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# Homology, Homotopy and Applications

## Volume 8 (2006)

### Number 1

### Categorical homotopy theory

Pages: 71 – 144

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

#### Author

#### Abstract

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.

#### Keywords

test categories, weak equivalence classes, cubical sets and presheaves

#### 2010 Mathematics Subject Classification

14F35, 18F20, 55P60