Title: Fibred sites and stack cohomology
Author: J.F. Jardine
AMS Classification numbers: 55P42, 18F20, 14A20

J.F. Jardine
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7
Canada
E-mail: jardine@uwo.ca

The usual notion of a site fibred over a stack is expanded to a
definition of a site C/A fibred over a presheaf of categories
A. Presheaves of simplicial sets on the site fibred over a presheaf of
categories A are contravariant enriched diagrams defined on A, taking
values in simplicial sets. The standard model structure for presheaves
of simplicial sets induces a coarse equivariant structure for enriched
contravariant A-diagrams. 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.