DOCSLIB.ORG

Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction Introduction My purpose in writing this exposition is to explain what adjoint functors are, prove a few of their properties, give a few examples, and to convince the reader that they are wonderful gems. Adjunctions elucidate the structure of mathematics by illuminating relationships between categories. In this section, I record any notation or results that I use later. • A 2 C means that A is an object of the category C. • Functors are sometimes named according to their formulae. In this case, the argument(s) are denoted with a dash: −. So the functor A 7! A ⊗ B for B fixed and A; B 2 C would be denoted − ⊗ B : C!C. By convention, when multiple dashes are used in the formula, they represent distinct arguments. • When discussing a hom functor (i.e. the functor Cop × C ! Set sending an ordered pair of objects to the set of morphisms pointing from the first to the second), it is useful to emphasize the category in which the objects live. For this reason, we will denote hom functors using the name of the category, so we have e.g. the hom functor C(−; −): Cop × C ! Set. Recall that the hom functor is functorial by pre and postcomposition in its two variables. •C D denotes the functor category of functors from D to C (with morphisms natural transformations). • For C a small category (i.e. one with a set, rather than a proper class, op of morphism), we identify PC def= SetC ; that is, PC is the category of presheaves on C. • A subcategory D ⊆ C is said to be full if for every A; B 2 D, f : A ! B a morphism of C, f is a morphism of D. • A presheaf on a category C is a functor Cop !Set. • We say that a presheaf F : Cop !Set is representable if it is isomorphic to a presheaf of the form C(−;A) for A 2 C. • Denote by Υ : C! PC the functor defined by A 7! C(−;A). This functor is known as the Yoneda embedding. • Let F : A!B, G; H : B!C, and I : C!D be functors, and let η : G ! H be a natural transformation. We define a new natural def transformation I ∗ η : IG ! IH by (I ∗ η)B = I(ηB) for B 2 B. Similarly, def we define a natural transformation η ∗ F : GF ! HF by (η ∗ F )A = ηF (A) for A 2 A. In the parlance of higher category theory, these constructions are known as \whiskering" η along I and F . 1 • Note that I do not explicitly articulate the duals to the results that I state. In particular, I've made the arbitrary choice to discuss left adjoints more than right adjoints. • I usually only define how functors act on objects. Technically, the definition of a functor requires a description of how it acts on morphisms as well. However, writers in this field tend to work from the assumption that the action of a functor on morphisms is obvious from its action on objects. Working out how a functor should act on morphisms given its action on objects is an excellent exercise for beginning category theorists. The following lemma and corollary will be used later in the paper and are important theorems of category theory, but they are also somewhat abstract, and may not feel intuitive at first blush. The reader should feel welcome to continue with the paper and return to these theorems later. Lemma (Yoneda) 1. For any presheaf F : Cop !Set and object A 2 C, there is an isomorphism F (A) =∼ PC(Υ(A);F ), and this isomorphism is natural in both F and A. This is a generalization of Cayley's Theorem when we consider groups as single-object categories. Proof. We can see this by considering idA 2 Υ(A). Given a natural transfor- mation f : Υ(A) ! F and a morphism g : B ! A in C, we have by naturality that fB(g) = F (g)(fA(idA)), so fA(idA) completely determines the natural transformation, and the function fA has codomain F (A). Corollary 2. The Yoneda embedding lives up to its name: it embeds C as a full subcategory of PC, identifying C with the subcategory of representable presheaves on C. Proof. PC(Υ(A); Υ(B)) =∼ Υ(B)(A) = C(A; B). Adjunctions Definition 3. A functor L : C!D is said to be left adjoint to a functor R : D!C if there exists a natural isomorphism D(L(−); −) =∼ C(−;R(−)). Note that these are functors Cop × D ! Set. In this case, L is a left adjoint functor.A right adjoint functor is the dual notion, so a functor is a left adjoint if it has a right adjoint and vice versa. We use the notation L a R to express \L is left adjoint to R". The reader should verify the following elementary examples. Example 4. Let C be a category with a terminal object and D be a category with an initial object. Let L : C!D be the constant functor defined by L : A 7! ; for all A 2 C and ; initial in D. Let R : D!C be the constant functor defined by R : B 7! ∗ for any B 2 D and ∗ terminal in C. Then L a R. 2 Example 5. Let F : Set !Grp be the functor sending a set to the free group it generates, and let U : Grp !Set be the functor sending a group to its underlying set. Then F a U. It is useful when reading mathematical texts that use category theory know that that when an object of some category is referred to as \free", this almost always means that it is in the essential image of some left adjoint functor (\in the essential image" means that the object is isomorphic to some object in the image). Example 6. Let Grph be the category of multi digraphs allowing vertices to be (multiply) self-adjacent. Let the functor U : Cat !Grph turn a small category into a multi digraph by taking the objects to be vertices, the arrows to be directed edges, and forgetting composition. U has a left adjoint which sends a graph to the free category it generates: the objects are the vertices, each edge determines an morphism, identity morphisms are added in (i.e. they are not presumed to be equal to the morphism determined by an edge witnessing a self-adjacency), and any finite sequence of \composable" edges determines a (new) morphism, with composition determined by concatenation of strings of \composable" edges. There is a similar adjunction between the category of small groupoids and the category of undirected multigraphs. Example 7. The functor U : T op !Set sending a topological space to its set of points has both a left and a right adjoint. The left adjoint sends a set to the discrete space with that set of points, while the right adjoint sends a set to the indiscrete space with that set of points. Example 8. For S a set, denote its powerset PS (there is a precise sense in which this notation agrees with the previously established notation for presheaf categories; perhaps I will discuss this further in a sequel note if there is interest in one, or certainly in person if anyone would like). We can regard PS as a category by positing the existence of a unique element of PS(U; V ) iff U ⊆ V . Then a function f : S ! T determines a functor f! : PS ! PT sending a subset to its image. We also have a functor f ∗ : PT ! PS sending a subset to its preimage. ∗ ∗ f! a f . But perhaps less obviously, f has a right adjoint as well. This is c c c the functor f∗ : PS ! PT defined by f∗ : U 7! (f(U )) where U denotes the complement of U. Example 9. Unsurprisingly, adjunctions arise in the context of other category- theoretic constructions, and often prove to be useful tools in these instances. Let C be a category with finite coproducts. For an object X 2 C, the under category, denoted C=X , is the category with objects morphisms X ! A for A 2 C and morphisms f : g ! h given by commutative triangles X g h f A B 3 f There is a forgetful functor CX= !C given by (X ! A) 7! A, and this functor has a left adjoint given by A 7! (X ! X t A). When C is T op and X is ∗, the point, T op∗= is the category of pointed spaces, and the left adjoint just entails adjoining a disjoint basepoint. Example 10. Let X and Y be spaces and let ShX and ShY be the categories of sheaves on X and Y respectively. Then a continuous map f : X ! Y determines a pushforward functor f∗ : ShX !ShY which is left adjoint to the pullback ∗ functor f : ShY !ShX . Proposition 11. A left adjoint functor has a unique right adjoint up to unique natural isomorphism. Proof. Suppose L a R and L a R0. Then for A 2 C;B 2 D, C(A; R(B)) =∼ D(L(A);B) =∼ C(A; R0(B)), so by the Yoneda Lemma, R(B) =∼ R0(B) and this isomorphism is natural in B, so it extends to a natural isomorphism R =∼ R0. Proposition 12. The composition of two left adjoint functors is a left adjoint. Proof. Take L1 : C!D, L2 : D!E with right adjoints R1, R2 respectively. ∼ ∼ Then E(L2L1A; B) = D(L1A; R2B) = C(A; R1R2B).
Load more

Recommended publications
  • Arxiv:1705.02246V2 [Math.RT] 20 Nov 2019 Esyta Ulsubcategory Full a That Say [15]
    WIDE SUBCATEGORIES OF d-CLUSTER TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the φ form φ∗ mod(Γ) in an essentially unique way, where Γ is a finite dimensional algebra and Φ −→ Γ is Φ an algebra epimorphism satisfying Tor (Γ, Γ) = 0. 1 Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d- kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G ) in an essentially φ Φ unique way, where Φ −→ Γ is an algebra epimorphism satisfying Tord (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories ℓ F ⊆ mod(Φ) over algebras of the form Φ = kAm/(rad kAm) . Dedicated to Idun Reiten on the occasion of her 75th birthday 1. Introduction Let d > 1 be an integer. This paper introduces and studies wide subcategories of d-abelian categories as defined by Jasso. The main examples of d-abelian categories are d-cluster tilting subcategories as defined by Iyama.
    [Show full text]
  • Category Theory for Autonomous and Networked Dynamical Systems
    Discussion Category Theory for Autonomous and Networked Dynamical Systems Jean-Charles Delvenne Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) and Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; [email protected] Received: 7 February 2019; Accepted: 18 March 2019; Published: 20 March 2019 Abstract: In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way. Keywords: ergodic theory; topological dynamics; control theory 1. Introduction The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics (studying continuous maps on topological spaces) and ergodic theory (studying probability measures invariant under the map) as two prominent examples. These theories have followed parallel tracks and built multiple bridges. For example, both have grown successful tools from the interaction with information theory soon after its emergence, around the key invariants, namely the topological entropy and metric entropy, which are themselves linked by a variational theorem. The situation is more complex and heterogeneous in the field of what we here call open dynamical systems, or controlled dynamical systems.
    [Show full text]
  • A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
    A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience.
    [Show full text]
  • Nearly Locally Presentable Categories Are Locally Presentable Is Equivalent to Vopˇenka’S Principle
    NEARLY LOCALLY PRESENTABLE CATEGORIES L. POSITSELSKI AND J. ROSICKY´ Abstract. We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopˇenka’s principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable. 1. Introduction Locally presentable categories were introduced by P. Gabriel and F. Ulmer in [6]. A category K is locally λ-presentable if it is cocomplete and has a strong generator consisting of λ-presentable objects. Here, λ is a regular cardinal and an object A is λ-presentable if its hom-functor K(A, −): K → Set preserves λ-directed colimits. A category is locally presentable if it is locally λ-presentable for some λ. This con- cept of presentability formalizes the usual practice – for instance, finitely presentable groups are precisely groups given by finitely many generators and finitely many re- lations. Locally presentable categories have many nice properties, in particular they are complete and co-wellpowered. Gabriel and Ulmer [6] also showed that one can define locally presentable categories by using just monomorphisms instead all morphisms. They defined λ-generated ob- jects as those whose hom-functor K(A, −) preserves λ-directed colimits of monomor- phisms. Again, this concept formalizes the usual practice – finitely generated groups are precisely groups admitting a finite set of generators. This leads to locally gener- ated categories, where a cocomplete category K is locally λ-generated if it has a strong arXiv:1710.10476v2 [math.CT] 2 Apr 2018 generator consisting of λ-generated objects and every object of K has only a set of strong quotients.
    [Show full text]
  • Notes and Solutions to Exercises for Mac Lane's Categories for The
    Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits .
    [Show full text]
  • Derived Functors and Homological Dimension (Pdf)
    DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic.
    [Show full text]
  • Arxiv:2008.00486V2 [Math.CT] 1 Nov 2020
    Anticommutativity and the triangular lemma. Michael Hoefnagel Abstract For a variety V, it has been recently shown that binary products com- mute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points π : Pt(C) → C, if and only if Gumm’s shifting lemma holds on pullbacks in V. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain cat- egorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal’tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corol- lary, every locally anticommutative variety V has directly decomposable congruence classes in the sense of Duda, and the converse holds if V is idempotent. 1 Introduction Recall that a category is said to be pointed if it admits a zero object 0, i.e., an object which is both initial and terminal. For a variety V, being pointed is equivalent to the requirement that the theory of V admit a unique constant. Between any two objects X and Y in a pointed category, there exists a unique morphism 0X,Y from X to Y which factors through the zero object. The pres- ence of these zero morphisms allows for a natural notion of kernel or cokernel of a morphism f : X → Y , namely, as an equalizer or coequalizer of f and 0X,Y , respectively. Every kernel/cokernel is a monomorphism/epimorphism, and a monomorphism/epimorphism is called normal if it is a kernel/cokernel of some morphism.
    [Show full text]
  • AN INTRODUCTION to CATEGORY THEORY and the YONEDA LEMMA Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transfo
    AN INTRODUCTION TO CATEGORY THEORY AND THE YONEDA LEMMA SHU-NAN JUSTIN CHANG Abstract. We begin this introduction to category theory with definitions of categories, functors, and natural transformations. We provide many examples of each construct and discuss interesting relations between them. We proceed to prove the Yoneda Lemma, a central concept in category theory, and motivate its significance. We conclude with some results and applications of the Yoneda Lemma. Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transformations 6 4. The Yoneda Lemma 9 5. Corollaries and Applications 10 Acknowledgments 12 References 13 Introduction Category theory is an interdisciplinary field of mathematics which takes on a new perspective to understanding mathematical phenomena. Unlike most other branches of mathematics, category theory is rather uninterested in the objects be- ing considered themselves. Instead, it focuses on the relations between objects of the same type and objects of different types. Its abstract and broad nature allows it to reach into and connect several different branches of mathematics: algebra, geometry, topology, analysis, etc. A central theme of category theory is abstraction, understanding objects by gen- eralizing rather than focusing on them individually. Similar to taxonomy, category theory offers a way for mathematical concepts to be abstracted and unified. What makes category theory more than just an organizational system, however, is its abil- ity to generate information about these abstract objects by studying their relations to each other. This ability comes from what Emily Riehl calls \arguably the most important result in category theory"[4], the Yoneda Lemma. The Yoneda Lemma allows us to formally define an object by its relations to other objects, which is central to the relation-oriented perspective taken by category theory.
    [Show full text]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
    N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y).
    [Show full text]
  • Coreflective Subcategories
    transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1).
    [Show full text]
  • Basic Category Theory and Topos Theory
    Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object .
    [Show full text]