The n-Category Café

Hi Alessandro!

Nice to meet you here. It was a real pleasure talking to you in Edinburgh. I am printing your article this very moment to read it now.

You write:

I was thinking that Hopkins-Singer in their take on differential cohomology theories have an approach based on Kan complexes: more precisely, the spectrum (“function space”) defining a differential $E$-theory, i.e. Cheeger-Simons in the case that $E$ is singular cohomology, has to be properly regarded as a Kan complex.

Is this somehow linked with what you have at hand, or is it mere coincidence?

I shouldn’t claim that I have fully absorbed their setup, but I think that the simplicial complex which appears in their definition 4.3, page 33 as the “differential function space complex” can be thought of as the $\infty$-groupoid of (something like, depending on the details of the choices made) $n$-gerbes with connection:

- objects are such $n$-gerbes with connection

- morphisms are “gauge transformations” between them (concordances, rather, actually)

- 2-morphisms are “gauge transformations of gauge transformations” (concordances of concordances)

and so on.

So there is clearly an $\infty$-groupoid in what they are looking at, naturally. It might even be a “smooth”, “Lie” $\infty$-groupoid in some sense, if one takes care of the details, though I am not sure if they discuss this anywhere, or if it matters for them.

And I don’t think they discuss differentiating this thing to a Lie $\infty$-algebroid. But maybe they do somewhere and I am just ignorant of that.

So, I’d say, yes, an $\infty$-groupoid in the guise of a Kan complex naturally arises in their discussion of differential cohomology, essentially because all “higher connection things” naturally live in some $n$-groupoid whose $k$-morphisms are order-$k$ gauge transformations.

That they also say something that would directly pertain to the issue of integrating Lie $\infty$-algebra to Lie $\infty$-groups or differentiating the latter to the former I currently can’t see. But I guess you didn’t actually mean to imply that they are.

Sorry, I always make this mistake; equation (2) above should read

where “$\circ$” is composition of linear endomorphisms of $\mathfrak{g}$.

(which we can think of as being equivalent to $P(g)$ due to translation invariance, right?)

Yes!

And thanks for pointing out that it is suggestive to assume that

- the two different ways of “integration without integration” of a Lie algebra

are not unrelated to

- the two different kinds of horizontal composition on paths in $G$ obtained in two different strict models of the String Lie 2-group.

I think you are onto something here. As soon as I am less tired, I’ll try to think about this.

Anyhow, do you mind explaining a tad more this co-pre-sheaf business? I couldn’t understand the last part of your notes.

Yeah, that last bit is a little terse, due to energy running low. It was supposed to remind us of the facts about $C^\infty$-algebras from chapter I, section 1 of Moerdijk & Reyes, Models for smooth infinitesimal analysis, the importance and relevance of which Todd was so kind to highlight in our discussion here – and then to indicate a strategy for how to extend that from algebras of functions to DGCAs of forms, something we failed to understand in that previous discussion.

I’ll talk about this in more detail in the next comment…

So here is the point of $C^\infty$-algebras:

let the category of test domains be open subsets of Euclidean spaces with smooth maps between them (or Euclidean spaces themselves, with smooth maps between them).

Then of course for every object $U$ in there we get the presheaf

$$Hom(-,U)$$

of functions into $U$ and the co-pre-sheaf

$$Hom(U,-)$$

of functions out of $U$.

We used to be tempted to restrict these co-pre-sheaves to their value on $\mathbb{R}$

$$C^\infty(U) := Hom(U,\mathbb{R})$$

That might be a mistake which keeps bugging us now.

If we don’t make that restriction, we notice that every $U$ has a co-pre-sheaf of functions out of it which happens to be a monoidal functor from test domains to sets:

(smooth) functions from $U$ with values in $V \times W$ consist of a function from $U$ to $V$ and a function from $U$ to $W$.

So we are lead to realize that we should say that a “smooth algebra” is any monoidal functor

$$S \to Set \,.$$

Given now a smooth space $X$ (a sheaf), we let its algebra of functions not be just the set

$$C^\infty(X) = Hom(X,C^\infty(-,\mathbb{R}))$$

but the co-pre-sheaf

$$C^\infty(X) : U \mapsto Hom(X,C^\infty(-,U)) \,.$$

Then Moerdijk-Reyes guarantee us that these $C^\infty$-algebras of functions on smooth spaces have all the nice properties the absence of which caused us a headache in our previous approaches.

But functions alone aren’t sufficient for what we are after here. We need the DGCAs of forms on a smooth space.

Last time we didn’t get to a conclusion for how to define these.

Now I am suggesting (but might be wrong) that we should proceed as follows:

we say a $C^\infty$DGCA is a DGCA $A$ together with a $C^\infty$-algebra $C$ such that $C(\mathbb{R})$ coincides with the degree 0 part of $A$.

(Or some slight variant of this.)

Then, indeed, we would be able assign to each smooth space not just the DGCA of forms on it, but the $C^\infty$DGCA of forms on it.

This looks like it must be part of the right answer. But it remains to check that this really helps.

Bruce Bartlett on February 26, 2008 9:51 PM2. wrote:

>The second way is on this page of Duistermaat and Heckmann’s book.

The book is by Duistermaat and Kolk.