### 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.

## Re: Bär on Fiber Integration in Differential Cohomology

So do I!

I’ve had a brief look at Kreck’s stuff because around the

Comparative Smootheologytime 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?