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Categories and Sheaves (Grundlehren der mathematischen Wissenschaften)
Product Review 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 are presented from scratch up to the most recent (sometimes new) results ." (Corrado Marastoni, Mathematical Reviews, Issue 2006 k) Product Description 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. Reader Reviews I'm a math and physics double major, and I've been interested in category theory for a while. This book does exactly what it intends to do, which the authors state in the preface is to "...present categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch and continuing with full proofs to an exposition of the most recent results in the literature and sometimes beyond." It's a typical advanced math book...i.e. a seeming grocery list of definitions and lemmas, then theorems and proofs. An advanced undergraduate may be interested, but it's a bit abstract (as all advanced math is!). The first chapter is a decent introduction to categories. I would advise learning linear algebra and abstract algebra before reading this chapter, and of course read this chapter before the rest of the book! Personally, without applications to reality, learning new math leaves me asking "So what?" The book has mathematical applications of category theory to "cast" abstract algebra and linear algebra in the language of categories instead of using the language of set theory. So if you don't know linear algebra and abstract algebra, you'll be left asking "So what?" Further, a lot of these concepts that are presented are hard...not in the sense "Solving this equation is hard!" But in the sense that it's deep, so it's hard like "Reading Hegel is hard!" Overall, I think it's a great book and worth it's money. I wouldn't advise getting it without good knowledge of abstract algebra (since then the notion of a category of groups, for example, would be meaningless without knowledge of what a group is!) or linear algebra (which helps with the notion of morphisms, etc.). Just my two cents... Comment | | (Report this)
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