nLab model structure on homotopical presheaves

Idea

For $V$ a sufficiently nice (monoidal) model category and $C$ a small category equipped with a Grothendieck topology $\tau$, there are left Bousfield localizations of the global model structure on functors $[C^{op}, V]$ whose fibrant objects satisfy descent with respect to ?ech cover?s or even hypercovers with respect to $\tau$.

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

• $V =$ SSet equipped with its Joyal model structure, modelling (∞,1)-categories

• $V = (n,r)\Theta Spaces$, the model structure on Theta spaces modelling weak (n,r)-categories;

• $V = dSet$, the model structure on dendroidal sets modelling (∞,1)-operads.

For these cases the local model structure on $V$-valued presheaves should model, respectively, $(n,r)$-category valued sheaves/stacks and $(\infty,1)$-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 ?ech 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.

Related concepts

• enriched Reedy category

• model structure on sSet-presheaves