## Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

### Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

#### July 18-21 2005, Australian National University, Canberra

Thu, 21 July: 14:30 - 15:30

##### Local test categories and equivariant homotopy theory

###### Cisinski, Denis-Charles (Université Paris 13)

Grothendieck's theory of local test categories is an attempt to give a general answer to the following question: when does the category of presheaves on a small category model the homotopy category of CW-complexes? The main well known example is the category of simplicial sets. Grothendieck conjectured that for any test category $A$ (that is a small category satisfying some nice combinatorial axioms) the category of presheaves on $A$ has a model structure Quillen equivalent with the one on simplicial sets, and that has now been proved. The relative version of this theory (also initiated by Grothendieck) gives rise to a lot of Quillen model structures on presheaf categories closely related to classical homotopy theory (or localized versions of it).

In this talk, I shall discuss how we can recover the equivariant homotopy theory (for a given presheaf of groups) from this view point. I shall also try to show how some ``higher Galois theory" yoga shows up naturally in this setting.