The Čech model structure on simplicial presheaves on a site $C$ is a model 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 global model structure on simplicial presheaves on $C$ by left Bousfield localizations at Cech covers: its fibrant objects are ∞-stacks that satisfy descent over Čech covers but not necessarily over hypercovers.
Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.
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.
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
be the corresponding Cech nerve, regarded as a simplicial presheaf on $C$. This comes canonically with a morphism
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.
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 poperties 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
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.
Mike Shulman: Two questions, one (hopefully) easy and one (perhaps) hard:
Is there a Quillen equivalent Čech model structure on simplicial sheaves? Can we just lift the model structure for simplicial presheaves along the sheafification adjunction?
Is there a characterization of the weak equivalences in either Čech model structure?
I am particularly interested in this for the following reason. According to Beke in Sheafifiable homotopy model categories, the weak equivalences in the local model structure on simplicial sheaves are precisely those maps $f\colon X\to Y$ of simplicial objects in the corresponding 1-topos of sheaves of sets such that the statement ”$f$ is a weak equivalence of simplicial sets” is true in the internal logic of the topos (at least, interperiting ”$f$ is a weak equivalence of simplicial sets” by one particular set of geometric sentences whose interpretation in $Set$ is equivalent to saying that a simplicial map is a weak equivalence). But if, as HTT teaches us, Čech descent is often to be preferred to hyperdescent, then we should be interested in Čech weak equivalences instead. So I would really like to know what it means for a map of simplicial sheaves to be a Čech weak equivalence, in the internal logic of the 1-topos of sheaves of sets. If nothing else, I think such a characterization would help me understand the real meaning of hypercompletion. But any sort of characterization of them would be better than none.
Urs Schreiber: below is a reply to the first question.
Mike Shulman: Thanks for attacking this. I thought I should also mention, for anyone listening in, that this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be.
check
We may form the transferred model structure on simplicial sheaves by transferring along the degreewise sheafification adjunction
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, sufficient conditions for this to be a model structure is that
the inclusion $Sh(C) \hookrightarrow PSh(C)$ preserves filtered colimits;
$sSh(C)$ has functorial fibrant replacement and functorial path objects for fibrant objects.
Since sheafification does preserve filtered colimits the first condition is satisfied degreewise and hence is satisfied.
Mike Shulman: I believe that sheafification preserves $\kappa$-filtered colimits for some sufficiently large $\kappa$, but if the site has covers of infinite cardinality, I don’t see why sheafification would preserve $\omega$-filtered colimits. But I think this is enough for the proof to work.
Since the small object argument holds in $sSh(C)$ for generating acyclic cofibrations we have functorial fibrant replacement. And a path object is obtained just by forming objectwise the standard path object in sSet, as in $[C^{op}, sSet]$.
Mike Shulman: The small object argument doesn’t automatically produce functorial fibrant replacements in this context… isn’t the whole question whether the map to the “fibrant replacement” is still a weak equivalence (in the underlying category)? I.e. whether $F(J)$-cell complexes are still weak equivalences.
A detailed though unfinished account of the Čech model structure is given in
But beware of this document is unfinished. Some aspects of this appeared in