The n-Category Café

Agreed - except that in the directed context it would be a functor from continuous paths to some category
not a groupoid
and
what actually happens in the topological
context is a homotopy coherent functor,
not a strict one,
thus detecting some higher order information
which may be hiding in the smooth context.

I wasn’t just saying that parallel transport in differential geometry is defined using a differential equation and thus works best for smooth paths - that’s sort of obvious.

I was saying that it seems impossible to extend the usual recipe for parallel transport from smooth paths to continuous ones in a continuous manner.

Maybe I should be boringly precise.

Suppose we have a trivial $G$-bundle over $M$. Then a smooth connection defines a smooth functor

$$f: P_{smooth}(M) \to G$$

from the smooth path groupoid of $M$ to $G$. I might like to extend this to a continuous functor

$$f: P_{cont}(M) \to G$$

from some topological groupoid of continuous paths in $M$ to $G$.

(I know exactly what I mean by $P_{smooth}(M)$, but not $P_{cont}(M)$, so we have to imagine a range of choices for how it’s defined.)

If a way to do this exists, it’s probably unique, since $P_{smooth}(M)$ should be dense in $P_{cont}(M)$ in any reasonable topology.

But, I don’t see why a way exists, in general! Consider smooth paths $\gamma_i$ converging to a continuous path $\gamma$. The question is whether

$f(\gamma_i)$

converges to anything.

But, I think there are examples where the $\gamma_i$ converge to $\gamma$ - in the $C^0$ topology, say - while $f(\gamma_i)$ keeps bouncing around madly, not converging to anything.

So, I don’t think we can extend $f$ from smooth paths to continuous ones in a continuous way - except in special cases, like when the original connection was flat. Then of course the limit is trivial!

Bismut clearly has some trick up his sleeve, but it seems to work “stochastically”: not for every path, just for almost every path in a Brownian motion.

You’ve got a good point: if we drop our insistence that

$$f: P_{cont}(M) \to G$$

be a continuous functor, and require it merely to be a homotopy-coherent functor (an “$A_\infty$ functor”), then things should get a lot easier.

My puzzlement comes from the seeming shortage of continuous

$$f: P_{cont}(M) \to G$$

that are actually functors on the nose. The only ones I see are those coming from flat connections!

There are papers […]

Thanks!

In chapter 1 of Baudoin’s book I see the kind of considerations that you develop in math/0612411.

Very interesting.

For example, parallel transport can be defined along any path of bounded variation […]

The parallel transport you have in mind here, is it that induced by an ordinary connection on a smooth bundle?

Until I have absorbed all that about stochastic parallel transport, maybe I could make my initial question more concrete:

As for instance discussed in C. Müller, C. Wockel, Equivalences of Smooth and Continuous Principal Bundles with Infinite-Dimensional Structure Group

smooth and topological bundles (without connection) are largely equivalent.

To a naive mind like mine, this seems to suggest that there should be a topological category of some kind of classes of continuous paths in base space, such that appropriate continuous functors on that are equivalent to the corresponding parallel transport functors in smooth bundles.