The n-Category Café

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 $BG$ 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 $BG$ 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 $BG$ 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.)

The following just occured to me, which puts my statements here into a larger perspective by relating the smooth space which I am calling $X_{\mathrm{CE}(g)}$ not just to the classifying space of $G$, but to the entire universal $G$-bundle.

Recall that I am pointing out (more or less explicitly so far) that for generalized smooth spaces $$X,$$ regarded as sheaves over smooth test domains, we can always form the (strict) fundamental $n$-groupoid $${\Pi }_n(X)$$ in the essentially obvious way by looking at sheaf homomorphisms $$[\mathrm{0,1}{]}^n\to X$$ and so on.

(In my latest article with Konrad we define in detail the groupoid $P_1(X)$ of thin-homotopy classes of paths of any smooth space. This has a straightforward generalization to all other flavors of path ($n$-)groupoids of $X$.)

For my following comment I’ll just need ${\Pi }_1(X)$, the ordinary fundamental 1-groupoid of $X$.

Now fix some Lie algebra $g$ and let $G$ be the simply connected Lie group integrating it.

Recall that I am writing $$X_{\mathrm{CE}(g)}$$ for the smooth space which is defined by the assignment $$U\mapsto {\mathrm{Hom}}_{\mathrm{DGCAs}}(\mathrm{CE}(g),{\Omega }^{•}(X))=\mathrm{set}\mathrm{of}\mathrm{flat}g-\mathrm{valued}1-\mathrm{forms}\mathrm{on}U$$ and that I am claiming that the fundamntal 1-groupoid of this smooth space is

$${\Pi }_1(X_{\mathrm{CE}}(g))=BG$$

by which I use to denote the one-object groupoid given by the group $G$. (This is supposed to be nothing by the de-nervified version of the standard construction described in On rational homotopy theory and Lie $n$-tegration).

Now let

$$G_{\mathrm{set}}$$

be the set underlying $G$ equipped with the discrete topology, regarded as the smooth space given by the assignment

$$U\mapsto \{\mathrm{constant}G-\mathrm{valued}\mathrm{functions}\mathrm{on}U\}=G_{\mathrm{set}}.$$

It is clear that

$${\Pi }_1(G_{\mathrm{set}})=\mathrm{Disc}(G),$$

where the right hand is supposed to denote the “discrete” category (in the cat sense, not in the topology sense) whose space of objects is $G$ and which has only identity morphisms.

Finally, write $G$ for the generalized smooth space given by $G$ itself.

Notice that

$${\Pi }_1(G)=\mathrm{Codisc}(G)=G//G=\mathrm{INN}(G)$$

is the pair groupoid of $G$, since $G$ is simply connected.

Now, notice that we canonically have morphisms of smooth spaces

$$G_{\mathrm{set}}\to G\to X_{\mathrm{CE}(g)}$$

The first map is just inclusion. The second map sends any function

$$g:U\to G$$

to its differential

$$\mapsto g^{-1}dg$$

which is indeed a flat $g$-valued 1-form.

Notice that all three smooth spaces we are talking about can be regarded as having functions that form a group. In that sense the above is actually a short exact sequence

$$G_{\mathrm{set}}\to G\to X_{\mathrm{CE}(g)}.$$

By the above, hitting this with ${\Pi }_1$ yields the short exact sequence of groupoids

$$G\to \mathrm{INN}(G)\to BG.$$

As discussed in some length at The inner automorphism 3-group of a strict 2-group

this is the groupoid version of the universal $G$-bundle.