equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Write $(\infty,0)Cat$ for the category ∞Grpd of $\infty$-groupoids regarded as an (∞,1)-category.
Let $S$ be a simplicial set (which in particular may be a quasi-category).
An $(\infty,1)$-presheaf on $S$ is an (∞,1)-functor
The (∞,1)-category of $(\infty,1)$-presheaves is the corresponding (∞,1)-category of (∞,1)-functors
$(\infty,1)$-presheaves can be presented by many different model categories, corresponding to several of the model structures for (∞,1)-categories. These include:
Various Quillen equivalences between these model structures are constructed in the references. For special cases of the domain $S$ there exist other model structures that are also Quillen equivalent to these, such as:
$(\infty,1)$-presheaf, (∞,1)-sheaf, ∞-stack ,
This is in Section 5.1 of
The various model structures, and their Quillen equivalences, can be found in the following references.
The global model structures on simplicial presheaves are a standard special case of the global model structures on functors. The fact that the injective model structure exists is a bit less classical; see injective model structure for references.
The model structure for right fibrations of quasicategories is constructed in Higher Topos Theory. It is shown there to be Quillen equivalent to the global model structure on simplicial presheaves by a straightening functor. Alternative proofs of such an equivalence can be found in
Gijs Heuts, Ieke Moerdijk, Left fibrations and homotopy colimits II, arXiv
Danny Stevenson, Covariant Model Structures and Simplicial Localization, arXiv
The latter also constructs the model structure on simplicial presheaves over simplices? and links it with Quillen equivalences to the other two.
The model structure on internal simplicial presheaves? is constructed in
The model structure for right fibrations of Segal spaces? is constructed in
and shown to be Quillen equivalent to both the model structure on internal simplicial presheaves? and the model structure for right fibrations of quasicategories.
Finally, the model structure on internal inverse diagrams? is constructed, and shown to be Quillen equivalent to the model structure for internal simplicial presheaves (hence all the others) in