The n-Category Café

For the charged particle in flat space, the naive path integral

is made rigorous by realizing that the combination

is to be interpreted as the Wiener measure $d\mu$ on the space of paths, such that we should read the above as

or rather

A similar formula is still true for particles on curved Riemannian spaces. Only that here the Wiener measure receives a certain deformation for nonvanishing Ricci curvature.

I expect that an analogous statement should hold for the string, but I don’t recall having seen it stated anywhere.

Is there a rigourous measure on the space of continuous maps $[0,1]^2 \to X$, analogous to the Wiener measure for paths $[0,1] \to X$, such that it would analogously yield the kinetic part of the string action, reducing the path integral for the charged string to

where now $\mathrm{hol}_\nabla$ would in genereal be the surface holonomy of a gerbe (possibly with boundary contribution) but which for the purpose of this question might be just the integral of a global 2-form.

In this week’s lecture note John describes

A coordinate-free definition of $p$-velocity.

I am trying to find an “implementation independent” definition of $p$-vector fields.

By this I mean a purely arrow-theoretic description, such that it reproduces the ordinary description when these arrows are interpreted internal to a suitable ambient category.

For $p=1$ I think I found a nice solution.

I am claiming that an “implementation independet” formulation of the concept of an ordinary vector field is the following #

Let $P_1(X)$ be a 1-category, supposed to model the idea of a category of paths in some space $X$.

Let $F(P_1(X))$ be the sub-category of $\Sigma(\mathrm{Aut}(P_1(X)))$ which has the single object $P(X)$ , whose morphisms are natural transformations of the form

and where composition of morphisms is horizonatal composition of these natural transformations.

For $R$ any group, I now say that an $R$-flow in $P_1(X)$ is a morphisms

I claim that if we internalize this arrow theory as follows, we recover the ordinary notion of (flow of a) vector field on a smooth space $X$:

Let $X$ be a smooth space and let $P_1(X)$ be the groupoid of thin-homotopy classes of paths in $X$. Let $R = \mathbb{R}$ and require all functors to be smooth.

Now the question is: how would we categorify this to describe not a flow of points, but a flow of strings.

I have one way how to do this, which is fine for most applications that I am currently dealing with. But I am hoping there is another way, one which would more directly be connected to the concept of 2-velocity that John describes in his lecture.

The one way I have is simply obtained by looking at 1-flows on something like the path space of $X$. This can be easily formulated in a “implementation independent”, i.e., in an arrow-theoretic way.

But what I would also like to have is something like this:

Replace $P_1(X)$ by a 2-category $P_2(X)$. Then consider the category $F(P_2(X))$ essentially as above, but now also including higher morphisms in the obvious way.

What next? How do we have to proceed such that internalizing in smooth path categories as before produces for us the notion of something dual to a 2-form field?