Simplicial Homotopy Theory

Goerss, Paul G.; Jardine, John F.

doi:10.1007/978-3-0348-8707-6

Cited by 494 publications

(450 citation statements)

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“…The article can be read independently of [30] and [31], though, of course, much of our inspiration derives from these two papers. The reader is advised to have some background in homotopy theory in the form of [8, and in the theory of triangulated categories in the form of [15,20], [21,. Also, we will frequently use results from [25].…”

Section: Theorem 3 (Zariski Excision) Let J : U ⊂ X Be Quasi-compact mentioning

confidence: 99%

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Schlichting

2009

Invent. math.

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We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove Additivity, Fibration and Approximation theorems for the hermitian K-theory of exact categories with weak equivalences and duality.

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Section: Theorem 3 (Zariski Excision) Let J : U ⊂ X Be Quasi-compact mentioning

confidence: 99%

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Schlichting

2009

Invent. math.

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“…This is a classical fact, for instance around Lemma 4.1 in chapter V of [GJ99]. In terms of the total simplicial set functor it is observed in Section 4 of [RS12].…”

Section: Presentation In Locally Fibrant Simplicial Sheavesmentioning

confidence: 86%

“…This result is discussed for instance in chapter V of [GJ99]; a new proof that the unit of the adjunction is a weak equivalence is given in [Ste2]. This morphism is the standard presentation of the universal G-principal simplicial bundle.…”

Section: Theorem 330 ([Q69]mentioning

confidence: 99%

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Principal $$\infty $$ ∞ -bundles: presentations

Nikolaus

Schreiber

Stevenson

2014

J. Homotopy Relat. Struct.

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We discuss two aspects of the presentation of the theory of principal ∞-bundles in an ∞-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves.First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal ∞-bundles over any object in the ∞-topos over C are classified by hyper-Čech-cohomology with coefficients in G. Then we show that over a site C with enough points, principal ∞-bundles in the ∞-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete ∞-groupoids) and the smooth site (in smooth ∞-groupoids, generalizing Lie groupoids and differentiable stacks).In the companion article [NSSc] we use these presentations for constructing classes of examples of (twisted) principal ∞-bundles and for the discussion of various applications.

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“…where sdX is the subdivision of X and h is the last vertex map, see [5,III.4]. If A is an object in C we consider the following object in ind-C…”

Section: 2mentioning

confidence: 99%