An $(\infty, 1)$-category is a generalization of ordinary category theory in which compositions are only well-defined up to homotopy, and which also capture the higher homotopies that arise as a result of this fact. $(infty, 1)$-categories are modeled using homotopy theory; three of the major extant models of $(\infty, 1)$ category are quasi-categories (Joyal/Lurie), complete Segal spaces (Rezk) and fibrant simplicial categories (Bergner).

 

The purpose of this talk is to discuss model structures on $(infty, 1)$-category valued presheaves on an arbitrary Grothendieck site, in which the weak equivalences are defined ‘stalkwise’. These extend, respectively the Joyal, Rezk and Bergner structures and are all Quillen equivalent. This class of model structures can be viewed as a generalization of the Jardine model structure on $(infty)$-groupoid valued presheaves, which was originally created to study descent theory.

One of the main technical tools to work with these model structures is the Barr cover of a topos. The Barr cover allow one to define the model structures for topoi which don’t have enough points. In addition, it allows us to replace pointwise (or stalkwise) reasoning with sectionwise reasoning.

 

As a consequence, we can show that various quasi-categorical constructions (the core, slice, mapping spaces, etc.) interact well with the local model structures we have produced, and that the usual Quillen equivalences extend to the presheaf level.