### Transgression of *n*-Transport and *n*-Connections

#### Posted by Urs Schreiber

Since it will play a role both for what is currently indicated in section 5 of the article on Lie $\infty$-algebra connections and their application to String- and Chern-Simons $n$-transport as well as for the next followup of my work with Konrad Waldorf, I am thinking again in more detail about

Trangression of $n$-transport and $n$-connections

Abstract.After going through some ground work concerning generalized smooth spaces and their differential graded commutative algebras of forms, I talk about the issue of transgression of transport $n$-functors and Lie $\infty$-valued $n$-connections to smooth mapping spaces.

This builds on the general idea of $n$-functorial transgression as the image of an internal hom as first voiced in this old comment and then later incorporated in the discussion of The charged $n$-particle and detailed a bit more in the entry Multiplicative Structure of Transgressed $n$-Bundles.

Currently only a few sketches are present in the above pdf, as I am going to develop this as we go along.

One important aspect, emphasized in the above abstract, is that the discussion greatly profits from a good general understanding of the relation between generalized smooth spaces and their differential graded-commutative algebras of differential forms. I started making comments on that here and now Todd Trimble thankfully chimed in by providing this detailed reply, which I will reproduce below.

But first, I’ll reproduce the introductory remarks from my notes to set the stage.

**Introduction**

I want to better understand the

$\;\;\bullet$ general systematics

and the

$\;\;\bullet$ specific details

of what it means to *transgress*

$\;\;\bullet$ transport $n$-functors [1, 2, 3 4]

and

$\;\;\bullet$ Lie $n$-algebra valued connections [5]

to mapping spaces.

This is essentially about understanding the
pull-push operation of $n$-transport and
$n$-connections on a “**tar**get space” $\mathrm{tar}$
from right to left through
a span
$\array{
&&& hom(par,tar) \otimes par
\\
&& \multiscripts{^{p_1}}{\swarrow}{} && \searrow^{ev}
\\
& hom(par,tar) &&&& tar
}$
to obtain an $n$-transport and $n$-connection on the
“**conf**iguration space” of maps
$\mathrm{conf} := \mathrm{hom}(\mathrm{par},\mathrm{tar})$
from some “**par**ameter space” $\mathrm{par}$ to
$\mathrm{tar}$.
But in fact it turns out that the “good” answer does
apparently not quite involve the naïve push-forward
along $p_1$, but a slight variant, which then amounts to
simply defining the transgression of the $n$-transport
or $n$-connection $\mathrm{tra}$ to be
$\mathrm{hom}(\mathrm{Id}_{\mathrm{par}}, \mathrm{tra})
\,.$

This difference to the naive definition of transgression
as direct push-pull through the above span actually
takes care of a fact neglected in standard discussions that
do not make the $n$-categorical nature of $n$-transport
manifest: namely that under transgression not only
the domain, but also the *co*domain of $n$-transport
and $n$-connections changes.

For instance, in the simplest kind of example, an ordinary abelian 2-connection is not really something taking values in $U(1)$, but rather something taking values in $\mathcal{B}U(1)$. Transgressing it to loop spaces by setting $\mathrm{par} = S^1$ in the above turns it into a 1-connection with values in $\mathrm{hom}(S^1, \mathcal{B}U(1))$, which is indeed $\Lambda \mathcal{B}U(1) = U(1)$ as it should be.

So this general notion of transgression is what shall be discussed here.

Before getting into the issue of transgression proper, I try to lay some necessary groundwork on the general concept of generalized smooth spaces and the differential graded-commutative algebras of differential forms on them.

Here I take “generalized smooth spaces” simply to be presheaves over manifolds. This is clearly the right ambient topos, in general, for any discussion of smooth parallel $n$-transport and smooth $n$-connections.

My tentative discussion of differential forms on such generalized smooth spaces, and the relation to general differential graded commutative algebras, is included here because I am not aware of a discussion of the necessary points in the literature. This may, however, well be – in parts or possibly even in total – just be due to my woeful ignorance.

Hopefully much of what I am trying to say concerning the general issue of smooth spaces versus differential graded algebras is actually well known, possibly in slightly different guise, in rational homtopy theory.

In any case, after having dealt to some extent with this groundwork, I’ll define in more detail the problem of transgression to be discussed here, and then start looking at concrete questions and specific examples.

**Todd’s first comment**

*Here is a copy of Todd’s comment on the relation between DGCAs and generalized smooth spaces, the discussion of which we should move to the comment section below. *

*This is what Todd wrote:*

Hi Urs,

Took a look at your comment and notes; here are some initial reactions.

First, you asked whether the formula

$U \mapsto hom(U \times X, Y)$

gives the ‘correct’ internal hom for smooth spaces; in this context I assume you’re asking whether this gives the correct exponential for cartesian closedness.

Yes, this formula is correct and works for general presheaf toposes, by an application of the Yoneda lemma. The result is standard and proved in many books on topos theory – the ones by Johnstone, by Mac Lane and Moerdijk, and by Freyd and Scedrov come to mind. I’m happy to go into more detail if you want.

Second: there is as you say a contravariant adjunction

$S^{\infty}(X, DGCA(A, \Omega^{\bullet}(-))) \cong DGCA(A, S^{\infty}(X, \Omega^{\bullet}(-))).$

This can be proved with the help of (again) the Yoneda lemma: since $X$ is a colimit of representables, one can easily reduce to the case where $X$ *is* a representable, and then check that case with the help of Yoneda. Notice that this type of adjunction has the general flavor of one coming from a Janusian/ambimorphic object ($\Omega^{\bullet}(-)$ having a kind of dual existence, one as a smooth space and another as a DGCA).

But, I don’t think this adjunction can be an equivalence, even if we restrict to locally quasi-free DGCA’s. The question seems to be whether $DGCA^{op}$ (or something like it) is equivalent to the topos $S^{\infty}$, and it’s sort of an interesting question because $DGCA^{op}$ *does* partake of some of the exactness properties satisfied by a topos. For one, it’s a lextensive category (it has finite pullbacks and finite coproducts which are disjoint and preserved under pullback) – lextensive categories are a very interesting and much-studied class of categories.

But I sort of doubt $DGCA^{op}$ or some easily identified full subcategory is *locally cartesian closed* (which it would be if it were a topos). This would mean that general colimits in this category are preserved under pulling back (one says “colimits are universal”), or that limits in $DGCA$ are preserved under pushing out. The pushout of a pair of morphisms

$B \leftarrow A \to C$

in $DGCA$ is given by $B \otimes_A C$; the question is whether $B \otimes_A -$ preserves limits of $A$-modules. Preservation of equalizers may be no big deal under some condition like “locally quasi-free” (although there one would have to watch out that objects satisfying that condition give a complete and cocomplete category – I’m not so sure about that), but $B \otimes_A -$ preserving *arbitrary* products, not just finite ones, looks like a much taller order.

(Another exactness condition to check has to do with whether there is an exact correspondence between epimorphisms and equivalence relations: whether every epi is the quotient of its kernel pair, and whether every equivalence relation is the kernel pair of its quotient. Never mind that for now.)

This is pretty much a gut reaction, and I feel it will probably read like a wet blanket reaction as well, which I really don’t mean. It’s possible that the desiderata of ‘internal homs’ could be relaxed a bit to stop short of actual cartesian closedness and still be interesting, but I just don’t know off hand. (By the way, I wanted to get back to you some time on calculations of internal homs of differential graded cocommutative coalgebras, but my first two attempts were obliterated by my 3-year-old daughter, and somehow I haven’t found much time for math during this holiday season.)

I would very much like to follow up some time on what you’re saying about universal bundles with connections.

## Re: Transgression of n-Transport and n-Connections

Todd,

I have now incorporated your statement that $\Omega^\bullet$ and $\mathrm{Hom}(-,\Omega^\bullet(-))$ form an adjunction.

What you say about how to put this into perspective, $\Omega^\bullet(-)$ being Janusian or ambimorphic etc, I find very intriguing, but haven’t yet incorporated into my notes, since I am not quite sure yet what to make of that. Clearly, I have a lot to learn here.

If and when you might be interested, I’d very much enjoy if you could have a look at the beginning of section 4, where I talk about transgression.

I am pretty sure I am onto something there, but it is also clear that there should be a much better abstract nonsense way to say what I am trying to say.

So here is the point:

usually, transgression is defined as pull-push from right to left through spans of the form $\array{ &&& hom(par,tar) \otimes par \\ && {}^{p_1}\swarrow && \searrow^{ev} \\ & hom(par,tar) &&&& tar } \,.$

I am claiming that if what we want to transgress is actually an $n$-functorial thing whose domain is $\mathrm{tar}$, then what we “really want” to regard as its transgression is instead its image under $\mathrm{hom}(\mathrm{par},-) \,.$

I know this is what we “really want” by looking at rather large classes of examples.

I might just be content with taking $\mathrm{hom}(\mathrm{par},-)$ by definition to be “my” notion of “good” transgression. But I want to clarify what’s going on, how this relates to the pull-push operation people usually consider.

At the beginning of that section 4 I present a long remark where I give my best attemt at clarifying the situation.

If what I do there is of any value at all, then certainly it is only scratching the surface of something. I have the suspicion you might maybe recognize some abstract nonsense more elegant and more useful which refines my discussion there.

As I said, if and when you are interested, this would be something I’d very much enjoy hearing your comment on.