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There seem to be some misleading remarks at Čech model structure on simplicial presheaves.
Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.
Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.
[…] 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 Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.
this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be
Based on the discussion here, it seems that the Čech model structure is not site-independent, even though it can be defined on the category of simplicial sheaves. A very strange state of affairs…
Thanks for catching that bit about Cech cohomology. I thought I had fixed that long ago, but apparently I didn’t in this entry. I have just removed that remark now.
Yes, the full story is that for Cech cohomology to compute the correct hom-spaces (in addition to the site being “small enough” that the coefficient complex of sheaves satisfies proper descent) that the Cech nerve needs to be cofibrant in the local projective Cech model structure. This is a generalized version of the familiar condition that the cover needs to be a “good cover” for Cech cohomology to compute sheaf cohomology.
Removed a query:
+–{: .query} 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:X→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
Removed a query:
+–{: .query} Mike Shulman: I believe that sheafification preserves κ-filtered colimits for some sufficiently large κ, but if the site has covers of infinite cardinality, I don’t see why sheafification would preserve ω-filtered colimits. But I think this is enough for the proof to work. =–
Removed a query:
+–{: .query} 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. =–
Wrote up a new proof of the existence of the transferred structure:
As discussed there, a necessary and sufficient condition for this to be a model structure is that
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.
And here are the leftovers from the previous proof:
the inclusion Sh(C)↪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.
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 [Cop,sSet].
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