A homotopy theory for stacks

By Sharon Hollander. 2001

126 citations recorded by the source corpus.

We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S2-nullification of Jardine’s model structure on sheaves of simplicial sets on C.

Subfields: Geometry and Topology, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/fc4c38ff06984531b18eb23525fa45e2784ffc15

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Citations in the corpus

Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, methodology): “And in fact, the model category theory developed for simplicial presheaves has been useful for attacking the theory of n-categories as we do here, and also for going from a theory of n-categories to a theory of n-stacks, as Hollander has done for 1-stacks [118] and as Hirschowitz and I did for n-sta”
The stack of formal groups in stable homotopy theory (2007, methodology): “as in [40, VI], as well as with the notion of lax/pseudo functor/presheaf on Aff with values in groupoids; see [ 12 ] and [43] for more details.”
Local Systems on Proper Algebraic V -manifolds (2010, background): “This is the same as the classical 2-category of 1-stacks over the site Sch, a compatibility well-known particularly from Hollander’s work [49].”
Homotopy theory of comodules over a Hopf algebroid (2003, background): “We now recall that an object M in a cocomplete category C is called -presented, for a regular cardinal , if C(M, ) commutes with -filtered colimits (See [ Bor94 , Section 6.4]).”
Duality for topological modular forms (2011, background): “blems by allowing points to have nontrivial automorphisms. The classical viewpoint is that a stack is a category ﬁbered in groupoids. One can also equivalently view a stack as a presheaf of groupoids [Hol08]. Deligne-Mumford stacks are particularlywell-behaved, because they can be studied by maps fr”
Higher and derived stacks: a global overview (2009, background): “[ Hol , Ja2]). Another model for St(C) is the model category of simplicial sheaves, originally introduced by A. Joyal and revisited by J. Jardine (see [Joy, Ja1]).”