nLab
sSet-site

Context

(,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

Enriched category theory

Contents

Idea

The notion of sSetsSet-site is the incarnation of the notion of (∞,1)-site when (∞,1)-categories are incarnated as simplicially enriched categories.

Definition

Definition

An sSetsSet-site is a simplicially enriched category CC together with the structure of a site on its homotopy category Ho(C)Ho(C).

This appears as (ToënVezzosi, def. 3.1.1)

Properties

Relation to (,1)(\infty,1)-sites

Proposition

Under the identification of simplicially enriched categories with models for (∞,1)-categories, sSetsSet-sites correspond to (∞,1)-sites.

Because, as discussed at (∞,1)-site, that is equivalently an (∞,1)-category equipped with the structure of a site on its homotopy category of an (∞,1)-category.

Relation to (,1)(\infty,1)-toposes

Proposition

For CC an sSetsSet-site, the local model structure on sSet-presheaves is a presentation of the (∞,1)-topos Sh (C)Sh_\infty(C) over the (∞,1)-site corresponding to CC

([C op,sSet] loc) Sh (C). ([C^{op}, sSet]_{loc})^\circ \simeq Sh_\infty(C) \,.

Examples

References

Revised on March 8, 2014 05:08:25 by Urs Schreiber (82.113.121.169)