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September 26, 2006

Fahrenberg and Raussen on Continuous Paths

Posted by Urs Schreiber

Suppose you want to transport something along some path through a space XX. Before you do so, you need to know what a path in XX is.

If XX is a smooth space, we tend to demand a path to be a smooth map

(1)IX, I \to X\,,

up to reparameterization. (II is the standard interval.)

What exactly is the analog of dividing out by reparameterization of paths in the case that XX is just a topological space?

Jim Stasheff, being interested in topological notions #, wondered why I kept going on about smooth paths # without ever talking about continuous paths. He was so kind to point me to the work

Ulrich Fahrenberg & Martin Raussen
Reparametrizations of Continuous Paths
Dept. of Mathematical Sciences, Aalborg University
Technical Report R-2006-22
(pdf)

where exactly this issue is investigated.

A central point of the discussion is the emphasis of the notion of a directed topological space, which is a space equipped with a collection of preferred paths that single out certain directions.

Under the term dd-space this has been proposed and developed in

Marco Grandis
Directed homotopy theory, I. The fundamental category
math.AT/0111048 .

In this context it is natural to restrict to those reparameterizations

(1)φ:II \varphi : I \to I

of the standard interval which are surjective (obviously), but also increasing (or rather: not decreasing).

Fahrenberg and Raussen study in detail which notion of path is obtained if we divide out maps

(2)γ:IX \gamma : I \to X

by this kind of continuous reparameterization.

A main result is that the space of general paths modulo these reparametrizations is homeomorphic to the space of regular paths modulo increasing auto-homeomorphisms of the interval. Here a regular path is one that never rests, i.e. one which is strictly increasing, in the obvious sense.

This is useful, because the space of regular paths in usually easier to deal with. Based on this, Martin Raussen develops a detailed theory of directed spaces

Martin Raussen
Invariants of directed spaces
Dept. of Mathematical Sciences, Aalborg University
Technical Report R-2006-28
(pdf).

Using this technology, one can go further and study directed nn-paths

(3)I nX I^n \to X

modulo continuous reparameterization, in a way analogous to what one would do for the smooth setup.

For 2-paths #, this will apparently be investigated in

Ulrich Fahrenberg
Homotopy of Squares in a Directed Hausdorff Space
(in preparation).

Posted at September 26, 2006 8:18 PM UTC

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11 Comments & 1 Trackback

Re: Fahrenberg and Raussen on Continuous Paths

One reason for being scared of parallel transport along continuous paths is that the differential equation describing parallel transport makes no obvious sense when the path fails to be differentiable:

dg(t)/dt=A(γ(t))g(t)d g(t)/d t = A(\gamma'(t)) g(t)

where gg is a GG-valued function of tt, AA is a Lie(G)\mathrm{Lie}(G)-valued 1-form on a manifold, and γ\gamma is a path in this manifold.

One can take any ordinary differential equation and integrate both sides to get an integral equation which has a better chance of making sense. In the present case this doesn’t make the derivative in γ(t)\gamma'(t) go away, but it means we don’t need γ(t)\gamma'(t) to be defined pointwise - something weaker will suffice to make the integral well-defined.

Still, there are serious issues to deal with, and I’ve never heard of anyone showing that a smooth connection allows one to define parallel transport for arbitrary continuous paths.

The closest thing I’ve heard is Bismut’s work on the heat equation for vector bundles, which got used in Getzler’s famous heat equation proof of the Atiyah-Singer index theorem.

Bismut showed how the solutions of this heat equation could be described using Brownian motion of particles having an internal degree of freedom that gets parallel transported as they wiggle around. Since Brownian motion is almost surely everywhere nondifferentiable, one really needs to think hard about how this parallel transport is defined. And, he managed to succeed.

However, I think the stochastic aspect was crucial here: he wasn’t trying to define parallel transport for a single nondifferentiable path, merely its average over a bunch of them!

Posted by: John Baez on September 29, 2006 5:25 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

Bismut showed how the solutions of this heat equation could be described using Brownian motion of particles having an internal degree of freedom that gets parallel transported as they wiggle around. Since Brownian motion is almost surely everywhere nondifferentiable, one really needs to think hard about how this parallel transport is defined. And, he managed to succeed.

That’s interesting. I didn’t know this.

One reason for being scared of parallel transport along continuous paths is that the differential equation describing parallel transport makes no obvious sense when the path fails to be differentiable

That’s true. But I think one may adopt the perspective that precisely for such reasons we may choose to regard equations as dg(t)/dt=A(γ(t))g(t)d g(t)/d t = A(\gamma' (t))g (t) as secondary, while regarding the parallel transport functor as primary. In terms of transport functors, we have a much wider choice of structures that we can put a “connection” on.

My understanding was that Jim Stasheff was interested in this particular aspect of using the concept of functorial transport in order to construct notions of connections in settings other than smooth spaces.

So here, for instance, if one understood this theory by Fahrenberg, Raussen, Grandis, etc. sufficiently, one could start working out what data a functor from continuous paths to some groupoid would consist of. One could study it’s local trivializations and all that using the general definitions - all applied now internal to this topological setup.

I don’t know if there would be any interesting applications for such a notion of continuous connections (although TFTs do come to mind). Maybe somebody else does.

Posted by: urs on September 29, 2006 5:52 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

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.

Posted by: jim stasheff on October 13, 2006 4:42 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

what actually happens in the topolgoical context is a homtopy coherent functor, not a strict one

But do I understand correctly that I can regard that homotopy coherent functor as a lax/pseudo functor from 1-paths to a suitable nn- (or maybe ω\omega-)category?

Posted by: urs on October 13, 2006 4:44 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

John’s point seems to be that `connections’
are defined in the differentiable category.
Fine, but my point is that parallel transport extends easily to the continuous
context and very nicely in (higher = homtopy coherent) categorical terms.

Posted by: jim stasheff on October 13, 2006 4:37 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

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 GG-bundle over MM. Then a smooth connection defines a smooth functor

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

from the smooth path groupoid of MM to GG. I might like to extend this to a continuous functor

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

from some topological groupoid of continuous paths in MM to GG.

(I know exactly what I mean by P smooth(M)P_{smooth}(M), but not P cont(M)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)P_{smooth}(M) should be dense in P cont(M)P_{cont}(M) in any reasonable topology.

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

f(γ i)f(\gamma_i)

converges to anything.

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

So, I don’t think we can extend ff 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)Gf: P_{cont}(M) \to G

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

My puzzlement comes from the seeming shortage of continuous

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

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

Posted by: John Baez on October 20, 2006 6:42 AM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Gauge Theory Kinematics
Weblog: The n-Category Café
Excerpt: A review of Fleischhack's discussion of a Weyl algebra on spaces of generalized connections.
Tracked: March 1, 2007 2:57 PM

Re: Fahrenberg and Raussen on Continuous Paths

I would like to cite the Fahrenberg-Raussen work in a paper on parallel transport along general notions of paths.

Does anyone know if the “technical reports” mentioned above have meanwhile appeared somewhere as referencable preprints?

Thanks for any hints!

Posted by: urs on April 11, 2007 11:03 AM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

There are papers by Terry Lyons and others on “rough paths”. His fundamental paper is “Differential equations governed by rough signals”, in Revista Math. Iberoamericana. He discusses exactly this type of issue in some generality.

For example, parallel transport can be defined along any path of bounded variation: the limit which was discussed above, actually exists. Bounded variation is more or less the same as p-Holder for some p>1/2. This allows for some truly nowhere differentiable paths. It all goes back to some work of L.C. Young in 1930’s. The relevant paper is “An inequality of the Holder type connected with the Stiltjes integral”, Acta Math. 1936.

What’s interesting is that there are stochastic processes (called fractional Brownian motions) whose all paths have bounded variation, and so one can integrate the holonomy over the probability measure on the space of Holder paths, without appealing to stochastic integration. One can read more in the book of F. Baudoin “Geometry of stochastic flows”.

Posted by: Mikhail Kapranov on April 11, 2007 10:48 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

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.

Posted by: urs on April 12, 2007 6:14 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

It is neat to see the discussion moving to the realm of stochastic paths. This could be of interest in mathematical finance :)

Of course, Urs can guess where my thoughts stray when stochastic calculus pops up. If you were to construct an n-diamond representation of the base manifold, a point particle would trace out stochastic paths in the continuum limit.

What you are discussing is probably totally unrelated though :o

Posted by: Eric on April 12, 2007 7:56 PM | Permalink | Reply to this

Re: Fahrenberg and Raussen on Continuous Paths

Yes, I meant the parallel transport induced by an ordinary smooth connection on a smooth bundle. It is just paths that are allowed to be Holder.

On the other hand, one can imagine some classes of “Holder manifolds” which are in between topological and differentiable ones. For such mflds one can probably speak about Holder paths and Holder connections.

There may be some (p,q) duality, perhaps for (1/p)+(1/q)=1 as common in L_p studies, Holder’s conditions etc. Like if paths are p-Holder, a connection is a q-Holder, then we have a good transport.

We can also consider the tensor category C of connections in vector bundles on X with given regularity conditions, say, smooth, or p-Holder. Then every point x defines a tensor functor from C to Vect, and we can call a formal path from x to y a natural isomorphism of two such functors. I don’t know of any precise statement in this direction: that every such transformation comes from an actual path of some kind.


Posted by: Mikhail Kapranov on April 12, 2007 10:23 PM | Permalink | Reply to this

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