Notes on Category Theory Robert L. Knighten November 9, 2007 c 2004{2007 by Robert L. Knighten All rights reserved Preface There are many fine articles, notes, and books on category theory, so what is the excuse for publishing yet another tome on the subject. My initial excuse was altruistic, a student asked for help in learning the subject and none of the available sources was quite appropriate. But ultimately I recognized the personal and selfish desire to produce my own exposition of the subject. Despite that I have some hope that other students of the subject will find these notes useful. Target Audience & Prerequisites Category theory can sensibly be studied at many levels. Lawvere and Schanuel in their book Conceptual Mathematics [47] have provided an intro- duction to categories assuming very little background in mathematics, while Mac Lane's Categories for the Working Mathematician is an introduction to categories for those who already have a substantial knowledge of other parts of mathematics. These notes are targeted to a student with significant \math- ematical sophistication" and a modest amount of specific knowledge. The sophistication is primarily an ease with the definition-theorem-proof style of mathematical exposition, being comfortable with an axiomatic approach, and finding particular pleasure in exploring unexpected connections even with un- familiar parts of mathematics Assumed Background: The critical specific knowledge assumed is a basic understanding of set theory. This includes such notions as subsets, unions and intersections of sets, ordered pairs, Cartesian products, relations, and functions as relations. An understanding of particular types of functions, particularly bi- jections, injections, surjections and the associated notions of direct and inverse images of subsets is also important. Other kinds of relations are important as well, particularly equivalence relations and order relations. The basic ideas regarding finite and infinite sets, cardinal and ordinal numbers and induction will also be used. All of this material is outlined in Appendix A on informal axiomatic set theory, but this is not likely to be useful as a first exposure to set theory. Although not strictly required some minimal understanding of elementary group theory or basic linear algebra will certainly make parts of the text much easier to understand. There are many examples scattered through the text which require some knowledge of other and occasionally quite advanced parts of mathematics. In iii iv PREFACE particular Appendix B (Catalog of Categories) contains a discussion of a large variety of specific categories. These typically assume some detailed knowledge of some parts of mathematics. None of these examples are required for under- standing the body of the notes, but are included primarily for those readers who do have such knowledge and secondarily to encourage readers to explore other areas of mathematics. Notation: The rigorous development of axiomatic set theory requires a very precise specification of the language and logic that is used. As part of that there is some concise notation that has become common in much of mathematics and which will be used throughout these notes. Occasionally, often in descriptions of sets, we will use various symbols from sentential logic particularly logical conjunction ^ for \and", logical disjunction _ for \or", implication ) for \implies" and logical equivalence () for \if and only if". We also use 8 and 9 from existential logic with 8 meaning \for all" and 9 meaning \there exists". Here is an example of the usage: For any sets A and B 8A 8B; A + B = fx :(x 2 A ^ x2 = B) _ (x2 = A ^ x 2 B)g from which we conclude 8A 8B; A + B = A ) A \ B = ; We have adopted two of Halmos' fine notational conventions here as well: the use of “iff" when precision demands \if and only if" while felicity asks for less; and the end (or absence) of a proof is marked with . Note on the Exercises There are 170 exercises in these notes, freely interspersed in the text. A list of the exercises is included in the front matter, just after the list of definitions. Although the main purpose of the exercises is to develop your skill working with the concepts and techniques of category theory, the results in the exercises are also an integral part of our development. Solutions to all of the exercises are provided in Appendix C, and you should understand them. If you have any doubt about your own solution, you should read the solution in the Appendix before continuing on with the text. If you find an error in the text, in the solutions, or just have a better solution, please send your comments to the author at [email protected]. They will be much appreciated. Alternative Sources There are many useful accounts of the material in these notes, and the study of category theory benefits from this variety of perspectives. In Appendix D are included brief reviews of the various books and notes, along with an indication of their contents. Throughout these Notes specific references are included for alternative dis- cussions of the material being treated, but no attempt has been made to provide attribution to original sources. Contents Preface iii Contents v List of Definitions xii List of Exercises xix Introduction 1 I Mathematics in Categories 3 I.1 What is a Category? . 3 I.1.1 Hom and Related Notation . 6 I.1.2 Subcategories . 8 I.1.3 Recognizing Categories . 9 I.2 Special Morphisms . 10 I.2.1 Isomorphisms . 11 I.2.2 Sections and Retracts . 12 I.2.3 Epimorphisms and Monomorphisms . 13 I.2.4 Subobjects and Quotient Objects . 20 I.3 Special Objects . 23 I.3.1 Products and Sums . 23 I.3.2 Final, Initial and Zero Objects . 36 I.3.3 Direct Sums and Matrices . 39 I.4 Algebraic Objects . 46 I.4.1 Magmas in a Category . 47 I.4.2 Comagmas in a Category . 50 I.4.2.1 Comagmas and Magmas Together . 53 I.4.3 Monoids in a Category . 56 I.4.4 Comonoids in a Category . 61 I.4.5 Groups in a Category . 64 II Constructing Categories 69 II.1 Duality and Dual Category . 69 II.2 Quotient Categories . 72 v vi Contents II.3 Product of Categories . 73 II.4 Sum of Categories . 74 II.5 Concrete and Based Categories . 74 II.6 Morphism Categories . 76 III Functors and Natural Transformations 79 III.1 What is a Functor? . 79 III.2 Examples of Functors . 82 III.2.1 Subcategories and Inclusion Functors . 82 III.2.2 Quotient Categories and Quotient Functors . 82 III.2.3 Product of Categories and Projection Functors . 82 III.2.4 Sum of Categories and Injection Functors . 83 III.2.5 Constant Functors . 84 III.2.6 Forgetful Functors . 85 III.2.7 The Product Functor . 85 III.2.8 The Sum Bifunctor . 86 III.2.9 Power Set Functor . 86 III.2.10 Monoid Homomorphisms are Functors . 86 III.2.11 Forgetful Functor for Monoid . 87 III.2.12 Free Monoid Functor . 87 III.2.13 Polynomial Ring as Functor . 88 III.2.14 Commutator Functor . 88 III.2.15 Abelianizer: Groups Made Abelian . 88 III.2.16 Discrete Topological Space Functor . 89 III.2.17 The Lie Algebra of a Lie Group . 90 III.2.18 Homology Theory . 90 III.3 Categories of Categories . 90 III.4 Digraphs and Free Categories . 92 III.5 Natural Transformations . 96 III.6 Examples of Natural Transformations . 98 III.6.1 Dual Vector Spaces . 98 III.6.2 Free Monoid Functor . 98 III.6.3 Commutator and Abelianizer . 99 III.6.4 The Discrete Topology and the Forgetful Functor . 100 III.6.5 The Godement Calculus . 101 III.6.6 Functor Categories . 102 III.6.7 Examples of Functor Categories . 102 III.6.8 Discrete Dynamical Systems . 104 IV Constructing Categories - Part II 107 IV.1 Comma Categories . 107 V Universal Mapping Properties 111 V.1 Universal Elements . 112 V.2 Universal Arrows . 114 V.3 Representable Functors . 114 Contents vii V.4 Initial and Final Objects . 115 V.4.1 Free Objects . 116 V.5 Limits and Colimits . 117 V.5.1 Cones and Limits . 121 V.5.2 Cocones and Colimits . 130 V.5.3 Complete Categories . 136 V.6 Adjoint Functors . 136 V.7 Kan Extensions . 145 V.7.1 Ends and Coends . 145 VI More Mathematics in a Category 147 VI.1 Relations In Categories . 147 VII Algebraic Categories 151 VII.1 Universal Algebra . 152 VII.2 Algebraic Theories . 152 VII.3 Internal Categories . 152 VIII Cartesian Closed Categories 155 VIII.1 Partial Equivalence Relations and Modest Sets . 155 IX Topoi 161 X The Category of Sets Reconsidered 163 XI Monoidal Categories 165 XII Enriched Category Theory 167 XIII Additive and Abelian Categories 169 XIV Homological Algebra 171 XIV.1 Introduction . 171 XIV.2 Additive Categories . 171 XIV.3 Abelian Categories . 171 XIV.4 Ext and Tor . 171 XIV.5 Category of Complexes . 171 XIV.6 Triangulated Categories . 171 XIV.7 Derived Categories . 171 XIV.8 Derived Functors . 171 XV 2-Categories 173 XVI Fibered Categories 175 Appendices 177 A Set Theory 179 viii Contents A.1 Extension Axiom . 180 A.2 Axiom of Specification . 181 A.2.1 Boolean Algebra of Classes . 182 A.3 Power-set Axiom . 185 A.4 Axiom of Pairs . 186 A.5 Union Axiom . 187 A.6 Ordered Pairs and Cartesian Products . 188 A.7 Relations . 190 A.8 Functions . 194 A.8.1 Families and Cartesian Products . 197 A.8.2 Images . 199 A.9 Natural Numbers . 199 A.10 Peano Axioms . 202 A.11 Arithmetic . 206 A.12 Order . 209 A.13 Number Systems .