Godement resolutions and sheaf homotopy theory

TL;DR

This paper explores conditions under which Godement resolutions can be used to develop sheaf homotopy theory, including fibrant models, sheaf cohomology, and derived functors, across various categories and sites.

Contribution

It identifies specific conditions on Grothendieck sites and coefficient categories that enable the use of Godement resolutions for sheaf homotopy theory.

Findings

Conditions for fibrant models via Godement resolutions

Criteria for defining sheaf cohomology using this approach

Extensions to derived functors in sheaf categories

Abstract

The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant models and hence to do sheaf homotopy theory. For instance, for which Grothendieck sites and coefficients we can define sheaf cohomology and derived functors through it.