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model structure on simplicial sheaves

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

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

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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.

Definition

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.

Theorem

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.

Properties

Theorem

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).

Proposition

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 .
Proof

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

Corollary

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) \,.
Proof

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.

References

The local model structure on simplicial sheaves was proposed in

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 (web)

Jardine’s lectures

  • J. 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

Revised on January 20, 2011 00:16:55 by Urs Schreiber (89.204.153.103)