### Categorical homotopy theory, by J.F. Jardine

This paper is an exposition and extension of the ideas and methods of
Cisinksi, set at the level of A-presheaves on a small Grothendieck
site, where A is an arbitrary test category in the sense of
Grothendieck. The model structures for the category of simplicial
presheaves and all of its localizations can be modelled by
A-presheaves in the sense that there is a corresponding model
structure for A-presheaves with an equivalent homotopy category. The
theory specializes, for example, to the homotopy theories of cubical
sets, cubical presheaves, and gives a cubical model for motivic
homotopy theory. The applications of Cisinski's ideas are explained
in some detail for cubical sets.

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