Fibred sites and stack cohomology

The usual notion of the site associated to a stack is expanded to a definition to a site C/A fibred over a presheaf of categories A on a site C. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site C/G is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on C/G. In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type.