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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.
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