### Diagrams and torsors, by J.F. Jardine

Suppose that A is a small presheaf of categories enriched in
simplicial sets on a small Grothendieck site. It is shown that the
homotopy theory of enriched A-diagrams taking values in simplicial
sets can be identified with the homotopy theory of simplicial
presheaves fibred over the diagonalized nerve dBA of A. One can also
identify the set [*,dBA] of morphisms in the simplicial presheaf
homotopy category with path components of the category of A-torsors,
suitably defined. These statements are special cases of localized
results which hold when the corresponding localized model structures
are proper. Examples of the latter include the motivic homotopy
category of Morel and Voevodsky, and so these results lead to a theory
of motivic A-torsors which is classifiable up to equivalence by a
family of morphisms in the motivic homotopy category.

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