  In this survey paper, we present \v{C}ech and sheaf cohomologies -- themes
that were presented by Koszul in University of S\~ao Paulo during his visit in
the late 1950s -- we present expansions for categories of generalized sheaves
(i.e, Grothendieck toposes), with examples of applications in other cohomology
theories and other areas of mathematics, besides providing motivations and
historical notes. We conclude explaining the difficulties in establishing a
cohomology theory for elementary toposes, presenting alternative approaches by
considering constructions over quantales, that provide structures similar to
sheaves, and indicating researches related to logic: constructive
(intuitionistic and linear) logic for toposes, sheaves over quantales, and
homological algebra.
