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.

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)

Masaki Kashiwara Professor at the Rims, Kyoto University
Plenary speaker ICM 1978
Invited speaker ICM 1990
http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/

Pierre Schapira, Professor at University Pierre et Marie Curie (Paris VI)
Invited speaker ICM 1990
http://www.math.jussieu.fr/~schapira/