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created stub for differential fivebrane structure
sounds easy, but due to lots of software trouble that took me a good bit of the afternoon! :-(
Hi Urs,
I guess in the last commutative diagram at the very end of differential fivebrane structures the upper left corner should be $Fivebrane$ and not $String$, right?
Also, you know (since we have already discussed this about twisted cohomology) that I don’t like the rightmost “picking a representative” map very much. I’ll try thinking about a reformulation which avoids this.
right?
yes, thanks, fixed.
Also, you know (since we have already discussed this about twisted cohomology) that I don’t like the rightmost “picking a representative” map very much. I’ll try thinking about a reformulation which avoids this.
If you can see another way to do this, that would be interesting. To me it seems the question would be to see to which extent the choice of points in connected components can be made $\infty$-functorially.
it seems to me we are defining the 6-groupoid of twisted differential fivebrane sructures as the set of all homotopy fibers of $\mathbf{H}_{conn}(X,\mathbf{B}String)\to \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$. This means that for each point in the target we consider the homotopy fiber of this morphism instead of the “actual” fiber. but the homotopy fiber is the “actual fiber” of a fibrant replacement, so the collection of all homotopy fibers is the collection of all fibers of a fibrant replacement. and this leads me to thinking that $Fivebrane_{diff,tw}(X)$ could be more intrinsically defined as a fibrant replacement of $\frac{1}{6}\hat{\mathbf{p}}_2:\mathbf{H}_{conn}(X,\mathbf{B}String)\to \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$.
I’ve looked at the twisted cohomology entry. In the idea section it is written
Given a morphism $f : \hat B \to B$ in an (∞,1)-topos $\mathbf{H}$, regarded as exhibiting a characteristic class on $\hat B$ with values in $B$, the $f$-twisted cohomology at stage $X$ is the fiber of $f$ over a given element of $B$.This is such that if $f : \hat B \to *$ is the terminal morphism, $f$-twisted cohomology is precisely the ordinary cohomology in $\mathbf{H}$ with coefficients in $\hat B$.
I’d like to replace that with
Given a morphism $f : \hat B \to B$ in an (∞,1)-topos $\mathbf{H}$, regarded as exhibiting a characteristic class on $\hat B$ with values in $B$, the coefficient object of $f$-twisted cohomology is a (any) fibrant replacement of $f : \hat B \to B$. This is such that if $\hat{B}$ is fibrant and $f : \hat B \to *$ is the terminal morphism, $f$-twisted cohomology is precisely the ordinary cohomology in $\mathbf{H}$ with coefficients in $\hat B$.
what do you think?
Hi Domenico,
I am not sure in which sense you think of anything using the term “fibrant replacement” as being more intrinsic. The notion of fibrant replacement is kind of by definition not intrinsic to $\infty$-category theory, but a hallmark of working in a model. And it depends on the model.
Particularly from the point of view of the $\infty$-category the fibrant replacemnt of any morphism is equivalent to that morphism, hence it seems to me there is no gain in defining twisted cohomology as the fibrant replacement of something.
What does have intrinsic meaning is the notion of homotopy fiber, fibration sequence and so on. For $A \to B$ a morphism in an $\infty$-category we can ask if it is the homotopy fiber of some morphism $B \to C$. And in applications this condition comes close to requiring $A \to B$ to be a “fibration” in the intuitive sense. For instance if $A \to B$ is a principal $\infty$-bundle, then it is also a homotopy fiber. All up to equivalence, of course.
I know that the idea that twisted cohomology and hence differential cohomology is defined in terms of homotopy fibers that are written down by choosing basepoints in each connected component tends to make people feel uneasy. But I am not sure that there is anything to be worried about. For homotopy fibers over any one base point I never saw any one feeling uneasy about the fact that this base point is only one of many in its connected component. Here for twisted cohomology we are simply taking homotopy fibers over all possible connected components, so we choose a point in each connected component. But that choice of point is irrelevant: the corresponding homotopy fibers do not depend on it, up to equivalence.
Maybe I am wrong, but from this point of view the definition appears entirely general: we have a morphism $A \to B$ and want to record all its homotopy fibers. Up to equivalence, there is one per connected component of $B$. So we choose one basepoint in each connected component and compute the homotopy fiber over that. Nothing depends on the choices made.
Moreover, nobody ever show himself or herself worried about this if we do this over just a single connected component and regard $B$ as a pointed object. All that is happening here is that we take the liberty of regarding $B$ as a pointed object in all possible ways. That should not be any more worriesome. In fact, that means that we are making fewer choices than if we equipped $B$ with a fixed structure of a pointed object.
For all this it seems to me to be a very natural thing to consider homotopy pullbacks of sections $\tau_{\leq 0} A \to A$ to the projections $\pi_0 : A \to \tau_{\leq 0} A$. But of course this feeling of mine is not a proof that there is not a better definition avaialble for the construction considered at twisted cohomology.
I see I was confused about this.. :(
yet, I still think we should avoid this choosing of points and think in more general terms: what I’m trying to say (but still in a very imprecise way) is that the twisted cohomology of $f:\hat{B}\to B$ should be the “homotopy fibre functor” defined by this morphism, rather than its value on a particular morphism (e.g., on a section $\tau_{\leq 0} B\to B$).
edit: and indeed we already have this point of view at twisted cohomology: it is where it is described in terms of $\mathbf{H}_{/B}$!
edit: so in the end I’d say just this: twisted cohomology of $f:\hat{B}\to B$ is the cohomology with coefficient object $f$ in the oo-over category $\mathbf{H}_{/B}$
also, the $\mathbf{H}_{/B}$ point of view seems to simplify the discussion at “Examples. Sections as twisted functions…” in twisted cohomology. Namely the relevany morphism there is the representation $\rho:\mathbf{B}G\to Vect$, and one is considering twisted cohomology $\mathbf{H}_{/Vect}(-,\rho)$; in particular the twisted cohomology of $const_k$ is the space of sections as discussed there.
Hi Domenico,
you are right, the over-category definition of twisted cohomology that we once noted there is nice. (I went now to the entry and polished it slightly).
In terms of that, the statement that we were talking about here amounts to saying that there are many possible lifts of $X \in \mathbf{H}$ to an object in $\mathbf{H}_{B}$ and that $\mathbf{H}_{/B}(X \to B, f)$ depends only on the connected component of these lifts.
Does this give a way to circumvent the issue that you were trying to avoid?
Hi Urs,
I see the difference between our points of view boild down to the following: I’m insisting that twisted cohomology is the cohomology of a cocycle $c:X\to B$, i.e., is $\mathbf{H}_{/B}(c,f)$, whereas you are interested in twisted cohomology of the object $X$ as the collection of all cohomologies $\mathbf{H}_{/B}(c,f)$ with $c\in\mathbf{H}(X,B)$. And from your point of view no doubt that seeing that $\mathbf{H}_{/B}(c,f)$ only depends on $[c]$ in $H(X,B)$ is a valuable point, reducing the quite badly defined “collection of all cohomologies” to the pullback along a section of $\mathbf{H}(X,B)\to H(X,B)$.
Yet, if we stress the $\mathbf{H}_{/B}(-,f)$ point of view, this “collection of all cohomologies” could be seen as the graph of a functor defined on $\mathbf{H}(X,B)$. This comes equipped with a natural projection to $\mathbf{H}(X,B)$ and seems to me a more intrinsic construction than picking representatives.
Domenico,
you are right, you have a very good point. What you emphasize indeed looks like a point of view that is better in several respects. I need to think about this a bit.
One important aspect about the definition of twisted cohomology in the form I used to use it is that it gives the correct long fiber sequence. (The one that I call the differential fiber sequence here in the context of differential cohomology). I should maybe think about how to express that in terms of $\mathbf{H}_{/B}(-,f)$.
Domenico,
another aspect that I believe you had been very right about and which I still need to sort out for myself is the intrinsic definition of $\mathbf{B}G_{conn}$. It fits into
$\array{ \mathbf{H}(X, \mathbf{B}G_{conn}) &\to& \prod_i \mathbf{H}(X, \mathbf{B}^{n_i} U(1)_{diff}) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\to & \prod_i\mathbf{H}(X, \mathbf{B}^{n_i} U(1)) }$and you suggested that it should be characterized by being the universal object with that property. That certainly sounds plausible. My problem is that I am having trouble computing the $\infty$-pullback of this diagram because our standard model of the bottom morphism by $\mathbf{B}G_{diff} \to \mathbf{B}^n U(1)_{diff}$ is not a fibraiton. So the $\infty$-pullback is not manifestly just the ordinary pullback. But it ought to be for $\mathbf{B}G_{conn}$ to come out right.
At differential string structure is described how to resolve the bottom morphism to a fibration in the case that we have just a single curvature class, by using the model from SSSIII. That works fine for constructing differential string structures. But I am not sure yet if or how this can be adapted to get the universal property of $\mathbf{B}G_{conn}$.
I have reworked the idea-section and definition section at differential fivebrane structure slightly, such as to make full use of the material that is now at smooth infinity-groupoid. In fact, I have verbatim copied over the stuff from differential string structure and just made adjustments in the degrees and prefactors.
Also, the writeup of statement and proof of the factorization of the differential Pontryagin class into a weak equivalence followed by a fibration that I now have at “differential string structure” goes through exactly verbatim now for the fivebrane structures. But I haven’t copied that over yet.
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