Categories for the Working MathematicianSpringer Science & Business Media, 17 avr. 2013 - 318 pages Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories. |
Table des matières
1 | |
Constructions on Categories | 31 |
Universals and Limits | 55 |
Adjoints | 79 |
Limits | 109 |
Monads and Algebras | 137 |
Monoids 161 566 | 161 |
Monoids | 170 |
Ends | 222 |
Coends | 226 |
Ends with Parameters | 228 |
Iterated Ends and Limits | 230 |
Kan Extensions | 233 |
Weak Universality | 235 |
The Kan Extension | 236 |
Kan Extensions as Coends | 240 |
Actions | 174 |
The Simplicial Category | 175 |
Monads and Homology | 180 |
Closed Categories | 184 |
Compactly Generated Spaces | 185 |
Loops and Suspensions | 188 |
Abelian Categories | 191 |
Additive Categories | 194 |
Abelian Categories | 198 |
Diagram Lemmas | 202 |
Special Limits | 211 |
Interchange of Limits | 214 |
Final Functors | 217 |
Diagonal Naturality | 218 |
Pointwise Kan Extensions | 243 |
Density | 245 |
All Concepts Are Kan Extensions | 248 |
Symmetry and Braiding in Monoidal Categories | 251 |
Monoidal Functors | 255 |
Strict Monoidal Categories | 257 |
The Braid Groups B and the Braid Category | 260 |
Braided Coherence | 263 |
Perspectives | 266 |
SingleSet Categories | 267 |
Examples of Bicategories | 283 |
Objects and Arrows | 293 |
303 | |
Autres éditions - Tout afficher
Expressions et termes fréquents
Ab-category abelian category abelian groups adjoint functor adjoint functor theorem adjunction F algebra arrow f assigns axioms bifunctor bijection binary braid CGHaus codomain coend coequalizer coherence theorem Colim colimits comma category commutative diagram composite construction coproduct counit defined definition dinatural dual elements equal equivalence example Exercises exists factors forgetful functor full subcategory function f functor category functor F given graph Hausdorff spaces hence hom-sets homomorphism identity arrow implies initial object inverse Kan extension kernel left adjoint Lemma Lim F limiting cone Mac Lane monad monic morphism natural isomorphism natural transformation pair of arrows parallel pair preorder preserves Proof Proposition prove pullback quotient R-Mod R-module right adjoint right Kan extension ring small hom-sets small set subobjects subset T-algebras tensor product theory topological space unique arrow universal arrow usual vector space vertex