Categories for the Working Mathematician

Front Cover
Springer Science & Business Media, Nov 11, 2013 - Mathematics - 262 pages
Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces.
 

Contents

Categories Functors and Natural Transformations
1
Constructions on Categories
31
Universals and Limits
55
Adjoints
77
7
78
10
88
Limits
105
Monads and Algebras
133
Diagonal Naturality
214
Ends
218
Coends
222
Ends with Parameters
224
Iterated Ends and Limits
226
Kan Extensions
229
Weak Universality
231
The Kan Extension
232

Monoids
157
13
173
Abelian Categories
187
16
190
Abelian Categories
194
Diagram Lemmas
198
Special Limits
207
Interchange of Limits
210
Final Functors
213
Kan Extensions as Coends
236
Pointwise Kan Extensions
239
Density
241
All Concepts are Kan Extensions
244
Table of Terminology
247
Bibliography
249
19
253
Index
255
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