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 defined 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 defined 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 Bousfield 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 infinity and explain the theorems that we have

obtained and some related conjectures.

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