Categories and Sheaves: An Introduction to Ind-Objects and Derived Categories (Grundlehren der mathematischen Wissenschaften Book 332)

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 basics, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond.The authors present 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. 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 π-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

Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.