Notes on Category Theory

420 Pages·2011·1.67 MB·English

by  Robert L. Knighten

Notes on Category Theory
Robert L. Knighten
June 12, 2011
(cid:13)c 2004–2011 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
personalandselfishdesiretoproducemyownexpositionofthesubject. 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 [75] 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
intersectionsofsets,orderedpairs,Cartesianproducts,relations,andfunctions
asrelations. Anunderstandingofparticulartypesoffunctions,particularlybi-
jections,injections,surjectionsandtheassociatednotionsofdirectandinverse
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 D 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 A (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: Therigorousdevelopmentofaxiomaticsettheoryrequiresavery
precisespecificationofthelanguageandlogicthatisused. Aspartofthatthere
issomeconcisenotationthathasbecomecommoninmuchofmathematicsand
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 ∀ and
∃fromexistentiallogicwith∀meaning“forall”and∃meaning“thereexists”.
Here is an example of the usage: For any sets A and B
∀A∀B, A+B ={x:(x∈A∧x∈/ B)∨(x∈/ A∧x∈B)}
from which we conclude
∀A∀B, 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
Thereare173exercisesinthesenotes,freelyinterspersedinthetext. Alist
of the exercises is included in the front matter, just after the list of definitions.
Althoughthemainpurposeoftheexercisesistodevelopyourskillworkingwith
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 B, 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
Therearemanyusefulaccountsofthematerialinthesenotes,andthestudy
ofcategorytheorybenefitsfromthisvarietyofperspectives. InAppendixCare
includedbriefreviewsofthevariousbooksandnotes, alongwithanindication
of their contents.
Throughout these Notes specific references are included for alternative dis-
cussions of the material being treated, and some references to original papers
and discussions of the history of category theory are provided but no attempt
has been made to provide any sort of definitive history.
MacLane-1988
Contents
Preface iii
Contents v
List of Definitions xiv
List of Exercises xx
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 . . . . . . . . . . . . . . . . . . . . . . . . 11
I.2.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . 11
I.2.2 Sections and Retracts . . . . . . . . . . . . . . . . . . . 13
I.2.3 Epimorphisms and Monomorphisms . . . . . . . . . . . 14
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 Groups in a Category . . . . . . . . . . . . . . . . . . . 61
II Constructing Categories 67
II.1 Duality and Dual Category . . . . . . . . . . . . . . . . . . . . 67
II.2 Quotient Categories . . . . . . . . . . . . . . . . . . . . . . . . 70
II.3 Product of Categories . . . . . . . . . . . . . . . . . . . . . . . 71
v
vi CONTENTS
II.4 Sum of Categories . . . . . . . . . . . . . . . . . . . . . . . . . 72
II.5 Concrete and Based Categories . . . . . . . . . . . . . . . . . 72
II.6 Morphism Categories . . . . . . . . . . . . . . . . . . . . . . . 74
III Functors and Natural Transformations 77
III.1 What is a Functor? . . . . . . . . . . . . . . . . . . . . . . . . 77
III.2 Examples of Functors . . . . . . . . . . . . . . . . . . . . . . . 80
III.2.1 Subcategories and Inclusion Functors . . . . . . . . . . 80
III.2.2 Quotient Categories and Quotient Functors . . . . . . 80
III.2.3 Product of Categories and Projection Functors . . . . 80
III.2.4 Sum of Categories and Injection Functors . . . . . . . 81
III.2.5 Constant Functors . . . . . . . . . . . . . . . . . . . . 82
III.2.6 Forgetful Functors . . . . . . . . . . . . . . . . . . . . 83
III.2.7 The Product Functor . . . . . . . . . . . . . . . . . . . 83
III.2.8 The Sum Bifunctor . . . . . . . . . . . . . . . . . . . . 84
III.2.9 Power Set Functor . . . . . . . . . . . . . . . . . . . . 84
III.2.10 Monoid Homomorphisms are Functors . . . . . . . . . 85
III.2.11 Forgetful Functor for Monoid . . . . . . . . . . . . . . 85
III.2.12 Free Monoid Functor . . . . . . . . . . . . . . . . . . . 85
III.2.13 Polynomial Ring as Functor . . . . . . . . . . . . . . . 86
III.2.14 Commutator Functor . . . . . . . . . . . . . . . . . . . 86
III.2.15 Abelianizer: Groups Made Abelian . . . . . . . . . . . 87
III.2.16 Discrete Topological Space Functor . . . . . . . . . . . 87
III.2.17 The Lie Algebra of a Lie Group . . . . . . . . . . . . . 88
III.2.18 Homology Theory . . . . . . . . . . . . . . . . . . . . . 88
III.3 Categories of Categories . . . . . . . . . . . . . . . . . . . . . 88
III.4 Digraphs and Free Categories . . . . . . . . . . . . . . . . . . 90
III.5 Natural Transformations . . . . . . . . . . . . . . . . . . . . . 94
III.6 Examples of Natural Transformations . . . . . . . . . . . . . . 96
III.6.1 Dual Vector Spaces . . . . . . . . . . . . . . . . . . . . 96
III.6.2 Free Monoid Functor . . . . . . . . . . . . . . . . . . . 96
III.6.3 Commutator and Abelianizer . . . . . . . . . . . . . . 98
III.6.4 The Discrete Topology and the Forgetful Functor . . . 99
III.6.5 The Godement Calculus . . . . . . . . . . . . . . . . . 99
III.6.6 Functor Categories . . . . . . . . . . . . . . . . . . . . 100
III.6.7 Examples of Functor Categories . . . . . . . . . . . . . 101
III.6.8 Discrete Dynamical Systems . . . . . . . . . . . . . . . 102
IV Constructing Categories - Part II 105
IV.1 Comma Categories . . . . . . . . . . . . . . . . . . . . . . . . 105
IV.1.1 Examples and Special Cases . . . . . . . . . . . . . . . 106
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
V.8 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
V.9 Algebras for a Monad . . . . . . . . . . . . . . . . . . . . . . . 146
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.1Partial 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.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.2Additive Categories . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.3Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.4Ext and Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.5Category of Complexes . . . . . . . . . . . . . . . . . . . . . . 171
XIV.6Triangulated Categories . . . . . . . . . . . . . . . . . . . . . 171
XIV.7Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.8Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . 171
XIV.9Abelian Categories and Topoi . . . . . . . . . . . . . . . . . . 171
XV n−Categories 173
XVI Fibered Categories 175
viii CONTENTS
Appendices 177
A Catalog of Categories 179
A.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.1.1 Set – sets . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.1.2 FiniteSet – finite sets . . . . . . . . . . . . . . . . . . 181
A.1.3 Rel – the category of sets and relations . . . . . . . . 181
A.1.4 RefRel – the category of sets and reflexive relations . 182
A.1.5 SymRel – the category of sets and symmetric relations 182
A.1.6 PSet – the category of sets and partial functions . . . 182
A.1.7 Set – the category of pointed sets . . . . . . . . . . . 182
∗
A.1.8 Ord – the category of ordinal numbers . . . . . . . . . 182
A.2 Semigroups, Monoids, Groups and Their Friends . . . . . . . . 182
A.2.1 Magma – the category of magmas . . . . . . . . . . . 182
A.2.2 Semigroup – the category of semigroups . . . . . . . 183
A.2.3 Monoid – monoids . . . . . . . . . . . . . . . . . . . . 183
A.2.4 CMonoid – commutative monoids . . . . . . . . . . . 186
A.2.5 Group – groups . . . . . . . . . . . . . . . . . . . . . 187
A.2.6 FiniteGroup – finite groups . . . . . . . . . . . . . . 188
A.2.7 Ab – Abelian groups . . . . . . . . . . . . . . . . . . . 188
A.2.7.1 TorAb – Torsion Abelian groups . . . . . . . 188
A.2.7.2 DivAb – Divisible Abelian groups . . . . . . . 188
A.2.7.3 TorsionFreeAb – Torsion Free Abelian groups 188
A.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A.3.1 Rig – rings without negatives . . . . . . . . . . . . . . 190
A.3.2 Rng – rings without identity . . . . . . . . . . . . . . 191
A.3.3 Ring – associative rings with identity . . . . . . . . . 191
A.3.4 CommutativeRing – commutative rings . . . . . . . 192
A.3.5 Field – fields . . . . . . . . . . . . . . . . . . . . . . . 192
A.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.4.1 Module – modules over a commutative ring . . . . . . 192
A.4.2 Matrices – matrices over a commutative ring . . . . . 193
A.4.3 Vect – vector spaces . . . . . . . . . . . . . . . . . . . 193
A.4.4 FDVect – finite dimensional vector spaces . . . . . . . 193
A.5 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.5.1 Algebra – associative algebras . . . . . . . . . . . . . 193
A.5.2 LieAlgebra – Lie algebras. . . . . . . . . . . . . . . . 194
A.5.3 Coalg – co-algebras . . . . . . . . . . . . . . . . . . . 194
A.5.4 Bialg – bialgebras . . . . . . . . . . . . . . . . . . . . 194
A.5.5 Hopf – Hopf algebras . . . . . . . . . . . . . . . . . . 195
A.6 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
A.6.1 Preorder – preorders . . . . . . . . . . . . . . . . . . 196
A.6.2 Poset – partially ordered sets . . . . . . . . . . . . . . 196
A.6.3 Lattice – lattices . . . . . . . . . . . . . . . . . . . . . 196
A.6.4 Full subcategories of equationally defined lattices . . . 199
CONTENTS ix
A.6.5 Boolean–thecategoriesofBooleanalgebrasandBoolean
rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A.6.5.1 Boolean – the category of complete Boolean
algebras . . . . . . . . . . . . . . . . . . . . . . 204
A.7 Ordered Algebraic Structures . . . . . . . . . . . . . . . . . . 204
A.7.1 OrderedMagma – Ordered Magmas. . . . . . . . . . 204
A.7.2 OrderedMonoid – Ordered Monoids . . . . . . . . . 204
A.7.3 OrderedGroup – Ordered Groups . . . . . . . . . . . 205
A.7.4 OrderedRig – Ordered Rigs . . . . . . . . . . . . . . 205
A.7.5 OrderedRing – Ordered Rings . . . . . . . . . . . . . 206
A.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
A.8.1 Graph – graphs . . . . . . . . . . . . . . . . . . . . . 207
A.8.2 Digraph – directed graphs . . . . . . . . . . . . . . . 207
A.9 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
A.9.1 Metric – metric spaces. . . . . . . . . . . . . . . . . . 207
A.9.2 Uniform – uniform spaces . . . . . . . . . . . . . . . . 207
A.9.3 Top – topological spaces . . . . . . . . . . . . . . . . . 208
A.9.4 Comp – compact Hausdorff spaces . . . . . . . . . . . 208
A.9.5 Kspace – K-spaces . . . . . . . . . . . . . . . . . . . . 208
A.9.6 Homotopy–thehomotopycategoryoftopologicalspaces208
A.9.7 HSpace – H-Spaces . . . . . . . . . . . . . . . . . . . 208
A.10 Simplicial Categories . . . . . . . . . . . . . . . . . . . . . . . 208
A.10.1 Simplicial – simplicial sets . . . . . . . . . . . . . . . 208
A.10.2 Kan – the homotopy category of Kan complexes . . . 210
A.11 Differential, Graded and Filtered Algebraic Gadgets . . . . . . 210
A.11.1 Graded Category . . . . . . . . . . . . . . . . . . . . . 210
A.11.2 GradedModule . . . . . . . . . . . . . . . . . . . . . 210
A.11.3 GradedRing . . . . . . . . . . . . . . . . . . . . . . . 210
A.11.4 ChainComplex– Chain complexes . . . . . . . . . . . 210
A.12 Topological Algebras . . . . . . . . . . . . . . . . . . . . . . . 211
A.12.1 TopGroup – topological groups . . . . . . . . . . . . 212
A.12.2 TopAb – Abelian topological groups . . . . . . . . . . 212
A.12.3 TopVect – topological vector spaces . . . . . . . . . . 212
A.12.4 HausdorffTopVect–Hausdorfftopologicalvectorspaces212
A.13 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.13.1 Banach – Banach spaces. . . . . . . . . . . . . . . . . 212
A.13.2 FDBanach – finite dimensional Banach spaces . . . . 212
A.13.3 BanachAlgebra . . . . . . . . . . . . . . . . . . . . . 212
A.13.3.1 C*-algebra . . . . . . . . . . . . . . . . . . . . 212
A.13.4 Hilbert – Hilbert spaces . . . . . . . . . . . . . . . . . 212
A.13.5 FDHilb – finite dimensional Hilbert spaces . . . . . . 213
A.14 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . 213
A.14.1 Manifold– smooth manifolds . . . . . . . . . . . . . . 213
A.14.2 LieGroup– Lie groups . . . . . . . . . . . . . . . . . . 213
A.15 Algebraic and Analytic Geometry . . . . . . . . . . . . . . . . 214
A.15.1 Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
x CONTENTS
A.15.2 RingedSpace . . . . . . . . . . . . . . . . . . . . . . . 214
A.15.3 Scheme – algebraic schemes . . . . . . . . . . . . . . . 214
A.16 Unusual Categories . . . . . . . . . . . . . . . . . . . . . . . . 215
A.17 Cat – small categories . . . . . . . . . . . . . . . . . . . . . . 215
A.18 Groupoid – groupoids . . . . . . . . . . . . . . . . . . . . . . 215
A.19 Structures as Categories . . . . . . . . . . . . . . . . . . . . . 215
A.19.1 Every set is a category . . . . . . . . . . . . . . . . . . 215
A.19.2 Every monoid is a category . . . . . . . . . . . . . . . 215
A.19.3 Monoid of Strings . . . . . . . . . . . . . . . . . . . . . 215
A.19.4 Every preorder is a category . . . . . . . . . . . . . . . 215
A.19.5 Every topology is a category . . . . . . . . . . . . . . . 216
A.20 Little Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 216
A.20.1 0 – the empty category . . . . . . . . . . . . . . . . . . 216
A.20.2 1 – the one morphism category . . . . . . . . . . . . . 216
A.20.3 2 – the arrow category . . . . . . . . . . . . . . . . . . 217
A.20.4 3 – the commutative triangle category . . . . . . . . . 217
A.20.5 The parallel arrows category . . . . . . . . . . . . . . . 217
B Solutions of Exercises 219
B.1 Solutions for Chapter I . . . . . . . . . . . . . . . . . . . . . . 220
B.2 Solutions for Chapter II . . . . . . . . . . . . . . . . . . . . . 261
B.3 Solutions for Chapter III . . . . . . . . . . . . . . . . . . . . . 263
B.4 Solutions for Chapter IV . . . . . . . . . . . . . . . . . . . . . 274
B.5 Solutions for Chapter V . . . . . . . . . . . . . . . . . . . . . 274
B.6 Solutions for Chapter VI . . . . . . . . . . . . . . . . . . . . . 275
B.7 Solutions for Chapter VII . . . . . . . . . . . . . . . . . . . . 276
B.8 Solutions for Chapter VIII . . . . . . . . . . . . . . . . . . . . 276
B.9 Solutions for Chapter IX . . . . . . . . . . . . . . . . . . . . . 276
B.10 Solutions for Chapter X . . . . . . . . . . . . . . . . . . . . . 276
B.11 Solutions for Chapter XI . . . . . . . . . . . . . . . . . . . . . 276
B.12 Solutions for Chapter XII . . . . . . . . . . . . . . . . . . . . 276
B.13 Solutions for Chapter XIII . . . . . . . . . . . . . . . . . . . . 276
B.14 Solutions for Chapter XIV . . . . . . . . . . . . . . . . . . . . 276
B.15 Solutions for Chapter XV . . . . . . . . . . . . . . . . . . . . 276
B.16 Solutions for Chapter XVI . . . . . . . . . . . . . . . . . . . . 276
B.17 Solutions for Appendix A . . . . . . . . . . . . . . . . . . . . . 276
C Other Sources 277
C.1 Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
C.1.1 IntroductiontoCategoriesandCategoricalLogic(Abram-
sky and Tzevelekos) [1]) . . . . . . . . . . . . . . . . . 278
C.1.2 Abstract and Concrete Categories: The Joy of Cats
(Ad´amek, Herrlich, and Strecker [3]) . . . . . . . . . . 278
C.1.3 Category Theory (Awodey [6]). . . . . . . . . . . . . . 279
C.1.4 CategoryTheory: AnIntroduction(HerrlichandStrecker
[52]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280