nLab
Čech model structure on simplicial sheaves

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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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(,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 Čech model structure on simplicial sheaves on a site CC is a model by simplicial sheaves for the topological localization of an (∞,1)-category of (∞,1)-presheaves on CC to the (∞,1)-category of (∞,1)-sheaves.

It is obtained from the the Čech model structure on simplicial presheaves on CC by transfer along the sheafification adjunction.

Further left Bousfield localization at “internal” weak homotopy equivalences leads from the Čech model structure to the model structure on simplicial sheaves that presents the hypercomplete (∞,1)-topos which is the hypercompletion of that presented by the Čech model structure.

Definition

Let CC be a small site, let sSh(C)sSh (C) be the category of simplicial sheaves on CC, and write [C op,sSet] Cˇ,proj[C^{op}, sSet]_{\check{C},proj} and [C op,sSet] Cˇ,inj[C^{op}, sSet]_{\check{C},inj} for the projective and injective Čech model structure on simplicial presheaves, respectively.

Definitions

The injective Čech model structure on simplicial sheaves on CC is the unique model structure on sSh(C)sSh (C) with the following properties:

  • The weak equivalences are the morphisms in sSh(C)sSh (C) that are weak equivalences in [C op,sSet] Cˇ,inj[C^{op}, sSet]_{\check{C},inj}.
  • The cofibrations are the monomorphisms.
  • The fibrations are the morphisms in sSh(C)sSh (C) that are fibrations in [C op,sSet] Cˇ,inj[C^{op}, sSet]_{\check{C},inj}.
Definition

The projective Čech model structure on simplicial sheaves on CC is the unique model category structure on sSh(C)sSh (C) with the following properties:

  • The weak equivalences are the morphisms in sSh(C)sSh (C) that are weak equivalences in [C op,sSet] Cˇ,proj[C^{op}, sSet]_{\check{C},proj}.
  • The trivial fibrations are the morphisms in sSh(C)sSh (C) that are trivial fibrations in the projective Čech model structure on [C op,sSet] Cˇ,proj[C^{op}, sSet]_{\check{C},proj}, i.e. the componentwise trivial Kan fibrations.

Constructions

To construct the injective Čech model structure on sSh(C)sSh (C), we use the following facts:

  • The inclusion sSh(C)[C op,sSet]sSh (C) \hookrightarrow [C^{op}, sSet] is fully faithful.
  • The left adjoint (i.e. sheafification) [C op,sSet]sSh(C)[C^{op}, sSet] \to sSh (C) preserves monomorphisms, and the class of monomorphisms in sSh(C)sSh (C) is closed under pushouts, transfinite composition, and retracts.
  • The adjunction unit is a natural weak equivalence with respect to the (injective) Čech model structure on [C op,sSet][C^{op}, sSet]: see Theorem A.2 in DHI.

We may then apply Kan’s recognition principle for cofibrantly generated model structures to transfer the injective Čech model structure from [C op,sSet][C^{op}, sSet] to sSh(C)sSh (C). By construction, the sheafification adjunction becomes a Quillen equivalence.

On the other hand, to construct the projective Čech model structure on sSh(C)sSh (C), we use Smith’s recognition principle for combinatorial model structures and build it like a mixed model structure.

References

Last revised on March 6, 2019 at 21:28:35. See the history of this page for a list of all contributions to it.