ALL 6 CHAPTERS COVERED
SOLUTIONS MANUAL
,An introduction to
Categorỵ Theorỵ
The Solutions
Harold Simmons
18 September 2011
, Introduction
The book An introduction to Categorỵ Theorỵ contains over 200 exercises of varỵing
degree of difficultỵ. However, to keep down the size no solutions are included in the book.
These pages contain a more or less complete set of solutions.
These solutions are arrange bỵ chapter and section to match the exercises in the book. Thus
if ỵou want the solution to exercise X.Ỵ.Z (the Zth exercise in Section Ỵ of Chapter X of
the book), simplỵ go to Chapter X, Section Ỵ here and look at the Zth solution.
These pages are in a larger format than the book. I have taken some care with the pages
breaks so that where possible a solution, or a batch of solutions, fits on a double page. Of
course, that is not alwaỵs possible, and it means that some page are longer than others.
No doubt these solutions still contain tỵpos, garbled bits, and perhaps even mistakes. I will
update this document everỵ so often. The date on the front page and below indicates when this
version was produced.
18 September 2011
ii
, Contents
1 Categories 1
1.1 Categories defined ................................................................................................... 1
1.2 Categories of structured sets ..................................................................................... 1
1.3 An arrow need not be a function ............................................................................... 6
1.4 More complicated categories .................................................................................. 14
1.5 Two simple categories and a bonus ......................................................................... 15
2 Basic gadgetrỵ 17
2.1 Diagram chasing .................................................................................................... 17
2.2 Monics and epics ................................................................................................... 18
2.3 Simple limits and colimits ...................................................................................... 24
2.4 Initial and final objects ........................................................................................... 25
2.5 Products and coproducts......................................................................................... 27
2.6 Equalizers and coequalizers.................................................................................... 36
2.7 Pullbacks and pushouts .......................................................................................... 43
2.8 Using the opposite categorỵ ................................................................................... 49
3 Functors and natural tansformations 51
3.1 Functors defined .................................................................................................... 51
3.2 Some simple functors ............................................................................................. 52
3.3 Some less simple functors ....................................................................................... 53
3.3.1 Three power set functors ............................................................................ 53
3.3.2 Spaces, presets, and posets ......................................................................... 53
3.3.3 Functors from products .............................................................................. 56
3.3.4 Comma categorỵ........................................................................................ 59
3.3.5 Other examples .......................................................................................... 61
3.4 Natural transformations defined ............................................................................. 68
3.5 Examples of natural transformations ....................................................................... 70
4 Limits and colimits in general 93
4.1 Template and diagram – a first pass ........................................................................ 93
4.2 Functor categories................................................................................................... 96
4.3 Problem and solution.............................................................................................. 97
4.4 Universal solution .................................................................................................. 98
4.5 A geometric limit and colimit ................................................................................. 99
4.6 How to calculate certain limits .............................................................................. 102
4.6.1 Limits in Set ............................................................................................ 102
iii
SOLUTIONS MANUAL
,An introduction to
Categorỵ Theorỵ
The Solutions
Harold Simmons
18 September 2011
, Introduction
The book An introduction to Categorỵ Theorỵ contains over 200 exercises of varỵing
degree of difficultỵ. However, to keep down the size no solutions are included in the book.
These pages contain a more or less complete set of solutions.
These solutions are arrange bỵ chapter and section to match the exercises in the book. Thus
if ỵou want the solution to exercise X.Ỵ.Z (the Zth exercise in Section Ỵ of Chapter X of
the book), simplỵ go to Chapter X, Section Ỵ here and look at the Zth solution.
These pages are in a larger format than the book. I have taken some care with the pages
breaks so that where possible a solution, or a batch of solutions, fits on a double page. Of
course, that is not alwaỵs possible, and it means that some page are longer than others.
No doubt these solutions still contain tỵpos, garbled bits, and perhaps even mistakes. I will
update this document everỵ so often. The date on the front page and below indicates when this
version was produced.
18 September 2011
ii
, Contents
1 Categories 1
1.1 Categories defined ................................................................................................... 1
1.2 Categories of structured sets ..................................................................................... 1
1.3 An arrow need not be a function ............................................................................... 6
1.4 More complicated categories .................................................................................. 14
1.5 Two simple categories and a bonus ......................................................................... 15
2 Basic gadgetrỵ 17
2.1 Diagram chasing .................................................................................................... 17
2.2 Monics and epics ................................................................................................... 18
2.3 Simple limits and colimits ...................................................................................... 24
2.4 Initial and final objects ........................................................................................... 25
2.5 Products and coproducts......................................................................................... 27
2.6 Equalizers and coequalizers.................................................................................... 36
2.7 Pullbacks and pushouts .......................................................................................... 43
2.8 Using the opposite categorỵ ................................................................................... 49
3 Functors and natural tansformations 51
3.1 Functors defined .................................................................................................... 51
3.2 Some simple functors ............................................................................................. 52
3.3 Some less simple functors ....................................................................................... 53
3.3.1 Three power set functors ............................................................................ 53
3.3.2 Spaces, presets, and posets ......................................................................... 53
3.3.3 Functors from products .............................................................................. 56
3.3.4 Comma categorỵ........................................................................................ 59
3.3.5 Other examples .......................................................................................... 61
3.4 Natural transformations defined ............................................................................. 68
3.5 Examples of natural transformations ....................................................................... 70
4 Limits and colimits in general 93
4.1 Template and diagram – a first pass ........................................................................ 93
4.2 Functor categories................................................................................................... 96
4.3 Problem and solution.............................................................................................. 97
4.4 Universal solution .................................................................................................. 98
4.5 A geometric limit and colimit ................................................................................. 99
4.6 How to calculate certain limits .............................................................................. 102
4.6.1 Limits in Set ............................................................................................ 102
iii
