model structure on homotopical presheaves


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



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

These model structures are expected to model VV-valued ∞-stacks on CC. This is well understood for the case V=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 VV 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 VV-valued presheaves should model, respectively, (n,r)(n,r)-category valued sheaves/stacks and (,1)(\infty,1)-operad valued sheaves/stacks.


The general localization result is apparently due to

which considers the ?ech cover?-localization assuming VV to be monoidal and

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

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