1998/07/09 by André Hirschowitz, Hirschowitz, André, Carlos Simpson +1 · 8 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.AT #math.CT
paper · pdf · doi:10.48550/arxiv.math/9807049
An error in Lemma 6.2 (pointed out by D. Dugger) is partially fixed and 6.1, 6.3, 10.11, 12.1, 14.4, 15.9, 19.4, 21.1 are modified accordingly; the descent 21.1 is now stated only for complexes bounded below
openalex publication_date 1998/07/09 · arxiv created 2001/03/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \Oo-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us.