The n-Category Café

Tom,

thanks a lot for these two pointers. I have looked at them quickly to get an impression.

The talk by you which you point to I like a lot. While I cannot quite see it in detail yet, this clearly has some reseamblance with what I tried to talk about.

The thing about motivic measure leaves me a bit puzzled for the moment, but we’ll see if that changes as time goes by…

It is maybe a sign that you were so quick to post a comment to my entry here (for which I am grateful): it was you and your theory of category cardinality which was very much in the back of my mind when I wrote the above.

You might recall that ever since I understood how your category cardinality is related to colimits ( $\sim$ integrals!) of functors over that category , as you explained here, I was trying to see if that helps to understand “path integrals” in quantum mechanics.

And in fact, apparently it does: it turns out that the the path integral measures both of Dijkgraaf-Witten theory # and of its higher dimensional analog # are precisely the “Leinster measure”, and also one gets the right lattice approximation to the quantum propagator of the free point particle by applying the Leinster measure idea to that situation (#).

While that is nice, it is a long way from understanding true path integrals in non-discrete setups.

But in the continuous cases, there turns out to be lots of nice ($n$-)category theory at work, too. In a way, it seems that what physicists invented under the name of BV-formalism is, secretly, a way to talk about measures on Lie $\infty$-groupoids.

After thinking about this for quite a bit, I began seeing a rather nice $n$-strcutural interpretation for this. I am still lacking one or two puzzle pieces, but the hints in the last section of arXiv:0801.3480 together with the remarks in those slides might give an idea.

Anyway, in that context it became ever more clear that the “path integral” is best understood as a a kind of transgression of Lie $\infty$-algebraic structures.

The discussion of integration of 1-forms in my above notes is supposed to be a warm-up for the full thing.

So while writing these notes, I was thinking to myself: “Very well, that looks nice. But is this still related to your other ideas about path integrals using the Leinster measure, or is this different?”

It looks very different. But maybe there is a relation to be found.

Hopefully.

Here lectures on motivic integration I found very usefull.

does your $G$ need to be a group?

It need not in the integral picture. But I don’t really know (and have the impression that nobody does, but see below) how to describe the differential version of smooth $n$-categories which are not $n$-groupoids.

Say you have a space $X$ and consider not its path groupoid, but its path category: objects are points, morphisms are paths modulo orientation-preserving diffeomorphisms. So composition of paths is associative, but no nontrivial path has an inverse.

Then hand me a smooth 1-functor from that path category to some smooth category (not necessarily a groupoid) and by applying the internal hom construction which I described, now using $\mathbf{B}\mathbb{N}$ where I used $\mathbf{B}\mathbb{Z}$ before, I can transgress the thing to the category of loops in $X$ and thereby integrate it (compute its holonomy, really).

So everything goes through as before. Same for higher $n$.

But if you then ask me to hit all this with some $Lie$-operation such as will send Lie $\infty$-groupoids to Lie $\infty$-algebras (described by Ševera in arXiv:math/0612349v1) then I don’t know what such an operation would send the non-groupoid Lie $n$-categories to.

(Well, I have a speculation, but not more, at this point.)

Quick comment about de Rham: if I is the integration operator, then

Id = 0 (Stokes thm)
dI = 0 (gradient of constant fcn = 0)

Thanks, Thomas. This looks like it is indeed closely related to the mechanism I was talking about.

What I am observing is: one sees that the integration functional $I : \Omega^n \to \Omega^0$ arises as constituting that particular 0-form on the space of $n$-forms which is “basic” (a “characteristic” 0-form, “invariant under vertical derivations”) in the sense of section 5.3.

I think what makes this work is a known mechanism, like you indicate. But it is nice to know that this can be understood as arising from such general principles.

Next I’d like to understand path integrals along these lines. But I am not there yet…

I just googled “integration without integration” and found that Louis Kauffman has a chapter

Vassiliev Invariants and Functional Integration Without Integration

Not sure if it is relevant, but thought I would provide the link just in case.