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the A is implicit. You can read it as being defined as the homotopy fiber of B -> C . The point of it is just to record which cohomology is being twisted.
ok. so it is actually as I was saying:
a way to remember what one would get in case c is the trivial cocycle
by the way, do you think the fibrant replacement suggestion can somehow be formalized into something meaningful, or you feel it is a completely wrong idea?
Sorry, I am not following that fibrant replacement remark. Could you say again in more detail what you have in mind?
One thing to keep in mind is that the idea of a universal case for twisted cohomology doesn't really exist in the usual sense. For example, the K-theory of a space, twisted by a PU-bundle on that space, is not just represented by the classifying space of K-theory, but also involves a classifying map for the PU-bundle. This may (or may not! I'm not sure) be pulled back from a universal such bundle on the classifying space of K-theory, which is pulled back along the canonical map KU -> K(Z,3) = BPU (not sure if the notation KU is correct, but you know what I mean) which is a Postnikov section, and the canonical PU bundle over BPU. Anyway, you probably know all this, it just helps me to get a handle on what you're talking about
well, I was thinking that a way to computing homotopy fibers of a morphism Y --> Z is to replace it with a fibration
Ah, I see. Yes, that's right. This is explained at homotopy pullback: we may compute the homotopy pullback of a diagram by computing the ordinary pullback of a weakly equivalent diagram that satisfies some nice properties. A sufficient property is: all three objects are fibrant and at least one of the morphisms is a fibration. In that case the ordinaty pullback is guaranteed to be weakly equivalent to the homotopy pullback
said in an very extreme way, in the npov, cohomology does not exist.
Well, it is an extreme case of the general principle that the nPOV simplifies concepts. Here it may be surprising just how simple the concept becomes, but it is still of interest to have the simple concept and all its nontrivial specific incarnations.
absolutely! and I love cohomology in all of its specific incarnations, an the unifying npov on them at the same time! what I wanted to stress is how much twisted cohomology seems to fail (at least at first sight) to fit the simple concept. not that twisted cohomology is something difficult, but it is at first sight something really different from other cohomologies. as David says,
One thing to keep in mind is that the idea of a universal case for twisted cohomology doesn’t really exist in the usual sense.
maybe it is just a matter of names.. one day something has been called twisted cohomology since its by hand construction looked like cohomology, but should have the hom-space pov on cohomoloy been more standard that day, maybe a different name would have been chosen, and this thread would not exist :-)
yet, I still try to think for a while to what universal twisted cohomology could be, before giving it up. and, ad David points out, I’ll have to test everything on twisted K-theory first.
how much twisted cohomology seems to fail (at least at first sight) to fit the simple concept.
well, it's more complicated, but still surprisingly simple, to my mind: we identify cohomology with hom-complexes. Then we are naturally interested in understanding morphisms of hom-complexes. These in turn are to a large degree characterized by their homotopy fibers. Which is the home of twisted cohomology. That is pretty nice, i think.
Is it worth saying that there may be sometimes a need to calculate some cohomology 'class'. The idea that cohomology (in the classical topological case) is the spectrum that represents it is not new (in that case) but what does it mean to calculate the cohomology of a 'space'? Surely the npov sheds light on the problem but pushes the work to something like the job of calculating homotopy classes of maps, and that is highly non-trivial. Similarly saying an oo-groupoid is a Kan complex is enlightening and at the same time, means you still have to work out how to 'calculate' with it. Perhaps extracting information from things like this is the next set of challenges for the npov!
Perhaps extracting information from things like this is the next set of challenges for the npov!
I see on the one hand the nPOV that tells you what to do, and on the other hand the concrete work you have to do it. The nPOV is not meant to make all computation superfluous. It is meant to organize it neatly. It provides a guide that ensures that if one has to compute something, at least one is sure about the nature of what it is one is computing. You still have to compute it, though. That's how I see it.
Then we are naturally interested in understanding morphisms of hom-complexes.
Exactly! one wants to study the morphism A –> B in H, and this is the same as studying the morphism H(-,A) –> H(-,B), and a lot of information can be read by studying the (homotopy) fibers of H(X,A) –> H(X,B) as X varies. I’m just saying that a ’single’ homotopy fiber is not interesting, is the ’collection’ of all homotopy fibres that matters.
when B is pointed, the morphism A –> B picks out a distinguished fibration sequence and we have (non-twisted) cohomology with coefficients in the (homotopy) fiber. but if we forget about the distinguished point in B, then what we get looks rather like a ’bundle’ of cohomologies over B: for every X-point of B, i.e., every cocylce c in H(X,B) there is the homotopy fiber of A –> B over that point. a couple of posts ago I was suggesting to think of this homotopy fiber as the ordinary fiber of a fibrant replacement, but now I see that the only fiber ’meaningful within (oo,1)-topos description of cohomology’ would be a homotopy fiber, so let us directly say fiber: the ’collection’ of all twisted cohomologies of X (with coefficients in the fiber of A –> B over the distinguished point of B), is the collection of all fibers of the morphism A –> B, i.e., it is the morphism A –> B itself.
in other words, in the npov, twisted cohomology does not exist. :-P
just joking here. but seeing things this way I now think the identity B –> B seen as a twisting cocycle for the twisted cohomology of B could have some relevance to the above.
Ah, now I see what you mean. Yes, I guess I very much agree with this.
So the total twisted cohomology is the homotopy pullback of the morphism that picks for each cohomology class in a (any) cocycle representative. The homotopy pullback of that along is the total twisted cohomology specified by the morphism . It comes with two projections: one to the underlying -cocycles, one to the twisting classes in -cohomology.
isn't this the graph of f?
I see what you mean. it is something like that, yes. Maybe closer to the graph of for fixed X.
I see what you are getting at, but the homotopy pullback of an identity is trivial: an equivalence.
In this business does play a role more than just being a spectator. It determines which twisting cocycles are available. The notion of "twisting cocycle" does not exist without having a domain.
But let me think what we can do to get what you want to get...
So suppose we could choose representatives of cocycles functorially as changes, so that we would get a transformation
where . If we could choose this, then I suppose we would take the homotopy pullback in the oo-category of oo-presheaves on , yes, of the diagram
which of course you are entitled to write simply as
Now I am not sure: do we expect to be able to choose cocycle representatives functorially? Hm...
Yeah, but the pullback of the identity will be an identity/equivalence. So that does not capture the information that I suppose you would want to capture.
Okay, sure.
Okay, right, "the fiber of the collection of -points of over a given -point of .
But maybe then actually proper topos-language is preferable: at stage , twisted cohomology is the (homotopy) fiber of over a given element of .
Yes.
Right, so who is going to put this into the entry now? :-)
I have to run and catch a bus, but maybe I can do it later...
a major clean up.
That might indeed be a good idea. Thanks!
Thanks, that looks good.
One remark I would have is that the "motivating example" at the beginning is not really an immediate motivation for the definition that follows, as it requires the argument following that in turn to see how it relates. Maybe it would be better to move that alltogether into the examples section.
Yes. Please feel free to replace the nLab entry with what you have. I have no time to work on this at the moment, but I do like what you did.
Thanks!
Now I did spent some time working more on twisted cohomology after all:
I edited the beginning, up to where the Examples-section begins:
added a sentence to the Idea-section
made the statements that follow the definition a series of formal propositions with formal proofs
polished the notation a bit here and there, for instance in changing to (for it not to collide with the -notation).
By the way, one reason (of several) why I have little right now is that I need to bring some stuff curently on my personal web into shape. Currently I am busy polishing and expanding the entry path oo-groupoid.
This has developed quite a bit, I think. I'd be interested in hearing your comments on this.
just let me know your favourite way of receiving comments: query boxes on the page? forum posts?
forum posts would be nice.
query boxes are fine, too, but I will tend to react to them quickly and then remove them. So maybe for a more coherent discussion forum posts would be better. Thanks.
I agree, twisted K-theory could be presented as the first example in the example section.
My faint memory says that the tensor product issues, hence product for classes, is more delicate in twisted K-theory than in the usual. Is this easy in this approach ?
similarly, whitehead towers of are obtained as fibration sequences
yes, very nice. I had made a remark exactly along these lines here, recently.
This needs to be polished and moved to Whitehead tower
it seems to me a beautyful example of how the nLab discussions are creating a common point of view on the subject. I really like this.
Yes, I am very much enjoying this, too. Glad to have you around here.
Added an Examples-section
and a brief followup-section
meant to put the construction by Ando-Blumberg-Gepner into perspective.
added to the section In terms of sections at twisted cohomology a pointer to the fully detailed proof of the statement stated there.
Inspired by discussion with Marc Hoyois in another thread I finally went and started trying to beautify the entry twisted cohomology a little.
So far I have
given it a better and considerably expanded Idea-section
given it a better and considerably expanded Definition-section
Especially the Definition section certainly could still do with more work, but for the moment I am out of steam.
Also the Examples-section should be cleaned up and more examples should be added. Maybe later. (Or maybe Marc feels energetic to add more! :-)
Added a section on effective computation of twisted cohomology using de Rham cohomology of the universal cover and the action of the fundamental group thereon:
The idea was inspired by the following MO answer by Peter Michor.
Would anyone know what to do if the fundamental group is not finite?
I see there’s a new article in this area:
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