model structure on homotopical presheaves
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
For a sufficiently nice (monoidal) model category and a small category equipped with a Grothendieck topology , there are left Bousfield localizations of the global model structure on functors whose fibrant objects satisfy descent with respect to Čech covers or even hypercovers with respect to .
These model structures are expected to model -valued ∞-stacks on . This is well understood for the case 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 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 -valued presheaves should model, respectively, -category valued sheaves/stacks and -operad valued sheaves/stacks.
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 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 to be monoidal.
Much of this was kindly pointed out by Denis-Charles Cisinski in discussion here.