Godement resolutions and sheaf homotopy theory

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.