Category Theory - Logic Matters

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Category Theory A Gentle Introduction

Peter Smith University of Cambridge

Version of January 29, 2018

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Peter Smith, 2018 This PDF is an early incomplete version of work still very much in progress. For the latest and most complete version of this Gentle Introduction and for related materials see the Category Theory page at the Logic Matters website. Corrections, please, to ps218 at cam dot ac dot uk.

Contents Preface

ix

1

The 1.1 1.2 1.3

categorial imperative Why category theory? From a bird’s eye view Ascending to the categorial heights

1 1 2 3

2

One 2.1 2.2 2.3 2.4 2.5 2.6

structured family of structures Groups Group homomorphisms and isomorphisms New groups from old ‘Identity up to isomorphism’ Groups and sets An unresolved tension

3

Categories defined 3.1 The very idea of a category 3.2 Monoids and pre-ordered collections 3.3 Some rather sparse categories 3.4 More categories 3.5 The category of sets 3.6 Yet more examples 3.7 Diagrams

17 17 20 21 23 24 26 27

4

Categories beget categories 4.1 Duality 4.2 Subcategories, product and quotient categories 4.3 Arrow categories and slice categories

30 30 31 33

5

Kinds of arrows 5.1 Monomorphisms, epimorphisms 5.2 Inverses 5.3 Isomorphisms 5.4 Isomorphic objects

37 37 39 42 44

4 4 5 8 11 13 16

iii

Contents 6

Initial and terminal objects 6.1 Initial and terminal objects, definitions and examples 6.2 Uniqueness up to unique isomorphism 6.3 Elements and generalized elements

46 47 48 49

7

Products introduced 7.1 Real pairs, virtual pairs 7.2 Pairing schemes 7.3 Binary products, categorially 7.4 Products as terminal objects 7.5 Uniqueness up to unique isomorphism 7.6 ‘Universal mapping properties’ 7.7 Coproducts

51 51 52 56 59 60 62 62

8

Products explored 8.1 More properties of binary products 8.2 And two more results 8.3 More on mediating arrows 8.4 Maps between two products 8.5 Finite products more generally 8.6 Infinite products

66 66 67 69 71 73 75

9

Equalizers 9.1 Equalizers 9.2 Uniqueness again 9.3 Co-equalizers

76 76 79 80

10 Limits and colimits defined 10.1 Cones over diagrams 10.2 Defining limit cones 10.3 Limit cones as terminal objects 10.4 Results about limits 10.5 Colimits defined 10.6 Pullbacks 10.7 Pushouts

83 83 85 87 88 90 91 94

11 The existence of limits 11.1 Pullbacks, products and equalizers related 11.2 Categories with all finite limits 11.3 Infinite limits 11.4 Dualizing again

96 96 100 102 103

12 Subobjects 12.1 Subsets revisited 12.2 Subobjects as monic arrows

104 104 105

iv

Contents 12.3 Subobjects as isomorphism classes 12.4 Subobjects, equalizers, and pullbacks 12.5 Elements and subobjects

106 107 109

13 Exponentials 13.1 Two-place functions 13.2 Exponentials defined 13.3 Examples of exponentials 13.4 Exponentials are unique 13.5 Further results about exponentials 13.6 Cartesian closed categories

110 110 111 113 116 117 119

14 Group objects, natural number objects 14.1 Groups in Set 14.2 Groups in other categories 14.3 A very little more on groups 14.4 Natural numbers 14.5 The Peano postulates revisited 14.6 More on recursion

123 123 125 127 128 129 131

15 Functors introduced 15.1 Functors defined 15.2 Some elementary examples of functors 15.3 What do functors preserve and reflect? 15.4 Faithful, full, and essentially surjective functors 15.5 A functor from Set to Mon 15.6 Products, exponentials, and functors 15.7 An example from algebraic topology 15.8 Covariant vs contravariant functors

135 135 136 138