nLab Čech model structure on simplicial presheaves

Idea

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

It is obtained from the global model structure on simplicial presheaves on $C$ by the left Bousfield localization at Čech covers: its fibrant objects are ∞-stacks that satisfy descent over Čech covers but not necessarily over hypercovers.

Further left Bousfield localization at hypercovers leads from the Čech model structure to the Joyal-Jardine local model structure on simplicial presheaves 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 and write $[C^{op}, sSet]_{proj}$ and $[C^{op}, sSet]_{inj}$ for the projective and injective global model structure on simplicial presheaves, respectively.

For $\{U_i \to V\}_i$ a covering family in the site $C$, let

$$C(\{U_i\}) := \left( \cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i \right)$$

be the corresponding Cech nerve, regarded as a simplicial presheaf on $C$. This comes canonically with a morphism

$$C(\{U_i\}) \to V$$

of simplicial presheaves, the corresponding Čech cover morphism .

Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.

Definition

The injective (projective) Čech model structure on simplicial presheaves $[C^{op},sSet]_{Cech}$ on $C$ is the left Bousfield localization of $[C^{op}, sSet]_{inj}$ ($[C^{op}, sSet]_{proj}$) at the set of Čech cover morphisms.

By the general properties of Bousfield localization this means that the fibrant-cofibrant objects $A$ of $[C^{op},sSet]_{Cech}$ are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the descent condition that for all covers $\{U_i \to V\}$ the morphism of simplicial sets

$$A(V) \cong [C^op,sSet](V,A) \to [C^{op},sSet](C(\{U_i\}), A)$$

is a weak equivalence in the standard model structure on simplicial sets.

This is the model for the $\infty$-analog of the sheaf condition, modelling the topological localization of an $(\infty,1)$-presheaf $(\infty,1)$-topos.

Model structures on simplicial sheaves

We may form the transferred model structure on simplicial sheaves by transferring along the degreewise sheafification adjunction

$$Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}} PSh(C) \,.$$

This defines fibrations and weak equivalences in $sSh(C)$ to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in $sPSh(C)_{Cech} = [C^{op},sSet]_{Cech}$.

As discussed there, a necessary and sufficient condition for this to be a model structure is that

• transfinite compositions of cobase changes of generating acyclic cofibrations are weak equivalences in $Sh(C)$.

Here the generating (acyclic) cofibrations in $Sh(C)$ are obtained by applying the associated sheaf functor to generating (acyclic) cofibrations in $PSh(C)$.

In the category $Sh(C)$, colimits like transfinite compositions and cobase changes are computed by applying the associated sheaf functor to the corresponding colimit in $PSh(C)$.

The latter colimit in $PSh(C)$ does yield a weak equivalence in $PSh(C)$ because $PSh(C)$ admits a model structure. By the 2-out-of-3 property, applying the associated sheaf functor yields a weak equivalence again.

References

A detailed though unfinished account of the Čech model structure is given in

• Daniel Dugger, Sheaves and homotopy theory (web, pdf)

But beware of this document is unfinished. Some aspects of this appeared in

• Daniel Dugger, Universal homotopy theories