### 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
## 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.