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model structure on homotopical presheaves

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Contents

Idea

For V a sufficiently nice (monoidal) model category and C a small category equipped with a Grothendieck topology τ, there are left Bousfield localizations of the global model structure on functors [C op,V] whose fibrant objects satisfy descent with respect to Čech covers or even hypercovers with respect to τ.

These model structures are expected to model V-valued ∞-stacks on C. This is well understood for the case V= SSet equipped with the standard model structure on simplicial sets modelling ∞-groupoids. In this case the resulting local model structure on simplicial presheaves is known to be one of the models for ∞-stack (∞,1)-toposes.

But the general localization procedure works for choices of V different from and more general than SSet with its standard model structure. In particular it should work for

For these cases the local model structure on V-valued presheaves should model, respectively, (n,r)-category valued sheaves/stacks and (∞,1)-operad valued sheaves/stacks.

References

The general localization result is apparently due to

  • Clark Barwick, On left and right model categories and left and right Bousfield localization Homology, Homotopy and Applications, vol. 12(2), 2010, pp.245–320 (pdf)

which considers the Čech cover-localization assuming V to be monoidal and

  • Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (pdf)

which apparently does the hypercover descent and without assuming V to be monoidal.

Much of this was kindly pointed out by Denis-Charles Cisinski in discussion here.

Revised on April 30, 2013 18:49:36 by Adeel Ahmad Khan (160.45.40.186)
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