Fibred sites and stack cohomology, by J.F. Jardine

The usual notion of the site associated to a stack is expanded to a definition of a site C/A fibred over a presheaf of categories A. All sites fibred over diagrams of schemes (including the etale site for a simplicial scheme) are examples of this construction. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent 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 a representing presheaf of groupoids.

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