Homology, Homotopy and Applications

Volume 8 (2006)

Number 1

Categorical homotopy theory

Pages: 71 – 144

DOI: https://dx.doi.org/10.4310/HHA.2006.v8.n1.a3


J. F. Jardine (Mathematics Department, University of Western Ontario, London, Ontario, Canada)


This paper is an exposition of the ideas and methods of Cisinksi, in the context of $A$-presheaves on a small Grothendieck site, where $A$ is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each 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 and 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.


test categories, weak equivalence classes, cubical sets and presheaves

2010 Mathematics Subject Classification

14F35, 18F20, 55P60

Published 1 January 2006