### Integration Without Integration

#### Posted by Urs Schreiber

In some comments to On Lie $N$-tegration and Rational Homotopy Theory, starting with this one, I began thinking about *defining* integration of forms over a manifold in terms of a mere passage to equivalence classes.

There is a big motivation here coming from the observation in Transgression of $n$-Transport and $n$-Connection, that fiber integration is *automatically induced* by hitting transport functors with inner homs.

We want the Lie $\infty$-algebraic version of this, in order to possibly understand how to perform the path integral of a charged $n$-particle coupled to a Lie $\infty$-algebraic connection as in the last section of $L_\infty$-connections and applications to String- and Chern-Simons $n$-transport (arXiv:0801.3480).

I think I made some progress with understanding this in more detail. I talk about that here:

*Integration without Integration* (pdf, 6 pages)

Abstract:On how transgression and integration of forms comes from internal homs applied on transport $n$-functors, on what that looks like after passing to a Lie $\infty$-algebraic description and how it realizes the notion ofintegration without integration.

While that is nice, I’d be grateful for further pointers to existing literature on “integration without integration”. I understand that there exist monographs on how to use that within the variational bicomplex of classical mechanics and hence, I suppose, say something about the path integral. But I haven’t yet managed to get my hands on these texts.

## Re: Integration without Integration

Urs wrote:

Two thoughts. First, your slogan reminds me of motivic integration; have you looked that up? There’s a nice paper by Thomas Hales, which as I remember is called “What is motivic measure?” and in the Bulletin of hte AMS.

Second, I found a small result that you might call integration for free (well, not quite free, but at a knock-down bargain price). There might be some connection.