### Differential Forms and Smooth Spaces

#### Posted by Urs Schreiber

As we have discussed at length in the entry on Transgression, differential graded commutative algebras (DGCAs for short) have a useful relation to presheaves on open subsets of Euclidean spaces (smooth spaces, for short):

- every smooth space $X$ has a DGCA $\Omega^\bullet(X)$ of differential forms on it;

- and every DGCA $A$ sits inside the algebra of differential forms of *some* smooth space $X_A$.

On top of that, every finite dimensional DGCA can be regarded as the Chevalley-Eilenberg algebra $CE(g)$ of *some* Lie $\infty$-algebroid $g$, which linearizes some Lie $\infty$-groupoid.

Here I want to talk about my expectation that

The smooth space $X_{CE(g)}$ associated to any Lie $\infty$-algebroid $g$ this way plays the role of the space $K(G,n)$ # of the Lie $n$-groupoid $G$ integrating $g$.

As motivation and plausibility consideration, recall that
in rational homotopy theory and notably in the work of Getzler and Henriques, one obtains a *simplicial space* $S^\bullet_g$ from $g$ by defining its collection of $n$-simplices to be the collection of $g$-valued forms on the standard $n$-simplex…

… and notice that with the above and using the Yoneda lemma, we can equivalently think of this as the collection of $n$-simplices *in* $X_g$:

$S^n_g = \mathrm{Hom}_{smooth spaces}( standard n-simplex in \mathbb{R}^n , X_{\mathrm{CE}(g)} ).$

Mapping simplices into a smooth space is like computing its *fundamental $\infty$-groupoid* $\Pi_\infty(X_{CE(g)})$, thought of as a Kan-complex. In simple situations, notably when $g$ is an ordinary Lie algebra, also the ordinary fundamental (1-)groupoid $\Pi_1(X_{CE(g)})$ is of interest. And I think
$\Pi_1(X_{CE}(g)) = \mathbf{B}G$
in this case, where the right hand side simply denotes the one-object groupoid with $G$ as its space of morphisms.

I am thinking, that hitting everything you see in sections 6 onwards in *Lie $\infty$-connections* (blog, pdf, arXiv) with $g \mapsto X_g \mapsto \Pi_\infty(X_g)$ should have various nice consequences.

I want to better understand *how nice exactly*. That involves better understanding the properties of these functors
$\array{
DGCAs &&\stackrel{\Omega^\bullet(--)}{\leftarrow} && smooth spaces
\\
& \searrow && \swarrow_{\Pi_{\infty}(--)}
\\
&& infty-groupoids
}$
in light of the above expectation.

All help is very much appreciated.

Posted at January 28, 2008 9:35 PM UTC
## Re: Differential Forms and Smooth Spaces

One thing I want to do — and I’d be glad to do it with you — is identify a class of ‘nice’ smooth spaces such that their deRham cohomology matches the real cohomology of their underlying topological space.

Then, given a ‘nice’ smooth group $G$, I would hope to show $B G$ is a ‘nice’ smooth space, whose cohomology is related to that of $G$ in the way we expect.

For example, starting from $\mathbb{Z}$ we should be able to build up smooth spaces $K(\mathbb{Z},n)$ by iterating this $B$ construction.

I’ve got some of this working using Chen spaces. For example, I know how to form a Chen space $B G$ from a Chen group $G$. (I usually say ‘smooth’ instead of ‘Chen’, but I want to emphasize that there are various competing formalisms and I’m using this specific one.)

I haven’t made progress on isolating the ‘nice’ Chen spaces. But Mostow has an important paper on getting a smooth space $B G$ for a smooth group $G$, and working out its deRham cohomology. He uses a different formalism. Gajer has also done important work on making $K(\mathbb{Z},n)$’s into diffeological spaces. So, I’m sure some idea like this will work. The challenge is to get a setup where it all works

smoothly.(Pardon the pun.)