### 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.

J.F. Jardine <jardine@uwo.ca>