Descente pour les n-champs (Descent for n-stacks)

TL;DR

This paper develops a comprehensive theory of n-stacks using model categories, providing definitions, characterizations, strictification, and descent results, with applications to complexes of sheaves of modules.

Contribution

It introduces a general framework for n-stacks via limits and descent, including strictification and new descent theorems, applicable to complexes of sheaves.

Findings

Defined n-stacks in terms of limits for generality

Established strictification of weak n-stacks

Proved descent for (n+1)-prestack of n-stacks

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.