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
Enriched category theory
The model structure on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site .
It generalizes the model structure on simplicial presheaves which is the special case obtained when happens to be just an ordinary category.
This means that in as far as the model structure on simplicial presheaves models ∞-stacks, the model structure on sSet-enriched categories model derived stacks.
The construction of the model structure on sSet-enriched categories closely follows the discussion of the model structure on simplicial presheaves, only that everything now takes place in enriched category theory.
Hence the ordinary category has as objects enriched functors and a morphism in is a natural transformation given by a collection of morphisms in sSet, for each object .
(global model structure)
Let be a simplicially enriched category.
The global projective model structure on
For an sSet-site, the local projective model structure on is the left Bousfield localization of at
This appears on (ToënVezzosi, page 14).
Over an unenriched site
It seems that the claim is that, indeed, in the special case that happens to be an ordinary category, the model structure on reproduces the projective local model structure on simplicial presheaves.
Presentation of -toposes
For an sSet-site regarded as an (∞,1)-site, the local model structure on is a presentation of the (∞,1)-category of (∞,1)-sheaves on , in that there is an equivalence of (∞,1)-categories
This is Lurie, prop. 22.214.171.124, remark 126.96.36.199.
Where a topos or (∞,1)-topos over an ordinary site encodes higher geometry, over a genuine sSet-site one speaks of derived geometry. An ∞-stack on such a higher site is also called a derived stack.
Therefore the model structure on -presheaves serves to model contexts of derived geometry. For instance over the etale (∞,1)-site.
The theory of model structures on -enriched presheaf categories was developed in
- Bertrand Toën, Gabriele Vezzosi, Segal topoi and stacks over Segal categories Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory , MSRI, Berkeley, January-May 2002 (arXiv:0212330)
The relation to intrinsically defined (∞,1)-topos theory is around remark 188.8.131.52 of