nLab model structure on sSet-enriched presheaves

Idea

The model structure $sPSh(C)$ on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site $C$.

It generalizes the model structure on simplicial presheaves which is the special case obtained when $C$ happens to be just an ordinary category.

This means that in as far as the model structure on simplicial presheaves models ∞-stacks, the model structure on sSet-enriched categories model derived stacks.

Definition

The construction of the model structure on sSet-enriched categories closely follows the discussion of the model structure on simplicial presheaves, only that everything now takes place in enriched category theory.

Notation and conventions

Regard the closed monoidal category sSet as a simplicially enriched category in the canonical way.

For $C$ any $sSet$-category, write $sk_1 C$ for its underlying ordinary category.

Write $[C^{op}, sSet]$ for the enriched functor category.

Hence the ordinary category $sk_1 [C^{op}, sSet]$ has as objects enriched functors $C^{op} \to sSet$ and a morphism $f : A \to B$ in $sk_1 [C^{op}, sSet]$ is a natural transformation given by a collection of morphisms $f_c : A(c) \to B(c)$ in sSet, for each object $c \in C$.

Definition

(global model structure)

Let $C$ be a simplicially enriched category.

The global projective model structure $sk_1[C^{op},sSet]_{proj}$ on $sk_1[C^{op}, sSet]$

• fibrations and weak equivalences are those transformations that are objectwise fibrations and weak equivalences in the standard model structure on simplicial sets.

Proposition

The global projective model structure on $sk_1 [C^{op}, sSet]$ makes the $sSet$-category $[C^{op}, sSet]$ a combinatorial simplicial model category.

Definition

For $C$ an sSet-site, the local projective model structure on $sk_1 [C^{op}, sSet]$ is the left Bousfield localization of $sk_1 [C^{op}, sSet]_{proj}$ at…

This appears on (ToënVezzosi, page 14).

Properties

Over an unenriched site

It seems that the claim is that, indeed, in the special case that $C$ happens to be an ordinary category, the model structure on $sPSh(C)_{proj}^{l loc}$ reproduces the projective local model structure on simplicial presheaves.

Presentation of $(\infty,1)$-toposes

Definition

For an sSet-site $C$ regarded as an (∞,1)-site, the local model structure on $[C^{op}, sSet]$ is a presentation of the (∞,1)-category of (∞,1)-sheaves on $C$, in that there is an equivalence of (∞,1)-categories

$$([C^{op}, sSet]_{loc})^\circ \simeq Sh_{(\infty,1)}(C) \,.$$

This is Lurie, prop. 6.5.2.14, remark 6.5.2.15.

Examples

Derived geometry

Where a topos or (∞,1)-topos over an ordinary site encodes higher geometry, over a genuine sSet-site one speaks of derived geometry. An ∞-stack on such a higher site is also called a derived stack.

Therefore the model structure on $sSet$-presheaves serves to model contexts of derived geometry. For instance over the etale (∞,1)-site.

(…)

Related concepts

• model topos

• enriched Reedy category

References

The theory of model structures on $sSet$-enriched presheaf categories was developed in

• Bertrand Toën, Gabriele Vezzosi, Topos Theory (arXiv:math/0207028)

• Bertrand Toën, Gabriele Vezzosi, Segal topoi and stacks over Segal categories Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory , MSRI, Berkeley, January-May

  2002 (arXiv:0212330)

The relation to intrinsically defined (∞,1)-topos theory is around remark 6.5.2.15 of

• Jacob Lurie, Higher Topos Theory