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Urs created submersion and I added a little more. Still a bit stubby, though.
added links (we have transversal maps, for instance).
We should say something about regular epimorphisms.
Edited transversal maps to change the direct sum sign to just a sum in the definition - as it should be.
which lead to […] which lead to […] which lead to […]
Thanks!
I added the POV to hypercohomology,
The very existence of the notion “hypercohomology” is a strong point in favor of the nPOV on cohomology.
Really cool stuff!
Yes. And in principle known since before 1973. A mystery why this isn’t taught in high school in the 21 century…
I added the example of Deligne cohomology to hypercohomology.
I like the discussion in Higher Topos Theory . But, as we have seen elsewhere, I am strange and different… ;-)
More seriously, I am not aware of many references that would describe Eilenberg-macLane objects in a higher sheaf topos really conceptually. Often it is just by decree that one takes these to be things represented by chain complexes concentrated in one degree. That’s fine of course, because that’s what it boils down to anyway. But a good conceptual description is in HTT.
I added an arrow-theoretic characterisation of transversal maps.
I put in a coordinate-dependent definition at submersion on the grounds that it applies in a more general situation.
Re #12: I’m a little baffled by this “slick category-theoretic” way to express transversality. As pointed out in a comment by Tom Goodwillie at MO (link visible for those with 10k of rep), if we have , then the pullback of and surely exists (the pullback of a mono against itself trivially exists in any category, taking the pullback projections to be identity maps). Moreover, the tangent bundle functor preserves the terminal object , so takes and to a map (necessarily monic, being a map out of the terminal) , and therefore the pullback diagram, pulling back along , is preserved by the tangent bundle functor.
Why does one need rep to see this comment? I can read the question, 6 comments on it, and an answer, nothing by Goodwillie. Are there hidden comments???
I'm trying to remember how that condition came about; I think that it's missing a clause.
@Toby: Sorry – the explanation is that the answer that I linked to, and which has Goodwillie’s comment, had been deleted; such answers on MO cannot be seen except by those with 10k of rep. (To such users the deleted answer appears in a pink box.) I just tried the link and was sent directly to the deleted answer; I guess other users are sent to the page anyway, but just can’t see the deleted answer I really meant to link to.
Yeah, that's what I see.
The answer was by me, and Goodwillie pointed out the obvious counterexample to the claim in my answer (which I think I got from the nLab page). My answer is:
Edit: This is wrong as it stands, I’m leaving it here in case someone (perhaps me) can fill in the missing detail.
Here’s a sketch.
Fix the map . The pullback of along exists if and only if is transversal to , namely given and such that , we have
(not direct sum, notice). (see, e.g. the nLab article ) (EDIT: is transversal to iff the pullback exists and is preserved by the tangent bundle functor!)
Thus for all pullbacks of to exist along arbitrary maps, including from , we must have , for all , i.e. is a submersion.
OK, so David thought that iff the pullback exists, I added the condition that the tangent bundle preserves it, and now we see that there is something more! I'm pretty sure that I teased this result out of Lang, so I should go look there again. In the meantime, there is a warning that the result is incomplete.
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