Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
A model structure of -sheaves is a model category presentation of the (2,1)-category of (2,1)-sheaves over some site or (2,1)-site.
There are several equivalent ways to set up a model category structure for -sheaves.
Suppose first that the (2,1)-site is just a 1-category, hence just a site.
The following definition first defines a model presentation for (2,1)-presheaves (1-truncated (∞,1)-presheaves) and then localizes at the covering morphisms in order to obtain the -sheaves.
Write Grpd for the category of small category groupoids and functors between them. Write for the natural model structure on groupoids.
Write for the projective model structure on functors on the functor category .
Let be the set of Cech nerve projections in for each covering family in the site .
Then let finally
be the left Bousfield localization at the set of morphisms .
The following definition first gives the presentations for (∞,1)-sheaves and then further restricts the 1-truncated objects in there, preseting the (n,1)-topos inside the full (∞,1)-topos over , for .
Write for a local model structure on simplicial presheaves on , the one which presents the (∞,1)-category of (∞,1)-sheaves on .
Let be the set of generating morphisms of weak equivalences on homotopy 1-types.
for the left Bousfield localization of the model structure for (∞,1)-sheaves at the morphisms . Then this is a model structure for -sheaves on .
These two model structures are equivalent:
be the nerve functor and its left adjoint . Postcomposition with this induces a Quillen adjunction
that is a Quillen equivalence.
This appears as (Hollander, theorem 5.4).
model structure for -sheaves
model structures for -sheaves
A model structure on presheaves of groupoids Quillen equivalent to the left Bousfield localization of the local model structure for (∞,1)-sheaves at morphisms that are weak equivalences of homtopy 1-types is in.
A discussion of -sheaves/stacks as 1-truncated objects in the full model structure for (∞,1)-sheaves is in
- J. F. Jardine, Stacks and the homotopy theory of simplicial sheaves , Homology Homotopy Appl. Volume 3, Number 2 (2001), 361-384. (project euclid)