Category theory. A gentle introduction

283 Pages·2016·1.416 MB·English

by  Smith P

Category Theory
A Gentle Introduction
Peter Smith
University of Cambridge
Version of February 8, 2016
(cid:13)cPeter Smith, 2016
This PDF is an early incomplete version of work still very much in
progress. For the latest and most complete version of this Gentle
IntroductionandforrelatedmaterialsseetheCategoryTheorypage
at the Logic Matters website.
Corrections, please, to ps218 at cam dot ac dot uk.
Contents
Preface ix
0 A very short introduction 1
0.1 Why categories? 1
0.2 What do you need to bring to the party? 2
0.3 Theorems as exercises 3
0.4 Notation and terminology 3
1 Categories defined 4
1.1 The very idea of a category 4
1.2 The category of sets 6
1.3 More examples 8
1.4 Diagrams 13
2 Categories beget categories 15
2.1 Duality 15
2.2 Subcategories, product and quotient categories 16
2.3 Arrow categories and slice categories 18
3 Kinds of arrows 22
3.1 Monomorphisms, epimorphisms 22
3.2 Inverses 24
3.3 Isomorphisms 27
3.4 Isomorphic objects 29
4 Initial and terminal objects 31
4.1 Initial and terminal objects, definitions and examples 32
4.2 Uniqueness up to unique isomorphism 33
4.3 Elements and generalized elements 34
5 Products introduced 36
5.1 Real pairs, virtual pairs 36
5.2 Pairing schemes 37
5.3 Binary products, categorially 41
iii
Contents
5.4 Products as terminal objects 44
5.5 Uniqueness up to unique isomorphism 45
5.6 ‘Universal mapping properties’ 47
5.7 Coproducts 47
6 Products explored 51
6.1 More properties of binary products 51
6.2 And two more results 52
6.3 More on mediating arrows 54
6.4 Maps between two products 56
6.5 Finite products more generally 58
6.6 Infinite products 60
7 Equalizers 61
7.1 Equalizers 61
7.2 Uniqueness again 64
7.3 Co-equalizers 65
8 Limits and colimits defined 68
8.1 Cones over diagrams 68
8.2 Defining limit cones 70
8.3 Limit cones as terminal objects 72
8.4 Results about limits 73
8.5 Colimits defined 75
8.6 Pullbacks 75
8.7 Pushouts 79
9 The existence of limits 81
9.1 Pullbacks, products and equalizers related 81
9.2 Categories with all finite limits 85
9.3 Infinite limits 87
9.4 Dualizing again 88
10 Subobjects 89
10.1 Subsets revisited 89
10.2 Subobjects as monic arrows 90
10.3 Subobjects as isomorphism classes 91
10.4 Subobjects, equalizers, and pullbacks 92
10.5 Elements and subobjects 94
11 Exponentials 95
11.1 Two-place functions 95
11.2 Exponentials defined 96
11.3 Examples of exponentials 98
11.4 Exponentials are unique 101
iv
Contents
11.5 Further results about exponentials 102
11.6 Cartesian closed categories 104
12 Group objects, natural number objects 108
12.1 Groups in Set 108
12.2 Groups in other categories 110
12.3 A very little more on groups 112
12.4 Natural numbers 113
12.5 The Peano postulates revisited 114
12.6 More on recursion 116
13 Functors introduced 120
13.1 Functors defined 120
13.2 Some elementary examples of functors 121
13.3 What do functors preserve and reflect? 123
13.4 Faithful, full, and essentially surjective functors 125
13.5 A functor from Set to Mon 127
13.6 Products, exponentials, and functors 128
13.7 An example from algebraic topology 130
13.8 Covariant vs contravariant functors 132
14 Categories of categories 134
14.1 Functors compose 134
14.2 Categories of categories 135
14.3 A universal category? 136
14.4 ‘Small’ and ‘locally small’ categories 137
14.5 Isomorphisms between categories 139
14.6 An aside: other definitions of categories 141
15 Functors and limits 144
15.1 Diagrams redefined as functors 144
15.2 Preserving limits 145
15.3 Reflecting limits 149
15.4 Creating limits 151
16 Hom-functors 152
16.1 Hom-sets 152
16.2 Hom-functors 154
16.3 Hom-functors preserve limits 155
17 Functors and comma categories 159
17.1 Functors and slice categories 159
17.2 Comma categories 160
17.3 Two (already familiar) types of comma category 161
17.4 Another (new) type of comma category 162
v
Contents
17.5 An application: free monoids again 163
17.6 A theorem on comma categories and limits 165
18 Natural isomorphisms 167
18.1 Natural isomorphisms between functors defined 167
18.2 Why ‘natural’? 168
18.3 More examples of natural isomorphormisms 171
18.4 Natural/unnatural isomorphisms between objects 176
18.5 An ‘Eilenberg/Mac Lane Thesis’? 177
19 Natural transformations and functor categories 179
19.1 Natural transformations 179
19.2 Composition of natural transformations 182
19.3 Functor categories 185
19.4 Functor categories and natural isomorphisms 186
19.5 Hom-functors from functor categories 187
19.6 Evaluation and diagonal functors 188
19.7 Limit functors 189
20 Equivalent categories 192
20.1 The categories Pfn and Set are ‘equivalent’ 192
(cid:63)
20.2 Pfn and Set are not isomorphic 194
(cid:63)
20.3 Equivalent categories 195
20.4 Skeletons and evil 198
21 The Yoneda embedding 201
21.1 Natural transformations between hom-functors 201
21.2 The Restricted Yoneda Lemma 204
21.3 The Yoneda embedding 205
21.4 Yoneda meets Cayley 207
22 The Yoneda Lemma 211
22.1 Towards the full Yoneda Lemma 211
22.2 The generalizing move 212
22.3 Making it all natural 213
22.4 Putting everything together 215
22.5 A brief afterword on ‘presheaves’ 216
23 Representables and universal elements 217
23.1 Isomorphic functors preserve the same limits 217
23.2 Representable functors 218
23.3 A first example 219
23.4 More examples of representables 221
23.5 Universal elements 222
23.6 Categories of elements 224
vi
Contents
23.7 Limits and exponentials as universal elements 226
24 Galois connections 227
24.1 (Probably unnecessary) reminders about posets 227
24.2 An introductory example 228
24.3 Galois connections defined 230
24.4 Galois connections re-defined 232
24.5 Some basic results about Galois connections 234
24.6 Fixed points, isomorphisms, and closures 235
24.7 One way a Galois connection can arise 237
24.8 Syntax and semantics briefly revisited 237
25 Adjoints introduced 239
25.1 Adjoint functors: a first definition 239
25.2 Examples 241
25.3 Naturality 245
25.4 An alternative definition 246
25.5 Adjoints and equivalent categories 251
26 Adjoints further explored 254
26.1 Adjunctions reviewed 254
26.2 Two more definitions! 255
26.3 Adjunctions compose 255
26.4 The uniqueness of adjoints 257
26.5 How left adjoints can be defined in terms of right adjoints 258
26.6 Another way of getting new adjunctions from old 262
27 Adjoint functors and limits 264
27.1 Limit functors as adjoints 264
27.2 Right adjoints preserve limits 266
27.3 Some examples 268
27.4 The Adjoint Functor Theorems 269
Bibliography 272
vii

Preface
This Gentle Introduction is work in progress, developing my earlier ‘Notes on
Basic Category Theory’ (2014–15).
The gadgets of basic category theory fit together rather beautifully in mul-
tiple ways. Their intricate interconnections mean, however, that there isn’t a
single best route into the theory. Different lecture courses, different books, can
quite appropriately take topics in very different orders, all illuminating in their
different ways. In the earlier Notes, I roughly followed the order of somewhat
over half of the Cambridge Part III course in category theory, as given in 2014
by Rory Lucyshyn-Wright (broadly following a pattern set by Peter Johnstone;
see also Julia Goedecke’s notes from 2013). We now proceed rather differently.
TheCambridge orderingcertainlyhasits rationale; butthealternativeordering
I now follow has in some respects a greater logical appeal. Which is one reason
for the rewrite.
Our topics, again in different arrangements, are also covered in (for example)
Awodey’s good but uneven Category Theory and in Tom Leinster’s terrific –
and appropriately titled – Basic Category Theory. But then, if there are some
rightlyadmiredtextsoutthere,nottomentionvarioussetsofnotesoncategory
theoryavailableonline(seehere),whyproduceanotherintroductiontocategory
theory?
Ididn’tintendto!Mygoalallalonghasbeentogettounderstandwhatlight
categorytheorythrowsonlogic,settheory,andthefoundationsofmathematics.
But I realized that I needed to get a lot more securely on top of basic category
theoryifIwaseventuallytopursuethesemorephilosophicalissues.Somyearlier
Notes began life as detailed jottings for myself, to help really fix ideas: and then
–ascanhappen–thewritinghassimplytakenonitsownmomentum.Iamstill
concentratingmostlyongettingthetechnicalitiesrightandpresentingthemina
pleasing order: I hope later versions will contain more motivational/conceptual
material.
What remains distinctive about this Gentle Introduction, for good or ill, is
that it is written by someone who doesn’t pretend to be an expert who usually
operatesattheveryfrontiersofresearchincategorytheory.Idohope,however,
that this makes me rather more attuned to the likely needs of (at least some)
beginners. I go rather slowly over ideas that once gave me pause, spend more
ix
Preface
time than is always usual in motivating key ideas and constructions, and I have
generally aimed to be as clear as possible (also, I assume rather less background
mathematicsthanLeinsterorevenAwodey).Wedon’tgetterriblyfar:however,
I hope that what is here may prove useful to others starting to get to grips
withcategorytheory.Myownexperiencecertainlysuggeststhatinitiallytaking
thingsatarathergentlepaceasyouworkintoafamiliaritywithcategorialways
of thinking makes later adventures exploring beyond the basics so very much
more manageable.
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