nLab (infinity,1)-presheaf

Definition

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

$$F : S^{op} \to (\infty,0)Cat \,.$$

The (∞,1)-category of $(\infty,1)$-presheaves is the corresponding (∞,1)-category of (∞,1)-functors

$$PSh(S) := Fun(S^{op}, (\infty,0)Cat) \,.$$

Remarks

• $(\infty,1)$-presheaves are the basic building block for the definition of (∞,1)-categories of (∞,1)-sheaves.

Model structures

$(\infty,1)$-presheaves can be presented by many different model categories, corresponding to several of the model structures for (∞,1)-categories. These include:

• global model structures on simplicial presheaves (injective and projective)
• model structure for right fibrations of quasicategories
• model structure on internal simplicial presheaves?
• model structure for right fibrations of Segal spaces?
• model structure on simplicial presheaves over simplices?

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:

• The Reedy model structure on simplicial presheaves, when $S$ is a Reedy category, or more generally a generalized Reedy category
• The analogous model structure on internal inverse diagrams?, when $S$ is an inverse EI (∞,1)-category.

Related concepts

• (0,1)-presheaf

• presheaf

• (2,1)-presheaf

• $(\infty,1)$-presheaf, (∞,1)-sheaf, ∞-stack ,

  • simplicial presheaf

  • model structure on functors

• (∞,n)-presheaf

References

$(\infty,1)$-categorical definition

This is in Section 5.1 of

• Jacob Lurie, Higher Topos Theory .

Model structures

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

• Geoffroy Horel, A model structure on internal categories in simplicial sets, TAC

The model structure for right fibrations of Segal spaces? is constructed in

• Pedro Boavida de Brito, Segal objects and the Grothendieck construction, arXiv

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

• Mike Shulman, Univalence for inverse EI diagrams, arXiv.