nLab Čech model structure on simplicial sheaves

Idea

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

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

Definitions

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

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

Definition

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

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

Constructions

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

• The inclusion $sSh (C) \hookrightarrow [C^{op}, sSet]$ is fully faithful.
• The left adjoint (i.e. sheafification) $[C^{op}, sSet] \to sSh (C)$ preserves monomorphisms, and the class of monomorphisms in $sSh (C)$ 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 $[C^{op}, sSet]$: 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 $[C^{op}, sSet]$ to $sSh (C)$. By construction, the sheafification adjunction becomes a Quillen equivalence.

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

References

• Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves (arXiv:math/020502).