Hypercovers and simplicial presheaves

By Daniel Dugger, Sharon Hollander, Daniel Isaksen. Mathematical Proceedings of the Cambridge Philosophical Society, 2002

258 citations recorded by the source corpus.

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75cc40bc988b65cb2e334219089895853376b175

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Citations in the corpus

Topological modular forms with level structure (2013, background): “Bywork ofDugger– Hollander–Isaksen,FI f is Jardine fibrant in the local model structure [12], and therefore also a fibrant replacement for F in our local model structure.”
Framed motives of algebraic varieties (after V. Voevodsky) (2014, methodology): “ry sPre•(Sm/k)Cech associated with Nisnevich topology is obtained from sPre•(Sm/k)proj by Bousﬁeld localization with respect to the set of all maps η: Uˇ∗→(−,X)+. It follows from [26, 5.10] (see also [4]) that sPre•(Sm/k)Cech coincides with the Nisnevich local model category sPre•(Sm/k)nis, with wea”
Higher Topos Theory (2006, background): “X is hypercomplete if and only if X satisfies a descent condition with respect to hypercoverings (other versions of this result can be found in [20] and [78]).”
Non-Archimedean analytic geometry as relative algebraic geometry (2013, background): “Stacks would be pre-stacks which satisfy descent with respect to hypercovers [16, 29, 30, 59, 60].”
Derived smooth manifolds (2008, background): “In [10], the authors show that Shv(X, sSets) is a localization of the injective model structure on sSets op at a certain set of morphisms (called the hypercovers) to obtain Shv(X, sSets).”
A commutative P^1-spectrum representing motivic cohomology over Dedekind domains (2012, background): “We localize at the set of maps hocolimn∈△op(Un ×QA)→ QA, where U● → X runs through a set of dense hypercovers (see [7]) of S and A through the set of domains and codomains of a set of generating cofibrations of C (QA denotes a cofibrant replacement of A).”
Higher and derived stacks: a global overview (2009, background): “in the following way. First of all for two categories Aand Bwith colimits we will denote by Homc(A,B) the category of functors commuting with colimits (”c” stands for ”continuous”). Also, recall from [DHI] the notion of an hypercovering in C(and noticed that an hypercovering in Cis not a simplicial ”
Duality for topological modular forms (2011, background): “scent would mean thatˇ the natural map F(U) →holimF(Cˇ•U) is an equivalence. As it turns out, this is not suﬃcient in all cases, but Cechˇ hyperdescent is. Dugger, Hollander, and Isaksen haveprovedin [DHI04] that Jardine’smodel categorymanner of recognizingsheaves is equivalent to the one using hype”
Butterflies I: Morphisms of 2-group stacks (2008, background): “3] when S has fiber products, but the argument goes through for generalized covers as well, or it follows more generally from the results of [15].”
Topological K-theory of complex noncommutative spaces (2012, background): “e have an equivalence ssp(E) ’colim( ‘ X2A C E(X) ssp(X) ‘ o Y! Z E(Z) ssp(Y)) o : 3.2 A1-etale model structure Let Cbe a Grothendieck site with topology denoted by ˝. Jardine (see [Jar87], and also [DHI04]) has proven the existence of a ˝-local model structure on the category SPr(C) of simplicial p”
Homotopical algebraic geometry. I. Topos theory. (2002, background): “The proof we present here is based on some arguments found in [S1], [H-S] and [DHI] (as well as on some hints from V.”
Stable Stems (2014, background): “The formal construction of motivic homotopy theory requires the heavy machinery of simplicial presheaves and model categories [33] [22] [12].”