Le localisateur fondamental minimal

Denis-Charles Cisinski

AMS Classification numbers 55U40

Institut de Mathématiques de Jussieu
Université Paris 7
Case 7012
2 place Jussieu
75251 Paris cedex 05 France

E-mail: cisinski@math.jussieu.fr

Abstract
Basic localizors were introduced by Grothendieck
in Pursuing Stacks. These are classes of arrows
in the category Cat of small categories
satisfying nice properties of descent (like
Quillen's theorem A).For example, every
cohomology theory defines a basic localizor.
In particular, classical weak equivalences
(i.e. those induced from the simplicial weak equivalences
from th nerve functor) form a basic localizor.
In this paper, we show Grothendieck's conjecture
that Cat's usual weak equivalences
are the smallest basic localizor. This gives in particular
a combinatorial/algebraic way to define classical
homotopy theory.