Vector Spaces..... 57 3.4.2 Alternate Framework using Quotient Categories........ 59 3.5 Proving Structural Equivalence...................... 62 3.5.1 Forward Structural Equivalence................. 62 3.5.2 Reverse Structural Equivalence.................. 72 iv 3.6 Bruhat Uniqueness Lemma........................ 75 3.7 Constructively Proving the Matrix Structural Theorem........ 77 3.7.1 Linear Algebra of Reduction................... 77 3.7.2 Matrix Structural Theorem via Reduction............ 81 4 An Isometry Theorem for Generalized Persistence Modules 87 4.1 Motivation.................................. 87 4.1.1 Algebraic Stability......................... 87 4.1.2 Connection to Finite-dimensional Algebras........... 88 4.2 A Particular Class of Posets........................ 90 4.3 Homomorphisms and Translations.................... 99 4.4 Isometry Theorem for Finite Totally Ordered Sets........... 110 4.5 Proof of Main Results............................ 120 4.6 Examples................................... 127 5 The Interleaving Distance as a Limit 137 5.1 Motivation.................................. 137 5.2 Restriction and Inflation.......................... 138 5.3 The Shift Isometry Theorem........................ 143 5.4 Interleaving Distance as a Limit...................... 150 5.5 Regularity.................................. 156 Bibliography 161 Vita 167 v ABSTRACT The purpose of this thesis is to advance the study and application of the field of persistent homology through both categorical and quiver theoretic viewpoints. While persistent homology has its roots in these topics, there is a wealth of material that can still be offered up by using these familiar lenses at new angles. There are three chapters of results. Chapter3 discusses a categorical framework for persistent homology that cir- cumvents quiver theoretic structure, both in practice and in theory, by means of viewing the process as factored through a quotient category. In this chapter, the widely used persistent homology algorithm collectively known as reduction is pre- sented in terms of a matrix factorization result. The remaining results rest on a quiver theoretic approach. Chapter4 focuses on an algebraic stability theorem for generalized persistence modules for a certain class of finite posets. Both the class of posets and their dis- cretized nature are what make the results unique, while the structure is taken with inspiration from the work of Ulrich Bauer and Michael Lesnick. Chapter5 deals with taking directed limits of posets and the subsequent expan- sion of categories to show that the discretized work in the second section recovers classical results over the continuum. vi Chapter 1 Introduction This research is motivated by the crossover between pure mathematics and real world problem solving. I have focused largely on topological methods of analyz- ing data, and in particular the device of persistent homology [ZC05a], [EH10]. At its inception, this field gathered algebraic topology, category theory, and quiver the- ory to provide a new theory of robust analysis of data that is independent of scale. Even in the short time since its inception, persistent homology has grown to give and take from a staggering number of disciplines—mathematical and otherwise. All the results featured in this manuscript are the result of collaborative efforts. The nature of the material and the coauthors with which it was derived is split into two groups. The first result (Chapter3) is from the paper [MPS17] and was written with my dissertation advisor Jan Segert and fellow graduate student Andrei Pavlichenko. We show that the category of filtered chain complexes is in fact Krull-Schmidt, and that this allows for decomposition before applying the homology functor. Fur- thermore, this abstract result—which we name the Categorical Structural Theo- rem—constitutes a non-constructive proof of the Matrix Structural Theorem, which is the matrix factorization result that is key to reduction, a common persistent ho- 1 mology algorithm in the literature. The Categorical Structural Theorem is the foundation for an alternate workflow for the persistent homology process that cir- cumvents quiver theoretic decomposition results entirely, using instead a quotient category to the category of filtered chain complexes. The last three chapters of results were all obtained in collaboration with post- doctoral researcher David Meyer. They stem from the papers [MM17a] (see Chap- ter4) and [MM17b] (see Chapter5), and Chapter 6 is the result of our ongoing re- search. All these results are related to algebraic stability between the interleaving metric on persistence modules and bottleneck metrics on barcodes. One overarch- ing goal of all three chapters is to bypass the wall raised by quiver theory on the road to multi-dimensional persistence: that the immaculate decomposition result over the An quiver fails for all but a very small list of additional quivers. In par- ticular, the posets that would be used for multi-dimensional persistence have, as quivers, representation theory that is known to be unsolvable. Through discretiza- tion of the usual R-indexed persistent homology and a restriction on the permitted category of indecomposables, we obtain algebraic stability results for a large class of finite posets that are not totally ordered. 2 Chapter 2 Preliminaries 2.1 Category Theory Definition 2.1.1. A category is: C a class of objects, Obj( ), which we will simply denote as itself, • C C a set of morphisms Hom(X; Y ) for all pairs of objects X; Y , • 2 C an identity morphism 1x Hom(X; X) for all X , and • 2 2 C a composition map • Hom(X; Y ) Hom(Y; Z) Hom(X; Z) × ! for all triplets X; Y; Z , such that 2 C 1. for all φ Hom(X; Y ); 1y φ = φ = φ 1x, and 2 ◦ ◦ 2. for all f g W X Y h Z; −! −! −! h (g f) = (h g) f. ◦ ◦ ◦ ◦ Definition 2.1.2. We define the following morphism types: 3 A category with zero morphisms is one
Load more

Recommended publications
  • Complete Objects in Categories
    Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion.
    [Show full text]
  • Separable Commutative Rings in the Stable Module Category of Cyclic Groups
    SEPARABLE COMMUTATIVE RINGS IN THE STABLE MODULE CATEGORY OF CYCLIC GROUPS PAUL BALMER AND JON F. CARLSON Abstract. We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. Contents Introduction1 1. Separable ring-objects4 2. The Kelly radical and the tensor6 3. The case of the group of prime order 14 4. The case of the general cyclic group 16 References 18 Introduction Since 1960 and the work of Auslander and Goldman [AG60], an algebra A over op a commutative ring R is called separable if A is projective as an A ⊗R A -module. This notion turns out to be remarkably important in many other contexts, where the module category C = R- Mod and its tensor ⊗ = ⊗R are replaced by an arbitrary tensor category (C; ⊗). A ring-object A in such a category C is separable if multiplication µ : A⊗A ! A admits a section σ : A ! A⊗A as an A-A-bimodule in C. See details in Section1. Our main result (Theorem 4.1) concerns itself with modular representation theory of finite groups: Main Theorem. Let | be a separably closed field of characteristic p > 0 and let G be a cyclic p-group. Let A be a commutative and separable ring-object in the stable category |G- stmod of finitely generated |G-modules modulo projectives.
    [Show full text]
  • Homological Algebra Lecture 1
    Homological Algebra Lecture 1 Richard Crew Richard Crew Homological Algebra Lecture 1 1 / 21 Additive Categories Categories of modules over a ring have many special features that categories in general do not have. For example the Hom sets are actually abelian groups. Products and coproducts are representable, and one can form kernels and cokernels. The notation of an abelian category axiomatizes this structure. This is useful when one wants to perform module-like constructions on categories that are not module categories, but have all the requisite structure. We approach this concept in stages. A preadditive category is one in which one can add morphisms in a way compatible with the category structure. An additive category is a preadditive category in which finite coproducts are representable and have an \identity object." A preabelian category is an additive category in which kernels and cokernels exist, and finally an abelian category is one in which they behave sensibly. Richard Crew Homological Algebra Lecture 1 2 / 21 Definition A preadditive category is a category C for which each Hom set has an abelian group structure satisfying the following conditions: For all morphisms f : X ! X 0, g : Y ! Y 0 in C the maps 0 0 HomC(X ; Y ) ! HomC(X ; Y ); HomC(X ; Y ) ! HomC(X ; Y ) induced by f and g are homomorphisms. The composition maps HomC(Y ; Z) × HomC(X ; Y ) ! HomC(X ; Z)(g; f ) 7! g ◦ f are bilinear. The group law on the Hom sets will always be written additively, so the last condition means that (f + g) ◦ h = (f ◦ h) + (g ◦ h); f ◦ (g + h) = (f ◦ g) + (f ◦ h): Richard Crew Homological Algebra Lecture 1 3 / 21 We denote by 0 the identity of any Hom set, so the bilinearity of composition implies that f ◦ 0 = 0 ◦ f = 0 for any morphism f in C.
    [Show full text]
  • Categories of Projective Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 110 (1996) 241-258 Categories of projective spaces Aurelio Carboni *, Marco Grandis Dipartimento di Matematica, Universitri di Genova, Via L.B. Alberti 4, 16132 Geneva, Italy Communicated by by F.W. Lawvere; received 1 December 1994 Abstract Starting with an abelian category d, a natural construction produces a category Pd such that, when d is an abelian category of vector spaces, llp& is the corresponding category of projective spaces. The process of forming the category P& destroys abelianess, but not completely, and the precise measure of what remains of it gives the possibility to reconstruct d out from PzZ, and allows to characterize categories of the form Pd, for an abelian d (projective categories). The characterization is given in terms of the notion of “Puppe exact category” and of an appropriate notion of “weak biproducts”. The proof of the characterization theorem relies on the theory of “additive relations”. 0. Introduction Starting with an abelian category &, a natural construction described in [5] produces a category P& such that, when & is an abelian category of vector spaces, Pd is the corresponding category of projective spaces. This construction is based on the subobject fimctor on d, which in the linear context we call “Grassmannian finctor”. Clearly, categories of the form P&’ for an abelian category ~4 are to be thought of as “categories of projective spaces”, and the natural question arises of their intrinsic characterization.
    [Show full text]
  • Math 256: Linear Algebra Dr
    Math 256: Linear Algebra Dr. Heather Moon contributing-Authors: Dr. Thomas J. Asaki, Dr. Marie Snipes, and Dr. Chris Camfield April 20, 2016 1 Contents 1 Systems of Equations 5 1.1 Review from your basic Algebra class . .5 1.1.1 Substitution . .5 1.1.2 Elimination . .5 1.2 Systems of Equations with more variables . .6 1.3 Using Matrices to solve systems of equations . .8 2 Matrix operations and Matrix Equations 12 2.1 Basic Operations on Matrices . 12 2.1.1 Matrix Addition . 13 2.1.2 Matrix Multiplication . 13 2.2 Matrix Equations and more . 14 2.3 Determinants . 18 3 Radiograpy and Tomography in Linear Algebra Lab #1 24 3.1 Grayscale Images . 24 4 Vector Spaces 26 4.1 Vectors and Vector Spaces . 27 4.2 Subspaces . 32 5 Span 35 5.1 Definition of Span . 36 5.2 Span as a Vector Space . 38 6 Linear Independence/Dependence 41 6.1 Linear Dependence Definitions . 41 6.2 Basis . 43 6.3 Dimension . 44 7 Coordinate Spaces 49 7.1 Coordinates in the Standard Basis . 49 7.2 Coordinates in Other Bases . 50 8 Transmission Radiography and Tomography A Simplified Overview 54 8.1 What is Radiography? . 54 8.1.1 Transmission Radiography . 55 8.1.2 Tomography . 55 8.2 The Incident X-ray Beam . 55 8.3 X-Ray Beam Attenuation . 56 8.4 Radiographic Energy Detection . 57 8.5 The Radiographic Transformation Operator . 58 8.6 Multiple Views and Axial Tomography . 58 8.7 Model Summary . 60 8.8 Model Assumptions .
    [Show full text]
  • Generalized 2-Vector Spaces and General Linear 2-Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 212 (2008) 2069–2091 www.elsevier.com/locate/jpaa Generalized 2-vector spaces and general linear 2-groups Josep Elgueta∗ Dept. Matematica` Aplicada II, Universitat Politecnica` de Catalunya, Spain Received 28 February 2007; received in revised form 28 October 2007 Available online 29 February 2008 Communicated by I. Moerdijk Abstract A notion of generalized 2-vector space is introduced which includes Kapranov and Voevodsky 2-vector spaces. Various kinds of such objects are considered and examples are given. The corresponding general linear 2-groups are explicitly computed in a special case which includes Kapranov and Voevodsky 2-vector spaces. c 2008 Elsevier B.V. All rights reserved. MSC: 18E05; 18D35; 18D20 1. Introduction In the last decade and a half, the need to introduce a categorical analog of the notion of vector space over a field K , usually called a 2-vector space over K , has become more and more obvious. This need was made explicit, for example, when looking for the right framework to study the generalizations of the Yang–Baxter equation to higher dimensions. It was indeed in this setting that the notion of (finite-dimensional) 2-vector space over a field K was first introduced by Kapranov and Voevodsky [14]. The main point in Kapranov and Voevodsky’s definition was to take the category VectK of finite-dimensional vector spaces over K as the analog of the field K and to define a (finite-dimensional) n ≥ 2-vector space over K as a “VectK -module category” equivalent in the appropriate sense to VectK for some n 0 (see [14] for details).
    [Show full text]
  • Arxiv:1409.5934V1 [Math.CT] 21 Sep 2014 I field a Fix O-Esrajnto.Teui for Unit the Adjunction
    REFLEXIVITY AND DUALIZABILITY IN CATEGORIFIED LINEAR ALGEBRA MARTIN BRANDENBURG, ALEXANDRU CHIRVASITU, AND THEO JOHNSON-FREYD Abstract. The “linear dual” of a cocomplete linear category C is the category of all cocontinuous linear functors C → Vect. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a correspond- ing copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category Qcoh(X) of quasi-coherent sheaves on a stack X is not dualizable if X is the classifying stack of a semisimple algebraic group in positive characteristic or if X is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if X is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset. 1. Introduction Fix a field K. For cocomplete K-linear categories C and D, let Hom(C, D)= Homc,K(C, D) denote the category of cocontinuous K-linear functors and natural transformations from C to D, and let C⊠D = C⊠c,KD denote the universal cocomplete K-linear category receiving a functor C×D →C⊠D which is cocontinuous and K-linear in each variable (while holding the other variable fixed). If C and D are both locally presentable (see Definition 2.1), then Hom(C, D) and C ⊠ D both exist as locally small categories.
    [Show full text]
  • Notes on Category Theory
    Notes on Category Theory Mariusz Wodzicki November 29, 2016 1 Preliminaries 1.1 Monomorphisms and epimorphisms 1.1.1 A morphism m : d0 ! e is said to be a monomorphism if, for any parallel pair of arrows a / 0 d / d ,(1) b equality m ◦ a = m ◦ b implies a = b. 1.1.2 Dually, a morphism e : c ! d is said to be an epimorphism if, for any parallel pair (1), a ◦ e = b ◦ e implies a = b. 1.1.3 Arrow notation Monomorphisms are often represented by arrows with a tail while epimorphisms are represented by arrows with a double arrowhead. 1.1.4 Split monomorphisms Exercise 1 Given a morphism a, if there exists a morphism a0 such that a0 ◦ a = id (2) then a is a monomorphism. Such monomorphisms are said to be split and any a0 satisfying identity (2) is said to be a left inverse of a. 3 1.1.5 Further properties of monomorphisms and epimorphisms Exercise 2 Show that, if l ◦ m is a monomorphism, then m is a monomorphism. And, if l ◦ m is an epimorphism, then l is an epimorphism. Exercise 3 Show that an isomorphism is both a monomorphism and an epimor- phism. Exercise 4 Suppose that in the diagram with two triangles, denoted A and B, ••u [^ [ [ B a [ b (3) A [ u u ••u the outer square commutes. Show that, if a is a monomorphism and the A triangle commutes, then also the B triangle commutes. Dually, if b is an epimorphism and the B triangle commutes, then the A triangle commutes.
    [Show full text]
  • Basics of Monoidal Categories
    4 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of think­ ing about category theory (which will be especially useful throughout these notes) is that category theory is a refinement (or “categorifica­ tion”) of ordinary algebra. In other words, there exists a dictionary between these two subjects, such that usual algebraic structures are recovered from the corresponding categorical structures by passing to the set of isomorphism classes of objects. For example, the notion of a (small) category is a categorification of the notion of a set. Similarly, abelian categories are a categorification 1 of abelian groups (which justifies the terminology). This dictionary goes surprisingly far, and many important construc­ tions below will come from an attempt to enter into it a categorical “translation” of an algebraic notion. In particular, the notion of a monoidal category is the categorification of the notion of a monoid. Recall that a monoid may be defined as a set C with an associative multiplication operation (x; y) ! x · y (i.e., a semigroup), with an 2 element 1 such that 1 = 1 and the maps 1·; ·1 : C ! C are bijections. It is easy to show that in a semigroup, the last condition is equivalent to the usual unit axiom 1 · x = x · 1 = x. As usual in category theory, to categorify the definition of a monoid, we should replace the equalities in the definition of a monoid (namely, 2 the associativity equation (xy)z = x(yz) and the equation 1 = 1) by isomorphisms satisfying some consistency properties, and the word “bijection” by the word “equivalence” (of categories).
    [Show full text]
  • Linear Algebra Jim Hefferon
    Answers to Exercises Linear Algebra Jim Hefferon ¡ ¢ 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯1 2¯ ¯3 1¯ ¡ ¢ 1 x · 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯x · 1 2¯ ¯ ¯ x · 3 1 ¡ ¢ 6 8 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯6 2¯ ¯8 1¯ Notation R real numbers N natural numbers: {0, 1, 2,...} ¯ C complex numbers {... ¯ ...} set of . such that . h...i sequence; like a set but order matters V, W, U vector spaces ~v, ~w vectors ~0, ~0V zero vector, zero vector of V B, D bases n En = h~e1, . , ~eni standard basis for R β,~ ~δ basis vectors RepB(~v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S M ⊕ N direct sum of subspaces V =∼ W isomorphic spaces h, g homomorphisms, linear maps H, G matrices t, s transformations; maps from a space to itself T,S square matrices RepB,D(h) matrix representing the map h hi,j matrix entry from row i, column j |T | determinant of the matrix T R(h), N (h) rangespace and nullspace of the map h R∞(h), N∞(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu µ upsilon υ epsilon ² nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8.
    [Show full text]
  • Reflecting Algebraically Compact Functors
    Reflecting Algebraically Compact Functors Vladimir Zamdzhiev Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France A compact T-algebra is an initial T-algebra whose inverse is a final T -coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes involving mixed-variance functors, such as function space. The construction of compact algebras is usually done in categories with a zero object where some form of a limit-colimit coincidence exists. In this paper we consider a more abstract approach and show how one can construct compact algebras in categories which have neither a zero object, nor a (standard) limit-colimit coincidence by reflecting the compact algebras from categories which have both. In doing so, we provide a constructive description of a large class of algebraically compact functors (satisfying a compositionality principle) and show our methods compare quite favorably to other approaches from the literature. 1 Introduction Inductive datatypes for programming languages can be used to represent important data structures such as lists, trees, natural numbers and many others. When providing a denotational interpretation for such languages, type expressions correspond to functors and one has to be able to construct their initial alge- bras in order to model inductive datatypes [9]. If the admissible datatype expressions allow only pairing and sum types, then the functors induced by these expressions are all polynomial functors, i.e., functors constructed using only coproducts and (tensor) product connectives, and the required initial algebra may usually be constructed using Adámek’s celebrated theorem [2].
    [Show full text]
  • Handout on Definition of a Vector Space
    Math 141 - General Vector Spaces Wed Mar 5 2014 Real Vector Spaces: Overview 3 Consider R = f(v1, v2, v3) j vi 2 Rg . • We saw that R3 consists of all linear combinations a^ı + b^ + c ^k where a, b and c are scalars. n o • R3 is an example of a vector space over R. We say that R3 is spanned by ^ı,^, ^k . • For any non-zero vector u we saw that W1 = fa u j a 2 Rg is a line through 0 . Sums and scalar multiples of vectors in W1 are again in the line W1. • For any non-zero vectors u and w we saw that W2 = fa u + b w j a, b 2 Rg is a plane through 0 . Sums and scalar multiples of vectors in W2 are again in the plane W2. 3 • W1 and W2 are called subspaces of R . These ideas are useful for studying other mathematical objects which can can also be viewed as vectors. p. 1 of 3 Math 141 - General Vector Spaces Wed Mar 5 2014 Definition of a Vector Space Let V be a nonempty set of objects (called vectors) on which two operations are defined: addition and scalar multiplication. Addition assigns to each pair of vectors u and v a vector denoted u + v. Scalar multiplication assigns to each scalar k and vector u a vector denoted ku. If for all vectors u, v, w and scalars k and m the following axioms (basic assumptions) are satisfied then we say that V is a vector space: 1.
    [Show full text]