nLab
(infinity,1)-presheaf

Context

(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Write (,0)Cat(\infty,0)Cat for the category ∞Grpd of \infty-groupoids regarded as an (∞,1)-category.

Let SS be a simplicial set (which in particular may be a quasi-category).

An (,1)(\infty,1)-presheaf on SS is an (∞,1)-functor

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

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

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

Remarks

Model structures

(,1)(\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 SS there exist other model structures that are also Quillen equivalent to these, such as:

References

(,1)(\infty,1)-categorical definition

This is in Section 5.1 of

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

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

  • 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

Revised on May 12, 2016 04:12:01 by Urs Schreiber (131.220.184.222)