nLab model structure on simplicial sheaves



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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.


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


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

This is (Jardine, theorem 5).

Call this the local injective model structure on simplicial sheaves.



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

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

with LL given by sheafification extends to a Quillen equivalence

(Li):Sh(C) loc Δ opLPSh(C) loc Δ op (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).


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

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

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


For DD a dense sub-site of CC we have an equivalence of (∞,1)-categories

Sh^ (,1)(C)Sh^ (,1)(D). \hat Sh_{(\infty,1)}(C) \simeq \hat Sh_{(\infty,1)}(D) \,.

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

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

This implies the claim with the above proposition.


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.