# nLab model structure on sSet-enriched presheaves

Contents

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

#### Enriched category theory

enriched category theory

# Contents

## 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 be $C$ a simplicially enriched category.

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

###### 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

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

## 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.

(…)

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

• 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