An introduction to Category Theory [draft]

436 Pages·2010·1.282 MB·English

by  Harold Simmons

An introduction to
Category Theory
with over 200 exercises
and solutions available
Harold Simmons
1 September 2010
This is the date this version was compiled

Contents
1 Introduction – How do I remove the number page viii
PART ONE DEVELOPMENT 1
1 Categories 3
1.1 Categories defined 3
Exercises 9
1.2 Categories of structured sets 10
Exercises 16
1.3 An arrow need not be a function 18
Exercises 23
1.4 More complicated categories 29
Exercises 32
1.5 Two simple categories and a bonus 33
Exercises 35
2 Basic gadgetry 36
2.1 Diagram chasing 36
Exercises 38
2.2 Monics and epics 39
Exercises 42
2.3 Simple limits and colimits 45
Exercises 47
2.4 Initial and final objects 47
Exercises 48
2.5 Products and coproducts 49
Exercises 57
2.6 Equalizers and coequalizers 58
iv Contents
Exercises 63
2.7 Pullbacks and pushouts 66
Exercises 72
2.8 Using the opposite category 73
Exercises 73
3 Functors and natural transformations 74
3.1 Functors defined 75
Exercises 77
3.2 Some simple functors 78
Exercises 80
3.3 Some less simple functors 81
3.3.1 Three power set functors 81
Exercises 83
3.3.2 Spaces, presets, and posets 83
Exercises 85
3.3.3 Functors from products 86
Exercises 87
3.3.4 Comma category 88
Exercises 89
3.3.5 Other examples 89
Exercises 89
3.4 Natural transformations defined 92
Exercises 94
3.5 Examples of natural transformations 95
Exercises 104
4 Limits and colimits in general 110
4.1 Template and diagram – a first pass 111
Exercises 116
4.2 Functor categories 116
Exercises 119
4.3 Problem and solution 120
Exercises 122
4.4 Universal solution 122
Exercises 126
4.5 A geometric limit and colimit 126
Exercises 131
4.6 How to calculate certain limits 132
4.6.1 Limits in Set 132
Exercises 136
Contents v
4.6.2 Limits in Pos 136
Exercises 139
4.6.3 Limits in Mon 140
Exercises 143
4.6.4 Limits in Top 143
Exercises 145
4.7 Confluent colimits in Set 145
Exercises 149
5 Adjunctions 150
5.1 Adjunctions defined 150
Exercises 154
5.2 Adjunctions illustrated 155
5.2.1 An algebraic example 156
Exercises 158
5.2.2 A set-theoretic example 158
Exercises 160
5.2.3 A topological example 160
Exercises 166
5.3 Adjunctions uncouple 166
Exercises 172
5.4 The unit and the co-unit 172
Exercises 176
5.5 Free and co-free constructions 176
Exercises 187
5.6 Contravariant adjunctions 188
Exercises 189
6 Posets and monoid sets 192
6.1 Posets and complete posets 192
Exercises 192
6.2 Two categories of complete posets 193
Exercises 195
6.3 Sections of a poset 195
Exercises 197
6.4 The two completions 197
Exercises 198
6.5 Three endofunctors on Pos 199
Exercises 200
6.6 Long strings of adjunctions 201
Exercises 203
vi Contents
6.7 Two adjunctions for R-sets 204
Exercises 206
6.8 The upper left adjoint 207
Exercises 210
6.9 The upper adjunction 211
Exercises 214
6.10 The lower right adjoint 214
Exercises 220
6.11 The lower adjunction 220
Exercises 223
6.12 Some final projects 223
References 225
Index 226
7 Stuff that has to be seen to 228
PART TWO SOLUTIONS 229
1 Categories 231
1.1 Categories defined 231
1.2 Categories of structured sets 231
1.3 An arrow need not be a function 237
1.4 More complicated categories 247
1.5 Two simple categories and a bonus 248
2 Basic gadgetry 249
2.1 Diagram chasing 249
2.2 Monics and epics 250
2.3 Simple limits and colimits 256
2.4 Initial and final objects 257
2.5 Products and coproducts 259
2.6 Equalizers and coequalizers 270
2.7 Pullbacks and pushouts 278
2.8 Using the opposite category 284
3 Functors and natural tansformations 285
3.1 Functors defined 285
3.2 Some simple functors 286
3.3 Some less simple functors 287
3.3.1 Three power set functors 287
3.3.2 Spaces, presets, and posets 288
3.3.3 Functors from products 292
Contents vii
3.3.4 Comma category 294
3.3.5 Other examples 297
3.4 Natural transformations defined 306
3.5 Examples of natural transformations 308
4 Limits and colimits in general 337
4.1 Template and diagram – a first pass 337
4.2 Functor categories 340
4.3 Problem and solution 342
4.4 Universal solution 344
4.5 A geometric limit and colimit 345
4.6 How to calculate certain limits 349
4.6.1 Limits in Set 349
4.6.2 Limits in Pos 350
4.6.3 Limits in Mon 353
4.6.4 Limits in Top 357
4.7 Confluent colimits in Set 357
5 Adjunctions 364
5.1 Adjunctions defined 364
5.2 Adjunctions illustrated 372
5.2.1 An algebraic example 372
5.2.2 A set-theoretic example 375
5.2.3 A topological example 377
5.3 Adjunctions uncoupled 381
5.4 The unit and the co-unit 386
5.5 Free and co-free constructions 392
5.6 Contravariant adjunctions 401
6 Posets and monoid sets 407
6.1 Posets and complete posets 407
6.2 Two categories of complete posets 408
6.3 Sections of a poset 409
6.4 The two completions 410
6.5 Three endofunctors on Pos 411
6.6 Long strings of adjunctions 413
6.7 Two adjunctions for R-sets 414
6.8 The upper left adjoint 414
6.9 The upper adjunction 418
6.10 The lower right adjoint 421
6.11 The lower adjunction 423
1
Introduction – How do I remove the
number
As it says on the box this book is an introduction to Category Theory.
Itgivesthedefinitionofthisnotion,goesthroughthevariousassociated
gadgetry such as functors, natural transformations, limits and colimits,
and then explains adjunctions. That material could probably be de-
veloped in 50 pages or so, but here it takes some 220 pages. That is
because there are many examles illustrating the various notions, some
ratherstraightforwardbutotherwithmorecontent.Therearealsoover
200 exercises.
Thebookisaimedprimarilyatthebeginninggraduatestudent.Thus
the book does not assume the reader has a broad knowledge of mathe-
matics. Most of the illustrations use rather simple ideas, but every now
and then a more advanced topic is mentioned.
The idea is that the book can be use by a single student or small
groupofstudentstolearnthesubjectontheirown.Thebookwillmake
a suitable text for a reading group.
The book is aimed primarily at the beginning graduate student, but
that does not mean that other students of professional mathemticians
will not find it useful.
Every mathematician should at least know of the existence of cate-
gory theory, and many will need to use categorical notions every now
and then. For those groups this is the book you should have. Other
mathematicians will use category theory every day. That group has to
learn the subject sometime, and this is the book to start that process.
The book has been developed over several years. Several 10 hours
courseshavebeentaught(notalwaysbyme)usingsomeofthematerial.
In 2007, 2008, and 2009 I gave a course over the web to about a dozen
different Universities in England. This was part of MAGIC, the
Introduction – How do I remove the number ix
Mathematics Acces Grid Instructional Collaboration
cooperative of quite a few University Departmants of Mathematics in
England and Wales. (The course is still being taught but someone else
has taken over the wand.)
As I said earlier there are over 200 exercises scattered throughout the
book. I have also written a more or less complete set of solutions to
these exercises. To keep the book reasonable short and the cost down
these solutions are not included here. (With the solutions inckuded the
book would be over 420 pages.) However, these solutions are available
Details needed
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ThebookisdividedintosixChapters,eachchapterisdividedintosev-
eral Sections, and a few of these are divided into Blocks (Subsections).
Each chapter contains a list of Items, that is Definitions, Lemmas, The-
orems, Examples, and so on. These are numbered by section. Thus item
X.Y.Z is an Chapter X, Section Y, and is the Zth item in that section.
Where a section is divided into blocks the items are still numbered by
the parent section.
Each section contains a selection of Exercises. These are numbered
separately throughout the section. Thus Exercise X.Y.Z is in Chapter
X, Section Y, and is the Zth exercise of that section. Again, where a
section is divided into blocks the exercises are still numbered by the
parent section.
Mention various people.