DOCSLIB.ORG

1 Natural Transformations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Natural Transformations Perry Hart K-theory reading seminar UPenn September 26, 2018 Abstract We introduce the concept of a natural transformation in category theory. Afterward, we describe equivalences and adjunctions. The main sources for this talk are the following. • nLab. • John Rognes’s Lecture Notes on Algebraic K-Theory, Ch. 3. • Peter Johnstone’s lecture notes for “Category Theory” (Mathematical Tripos Part III, Michaelmas 2015), Ch. 1. 1 Natural transformations Let C and D be categories and F and G be functors C → D.A natural transformation φ : F ⇒ G is a function A 7→ fA from ob C to mor D such that fA is a map F (A) → G(A) and the following diagram commutes for any morphism h : A → B in C . FA F h FB fA fB GA GB Gh In symbols, this may be written as fBh∗ = h∗fA, where fA is called a component of φ. −1 Note 1.1. If every fA is an isomorphism, then the maps (fA) define a natural transformation G ⇒ F . If each fA is an isomorphism, then we say that φ is a natural isomorphism. Note that if D is a groupoid (i.e., a category in which every morphism is an isomorphism), then φ must be a natural isomorphism. Let F , G, and H be functors C → D. The identity natural transformation IdF : F ⇒ F is given by A 7→ IdF (A). Moreover, given natural transformations φ : F → G and ψ : G → H, define the composite natural transformation ψ ◦ φ by A 7→ (ψ ◦ φ)A := ψA ◦ φA. Lemma 1.2. A natural transformation φ : F ⇒ G is a natural isomorphism iff it has an inverse φ−1 : G ⇒ F . Proof. This follows from Note 1.1 along with our definition of a composite natural transformation. Example 1.3. 1. Let R and S be commutative rings. Any ring homomorphism f : R → S induces a ring homomorphism GLn(f) : GLn(R) → GLn(S) satisfying f(det(A)) = det GL(f)(A) . n 1 ∗ ∗ By viewing GLn and R 7→ R as functors from Ring to Grp and detR : GLn(R) → R as a mor- ∗ ∗ phism in Grp, we see that detR defines a natural transformation φ : GLn ⇒ f where f denotes ∗ ∗ f R∗ : R → S . GLn(f) GLn(R) GLn(S) detR detS R∗ S∗ f ∗ 2. Consider the power set functor P : Set → Set defined on objects by A 7→ P(A) and on morphisms g by Pg(S) = g(S). Then the function fA : A → P(A) given by a 7→ {a} defines a natural transformation φ : IdSet ⇒ P. ab 3. Set C = D = Grp, F = IdC , and G = (−) . Then given a group H, the natural projection ab f : H H induces a natural transformation φ : F ⇒ G. 4. We can view preorders (P, ≤) and (Q, ≤) as small categories and functors F, G : P → Q as order- preserving functions. Then there is a unique natural transformation φ : F ⇒ G iff F (x) ≤ G(x) for every x ∈ P . 5. The inversion isomorphism from a group G to its opposite group Gop defines a natural transformation op op φ : IdGrp ⇒ ((−) : Grp → Grp). In this sense, G is naturally isomorphic to G . Definition 1.4. Let C and D be categories with C small. The functor category Fun(C , D) := D C has functors F : C → D as objects and natural transformations as morphisms. Remark 1.5. Any Grothendieck universe models ZFC, in particular Replacement. This ensure that for any two functors F, G : C → D, the class of natural transformation φ : F ⇒ G is a set. This means that Fun(C , D) is locally small, a condition of our definition of a category. Definition 1.6. Given a category C , the arrow category Ar(C ) of C has as objects morphisms f : X0 → X1 in C and as morphisms M :(f : X0 → X1) → (g : Y0 → Y1) the pairs (M0,M1) of morphisms M0 : X0 → Y0 and M1 : X1 → Y1 such that f X0 X1 M0 M1 Y0 g Y1 commutes. Note 1.7. 1. Ar(C ) =∼ Fun([1], C ). 2. Fun(C × D, E ) =∼ Fun(C , Fun(D, E )). 2 2 Equivalences Usually, it is useful to make our notion of sameness between categories weaker than isomorphism. Definition 2.1. A functor F : C → D is an equivalence if there is a functor G : D → C , called the ∼ ∼ quasi-inverse of F , such that F ◦ G = IdC and G ◦ F = IdD . In this case, we say that F and G are equivalent categories. Moreover, we say that a property of C is categorical if it is invariant under categorical equivalence. Example 2.2. Let k be a field. Let the category Matk have natural numbers as objects and morphisms n → p given by p × n matrices over k. Let fdMod denote the category of finite-dimensional vector spaces with linear maps as morphisms. These two categories are equivalent. Indeed, send the natural number n to kn in one direction and the space V to dim V in the other direction. Definition 2.3. A functor F : C → D is essentially surjective if for each object Z of D, there is some object Y of C such that F (Y ) =∼ Z. Theorem 2.4. A functor is an equivalence iff it is full, faithful, and essentially surjective.1 Definition 2.5. A skeleton of C is a full subcategory C 0 ⊂ C such that each element of ob C is isomorphic to exactly one element of ob C 0. An application of Theorem 2.4 yields the following result. Lemma 2.6. Let C 0 be a skeleton of C . Then the inclusion functor C 0 ,→ C is an equivalence. Lemma 2.7. Any two skeleta C 0, C 00 ⊂ C are isomorphic. 0 00 ∼ Proof. Define F : C → C on objects by F (X) = Y where X = Y via a chosen isomorphism hX and on −1 −1 00 0 morphisms f ∈ C (X, Y ) by F (f) = hY ◦ f ◦ (hX ) . To get F , define G : C → C by similarly choosing −1 00 an isomorphism (hX ) for each X ∈ ob C . Remark 2.8. Both Lemma 2.6 and Lemma 2.7 are logically equivalent to the axiom of choice, as is the statement that every category admits a skeleton. 3 Adjunctions Definition 3.1. op 1. Let Z ∈ ob C . Define the contravariant functor YZ : C → Set on objects by Y 7→ C (Y, Z) and on morphisms by sending f : X → Y in C to the map f ∗ : C (Y, Z) → C (X, Z) given by g 7→ gf. We call C (−,Z) := Y Z the set-valued functor represented by Z in C . 2. Let X ∈ ob C . Define the functor Y X : C → Set on objects by Y 7→ C (X, Y ) and on morphisms by sending g : Y → Z to the map g∗ : C (X, Y ) → C (X, Z) given by f 7→ gf. We call C (X, −) := Y X the set-valued functor corepresented by X in C . 1Theorem 3.2.10 (Rognes). 3 A functor of the form C × C 0 → D is called a bifunctor. In particular, define C (−, −): C op × C → Set on objects by (X, X0) → C (X, X0) and on morphisms by sending (f, f 0):(X, X0) → (Y, Y 0) to the map C (f, f 0): C (X, X0) → C (Y, Y 0) given by g 7→ f 0gf. Let C and D be categories and F : C → D and G : D → C be functors. Definition 3.2 (Kan). Consider the set-valued bifunctors D(F (−), −), C (−,G(−)) : C op × D → Set. An adjunction from F to G is a natural isomorphism φ : D(F (−), −) ⇒ C (−,G(−)). If such a φ exists, then we say that (F, G) is an adjoint pair (of functors). Note that φ is natural in the sense that for any map c : X0 → X in C and d : Y → Y 0 in D, the square φX,Y D(F X, Y ) C (X, GY ) ∗ ∗ c d∗ c d∗ D(FX0,Y 0) C (X0, GY 0) φX0,Y 0 commutes in Set. Example 3.3. Let (P, ≤) and (Q, ≤) be preorders. An adjoint pair (F : P → Q, G : Q → P ) is precisely a pair of order-preserving functions such that F x ≤ y ⇐⇒ x ≤ Gy for all x ∈ P and y ∈ Q. In order theory, such a pair is called a Galois connection. Proposition 3.4. Left and right adjoints are both unique up to unique isomorphism. Terminology. We call F the left adjoint to G and G the right adjoint to F . In symbols, F a G. Note 3.5. It is straightforward to check that any adjoint triple F a G a H yields two new adjunctions: GF a GH FG a HG Definition 3.6. Given an adjunction φ : D(F (−), −) ⇒ C (−,G(−)), define the unit morphism ηX = φX,F X (IdFX ) ∈ C (X, GF (X)) and the counit morphism −1 Y = φGY,Y (IdGY ) ∈ D(FG(Y ),Y ). Lemma 3.7. The unit morphisms (ηX )X∈ob C define a natural transformation η : IdC ⇒ GF , and the counit morphisms (Y )Y ∈ob D define a natural transformation : FG ⇒ IdD . 4 Proof. For simplicity, let us just prove that is a natural transformation. We must check that FG(y) FG(Y ) FG(Y 0) Y Y 0 0 Y y Y commutes for any map y : Y → Y 0 in D. By the naturality of φ, we have that −1 y ◦ Y = y ◦ φ (IdGY ) −1 = φ (Gy ◦ IdGY ) −1 = φ (IdGY 0 ◦Gy) −1 = φ (IdGY 0 ) ◦ FG(y) = Y 0 ◦ FG(y), as required.
Load more

Recommended publications
  • Math 601: Algebra Problem Set 2 Due: September 20, 2017 Emily Riehl Exercise 1. the Group Z Is the Free Group on One Generator
    Math 601: Algebra Problem Set 2 due: September 20, 2017 Emily Riehl Exercise 1. The group Z is the free group on one generator. What is its universal property?1 Exercise 2. Prove that there are no non-zero group homomorphisms between the Klein four group and Z=7. Exercise 3. • Define an element of order d in Sn for any d < n. • For which n is Sn abelian? Give a proof or supply a counterexample for each n ≥ 1. Exercise 4. Is the product of cyclic groups cyclic? If so, give a proof. If not, find a counterexample. Exercise 5. Prove that if A and B are abelian groups, then A × B satisfies the universal property of the coproduct in the category Ab of abelian groups. Explain why the commutativity hypothesis is necessary. Exercise 6. Prove that the function g 7! g−1 defines a homomorphism G ! G if and only if G is abelian.2 Exercise 7. (i) Fix an element g in a group G. Prove that the conjugation function x 7! −1 gxg defines a homomorphism γg : G ! G. (ii) Prove that the function g 7! γg defines a homomorphism γ : G ! Aut(G). The image of this function is the subgroup of inner automorphisms of G. (iii) Prove that γ is the zero homomorphism if and only if G is abelian. Exercise 8. We have seen that the set of homomorphisms hom(B; A) between two abelian groups is an abelian group with addition defined pointwise in A. In particular End(A) := hom(A; A) is an abelian group under pointwise addition.
    [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]
  • 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]
  • Categories, Functors, Natural Transformations
    CHAPTERCHAPTER 1 1 Categories, Functors, Natural Transformations Frequently in modern mathematics there occur phenomena of “naturality”. Samuel Eilenberg and Saunders Mac Lane, “Natural isomorphisms in group theory” [EM42b] A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G E as a normal subgroup and a surjective homomorphism → E H that displays H as the quotient group E/G. This data is typically displayed in a diagram of group homomorphisms: 0 G E H 0.1 → → → → A pair of group extensions E and E of G and H are considered to be equivalent whenever there is an isomorphism E E that commutes with the inclusions of G and quotient maps to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian group extensions E of H by G defines an abelian group Ext(H, G). In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which 1 he computed for a prime p that Ext(Z[ p ]/Z, Z) Zp, the group of p-adic integers, where 1 Z[ p ]/Z is the Prüfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times inside the preceding torus. In teasing apart this connection, the pair of them discovered what is now known as the universal coefficient theorem in algebraic topology, which relates the homology H and cohomology groups H∗ ∗ associated to a space X via a group extension [ML05]: n (1.0.1) 0 Ext(Hn 1(X), G) H (X, G) Hom(Hn(X), G) 0 .
    [Show full text]
  • Math 395: Category Theory Northwestern University, Lecture Notes
    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
    [Show full text]
  • Arxiv:2103.17167V1 [Math.DS] 31 Mar 2021 Esr-Hoei Opeiyi Ie Ae Nteintrodu the in Later Given Is Complexity Measure-Theoretic Preserving 1.1
    AN UNCOUNTABLE FURSTENBERG-ZIMMER STRUCTURE THEORY ASGAR JAMNESHAN Abstract. Furstenberg-Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure pre- serving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogues of those algebraic and geometric descrip- tions are sought. Although the unconditional dichotomy and the characteriza- tions are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying group and spaces. The aim of this article is to remove these restrictions in the relative sit- uation and establish a Furstenberg-Zimmer structure theory in full generality. To achieve the desired generalization we had to change our perspective from systems defined on concrete probability spaces to systems defined on abstract probability spaces, and we had to introduce novel tools to analyze systems rel- ative to a factor. However, the change of perspective and the new tools lead to some simplifications in the arguments and the presentation which we arrange in a self-contained manner in this work. As an independent byproduct, we es- tablish a new connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi. 1. Introduction 1.1. Motivation. In the pioneering work [15], Furstenberg applied the count- able1 Furstenberg-Zimmer structure theory, developed by him in the same article and by Zimmer in [43, 44] around the same time, to prove multiple recurrence for Z-actions and show that this multiple recurrence theorem is equivalent to arXiv:2103.17167v1 [math.DS] 31 Mar 2021 Szemerédi’s theorem [39].
    [Show full text]
  • Categories and Natural Transformations
    Categories and Natural Transformations Ethan Jerzak 17 August 2007 1 Introduction The motivation for studying Category Theory is to formalise the underlying similarities between a broad range of mathematical ideas and use these generalities to gain insights into these more specific structures. Because of its formalised generality, we can use categories to understand traditionally vague concepts like \natural" and \canonical" more precisely. This paper will provide the basic concept of categories, introduce the language needed to study categories, and study some examples of familiar mathematical objects which can be categorized. Particular emphasis will go to the concept of natural transformations, an important concept for relating different categories. 2 Categories 2.1 Definition A category C is a collection of objects, denoted Ob(C ), together with a set of morphisms, denoted C (X; Y ), for each pair of objects X; Y 2 Ob(C ). These morphisms must satisfy the following axioms: 1. For each X; Y; Z 2 Ob(C ), there is a composition function ◦ : C (Y; Z) × C (X; Y ) ! C (X; Z) We can denote this morphism as simply g ◦ f 2. For each X 2 Ob(C ), there exists a distinguished element 1X 2 C (X; X) such that for any Y; Z 2 Ob(C ) and f 2 C (Y; X), g 2 C (X; Z) we have 1X ◦ f = f and g ◦ 1X = g 3. Composition is associative: h ◦ (g ◦ f) = (h ◦ g) ◦ f Remark: We often write X 2 C instead of X 2 Ob(C ) Remark: We say a category C is small if the collection Ob(C ) forms a set.
    [Show full text]
  • Mathematics and the Brain: a Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness
    entropy Review Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness 1,2,3, , 4,5,6,7, 8, Georg Northoff * y, Naotsugu Tsuchiya y and Hayato Saigo y 1 Mental Health Centre, Zhejiang University School of Medicine, Hangzhou 310058, China 2 Institute of Mental Health Research, University of Ottawa, Ottawa, ON K1Z 7K4 Canada 3 Centre for Cognition and Brain Disorders, Hangzhou Normal University, Hangzhou 310036, China 4 School of Psychological Sciences, Faculty of Medicine, Nursing and Health Sciences, Monash University, Melbourne, Victoria 3800, Australia; [email protected] 5 Turner Institute for Brain and Mental Health, Monash University, Melbourne, Victoria 3800, Australia 6 Advanced Telecommunication Research, Soraku-gun, Kyoto 619-0288, Japan 7 Center for Information and Neural Networks (CiNet), National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan 8 Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan; [email protected] * Correspondence: georg.northoff@theroyal.ca All authors contributed equally to the paper as it was a conjoint and equally distributed work between all y three authors. Received: 18 July 2019; Accepted: 9 October 2019; Published: 17 December 2019 Abstract: Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, we provide a novel mathematical framework of category theory (CT), in which we can define and study the sameness between different domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the relationships between various domains of phenomena.
    [Show full text]
  • Part 1 1 Basic Theory of Categories
    Part 1 1 Basic Theory of Categories 1.1 Concrete Categories One reason for inventing the concept of category was to formalize the notion of a set with additional structure, as it appears in several branches of mathe- matics. The structured sets are abstractly considered as objects. The structure preserving maps between objects are called morphisms. Formulating all items in terms of objects and morphisms without reference to elements, many concepts, although of disparate flavour in different math- ematical theories, can be expressed in a universal language. The statements about `universal concepts' provide theorems in each particular theory. 1.1 Definition (Concrete Category). A concrete category is a triple C = (O; U; hom), where •O is a class; • U : O −! V is a set-valued function; • hom : O × O −! V is a set-valued function; satisfying the following axioms: 1. hom(a; b) ⊆ U(b)U(a); 2. idU(a) 2 hom(a; a); 3. f 2 hom(a; b) ^ g 2 hom(b; c) ) g ◦ f 2 hom(a; c). The members of O are the objects of C. An element f 2 hom(a; b) is called a morphism with domain a and codomain b. A morphism in hom(a; a) is an endomorphism. Thus End a = hom(a; a): We call U(a) the underlying set of the object a 2 O. By Axiom 1, a morphism f f 2 hom(a; b) is a mapping U(a) −! U(b). We write f : a −! b or a −! b to indicate that f 2 hom(a; b). Axiom 2 ensures that every object a has the identity function idU(a) as an endomorphism.
    [Show full text]
  • Category Theory Course
    Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph.................
    [Show full text]
  • A Concrete Introduction to Category Theory
    A CONCRETE INTRODUCTION TO CATEGORIES WILLIAM R. SCHMITT DEPARTMENT OF MATHEMATICS THE GEORGE WASHINGTON UNIVERSITY WASHINGTON, D.C. 20052 Contents 1. Categories 2 1.1. First Definition and Examples 2 1.2. An Alternative Definition: The Arrows-Only Perspective 7 1.3. Some Constructions 8 1.4. The Category of Relations 9 1.5. Special Objects and Arrows 10 1.6. Exercises 14 2. Functors and Natural Transformations 16 2.1. Functors 16 2.2. Full and Faithful Functors 20 2.3. Contravariant Functors 21 2.4. Products of Categories 23 3. Natural Transformations 26 3.1. Definition and Some Examples 26 3.2. Some Natural Transformations Involving the Cartesian Product Functor 31 3.3. Equivalence of Categories 32 3.4. Categories of Functors 32 3.5. The 2-Category of all Categories 33 3.6. The Yoneda Embeddings 37 3.7. Representable Functors 41 3.8. Exercises 44 4. Adjoint Functors and Limits 45 4.1. Adjoint Functors 45 4.2. The Unit and Counit of an Adjunction 50 4.3. Examples of adjunctions 57 1 1. Categories 1.1. First Definition and Examples. Definition 1.1. A category C consists of the following data: (i) A set Ob(C) of objects. (ii) For every pair of objects a, b ∈ Ob(C), a set C(a, b) of arrows, or mor- phisms, from a to b. (iii) For all triples a, b, c ∈ Ob(C), a composition map C(a, b) ×C(b, c) → C(a, c) (f, g) 7→ gf = g · f. (iv) For each object a ∈ Ob(C), an arrow 1a ∈ C(a, a), called the identity of a.
    [Show full text]