On sheaf cohomology and natural expansions

TL;DR

This survey explores sheaf and Čech cohomologies, their expansions to generalized sheaves like Grothendieck toposes, and discusses challenges and alternative approaches for cohomology theories in elementary toposes and related logical frameworks.

Contribution

It introduces expansions of sheaf cohomology to Grothendieck toposes and discusses alternative methods for elementary toposes using quantales, with applications and historical context.

Findings

Expanded cohomology concepts to Grothendieck toposes

Identified difficulties in defining cohomology for elementary toposes

Proposed alternative approaches using quantales and logical frameworks

Abstract

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.