# nLab model structure on simplicial sheaves

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

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

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

The Joyal- model structure on simplicial sheaves over a site is a model category presentation of the hypercomplete (∞,1)-topos over that site.

It is Quillen equivalent to the Jardine-local model structure on simplicial presheaves.

There is also a Čech model structure on simplicial sheaves (see there) modelling not the hypercompletion but just the topological localization of the (∞,1)-category of (∞,1)-presheaves.

## Definition

Let $C$ be a small site. Write $Sh(C)$ for the category of sheaves on $C$ and $Sh(C)^{\Delta^{op}}$ for the category of simplicial objects in $Sh(C)$: the category of simplicial sheaves over $C$.

###### Theorem

There is a left proper simplicially enriched model category structure on $Sh(C)^{\Delta^{op}}$ such that

This is (Jardine, theorem 5).

Call this the local injective model structure on simplicial sheaves.

## Properties

###### Theorem

The geometric embedding of a category of sheaves into its category of presheaves

$(L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$

with $L$ given by sheafification extends to a Quillen equivalence

$(L \dashv i) : Sh(C)^{\Delta^{op}}_{loc} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)^{\Delta^{op}}_{loc}$

between the above local model structure on simplicial sheaves and the injective hyperlocal Jardine-model structure on simplicial presheaves.

This is (Jardine, theorem 5).

###### Proposition

The simplicial combinatorial model category $Sh(C)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion $\hat Sh_{(\infty,1)}(c)$ of the (∞,1)-category of (∞,1)-sheaves on $C$:

$\hat Sh_{(\infty,1)}(C) \simeq (Sh(C)^{\Delta^{op}}_{loc})^\circ .$
###### Proof

The proof is spelled out at hypercomplete (∞,1)-topos.

###### Corollary

For $D$ a dense sub-site of $C$ we have an equivalence of (∞,1)-categories

$\hat Sh_{(\infty,1)}(C) \simeq \hat Sh_{(\infty,1)}(D) \,.$
###### Proof

By the comparison lemma at dense sub-site we have already an equivalence of categories

$Sh(C) \simeq Sh(D) \,.$

This implies the claim with the above proposition.

## References

The local model structure on simplicial sheaves was proposed in

This is, with BrownAHT, among the first proposals for models for ∞-stacks which eventually came to be used in the theory of (∞,1)-toposes.

Discussion of the model structure:

• Sjoerd Crans, Quillen closed model structure for sheaves, J. Pure Appl. Algebra 101 (1995), 35-57 (pdf)

Jardine’s lectures

discuss the Quillen equivalence between the model structure on simplicial sheaves and the model structure on simplicial presheaves.

Wendt discusses “the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves” in

Last revised on June 19, 2022 at 13:55:02. See the history of this page for a list of all contributions to it.