## February 26, 2008

### Impressions on Infinity-Lie Theory

#### Posted by Urs Schreiber

While talking to people who will hold still and listen, like when I talk to Bruce Bartlett or to Danny Stevenson, I realized that while I talked about integration theory of Lie $\infty$-algebras here and there, I might not have gotten my point across succinctly.

So, as a kind of meta-response to some aspect of Bruce’s highly appreciated exegesis. I have now prepared a manifesto:

Impressions on $\infty$-Lie theory
pdf (6 pages)

Abstract. I chat about some of the known aspects of the categorified version of Lie theory – the relation between Lie $\infty$-algebras and Lie $\infty$-groups – indicate how I am thinking about it, talk about open problems to be solved and ideas for how to solve them.

This starts with very roughly reviewing the basic ideas underlying the work by Getzler, Henriques and Ševera, then exhibits the “shift in perspective” which I keep finding helpful and important, mentions applications like the strict integration of the String Lie 2-algebra using strict path 2-groupoids and ends by quickly sketching how that relates the “fundamental problem of Lie $\infty$-theory” to the relation between smooth spaces and “$C^\infty$DGCAs” which we are having a long discussion about with Todd Trimble and Andrew Stacey, scattered over various threads (notably here, and here).

At the end of the notes I indicate how I am currently trying to address this issue. But I need more time to work this out. Comments are appreciated.

Posted at February 26, 2008 5:16 PM UTC

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### Re: Impressions on Infinity-Lie Theory

Hi Urs,

I was thinking that Hopkins-Singer in their take on differential cohomology theories have an approach based on Khan 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 Khan complex.
Is this somehow linked with what you have at hand, or is it mere coincidence?
Unfortunately I can’t see it through, as I seriously lack the technical expertise, and HS paper sure is damn hard to penetrate.

Posted by: Alessandro on February 26, 2008 6:24 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

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.

Posted by: Urs Schreiber on February 26, 2008 9:08 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

Hi Urs,

yes, you’re right, I wasn’t inferring they have something to say about integrating Lie ∞-algebra, but I was curios to know if their Khan complex could be interpreted via n-gerbes with connections, and as you say it does.
Thanks!

Posted by: Alessandro on February 26, 2008 9:35 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

Alessandro wrote:

I was thinking that Hopkins-Singer in their take on differential cohomology theories have an approach based on Khan complexes…

You must mean ‘Kan complex’: a Khan complex is the Asian analogue of a Napoleon complex.

Posted by: John Baez on February 26, 2008 9:38 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

Hi Urs, this is useful thanks. With regards to your view that the candidate formula

(1)$BG_\mu := \Pi_2^{str}(X_{\mathfrak{g}_\mu})$

for the strict version of the Lie 2-group integrating the String Lie 2-algebra being the same as that in in the previous work of yourself, Danny, John and Alissa, except for a posssibly different law for horizontal composition, I have the following comment:

As you know, I’m fascinated by the ‘elementary’ integration procedure to get a Lie group from a Lie algebra. I commented before that there are essentially two different ways of doing it, and I think this is the same thing which crops up in your String 2-group case!

We start with a Lie algebra $\mathfrak{g}$. Both methods will produce a simply-connected Lie group $G$ integrating $\mathfrak{g}$ by introducing a composition law on the collection $P(\mathfrak{g})$ of all paths in $\mathfrak{g}$, and then then they will give an equivalence relation on these paths such that $P(\mathfrak{g})/\sim$ is our group $G$. Goodness knows how these two are related, I’d love it if someone could explain that.

1. The first way is your ‘Weinstein’ method via DGCA’s, which is the same as Graeme Segal’s. You consider the collection of $\mathfrak{g}$-valued 1-forms on $[0,1]$ (which we can think of as being equivalent to $P(\mathfrak{g})$ due to translation invariance, right?). Composition is just concatenation of paths. You quotient out by ‘DGCA homotopy’ - which means two paths are equivalent if there is a path-of-paths gonig between them satisfying the Maurer-Cartan equation.

2. The second way is on this page of Duistermaat and Heckmann’s book. To express it, first we need the following.

If $\gamma$ is a path in $\mathfrak{g}$, then define $A_\gamma(t)$, the “Lie bracket unfolded along $\gamma$”, as the path of linear endomorphisms of $\mathfrak{g}$ defined as the solution to the differential equation

(2)$\frac{dA}{dt} = [\gamma(t), A(t)], \quad A(0) = id.$

Given two paths $\gamma_1, \gamma_2 \in P(\mathfrak{g})$, we define their product as the path

(3)$(\gamma_1 * \gamma_2)(t) := \gamma_1(t) + A_\gamma (t)(\gamma_2(t)).$

Observe that this is not composition of paths. It’s sort of an adjoint-twisted addition of paths. At time $t=0$, $\gamma_1 * \gamma_2 (0) = \gamma_1(0) + \gamma_2(0)$… after that it gets twisted.

There’s a way to characterize the group $\Lambda(\mathfrak{g})^0$ of contractible loops as sitting inside $P(\mathfrak{g})$, and the Lie group we want is the quotient group

(4)$G = P(\mathfrak{g}) / \Lambda^0(\mathfrak{g}).$

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

Posted by: Bruce Bartlett on February 26, 2008 9:51 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

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

(1)$\frac{dA}{dt} = ad\gamma(t) \circ A(t), \quad A(0) = id$

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

Posted by: Bruce Bartlett on February 26, 2008 9:57 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

(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.

Posted by: Urs Schreiber on February 26, 2008 10:09 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

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.

Posted by: Urs Schreiber on February 26, 2008 10:23 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

(1)$Smooth spaces \leftrightarrow DGCAs.$

The beauty of it is that the left hand side has to do with calculus and ‘smoothness’, while the right hand side is formulated entirely algebraically. If you allow ‘sheaf-like’ things to be included in the definition of the right hand side, we’ll just be saying

(2)$calculus = calculus$

which isn’t much fun.

Posted by: Bruce Bartlett on February 27, 2008 10:35 AM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

Good enough? if not, why not?

Posted by: jim stasheff on February 27, 2008 2:24 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

How about simplicial manifolds? Good enough? if not, why not?

In rational homotopy theory we do have that

$DGCAs \to simplicial spaces \to DGCAs$

is an isomorphism in cohomology. I should just be content with that.

But, as I tried to point out I have that feeling that we should rather be looking at

$DGCAs \to smooth spaces \to DGCAs$

and regard the appearance of simplicial spaces here as the result of the act of forming fundamental $\infty$-groupoids of smooth space, something we should, as I argued, retain the liberty for of not necessarily doing.

I might be just fighting a pointless and already lost battle. But to find out, I will first have to lose that.

Posted by: Urs Schreiber on February 27, 2008 5:20 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

$calculus = calculus$ which isn’t much fun.

Currently I am less prejudiced about what I want things to look like philosophically speaking, and more interested in getting everything to work the way it should. I am not sure that

$calculus \leftrightarrow algebra$

is exactly the right way to look at what we are dealing with.

I suppose that if we want to fit the discussion into a grand philosophical framework, we should follow the lore of the master himself, Lawvere, as recalled by one of his prophets here: this says that the duality we are talking about is

$space \leftrightarrow quantity \,.$

Notice, by the way (this I failed, as you may now recall, to say when we talked about this in Sheffield), that apart from the possible co-pre-sheaf structure on DGCAs, we want to be thinking of all our DGCAs here as smooth (vector) spaces equipped with the (smooth) structure of a DGCA, due to the fact that we can regard

$\Omega^\bullet(X) = Hom_{S^\infty}(X,\Omega^\bullet(-))$

as equipped with the smooth structure given by realizing this as the internal hom sheaf

$\Omega^\bullet(X) = hom_{S^\infty}(X,\Omega^\bullet(-)) : U \mapsto Hom_{S^\infty}(X \times U , \Omega^\bullet(-)) \,.$

(This then allows to talk about smooth maps $\Omega^\bullet(X) \to \mathbb{R}$, something I failed to say when you asked me about it in Sheffield.)

Posted by: Urs Schreiber on February 27, 2008 5:13 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

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.

Posted by: Maarten Bergvelt on February 27, 2008 6:54 PM | Permalink | Reply to this

### Re: Impressions on Infinity-Lie Theory

Indeed… silly mistake.

Posted by: Bruce Bartlett on February 27, 2008 7:09 PM | Permalink | Reply to this
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