Title: Descente pour les $n$-champs (Descent for $n$-stacks)
Authors: Andr\'e Hirschowitz, Carlos Simpson
Authors' addresses:
Universit\'e de Nice-Sophia Antipolis, Parc Valrose,
06108 Nice cedex 2, France;
Laboratoire Emile Picard, Universit\'e Toulouse 3, 31062 Toulouse cedex,
France
Authors' email addresses:
ah@math.unice.fr;
carlos@picard.ups-tlse.fr
Subj-class: Algebraic Geometry; Algebraic Topology; Category Theory
Abstract:
We develop the theory of $n$-stacks (or more generally Segal $n$-stacks
which are $\infty$-stacks such that the morphisms are invertible above
degree $n$). This is done by systematically using the theory of
closed model categories (cmc). Our main results are: a definition of
$n$-stacks in terms of limits, which should be perfectly general for stacks
of any type of objects; several other characterizations of $n$-stacks in
terms of ``effectivity of descent data''; construction of the stack
associated to an $n$-prestack; a strictification result saying that
any ``weak'' $n$-stack is equivalent to a (strict) $n$-stack; and a
descent result saying that the $(n+1)$-prestack of $n$-stacks (on a
site) is an $(n+1)$-stack. As for other examples, we start from a
``left Quillen presheaf'' of cmc's and introduce the associated
Segal $1$-prestack. For this situation, we prove a general descent result,
giving sufficient conditions for this prestack to be a stack.
This applies to the case of complexes, saying how complexes of
sheaves of $\Oo$-modules can be glued together via quasi-isomorphisms.
This was the problem that originally motivated us.