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Categories and sheaves

Author: Masaki Kashiwara; Pierre Schapira
Publisher: Berlin ; New York : Springer, ©2006.
Series: Grundlehren der mathematischen Wissenschaften, 332.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:

Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner. It presents the general  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Masaki Kashiwara; Pierre Schapira
ISBN: 3540279490 9783540279495
OCLC Number: 61703264
Description: x, 497 pages : illustrations ; 25 cm.
Contents: 1 The Language of Categories 9 --
1.1 Preliminaries: Sets and Universes 10 --
1.2 Categories and Functors 11 --
1.3 Morphisms of Functors 19 --
1.4 The Yoneda Lemma 23 --
1.5 Adjoint Functors 27 --
2 Limits 35 --
2.1 Limits 36 --
2.3 Kan Extension of Functors 50 --
2.4 Inductive Limits in the Category Set 54 --
2.5 Cofinal Functors 57 --
2.6 Ind-lim and Pro-lim 59 --
2.7 Yoneda Extension of Functors 62 --
3 Filtrant Limits 71 --
3.1 Filtrant Inductive Limits in the Category Set 71 --
3.2 Filtrant Categories 78 --
3.3 Exact Functors 81 --
3.4 Categories Associated with Two Functors 87 --
4 Tensor Categories 93 --
4.1 Projectors 93 --
4.2 Tensor Categories 96 --
4.3 Rings, Modules and Monads 103 --
5 Generators and Representability 113 --
5.1 Strict Morphisms 113 --
5.2 Generators and Representability 117 --
5.3 Strictly Generating Subcategories 121 --
6 Indization of Categories 131 --
6.1 Indization of Categories and Functors 131 --
6.2 Representable Ind-limits 138 --
6.3 Indization of Categories Admitting Inductive Limits 139 --
6.4 Finite Diagrams in Ind(C) 142 --
7 Localization 149 --
7.1 Localization of Categories 149 --
7.2 Localization of Subcategories 158 --
7.3 Localization of Functors 159 --
7.4 Indization and Localization 161 --
8 Additive and Abelian Categories 167 --
8.1 Group Objects 167 --
8.2 Additive Categories 169 --
8.3 Abelian Categories 175 --
8.4 Injective Objects 186 --
8.5 Ring Action 188 --
8.6 Indization of Abelian Categories 193 --
8.7 Extension of Exact Functors 197 --
9 [pi]-accessible Objects and F-injective Objects 215 --
9.1 Cardinals 216 --
9.2 [pi]-filtrant Categories and [pi]-accessible Objects 217 --
9.3 [pi]-accessible Objects and Generators 223 --
9.4 Quasi-Terminal Objects 228 --
9.5 F-injective Objects 231 --
9.6 Applications to Abelian Categories 235 --
10 Triangulated Categories 241 --
10.1 Triangulated Categories 242 --
10.2 Localization of Triangulated Categories 248 --
10.3 Localization of Triangulated Functors 253 --
10.4 Extension of Cohomological Functors 256 --
10.5 The Brown Representability Theorem 258 --
11 Complexes in Additive Categories 269 --
11.1 Differential Objects and Mapping Cones 270 --
11.2 The Homotopy Category 272 --
11.3 Complexes in Additive Categories 278 --
11.4 Simplicial Constructions 282 --
11.5 Double Complexes 285 --
11.6 Bifunctors 289 --
11.7 The Complex Hom 292 --
12 Complexes in Abelian Categories 297 --
12.1 The Snake Lemma 297 --
12.2 Abelian Categories with Translation 300 --
12.3 Complexes in Abelian Categories 302 --
12.4 Example: Koszul Complexes 306 --
12.5 Double Complexes 313 --
13 Derived Categories 319 --
13.1 Derived Categories 319 --
13.2 Resolutions 325 --
13.3 Derived Functors 329 --
13.4 Bifunctors 337 --
14 Unbounded Derived Categories 347 --
14.1 Derived Categories of Abelian Categories with Translation 347 --
14.2 The Brown Representability Theorem 353 --
14.3 Unbounded Derived Category 354 --
14.4 Left Derived Functors 357 --
15 Indization and Derivation of Abelian Categories 369 --
15.1 Injective Objects in Ind(C) 369 --
15.2 Quasi-injective Objects 372 --
15.3 Derivation of Ind-categories 374 --
15.4 Indization and Derivation 381 --
16 Grothendieck Topologies 389 --
16.1 Sieves and Local Epimorphisms 389 --
16.2 Local Isomorphisms 394 --
16.3 Localization by Local Isomorphisms 399 --
17 Sheaves on Grothendieck Topologies 405 --
17.1 Presites and Presheaves 405 --
17.2 Sites 411 --
17.3 Sheaves 414 --
17.4 Sheaf Associated with a Presheaf 417 --
17.5 Direct and Inverse Images 423 --
17.6 Restriction and Extension of Sheaves 424 --
17.7 Internal Hom 429 --
18 Abelian Sheaves 435 --
18.1 R-modules 435 --
18.2 Tensor Product and Internal Hom 438 --
18.3 Direct and Inverse Images 442 --
18.4 Derived Functors for Hom and Hom 444 --
18.5 Flatness 445 --
18.6 Ringed Sites 449 --
18.7 Cech Coverings 455 --
19 Stacks and Twisted Sheaves 461 --
19.1 Prestacks 461 --
19.2 Simply Connected Categories 466 --
19.3 Simplicial Constructions 467 --
19.4 Stacks 470 --
19.5 Morita Equivalence 474 --
19.6 Twisted Sheaves 477.
Series Title: Grundlehren der mathematischen Wissenschaften, 332.
Responsibility: Masaki Kashiwara, Pierre Schapira.
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From the reviews: "This book of Kashiwara and Schapira, recognized specialists in algebraic analysis, is a detailed full-scale exposition of categories, homological algebra and sheaves. These notions Read more...

 
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Linked Data


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