viii CONTENTS
3.1.1 Diagram chasing 38
3.1.2 Subcategories, isomorphisms, monics and epis 39
3.2 Examples 40
3.2.1 Sets and finite sets 40
3.2.2 Graphs 40
3.2.3 Finite categories 41
3.2.4 Relations and partial orders 41
3.2.5 Partial orders as categories 42
3.2.6 Deductive systems 42
3.2.7 Universal algebra: terms, algebras and equations 43
3.2.8 Sets with structure and structure-preserving arrows 46
3.3 Categories computationally 47
3.4 Categories as values 49
3.4.1 The category of finite sets 49
3.4.2 Terms and term substitutions: the category T
Ω
F in
50
3.4.3 A finite category 52
3.5 Functors 53
3.5.1 Functors computationally 54
3.5.2 Examples 54
3.6 Duality 55
3.7 An assessment
∗
57
3.8 Conclusion 60
3.9 Exercises 60
4 Limits and Colimits 65
4.1 Definition by universality 67
4.2 Finite colimits 68
4.2.1 Initial objects 69
4.2.2 Binary coproducts 70
4.2.3 Coequalizers and pushouts 72
4.3 Computing colimits 74
4.4 Graphs, diagrams and colimits 79
4.5 A general construction of colimits 82
4.6 Colimits in the category of finite sets 88
4.7 A calculation of pushouts 90
4.8 Duality and limits 93
4.9 Limits in the category of finite sets 95
4.10 An application: operations on relations 97
4.11 Exercises 100
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