Ebook: Categories and Sheaves
- Tags: Category Theory Homological Algebra, Manifolds and Cell Complexes (incl. Diff.Topology)
- Series: Grundlehren der mathematischen Wissenschaften 332
- Year: 2006
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.
This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.
The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.
This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.
The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.
This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.
The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.
Content:
Front Matter....Pages I-X
Introduction....Pages 1-8
The Language of Categories....Pages 9-34
Limits....Pages 35-69
Filtrant Limits....Pages 71-91
Tensor Categories....Pages 93-111
Generators and Representability....Pages 113-130
Indization of Categories....Pages 131-147
Localization....Pages 149-165
Additive and Abelian Categories....Pages 167-213
?-accessible Objects and F-injective Objects....Pages 215-240
Triangulated Categories....Pages 241-268
Complexes in Additive Categories....Pages 269-296
Complexes in Abelian Categories....Pages 297-318
Derived Categories....Pages 319-345
Unbounded Derived Categories....Pages 347-368
Indization and Derivation of Abelian Categories....Pages 369-387
Grothendieck Topologies....Pages 389-403
Sheaves on Grothendieck Topologies....Pages 405-433
Abelian Sheaves....Pages 435-460
Stacks and Twisted Sheaves....Pages 461-481
Back Matter....Pages 483-497
Categories and sheaves, which emerged in the middle of the last century as an enrichment for the concepts of sets and functions, appear almost everywhere in mathematics nowadays.
This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.
The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization. Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.
Content:
Front Matter....Pages I-X
Introduction....Pages 1-8
The Language of Categories....Pages 9-34
Limits....Pages 35-69
Filtrant Limits....Pages 71-91
Tensor Categories....Pages 93-111
Generators and Representability....Pages 113-130
Indization of Categories....Pages 131-147
Localization....Pages 149-165
Additive and Abelian Categories....Pages 167-213
?-accessible Objects and F-injective Objects....Pages 215-240
Triangulated Categories....Pages 241-268
Complexes in Additive Categories....Pages 269-296
Complexes in Abelian Categories....Pages 297-318
Derived Categories....Pages 319-345
Unbounded Derived Categories....Pages 347-368
Indization and Derivation of Abelian Categories....Pages 369-387
Grothendieck Topologies....Pages 389-403
Sheaves on Grothendieck Topologies....Pages 405-433
Abelian Sheaves....Pages 435-460
Stacks and Twisted Sheaves....Pages 461-481
Back Matter....Pages 483-497
....