Strict n-categories as models for homotopy types (joint work with Dimitri Ara)

Quillen realized in the sixties that small categories modelize
homotopy types. More precisely, he proved that the Gabriel-
Zisman localization of the category Cat of small categories by the
weak equivalences defined by the Grothendieck nerve is equivalent
to the homotopy category of simplicial sets. He also proved some
important properties of weak equivalences in Cat known as theorem
A and theorem B. Later, Thomason defined a Quillen model structure
on Cat and a Quillen equivalence of this structure with the
Quillen model structure on simplicial sets. In "Pursuing Stacks"
Grothendieck introduced the notion of basic localizers in Cat, based
on Quillen's theorem A, and Cisinski proved that basic localizers
classify left Bousfield localizations of the homotopy category
of spaces. In this lecture, I will report on a long run project with
Dimitri Ara to extend all these results to strict n-categories including
n equal to infinity and explain the theorems that we have
obtained and some related conjectures.