nLab
sSet-site

Idea

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

Definition

Definition

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

This appears as (ToënVezzosi, def. 3.1.1)

Properties

Relation to $(\mathrm{\infty },1)$-sites

Proposition

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

Proposition

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

$$([C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{loc}}{)}^{\circ }\simeq {\mathrm{Sh}}_{\mathrm{\infty }}(C).$$([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, Homotopical Algebraic Geometry I: Topos theory (arXiv)