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at hypercover I have slightly expanded the remark about the examples: I have created a subsection “Examples” and described what degreewise the conditions on a hypercover $Y \to X$ are in the case that $X$ is simplicially constant.
added to hypercover a brief section with the main statement from Dugger-Hollander-Isaksen on hypercovers over “Verdier sites” here.
Something really basic: given a coverage $C$ with associated Grothendieck topology $T$, every $T$-hypercover should have a refinement by a $C$-hypercover – right?
Hmm, I can think of how to build one inductively and then this should work for truncated hypercovers, but I’m not sure about non-truncated ones. (Further answers will have to wait till tomorrow)
Thanks, good point. Hm.
According to the cited definition of hypercover, only the Grothendieck topology matters, so the two notions should be the same. No? (The point is that the notion of local epimorphism depends only on the Grothendieck topology.)
I should clarify, I was using the term in the sense of Dugger-Hollander-Isaksen which incorporates part of “split hypercover”. David seemed to have guessed correctly what I mean, but to clarify:
given a hypercover which is degreewise a coproduct of domains of morphisms in covering families of the Grothendieck topology $T$ coming from a coverage $C$, may we refine by a hypercover which is degreewise a coproduct of domains of morphisms in covering families of $C$?
It should be clear that one may just iteratively choose $C$-covers. But as David says, maybe there is a subtlety in “taking the limit”?
I’m afraid that’s not any clearer. Every representable occurs as a domain of a morphism in a covering family, namely the trivial one. Moreover, the DHI definition of “(split) hypercover” depends only on the Grothendieck topology; and cofibrancy in the projective model structure does not depend on the choice of coverage at all, so the cofibrant hypercovers remain the same, if that is what you are interested in.
Not for a coverage.
As an example: consider the category of complex manifolds with coverage-ing families the covers by polydiscs. Any complex manifold may be covered by just these. Now given any hypercover which is degreewise a coproduct of complex manifolds (or of Stein spaces, if you wish), may one refine by a hypercover which is degreewise a coproduct of just polydiscs?
The analogous statement in (DHI02) (which assumes existence of more pullbacks in the site than I want to assume, I don’t want to assume a Verdier site here) is theorem 8.6. That sets up the expected induction and then the proof ends.
Hmmm. In some sense, what you are doing there is shrinking the underlying category of the site. Regardless, if that’s all you are interested in, then there is no problem: every hypercover can be refined in the way you want. In the usual proof (e.g. Lemma 8.2.20 in my notes) that every local trivial fibration (of a representable) can be refined by a hypercover, in the step where we form a coproduct of representables, replace each representable with a coproduct of distinguished ones.
In some sense, what you are doing there is shrinking the underlying category of the site.
That’s exactly what I am after, yes. The $\infty$-version of “dense subsite”.
When dealing with cohesion it turned out to be a nice trick to shrink the category of manifolds to the category of Cartesian spaces, i.e. to discs. (Since in that case one has good open covers, I could avoid working with (and, worse, thinking about) hypercovers.) Now I want to shrink a category of analytic manifolds to that of just polydiscs.
Thanks a lot for the pointer to your notes! I’ll have a look now.
Okay, I have used this to jot down the intended proof that $\mathbb{C}Anlytic\infty Grpd$ is cohesive, here.
This needs more polishing, but I have to dash off now.
Any interesting spectra in $\mathbb{C}Anlytic\infty Grpd$? What’s the equivalent of ’differential’ as in differential cohomology/refinement? Analytic?
The holomorphic de Rham complex and Deligne complex are objects in CplxAnalyticInftyGrpd, representing what would alternatively be called Dolbeault cohomology, holomorphic differential cohomology or the like. (One should recall at this point that this holomorphic version was what Deligne and Beilinson originally introduced in the early 1970s, long before Brylinski and others popularized the smooth differential geometric version.)
Crucially, the discussion at intermediate Jacobian lives here, that’s what I was after now.
(Telegraphic since i am on my phone, more later.)
Zhen Lin,
your lemma 8.2.20 proceeds by induction, as does theorem 8.6 in DHI, as mentioned above. This clearly gives the answer for n-truncated hypercovers for any finite $n$. Does one not need to add some argument that the construction also works for the untruncated case? This is what David alluded to in #4.
I don’t see any problem. See, for instance, Propositions 5.4 and 6.4 in [DHI, 2004].
Okay, I am seeing there… let’s agree on a version, it seems the numbering changed. How about we look at math/0205027v2. Do you mean to point to 5.4 and 6.4 there?
The induction causes no trouble if it doesn’t touch the earlier stages of the construction, which does happen when trying to refine a hypercover by the nerve of an ordinary cover, but I suspect not in this case. In fact that’s the point of a hypercover, you just use what you have in lower degrees and fix up where you are up to.
@Urs
I was referring to the published version. In the arXiv v2 version, it’s Propositions 5.5 and 6.6.
Thanks, right. I was being stupid here. Sorry.
Once I am no longer just on my phone, I should next look more into if this way we indeed get a split hypercover…
Ah, of course splitness is automatic by the construction.
I have added the statement to the entry, see at hypercover – Properties – Existence and refinement – Over general sites.
Please check and please feel invited (I’ll keep reiterating that) to expand.
Please bear in mind that I update my notes from time to time, and sometimes the numbering gets changed; but I keep all versions available online, so as long as you include the date/version, there should be no confusion.
My theorem 8.2.14 does not quite say that local trivial Kan fibrations have the local lifting property; rather it corresponds to Theorem 6.15 in [DI, 2002]. Perhaps what you want is Theorem 7.2 in [DI, 2002]?
Okay, I have added the date to all the pointers.
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