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model structure on simplicial sheaves

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

• cofibrations are precisely the objectwise cofibrations (monomorphisms) of simplicial sets;

• weak equivalences are the local weak equivalences of the underlying simplicial presheaves as defined at model structure on simplicial presheaves.

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.

Related concepts

• Čech model structure on simplicial sheaves

• model structure on cubical presheaves

References

The local model structure on simplicial sheaves was proposed in

• Andre Joyal, Letter to Alexander Grothendieck, 11.4.1984, (pdf scan).

This is with BrownAHT among the first proposals for models for ∞-stack.

A discussion can be found in

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

Jardine’s lectures

• John Frederick Jardine, Field Lectures: Simplicial presheaves (pdf)

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

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves, arxiv/1009.2930