### Questions on *n*-Curvature

#### Posted by John Baez

I’d like to ask Urs some questions about *n*-curvature. I thought I’d ask them publicly, because it might help other people learn about this subject.

Dear Urs:

I know that once you were very eager to understand Breen-Messing data in terms of $n$-transport. All of a sudden I’m very interested in applying this idea to some problems. So I’m trying to find the places where you worked out this idea most fully and explicitly — and of course I’m embarrassed that I didn’t pay more attention a couple of years ago.

Anyway, here’s a pile of questions. I bet I’m confused about some things, so some of them won’t make sense. Feel free to answer as many or as few as you want!

Take a trivial $X$-2-bundle for some Lie 2-group $X$ described in terms of a crossed module $(G,H,t,\alpha)$. By **Breen-Messing data** I simply mean a $Lie(G)$-valued 1-form $A$ and a $Lie(H)$-valued 2-form $B$.

In your notes you seem to say this Breen-Messing data gives well-defined $\Sigma(Inn(X))$-valued 2-transport, where $Inn(X)$ is the inner automorphism 3-group of $X$. This is already somewhat confusing to me since I normally expect $n$-transport when I have an $n$-group, but now you’re talking about $2$-transport for a $3$-group! But I guess it makes perfect sense to talk about a 3-functor

$P_2(M) \to \Sigma(Inn(X))$

from the path 2-groupoid $P_2(M)$ of a manifold $M$ to $\Sigma(Inn(X))$, which is $Inn(X)$ regarded as a 3-groupoid with one object. That’s what you mean, right? How strict is this 3-functor supposed to be? In particular, does the 3-morphism part of $\Sigma(Inn(X))$ get involved here?

Anyway, what I really want to know is some very explicit stuff:

What does a general one of these 3-functors

$P_2(M) \to \Sigma(Inn(X))$

look like in terms of differential forms? Presumably it consists of a 1-form, a 2-form and a 3-form valued in certain Lie algebras, satisfying certain equations? What are these Lie algebras in terms of the crossed module description of $X$? From your note it sounds like we have a 1-form valued in the Lie algebra of $G$ and a 2-form valued in the Lie algebra of the semidirect product of $G$ and $H$. If we’re given Breen–Messing data, what are the formulas for these Lie-algebra-valued forms? I think I can guess *that*.

## Re: Questions on n-Curvature

Thanks, John!

This comes in at a moment where I will have to run soon. So I will postpone detailed replies.

For the moment, all I do is point out that I started developing an exposition of the general theory on my personal $n$Lab web.

This answers many of your questions. Others are scttered through my articles and notes, but I will insert them into the web eventually. (Tomorrow, if I can negotiate that.)

The entry point is here:

When you read that overview page to the end, you will see that it derives from some very general abstract nonsense the theorem that characterizes cocycles in differential nonabelian cohomology by

The upshot is that Cartan-Ehresmann $\infty$-connections are the structures introduced and studied in

Where much of the details that you are looking for are given. (Search the document for $inn(\mathfrak{g})$).

There we just sketched the idea of how these dg-structures come from principal $\infty$-bundles with connection. The writeup on the Lab now gives the derivation.

For the particular case that you are asking about we have a structure Lie 2-group $G$ with $\mathfrak{g}$ its Lie 2-algebra. The differential form data of a $G$-2-bundle $P\to X$ with connection is an $L_\infty$-algebra valued differential form given by a morphism of $\infty$-Lie algebroids

$\Pi^{inf}(P) \to cone(\mathfrak{g})$

with $\Pi^{inf}(P)$ the infinitesimal path $\infty$-groupoid

which is dually a dga-morphism

$\Omega^\bullet(P) \leftarrow W(\mathfrak{g}) : A$

from the Weil-algebra of the Chevalley-Eilenberg-algebra of $\mathfrak{g}$.

Unwrapping what this is we find that it consists of

- a $\mathfrak{g}_1$-valued connection 1-form $a$

- a $\mathfrak{g}_2$-valued connection 2-form $B$

- a $\mathfrak{g}_1$-valued curvature 2-form $\beta$

- a $\mathfrak{g}_2$-valued curvature 3-form $H$

with

$\beta = d_a + [a\wedge a] + \delta_* B$

$H = d_a B \,.$

This is the data that you are looking for. The equivariance/cocycle data that this has to satisfy to qualify as connection data on $P$ is the first and second $\infty$-Ehresmann condition given by the requirement that the above morphism $A$ is required to fit into a diagram

$\array{ \Omega^\bullet_{vert}(P) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& flat vertical \mathfrak{g}-valued differential form \\ \uparrow && \uparrow &&& first Ehresmann condition \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) &&& \mathfrak{g}-valued connection form on total space \\ \uparrow && \uparrow &&& second Ehresmann condition \\ \Omega^\bullet(X) &\stackrel{P(F_A)}{\leftarrow}& inv(\mathfrak{g}) &&& characteristic forms on base space }$