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.

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