nLab sSet-site

Idea

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

Definition

Definition

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

This appears as (Toën & Vezzosi 005, def. 3.1.1)

Properties

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

Proposition

Under the identification of simplicially enriched categories with models for (∞,1)-categories, $sSet$-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 $(\infty,1)$-toposes

Proposition

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

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

Examples

• simplicial Stein site

Related concepts

• site

• 2-site

• (∞,1)-site

  • model site, simplicial site

  • model topos

References

• Bertrand Toën, Gabriele Vezzosi, p. 4 and Def. 3.1.1 of: Homotopical Algebraic Geometry I: Topos theory, Advances in Mathematics 193 2 (2005) 257-372 [arXiv:0207028]