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.

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