## The cubical enrichment of Cisinski model categories

### Alexander Campbell – 7 November 2018

In his book on presheaves as models for homotopy types, Cisinski leaves as an exercise to the reader to show that a (presheaf) topos equipped with a Cisinski model structure (i.e. a combinatorial model structure in which the cofibrations are the monomorphisms), determined by an exact cylinder and a class of anodyne extensions, admits a canonical enrichment over the monoidal model category of cubical sets. In this talk, we will carry out this exercise and use these cubical enrichments to explain some of the basic features of the theory of Cisinski model structures, such as the notion of exact cylinder, the construction of classes of anodyne extensions, and the relationship between localisers and Bousfield localisations.

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