@TECHREPORT{Hirschowitz_descentepour, author = {André Hirschowitz and Carlos Simpson}, title = {Descente pour les n-champs}, institution = {}, year = {} }

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Abstract

We develop the theory of n-stacks (or more generally Segal n-stacks which are ∞-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 O-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us. Résumé