November 26, 2008

Bär on Fiber Integration in Differential Cohomology

Posted by Urs Schreiber

After the workshop in Göttingen yesterday I visited Zoran Škoda at the MPI in Bonn, where we worked on differential nonabelian cohomology. In the evening there was a talk by Christian Bär over at the university:

Christian Bär
Fiber integration and Cheeger-Simons characters
(my pdf notes taken during the talk)

As far as I am aware Christian Bär and his collaborators had been trying to understand fiber integration in Deligne cohomology (which is the diffential (= “with connection”) version of $U(1)$ Čech cohomology, one model for abelian differential cohomology) for a while, in particular in order to understand the transgression map which sends a degree $n$-structure – such as a gerbe – on a space $X$ to a degree $(n-k)$-structure – such as a line bundle – on $k$-fold loop space $L^k X$. Doing this with concrete Deligne cocycles leads to rather intricate combinatorics. To circumvent this Christian Bär in joint work with Christian Becker now passed from Deligne cohomology to the equivalent Cheeger-Simons differential characters.

A Cheeger-Simons differential $(n+1)$ character $(h,\omega)$ is defined almost like a parallel transport $n$-functor # $h$ with values in $\mathbf{B}^n U(1)$ (only that the $n$-categorical aspects are not made explicit, which works due to the abelianness of $U(1)$): it is an assignment of elements in $U(1)$ to smooth singular $n$-cycles satisfying a certain smoothness condition controlled by demanding that there is a smooth curvature $(n+1)$-form $\omega$ of this parallel transport.

There is an obvious naïve guess for what the result of fiber integrating such a differential character should be: evaluated on a cycle in the base of a fibration it should yield the original character evaluated on the preimage of this cycle in the fibers. The problem with this naïve idea is that this preimage of a smooth cycle is of course not generally a well behaved smooth cycle itself.

The solution to this problem that Christian Bär indiated in his talk is that one generalizes the domain of Cheeger-Simons differential characters to something more general than smooth cycles, i.e. smooth images of smooth manifolds: he introduces a generalization called geometric chains. One example of a model for these is, he says, obtained by using M. Kreck’s stratifolds as a generalization of smooth manifolds.

Stratifolds are a way to talk about generalized smooth spaces using the out-plot perspective which we discussed at some length in Comparative Smootheology (II, III): a generalized smooth structure on a topological space is specified by declaring subsets of the set of continuous functions out of the space to be smooth. Kreck has been developing Differential Algebraic Topology using stratifolds.

Recently I have had some very nice discussion with Matthias Kreck about the relation of that concept to the diffeological or Chen-smooth spaces surveyed and discussed in A convenient category of smooth spaces: this is based on the in-plot perspective where a generalized smooth space is characterized in terms of the maps into it which are supposed to me smooth.

For instance, working with that concept of topological spaces, Konrad Waldorf and I show (section 4) that the trangression map on differential characters regarded as parallel transport $n$-functors $h$ on $X$ to the space of maps from $\Sigma$ into $X$ is nothing but the internal hom $hom(\Sigma, h)$ in the category of $n$-categories internal to diffeological spaces. This makes the nice aspects of the transgression operation, such as its naturality in $\Sigma$ and $h$ pretty manifest.

The in-plot perspective has the advantage that it connects to the general idea of allowing a generalized structure to be a sheaf on the category of not-so-general structures: namely a rule which assigns to each not-so-general test structure that collection of maps from this test domain into our would-be general structure which we want to consider to be structure preserving – for instance smooth.

Possibly more widely familiar these days than the simple idea of regarding certain sheaves on the category of manifolds as diffeological spaces is the next higher generalization of this concept, where one keeps track of possibly different but isomorphic ways to probe general structures by not-so-general structures: such a more general sheaf is of course a stack.

These terms are unfortunately far from evocative. Over dinner tonight we talked about whether or not out-plot models like locally ringed spaces such as stratifolds give a a better picture to generalized differential geometry than in-plot pictures such as diffeological spaces and more generally sheaves and stacks on the category of manifolds.

One of my concluding impressions was that – apart from the practically relevant observation that to date not all theorems one would eventually hope for have already been generalized to both contexts – a language barrier impedes communication and progress.

I, for one, need to learn more about the theory of stratifolds.

Posted at November 26, 2008 7:28 AM UTC

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Re: Bär on Fiber Integration in Differential Cohomology

I, for one, need to learn more about the theory of stratifolds.

So do I!

I’ve had a brief look at Kreck’s stuff because around the Comparative Smootheology time I was also talking to a couple of people who were thinking about stratified spaces and wondering whether to use stratifolds. (Of course, I was making the case for Frölicher spaces!) I only got as far as deciding not to include them in the Comparative Smootheology paper - the reason being that they are examples of Sikorski spaces and don’t form a particularly nice subcategory.

However, that’s not to say that they are not interesting! So why not induce Kreck to write a guest post and then we can all learn more about them?

Posted by: Andrew Stacey on November 26, 2008 8:18 AM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

I only got as far as deciding not to include them in the Comparative Smootheology paper - the reason being that they are examples of Sikorski spaces and don’t form a particularly nice subcategory.

This relates to one of the points we discussed more intensively over dinner (though far from exhaustively, I’d think):

one has to be well aware of a complementarity in this business between general abstract structure and concrete realizations, the complementarity between

- nice categories of possibly wild objects

and

- less nice categories of more tame objects.

This is most vividly exemplified by the obvious observation that generalized smooth space is, strictly speaking, a misnomer for many of these structures, since they include as a full subcategory the category of (just) topological spaces.

Indeed, (pre)sheaves on topological spaces equipped with structure $S$ (for instance smooth structure), form a category of generalized space which carry at most a generalized $S$-structure. But maybe less structure.

I can’t seem to think of a better short term for “diffeological spaces” which would still convey the idea of a generalized smooth space while at the same time alert people of the above subtlety, but I can understand the complaint about the “diffeological” terminology.

On the other hand, it needs to be emphasized that the issue of the most evocative terminology is quite distinct of the technical issue at hand: there is really no dichotomy between, say, stratifolds on the one hand and more general “smooth spaces” – Sikorski spaces in this case – on the other: rather the latter contains the former and the simple obervation is that for some operations on tame objects it can be helpful to allow ourselves in intermediate steps to pass through a more general world of spaces.

This is of course the old observation underlying the development of schemes. But the corresponding discussion deserves to be had among differential geometers, too. Now seems to be the time.

Posted by: Urs Schreiber on November 26, 2008 8:47 AM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

Thanks for the invitation to write something. There are several aspects which are related, but rather different. One is my general concern about the development of certain areas of mathematics which make it for outsiders very difficult to follow. Perhaps I’m the only one who leaves at least half of the colloquium talks with the feeling, having understood almost nothing. This is often related to the fact that a language is used which needs a long time to learn it and a much longer time to be able to speak and understand it. The first area for which I experienced this for all my mathematical life is algebraic geometry. I made - when I was young - several attempts to learn the modern language, but I never got so much used to it that I can follow a colloquium talk in that area. But now, my own area, topology, is developing in a similar direction and I often leave our Oberseminar with this feeling. Consider for example the fascinating work by Jacob Lurie. Not to be misunderstood: I think this is a very interesting development, but hard to learn (at least for an old man). I know that Hilbert addressed this problem in his 1900 speech and expressed his conviction that this problem would resolve itself since after a while a simplification would be found making things accessible for everybody again. I don’t know what Hilbert means by “after a while”, but 50 years of modern algebraic geometry is a long time and have the impression that here Hilbert was wrong.

The second aspect is more concrete and less important. I have read several articles about diffeological spaces or similar concepts where the people call this world part of differential geometry or differential topology. Of course, names are “Schall und Rauch”, but they have their right, since communication needs content associated to names. And the word “differential” is traditional reserved for mathematical objects where one can speak about the derivative of a map and that it locally describes the map.

More on the mathematical side, it is of course potentially useful to generalize smooth manifolds and the idea to use topological spaces with distinguished maps from subsets of R^n (diffeological spaces) or to R (Sikorski spaces) is very natural. But I find it too general (both concepts are by now rather old and not very much was done with them in this generality). If we look back in the history we can see that successful generalizations of concepts often went on slowly. I recently heard a nice talk by Volker Puppe on the occasion of Dold’s 80th birthday about the development of the concept of fibrations starting from fibre bundles to the comparatively abstract quasi fibrations (invented by Dold if I am not mistaken). Every step in the chain of generalizations came with spectacular new results which completely can be formulated in the old language, the new concepts serve as tools in the proof (for example the famous Dold-Thom theorem). I think we can learn here from these masters. Of course I ask myself, whether I am doing any better with my stratifolds, and I am not sure. But at least I have a few small results which address classical problems like for example the famous question of finding geometric representatives for homology classes (the Steenrod representation problem, which homology classes are the image of the fundamental class of a smooth manifold under a continuous map). If one allows stratifolds instead of manifolds (and stratifolds are in a sense very close to smooth manifolds since most basic concepts of differential topology like Sard’s theorem, transversality … work there), the answer to the Steenrod problem is yes.

A word about Stacey’s comment: “I only got as far as deciding not to include them in the Comparative Smootheology paper - the reason being that they are examples of Sikorski spaces and don’t form a particularly nice subcategory.” What is the criterion for “nice”? For me the stratifolds have nice aspects like beeing so close to differential topology (one can also do basics of differential geometry, but there I have nothing serious to say) or by giving an answer to Steenrod’s problem or by the fact that many interesting spaces like algebraic varieties are stratifolds and that I find it hard to imagine a differentiable world which comes closer to varieties than stratified spaces.

A word at the end. I remember, when I was young, that several old mathematicians could not understand the world any more and complained about the modern abstract nonsense. My position is probably just a proof that I am now also old and thus should not ne taken so seriously.

Posted by: Matthias Kreck on November 26, 2008 10:27 AM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

When I saw the term ‘stratifold’ for some reason I thought it would be an amalgam between stratified space and orbifold, rather than manifold. Next step then orbistratifolds.

Posted by: David Corfield on November 26, 2008 11:51 AM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

There is now an entry generalized smooth spaces at the $n$Lab.

Posted by: Urs Schreiber on December 1, 2008 10:18 PM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

You should also have something on Hofer’s polyfolds…

Posted by: Eugene Lerman on December 3, 2008 3:57 PM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

You should also have something on Hofer’s polyfolds…

Posted by: Urs Schreiber on December 3, 2008 4:11 PM | Permalink | Reply to this

Re: Bär on Fiber Integration in Differential Cohomology

You should also have something on Hofer’s polyfolds…

Eugene sent a list of references by email. I have put them on the new page polyfolds.

But it’s just a list of references so far. Somebody should write a few sentences on the basic idea.

Posted by: Urs Schreiber on December 3, 2008 6:56 PM | Permalink | Reply to this

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