Kock on 1-Transport

September 8, 2006

Posted by Urs Schreiber

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A few months ago, I had a very inspiring e-mail conversation with Anders Kock on the concepts of connection, parallel transport and holonomy.

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I had learned of Anders Kock’s work through the central role it played in Breen & Messing’s work on connections in nonabelian gerbes in terms of combinatorial differential forms.

Since John and I were thinking about another approach # describing such 2-transport, I was interested in understanding which use we could make of synthetic reasoning.

From our point of view, a principal $n$ -connection should be a suitably well-behaved $n$ -functor with domain some $n$ -category of “ $n$ -paths in the base” and target some $n$ -groupoid (of morphisms between fibers of an $n$ -bundle, for instance).

At that time we were mainly concentrating on modelling $n$ -paths as thin-homotopy classes of smooth maps from $n$ -cubes into the base space. For $n=1$ it is an old theorem that, with a suitable notion of smoothness and target the transport groupoid of a principle bundle, such functors are in bijection with connections on that bundle.

We were essentially looking for a categorified version of this statement. But it was rather obvious that working with these classes of $n$ -paths up to thin homotopy was rather cumbersome.

So, I was thinking, it would be nice if there were a description of smooth $n$ -transport which was closer to the way physicists handle such things intuitively anyway, namely in terms of “small infinitesimal straight pieces” of $n$ -paths. It seemed to me that the formalism used in particular by Anders Kock might be a way to make that approach precise.

Apparently partly in reaction to this motivation, Anders Kock has now written up a paper which is intended as a first step in this program:

Anders Kock
Connections and path connections in groupoids
preprint.

As a preparation for underanding the case $n \gt 1$ , this work formalizes the notion of a connections in a fiber bundle, and the notion of holonomy of a connection, in synthetic differtial geometric terms.

The main concept that Anders Kock introduces and uses is that of a connection as a morphism of graphs.

The idea is (in my heuristic paraphrasing) to pass from smooth transport functors on categories to their differential version, which are mere morphisms on graphs. Heuristically, by passing to infinitesimals, we forget how to compose morphisms, since such composition is a first step towards “integrating” many infinitesimal paths to one composite finite path.

Please take notice that this is not how Anders Kock describes it. But I do think it gives the right idea of the spirit of his definitions.

A graph map need not respect composition (when applied to the underlying graph of a category), but it may. If it does, we say it is flat.

This is best understood my noting that what played the role of the groupoid of thin-homotopy classes of paths in my above description, is now, roughly, the graph of “infinitesimally close points” in the base space. A connection should associate to each such pair the infinitesimal parallel transport along a small path between the two points.

While such graph morphisms describing connections usually are not flat, they all become flat (under sufficiently nice conditions) when pulled back to finite paths.

Namely, we may imagine mapping the interval into our base space and pulling all the structure we had over there back to our interval. Any bundle will thus become trivial (trvially so). Its connection necessarily becomes flat, meaning that the pulled-back connection graph morphism now in fact is an honest functor.

Anders Kock calls this a partial integration of the connection along the given path. It gives rise to the notion of holonomy or parallel transport along finite paths. Thus one makes contact with the description of connections that I described at the beginning.

I am very much delighted to see these ideas in (electronic) print. I am sure there is a nice generalization of these concepts that allows one to treat smooth $n$ -transport for $n \gt 1$ in the same fashion

In fact, I have thought about this issue, too, since then.

A while ago I made a remark on how the results of my work with John ( $n=2$ ) should look like in synthetic language.

There are meanwhile also more detailed notes on this here.

A general remark I would like to make in closing is this:

I believe that an equivalent way to think of morphisms of graphs, with domain the (infinitesimal) pair groupoids, with target a given groupoid, is to think of pseudofunctors to a 2-groupoid extending the original groupoid.

I have more details on that idea here.

Posted at September 8, 2006 4:48 PM UTC

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7 Comments & 3 Trackbacks

Re: Kock on 1-Transport

I think you know what I am thinking, but I will spare the electrons :)

Posted by: Eric on September 9, 2006 5:39 AM | Permalink | Reply to this

synthetic Dirac operators

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I make some comments on what Eric is alluding to.

We have a pretty good idea how to grasp the concept of a connection in terms of functorial language.

Quillen suggested to think of Dirac operators as quantized connections.

For many applications, notably in physics #, we are interested in connections mainly in their incarnation as Dirac operators.

What is a good synthetic concept of Dirac operators?

Consider this:

(The following is pseudocode. It won’t compile.)

Let $X$ be some collection of points. Let there be a unique 1-morphism

(1)

x \to y

whenever $x$ and $y$ are infinitesimally close.

Let there be a unique 2-morphism for every infinitesimal square

(2)

\array{ x &\to & y_1 \\ \downarrow &\Downarrow& \downarrow \\ y_2 &\to & z } \,,

and so on for higher dimensional hypercubes.

A square as above is uniquely determined, in particular, by its source 1-morphism

(3)

\array{ x &\to & y_1 \\ && \downarrow \\ && z } \,.

In the same way, the inverse 2-morphism of this square is specified by its source morphism

(4)

\array{ x \\ \downarrow \\ y_2 &\to & z } \,.

Using this abbreviated notation for 2-morphisms, we have

(5)

\array{ x &\to & y_1 \\ && \downarrow \\ && z } = \left( \array{ x \\ \downarrow \\ y_2 &\to & z } \right)^{-1} \,.

In particular, if $y_1$ and $y_2$ coincide we have a “degenerated 2-cell” which is an identity 2-morphism. Let me write

(6)

\array{x &\to & y &\to& z } := \mathrm{Id}_{\array{x &\to & y &\to& z }} \,,

where, by the above abuse of notation, on the left the morphism is shorthand for a 2-morphism.

Using this notation suggests to equip the collection of morphisms with the structure of a graded algebra over some field. The product is by concatenation. It sends an $n$ -morphism and an $m$ -morphism to an $(n+m)$ -morphism.

For instance (the only case I can draw without providing you a pdf) the product of the 1-morphism

(7)

\array{ x \to y_1 }

with the 1-morphism

(8)

\array{ y_1 \\ \downarrow \\ z }

is the 2-morphism from above.

Notice that this is not commutative. The above two morphisms don’t match in the reverse order, so the reverse product vanishes.

So the algebra I am describing here has a vague resemblence to the category algebra of 1-morphisms, but it is quite different.

It is something like an algebra of differential forms under the wedge product.

There is naturally an operator of degree 1 acting on the algebra, namely the supercommutator with the “sum of all” 1-morphisms.

This operator is like the exterior derivative on differential forms.

Moreover, in the same vein there is an analogue of the Hilbert space of differential forms with inner product the Hodge inner product $\int \alpha \wedge * \beta$ .

Taking the sum of our “exterior derivative” with its adjoint under this inner product, we obtain a notion Dirac operator in our context.

Now finally, the relation to “quantized connections”:

We should think of the above algebra as an algebra of “functions” from $n$ -morphisms to our field $K$ .

Better, maybe, we should think of these “functions” as morphisms of $n$ -graphs. Domain is the $n$ -graph of 1-, 2-,… $n$ -morphisms. Codomain is the underlying $n$ -graph of the $n$ -group which looks like $K$ in each degree.

So in particular, we may consider a trivial connection in a trivial $K^\times$ -bundle as corresponding to the element of our algebra which sends every 1-morphism to $1 \in K$ . Let me write $\nabla_0$ for this connection.

Then, by the above, the supercommutator with $\nabla_0$ is the exterior derivative.

Next, we want to pass from the abelian $n$ -group with $K$ in each degree to more general codomains.

I’ll just indicate some aspects while restricting to $n=2$ .

So let $\nabla$ be a connection on a trivial $G$ -bundle over $X$ . Let $G \stackrel{\mathrm{Id}}{\to} G$ be the strict 2-group of the respective crossed module.

$\nabla$ is the 1-functor given by

(9)

(x\to y) \mapsto (\bullet \overset{\mathrm{tra}_\nabla(x,y)}{\to} \bullet) \,.

Forming the supercommutator of $\nabla$ with itself, we get the 2-functor

(10)

\left( \array{ x & \to & y_1 \\ \downarrow &\Downarrow & \downarrow \\ y_2 & \to & z } \right) \mapsto \array{ \bullet & \overset{\mathrm{tra}_{\nabla}(x,y_1)}{\to}& \bullet \\ \downarrow &\Downarrow \mathrm{tra}_\nabla(x,y_2,z,y) & \downarrow \\ \bullet & \overset{\mathrm{tra}_{\nabla}(y_2,z)}{\to}& \bullet } \,.

Everything is understood to first order in infinitesimal quantities.

So we now again form a Hilbert space as above, take the adjoint of $[\nabla,\cdot]$ and obtain something like Dirac operator this way.

Sorry, this was very sketchy.

Some of the pictures I tried to draw are like those at the end of my $n$ -curvature notes (pdf, html.)

Posted by: urs on September 9, 2006 7:51 PM | Permalink | Reply to this

Re: synthetic Dirac operators

For what its worth, my opinion is that the graph stuff we worked on provides a simple framework that contains enough structure to be able to work out n-transport stuff. It has a nicely defined continuum limit in which you should expect things to work out transparently. Synthetic geometry seems to provide an interesting middle ground between the continuum and discrete. My goal all along has been trying to develop a robust “discrete” framework in which to think about spaetime. Too bad I’m not smart enough, but what keeps me from giving up is little more than a belief that things should be simple and from our experience, formulating things discretely provides massive simplifications, e.g. even I can understand it.

Posted by: Eric on September 9, 2006 8:22 PM | Permalink | Reply to this

Re: synthetic Dirac operators

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a simple framework that contains enough structure to be able to work out $n$ -transport stuff

Yes. I tried to indicate some steps that need to be taken to fully realize this.

Back then, we didn’t really get a handle on the category-theory behind it. We made the right observation on how gauge connections and curvature 2-forms arise, but didn’t even ever think of identifying our “graph operator” with a functor, properly.

In a sense, we were lucky that the 2-group $G \stackrel{\mathrm{Id}}{\to} G$ is so simple.

I have been beginning to make attempts to improve on that situation. My above comment was supposed to give some hints towards what I believe is needed on top of what we already had.

So, first of all, what is required is an understanding on how the discrete $p$ -forms that we used were really some sort of maps with domain $p$ -morphisms of a $n$ -category of $n$ -cubes.

The all-important condition

(1)

\array{ x &\to & y_1 \\ && \downarrow \\ && z } \;\; = \;\; - \;\; \array{ x \\ \downarrow \\ y_2 &\to & z } \,.

that we used is then really seen to be an expression for the fact that our $n$ -category of $n$ -paths is a strict $n$ -groupoid

(2)

\array{ x &\to & y_1 \\ && \downarrow \\ && z } = \left( \array{ x \\ \downarrow \\ y_2 &\to & z } \right)^{-1} \,,

as I have tried to indicate above.

Taking this point of view, and multiplying discrete differential $n$ -group-valued differential forms in this sense by using horizontal composition of the image of the respective functors, does indeed yield a very nice calculus for computation of quantities in $n$ -transport theory.

I tried to indicate that, too, but I expect what I wrote was too cryptic to be intelligible.

For a slightly better impression of what the sentence above should mean, you might have a look at the figure 1 of some notes that I just typed:

tiny notes on tiny cubes

The nice thing about our formalism is, then, that it indicates a way how to pass from connections to “quantized connections”, i.e. to Dirac operators. That, too, needs to be described in more detail.

I think there should be a nice description of 2-Dirac operators along these lines by passing from the strict $n$ -groupoid of $n$ -paths to the $(n-1)$ -groupoid of $n$ -paths by forgetting 1-morphisms. This makes little squares the elementary building blocks, which give rise to something as 1-forms on path space. And in two different ways so, which should account for the fact that 2-Dirac operators come in pairs.

All this needs to be spelled out.

Posted by: urs on September 11, 2006 3:24 PM | Permalink | Reply to this

Re: synthetic Dirac operators

See? Wasn’t that easy? Even I understood it :)

Posted by: Eric on September 14, 2006 7:57 AM | Permalink | Reply to this

Re: Kock on 1-Transport

Dear Anders,

I am quite intrigued by your Connections…in groupoids. Long ago I worked at the opposite extreme: parallel transport in fibrations, unique only up to strong homotopy.
Also p.5 (4) reminds me of H. Cartan’s discussion of characteristic classes in terms purely of algebra; flatness being a morphism of …
Unique partial integrals are precisely the main difference between the smooth case with connection and the non-unique path lifting in fibrations.

Posted by: jim stasheff on September 10, 2006 2:40 AM | Permalink | Reply to this

Re: Kock on 1-Transport

Note for readers that Jim Stasheff put Homotopy Transition Cocycles on the ArXiv today.

Posted by: David Corfield on September 11, 2006 3:42 PM | Permalink | Reply to this

Read the post Quantum n-Transport
Weblog: The n-Category Café
Excerpt: An attempt to understand the path integral for an n-dimensional field theory as a coproduct operation over transport n-functors.
Tracked: September 14, 2006 2:12 PM

Read the post Differential n-Geometry
Weblog: The n-Category Café
Excerpt: A quest for arrow-theoretic differential geometry.
Tracked: September 20, 2006 9:18 PM

Read the post Kock on Higher Connections
Weblog: The n-Category Café
Excerpt: Anders Kock gives a synthetic differential description of parallel n-transport using strict n-fold categories.
Tracked: May 31, 2007 12:38 PM

The n-Category Café

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September 8, 2006