Category theory

Description

This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of Category Theory understandable to this broad readership. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided; a must for computer scientists, logicians and linguists! Preface......Page 6 Contents......Page 9 1.1 Introduction......Page 13 1.2 Functions Of Sets......Page 15 1.3 Definition Of A Category Definition 1.1. A Category Consists Of The Following Data:......Page 16 1.4 Examples Of Categories......Page 17 1.5 Isomorphisms Definition 1.3. In Any Category C, An Arrow F : A B Is Called An......Page 23 1.6 Constructions On Categories......Page 25 1.7 Free Categories Free Monoid. Start With An چalphabetژ A Of چlettersژ (a Set)......Page 28 1.8 Foundations: Large, Small, And Locally Small......Page 33 1.9 Exercises......Page 35 2.1 Epis And Monos......Page 37 2.2 Initial And Terminal Objects......Page 40 2.3 Generalized Elements......Page 41 2.4 Sections And Retractions......Page 45 2.5 Products......Page 46 2.6 Examples Of Products......Page 48 2.7 Categories With Products......Page 53 2.8 Hom-sets......Page 54 2.9 Exercises......Page 57 3.1 The Duality Principle......Page 59 3.2 Coproducts......Page 61 3.3 Equalizers......Page 66 3.4 Coequalizers......Page 69 3.5 Exercises......Page 75 4.1 Groups In A Category......Page 77 4.2 The Category Of Groups......Page 80 4.3 Groups As Categories......Page 82 4.4 Finitely Presented Categories......Page 85 4.5 Exercises......Page 86 5.1 Subobjects......Page 89 5.2 Pullbacks......Page 92 5.3 Properties Of Pullbacks......Page 96 5.4 Limits......Page 101 5.5 Preservation Of Limits......Page 106 5.6 Colimits......Page 107 5.7 Exercises......Page 114 6.1 Exponential In A Category......Page 117 6.2 Cartesian Closed Categories Definition 6.2. A Category Is Called Cartesian Closed If It Has All “nite Products......Page 120 6.3 Heyting Algebras......Page 125 6.4 Equational Definition......Page 130 6.5 Lambda-calculus......Page 131 6.6 Exercises......Page 135 7.1 Category Of Categories......Page 137 7.2 Representable Structure......Page 139 7.3 Stone Duality......Page 143 7.4 Naturality......Page 145 7.5 Examples Of Natural Transformations......Page 147 7.6 Exponentials Of Categories......Page 151 7.7 Functor Categories......Page 154 7.8 Equivalence Of Categories......Page 158 7.9 Examples Of Equivalence......Page 162 7.10 Exercises......Page 167 8.1 Set-valued Functor Categories......Page 171 8.2 The Yoneda Embedding......Page 172 8.3 The Yoneda Lemma Lemma 8.2.op (yoneda). Let C Be Locally Small. For Any Object C C And Functor......Page 174 8.4 Applications Of The Yoneda Lemma......Page 178 8.5 Limits In Categories Of Diagrams......Page 179 8.6 Colimits In Categories Of Diagrams......Page 180 8.7 Exponentials In Categories Of Diagrams......Page 184 8.8 Topoi......Page 186 8.9 Exercises......Page 188 9.1 Preliminary Definition......Page 191 9.2 Hom-set Definition......Page 195 9.3 Examples Of Adjoints......Page 199 9.4 Order Adjoints......Page 203 9.5 Quanti.ers As Adjoints......Page 205 9.6 Rapl......Page 209 9.8 Adjoint Functor Theorem......Page 222 9.9 Exercises......Page 231 10.1 The Triangle Identities......Page 235 10.2 Monads And Adjoints......Page 237 10.3 Algebras For A Monad Proposition 10.6. Every Monad Arises From An Adjunction. More Precisely,......Page 241 10.5 Algebras For Endofunctors......Page 248 10.6 Exercises......Page 256 References......Page 261 Index......Page 263