model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -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 ?ech cover?s 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
SSet equipped with its Joyal model structure, modelling (∞,1)-categories
, the model structure on Theta spaces modelling weak (n,r)-categories;
, the model structure on dendroidal sets modelling (∞,1)-operads.
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
which considers the ?ech cover?-localization assuming to be monoidal and
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.
Last revised on April 30, 2013 at 18:49:36. See the history of this page for a list of all contributions to it.