Propriétés universelles et
extensions de Kan dérivées

Denis-Charles Cisinski

AMS Classification numbers 55U3, 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
We show that for all small category A,
the derivator associated to the homotopy theory
of presheaves in categories (or in simplicial sets)
on A is the solution of a universal problem (and
a similar statement about the pointed versions
of such derivators is proved). When A is the
final category, this shows that the derivator HOT
associated to the classical homotopy theory is
canonically endowed with a monoidal structure,
and that every derivator admit a canonical
action of HOT. As every model category defines
a derivator, Hovey's homotopy coherence
conjectures are then a consequence of these
constructions.