2-Connections on 2-Bundles

December 2, 2005

Posted by Urs Schreiber

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We have a new, prettified and more concise version of our work on 2-connections in 2-bundles, prepared for the proceedings of the Streetfest conference last summer.

J. Baez & U. S.
Higher Gauge Theory
math.DG/0511710

Compared to our previous preprint the discussion has been made more transparent. The interpretation of 2-transitions of 2-connections as 2-functor $p$ -morphisms is now included (as discussed here).

For a nice overview of the key ideas see the transparencies of the talk that John is giving next Sunday at the Union College Math Conference.

Update (May, 28, 2006): We have now received the (anonymous) referee report on that paper. Here it is

Report on the paper “Higher Gauge Theory” by John Baez and Urs Schreiber.

This paper is an introduction to ongoing research by the authors of higher dimensional generalization of gauge theory (see [BS]). If to think of gauge theory as the study of parallel transport of 0- dimensional objects (points) the higher gauge theory will deal with parallel transport of 1-dimensional gadgets (strings, paths). The authors “categorify” such standard ingredients of the gauge theory as structure group G, principal G-bundle, connection on it. For some of them category analogs are well know, for others the authors work out what they should be to fit nicely in the gauge picture. The testing ground is the “holonomy functor” which is the assignment to a path in the base of a principal G-bundle with connection the holonomy along it. The higher version of this is the holonomy of paths between paths taking values in the structure 2-group. It should be noted that the whole picture is “smooth” meaning that everything is “internalized” in the category of smooth spaces.

The paper is very well written. The definitions and constructions are carefully motivated and explained. The (mostly omitted) proofs can be found in the preprint [BS]. Aimed at “categorically oriented” mathematical public the paper will fit perfectly into the volume and should be published in its present form.

Posted at December 2, 2005 2:37 PM UTC

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Re: 2-Connections on 2-Bundles

Hi Urs,

The premise that gauge theory is a theory of transporting points along paths in spacetime and that higher gauge theory is about transporting paths along surfaces in spacetime is pretty neat and makes perfect sense. But if you had asked me to develop a mechanism for transporting paths, the first thing that would come to my mind would be to draw disjoint paths and then draw a rectangular sheet connecting the two paths.

This might be a silly question, but why does it seem like you are holding the ends of the paths fixed so that you get these (American) football shaped surfaces, i.e. why are you restricting to a Path 2-Groupoid? Is that necessary? That is not what you do on loop space, right?

Just curious :)

Eric

Posted by: Eric on December 3, 2005 6:43 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

why does it seem like you are holding the ends of the paths fixed

Very good question. We don’t talk about that in the paper, but maybe an exposition is due, since I am being asked this in every second talk I give.

Your suggestion to

draw disjoint paths and then draw a rectangular sheet connecting the two paths

is indeed very natural. It leads to a notion of higher category known as double category.

A double category itself is in a sense the most natural way to define a higher category. Namely a double category is a category object internal to the category of categories.

This means a double category is something with

- a category of objects (think of the morphisms in this category of objects as vertical arrows)

- a category of morphisms between these arrows (think of these as squares between two vertical arrows)

- source and target functors (which assign to a rectangular array of squares the vertical arrows forming their left and right boundary)

- a composition functor, which gives just the horizontal composition of squares

Now, the 2-categories which you have seen, where 2-morphisms don’t look like squares but like 2-dimensional american footballs (called bigons) are evidently a special case of double categories, namely one where all morphisms in the category of objects are identity morphisms! I.e., where all vertical arrows have really no extension.

Hence the question is if one looses available freedom by restricting from double categories to 2-categories.

The answer to that is no…

…under the assumption that

- the horizontal and the vertical arrows in the double category are ‘of the same sort’ (this case is called edge symmetry and is clearly what we would want for applications like in higher gauge theory)

- and that the double categories in question have what is called a thin structure.

In this case it is a theorem that working with double categories is ‘the same’ as working with 2-categories.

Of course, as always in category theory, if two structures are ‘the same’ it really means that their respective categories are equivalent. Hence the theorem is

The category of edge symmetric double categories with thin structure is equivalent to that of 2-categories.

That’s theorem 5.3. in

R. Brown & G. Mosa
Double Categories, 2-Categories, Thin Structures and Connections
Theor. Appl. Cat. 5 7 (1999) 163-175 .

All technical fine print can be found there.

The theorem is not original to that paper, but in this paper a nice and accessible and mostly graphical proof is given. The authors present a neat graphical plumbing-like notation for computation in double categories.

Since this is a highly technical issue, and due to the assumptions necessary to establish this, I cannot really exclude that one day one finds that for some aspects of string transport one needs double categories, after all.

Note though that in string transport, at the heurstic level, it’s all just about if we want to single out two or four arbitrary points on the boundary of a piece of worldsheet. Composition by degenerate line-like pieces of worldsheet allows to move these points around at will (this is called whiskering), so it should not matter.

Currently many people are looking at CFT in terms of 1-categories of bordisms. There objects are strings and 1-morphisms are worldsheets bounded by these strings. Stolz and Teichner have pointed out that this too coarse a picture. That one should take objects to be points, morphisms to be pieces of string and 2-morphisms to be pieces of worldsheet between these, such that the bordism picture is reobtained as a special configuration. They suggest hence to use 2-categories the way we are using them, too.

On the other hand, I have heard the rumour that one reason their work remains unpublished is the technical difficulty of how to cleanly define the horizontal composition of such bigons that carry addition structure, like conformal structure. Since horizontal composition in a 2-category takes place at a single point (instead of over smooth interval like it would be the case if we used double categories) it is non-obvious how to glue things like conformal structure in these points.

Hence maybe one should take this as an indicatation that double categories will be necessary after all.

I guess somebody would have to sit down and think about it…

Posted by: Urs on December 4, 2005 6:23 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Hi Urs,

Thanks for that explanation.

It has been ages since I looked at it and my memory isn’t that good to start with, but what you described as a double category gives a sense of deja vu. Did Ross Street do that ages ago? Or was he working with simplices?

I like to throw darts one after another. Mostly I hit nothing but the wall, but sometimes I hit something that scores a few points. So let me throw another dart out there. The picture that arises from your description of a double category seems to be essentially an n-diamond. We’ve (you’ve) already shown that lattice gauge theory works out in a completely transparent and simple way using discrete calculus on a diamond graph. It seems to me (and this is what I’ve been trying unsuccessfully to enunciate for a long time) that if we augment the gauge theory of our paper with group algebra value 2-forms, then we would have something that would seem to have the right to be call “discrete higher gauge theory” and we already know that it would have the correct continuum limit. I’m pretty sure that will work out, but I can easily imagine that it would simply reproduce the work others have done on higher lattice gauge theory. Nevertheless, you never know. It would be a fun exercise nonetheless. Unfortunately, it is probably beyond my meager abilities.

Cheers!
Eric

PS: My interest in this is not purely academic. Remember that email we received a long time ago about using techniques from higher gauge theory to model the yield curve? :)

Posted by: Eric on December 4, 2005 7:32 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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Did Ross Street do that ages ago?

Could be, don’t know. With a little luck John sees this and gives you a better reply.

All I know is that double categories were first identified by Ehresmann. He was the one who came up with the idea of internalization, where you externalize any concept by defining it in terms of abstract diagrams, and then internalize it in some category by thinking of all these diagrams as diagrams in that category. As I said, when you define a category itself by abstract diagrams[1] and then internalize these diagrams in $Cat$ , you get the concept of a double category. When you repeat this process you get $n$ -fold categories, or whatever they are called.

The picture that arises from your description of a double category seems to be essentially an n-diamond.

I see what you mean. On the other hand, a 2-morphism in a 2-category is in a sense closer to the concept of a 2-diamond, actually. Just draw the source and target morphisms of the bigon not as round arcs, but as segments of straight lines with a rectangular kink in the middle (it’s just notation, after all!).

I have some diagrams where I compute the surface holonomy of a torus in an orbifold using such ‘rectangular’ bigons and it does look precisely like the 2-diamond graph that you are probably thinking of. I cannot make these public yet, but as soon as I can I will show you.

if we augment the gauge theory of our paper with group algebra value 2-forms

Yes, true, that should work and should be interesting.

I am beginning, slowly but surely, to delve into synthetic constructions. Right now I am in the process of describing 2-connections on nonabelian bundle gerbes by ‘synthetically differentiated’ 2-functors. That involves essentially the step that you mentioned, where the graph in question is that of ‘infinitesimal neighbours’.

As soon as I have something presentable, I’ll show you the details.

using techniques from higher gauge theory to model the yield curve

Yes, I remember. I never understood if in that context one really wants to integrate along a path in the space of yield curves. Does one? Is there really a quantity associated to a fixed yield curve and one wants to know how this quantity changes as the yield curves is deformed?

If yes, then that might be a surface transport. Is it?

[1]

In case anyone wonders: by defining a category by abstract diagrams I mean the following:

a category $C$ is a tuple $(Obj (C), Mor (C))$ of abstract objects together with arrows

(1)

Obj (C) \overset{i}{\to} Mor (C)

(the identity morphism-assignment)

(2)

Mor (C) \overset{s, t}{\to} Obj (C)

(source and target)

(3)

Mor (C)_{t} \times_{s} Mor (C) \overset{\circ}{\to} Mor (C)

(composition of morphisms)

and diagrams like

(4)

\begin{matrix} Mor (C)_{t} \times_{s} Mor (C)_{t} \times_{s} Mor (C) & \overset{\circ \times Id}{\to} & Mor (C)_{t} \times_{s} Mor (C) \\ Id \times \circ ↓ & ↓ \circ \\ Mor (C)_{t} \times_{s} Mor (C) & \vec{\circ} & Mor (C) \end{matrix}

(associativity)

as well as diagrams expressing the property of the identity morphisms. When one thinks of this in $Set$ it is an ordinary category. When one thinks of this in $Cat$ one gets a double category.

Posted by: Urs on December 5, 2005 12:05 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Your suggestion to
>draw disjoint paths and then draw a rectangular sheet connecting the two paths
is indeed very natural. It leads to a notion of higher category known as double category.

Interesting. It sounds like double categories would be the right setting for 2-gauge theories in my sense. Not that it helps me much, since I find it very difficult to understand, and let alone formulate, the kind of stringent language used by mathematicians. With considerable awe and envy I observe how quickly young people like yourself absorb and internalize modern math.

- the horizontal and the vertical arrows in the double category are ‘of the same sort’ (this case is called edge symmetry and is clearly what we would want for applications like in higher gauge theory)
- and that the double categories in question have what is called a thin structure.
In this case it is a theorem that working with double categories is ‘the same’ as working with 2-categories.

So what is a thin structure, and why do we want it?

A quick glance in the Brown and Mosa paper reveals that a thin structure is equivalent to a connection pair. Perhaps you can formulate something like 2-gauge theories in the plaquette-gluing sense with double categories without connection pairs.

Posted by: Thomas Larsson on December 5, 2005 9:57 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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It sounds like double categories would be the right setting for 2-gauge theories in my sense.

Yes, maybe. You would probably want to take the little plaquettes that you are considering and then take the double category generated by these under composition. (Sort of the 2-analog of the path category of a graph.)

I might not fully recall the details of the construction that you have proposed. I guess you want a single object, 1-morphisms to be vector spaces, composition of 1-morphisms to be the tensor product of vector spaces and 2-morphisms to be linear maps between these vector spaces.

That should give a decent double category.

In fact, I think when you pass to the 2-category equivalent to this double category you end up with the monoidal category $Vect$ of vector spaces, but regarded as a 2-category with a single object. (Where the tensor product, as above, corresponds to horizontal composition.)

I have a pre-pre-print showing that 2-functors from surfaces to $Vect$ -regarded-as-a-2-category-with-a-single-object yield bundle gerbes with connective structure, when ‘pre-trivialized’.

So what is a thin structure, and why do we want it?

[…] a thin structure is equivalent to a connection pair.

A thin structure on a double category $D$ in this context is a morphism of double categories (a ‘double functor’) from the double category of commuting squares in the edge category of $D$ to $D$ itself.

In plain english this means that whenever four 1-edges of a double category form a commuting square, then there is at least one 2-morphism (square/plaquette) ‘filling’ this square, i.e. with these 1-morphisms as boundary.

This is a very natural requirement. It essentially says that all ‘identity 2-morphisms’ do exist.

Brown and Mosa re-express this condition in terms of connection pairs, but I think the word ‘connection’ here is not related to the same word as it appears in ‘bundle with connection’.

Posted by: Urs on December 5, 2005 12:30 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Eric wrote:

[…] why does it seem like you are holding the ends of the paths fixed […]?

Very good question. We don’t talk about that in the paper, but maybe an exposition is due, since I am being asked this in every second talk I give.

Hmm… nobody ever asked me about this until Stasheff just did in his email. Maybe mathematicians are more willing to accept this as an arbitrary detail of the formalism - and less interested in actual strings moving along, without their ends pinned down.

Anyway, it’s a good question. As Urs explained, if you prefer squares to bigons you can use double categories instead of 2-categories. But, there’s an obvious way to think of a square as a special sort of bigon. So, any 2-category gives a double category where the category of vertical edges is the same as the category of vertical edges.

So, I’m pretty sure it’s no big deal. You can translate easily between the two setups. 2-categories are simple; they can handle arbitrary polygons; so we use 2-categories.

But, I guess we’d really better explain this in our paper, at least briefly.

Urs writes:

Currently many people are looking at CFT in terms of 1-categories of bordisms. There objects are strings and 1-morphisms are worldsheets bounded by these strings. Stolz and Teichner have pointed out that this too coarse a picture. That one should take objects to be points, morphisms to be pieces of string and 2-morphisms to be pieces of worldsheet between these, such that the bordism picture is reobtained as a special configuration. They suggest hence to use 2-categories the way we are using them, too.

On the other hand, I have heard the rumour that one reason their work remains unpublished is the technical difficulty of how to cleanly define the horizontal composition of such bigons that carry addition structure, like conformal structure. Since horizontal composition in a 2-category takes place at a single point (instead of over smooth interval like it would be the case if we used double categories) it is non-obvious how to glue things like conformal structure in these points.

Stolz and Teichner published a version of their paper in the Festschrift for Graeme Segal. But, I’m not sure they’ve resolved this issue yet. I’ve spoken to both of them about it, and we’ve discussed the relative merits of 2-categories, bicategories and double categories - but in their setup, where the surfaces need conformal structures, all these approaches pose considerable technical challenges.

You explained the problem with 2-categories or bicategories; one problem with double categories is that composition of morphisms must be strictly associative, which forces one to use unparametrized edges, which (I think) makes gluing complex structures along these edges tricky.

I now suspect that Dominic Verity’s “double bicategories” may be helpful; I’m busy working with my student Jeff Morton on applications of these to extended TQFTs.

Posted by: john baez on December 5, 2005 3:52 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Any reason you can’t generalize to A-infinity structures?

Posted by: Aaron Bergman on December 5, 2005 4:01 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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Any reason you can’t generalize to A-infinity structures?

For me the reason is simply that I am not familar with them.

I have heard the basic idea (associativity up to $\dots$ up to $\dots$ , $\dots$ ), but not much more.

I’d be grateful for good references.

Meanwhile I would like to turn the question around. What would you think is the promising generalization of the following, in regard of string applications like Costello’s work and related stuff:

I know that an abelian bundle gerbe with connection is something obtained from a 2-functor from 2-paths to $Vect$ , where $Vect$ is regarded as a 2-category with a single object.

I am now also pretty sure that a nonabelian bundle gerbe with connection is something obtained from a 2-functor of the above sort where now $Vect$ is replaced by $BiTor (G)$ , the weak monoidal category of bitorsors over $G$ , also regarded as a 2-category, with horizontal composition the ‘tensor product’ of bitorsors and 2-morphisms being bitorsors morphisms.

This has an obvious generalization. Take any algebra $A$ and let the target for a 2-transport 2-functor of the above sort be the monoidal category of bimodules over that algebra, regarded as a 2-category with a single object.

This again has an obvious generalization: Allow for different objects and consider the 2-category whose objects are algebras $A$ (von Neumann type ${III}_{1}$ -factors, perhaps… ;-) , whose 1-morphisms are $A - B$ -bimodules and whose 2-morphisms are bimodule morphisms.

I guess we have talked about similar things before.

Now, all the 2-categories appearing here are weak in that the composition of their 1-morphisms is associative (only) up to 2-isomorphism.

Hence I am wondering. Where does $A_{∞}$ enter the game?

Posted by: Urs on December 5, 2005 4:41 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

For me the reason is simply that I am not familar with them.

I have heard the basic idea (associativity up to … up to …, …), but not much more.

I’d be grateful for good references.

Associativity up to homotopy (in the algebraic sense). In other words, there is a differential on the algebra and one has that the product is associative up to a commutator with this differential.

I’ve been trying to learn them from the notes on B. Keller’s webpage. There are a number of string articles on them. You can try stuff by Aspinwall, Lazaroiu or Tomasiello on the physics side.

As for the rest, I would think that an A_oo structure might show up when your homotopies only fit together up to homotopy.

Posted by: Aaron Bergman on December 5, 2005 5:25 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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there is a differential on the algebra

Hm, I know what an $A_{∞}$ -algebra is. So maybe an $A_{∞}$ -category is to an $A_{∞}$ -algebra just like an ordinary category is to an ordinary algebra (namely something that generalizes a monoid to something with many ‘objects’)?

notes on B. Keller’s webpage.

Thanks!!

Posted by: Urs on December 5, 2005 5:39 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

A category is secretly an algebra. IIRC, an A_oo category is a ‘category’ that’s secretly an A_oo algebra. A proper definition is in some of Keller’s notes.

I haven’t looked at it yet, but Costello recommended Seidel’s book, so you might want to check that out.

Posted by: Aaron Bergman on December 5, 2005 5:53 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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A category is secretly an algebra.

I guess what you have in mind is identifying a category with its ‘path algebra’? (Product of morphisms in the algebra being their composition in the category if defined and $0$ otherwise.)

Seidel’s book,

Thanks again.

Posted by: Urs on December 5, 2005 6:07 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Aaron writes:

A category is secretly an algebra. IIRC, an A_oo category is a ‘category’ that’s secretly an A_oo algebra.

Right. Any intelligent being who

1) knows about A-infinity algebras

and

2) knows how a category can be turned into an algebra,

can reinvent the definition of an A-infinity category using this analogy:

algebra : category :: A-infinity algebra : A-infinity category

About point 2), Urs asks:

I guess what you have in mind is identifying a category with its ‘path algebra’? (Product of morphisms in the algebra being their composition in the category if defined and 0 otherwise.)

Exactly! Personally I call this the “category algebra” of our category, since a special case is the familiar “group algebra” of a group - a group being a category with one object and all morphisms invertible.

But, “path algebra” is also a fine name, especially when the morphisms in our category are paths in some space. You considered examples like this in your work with Eric on discrete differential geometry.

Anyway, let’s summarize. We have inclusions

[groups] -> [categories] -> [A-infinity categories]

and the functor

GrpAlg: [groups] -> [algebras]

extends to a functor

CatAlg: [categories] -> [algebras]

which in turn extends to a functor

A-infinityCatAlg: [A-infinity categories] -> [A-infinity algebras]

Of course none of this will help the poor readers who are unfamiliar with A-infinity algebras, but those people should read Stasheff’s “operadchik” notes or some other intro to A-infinity stuff.

Posted by: john baez on December 7, 2005 5:37 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

If I understand Keller correctly, viewing the A_oo structure as an algebra over an operad has a stricter notion of a morphism than we want.

Posted by: Aaron Bergman on December 7, 2005 6:14 AM | Permalink | Reply to this

A_oo

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Exactly!

Thanks! It’s one thing to guess, another to see the guess being confirmed. :-)

Here is something I still cannot quite guess: what’s the relation between $A_{∞}$ -categories and $n$ -categories/ $ω$ -categories?

Do $A_{∞}$ -categories know about $n$ -categories? If not, do we need a concept of $A_{∞} - n$ -category?

I can turn this question around. I guess we can form the ‘algabra’ associated to a 2-category. It should be the vector space spanned by all 2-morphisms in the 2-category. On this space there should be two product structures, one coming from horizontal, one from vertical composition of 2-morphisms, such that they satisfy the exchange law.

Can one make such a 2-category-algebra $A_{∞}$ ??

I’d guess so. Horizontal and vertical composition each give an algebra for themselves, which could be replaced by $A_{∞}$ -algebras each. Hence it does not sound unreasonable that a 2-category- $A_{∞}$ algebra would be a pair of two $A_{∞}$ -algebra structures on a given space such that an $A_{∞}$ version of the exchange law holds.

Or something like that.

Posted by: Urs on December 7, 2005 9:53 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Urs wanted some references on the A-infinity operad and its applications. But, for gluing Riemann surfaces with corners, Stolz and Teichner probably need some other operad. So, it’s probably good to learn more generally about operads and their many applications to string theory.

So, here’s what I recommend:

James Stasheff, Hartford/Luminy talks on operads, available at http://www.math.unc.edu/Faculty/jds/operadchik.ps

and for more detail:

Martin Markl, Steve Schnider and Jim Stasheff, Operads in Algebra, Topology and Physics, AMS, Providence, Rhode Island, 2002.

Stasheff was essentially the one who invented the A-infinity operad (though Peter May invented the general concept of “operad”), so you’ll see a lot about that example here, together with a bunch more that are closely related to string theory.

Unfortunately, the book gives a definition of operad with one axiom missing - it’s not really a big deal, but it’s slightly embarrassing. You can find the missing axiom and my own mini-introduction to operads here:

John Baez, This Week’s Finds in Mathematical Physics, week191

From an n-categorical viewpoint, the A-infinity operad is how you systematically figure out the pentagon identity satisfied by the associator in a monoidal category, the next identity satisfied by the “pentagonator” in a monoidal 2-category, and so on… forever!

Posted by: john baez on December 6, 2005 5:51 PM | Permalink | Reply to this

A_infty

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Thanks for all the references.

I am busy right now, so I haven’t yet looked at much of this material. But in fact I think I do know what an operad is, we recently had a seminar about it. I also know what an $A_{∞}$ -algebra is.

What I still need to find out is what an $A_{∞}$ -category is!

Given Aaron’s comments I can of course roughly guess the idea.

Posted by: Urs on December 6, 2005 5:59 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Aaron wrote:

John Baez wrote:

You explained the problem with 2-categories or bicategories; one problem with double categories is that composition of morphisms must be strictly associative, which forces one to use unparametrized edges, which (I think) makes gluing complex structures along these edges tricky.

Any reason you can’t generalize to A-infinity structures?

The A-infinity operad describes all possible ways of composing paths, taking reparametrization into account, so it’s relevant. But Stolz and Teichner really need something more: maybe an operad that describes all possible ways of gluing together “Riemann surfaces with corners” to get new Riemann surfaces. This would probably involve keeping track of the parametrizations of the surfaces’ edges, but also much more.

People have developed lots of operads for describing the gluing of Riemann surfaces, but not yet (as far as I know!) involving corners.

Posted by: john baez on December 6, 2005 5:17 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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I’d be curious what this extra structure is. Should I think of the corners because you want the boundaries to have boundaries?

I’m not sure why you want an operad for the whole gluing shebang? All you need to do is to define the source (bordism) category for the (C|T)FT functor, right? I would think the $A_{∞}$ structure would help you along in gluing the parametrized boundaries, but you wouldn’t need such a thing for gluing the Riemann surfaces together.

Posted by: Aaron Bergman on December 7, 2005 5:55 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

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Should I think of the corners because you want the boundaries to have boundaries?

I’d think so. Like in section 3.3 of Lauda&Pfeiffer.

I’m not sure why you want an operad for the whole gluing shebang? All you need to do is to define the source (bordism) category for the (C|T)FT functor, right? I would think the $A_{∞}$ structure would help you along in gluing the parametrized boundaries, but you wouldn’t need such a thing for gluing the Riemann surfaces together.

The issue is that in constructions like those by Stolz&Teichner one does not work with a cobordism (1-)category. Instead this 1-category is refined to a 2-category of bigons.

Posted by: Urs on December 7, 2005 10:17 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Aaron wrote:

Should I think of the corners because you want the boundaries to have boundaries?

Right. Segal originally hoped that elliptic cohomology amounts to a way of studying a topological space X by mapping Riemann surfaces with boundary into X. Such maps organize themselves into a category, and he hoped that a cycle in elliptic cohomology was a nice functor from this category into Vect. In the case where X is a point, such nice functors are just conformal field theories. In general, Segal called them elliptic objects.

But, Stolz and Teichner noticed that you can’t prove the excision axiom for elliptic cohomology if you use this approach! The excision axiom is one of the axioms for a generalized cohomology theory; it says how the cohomology of a big space X is related to the cohomologies of little open sets covering X.

The problem is that when you have a map from a Riemann surface with boundary into X, it doesn’t give maps from smaller Riemann surfaces with boundary into the little open sets covering X. Why not? Because when you chop a Riemann surface with boundary into pieces, the pieces aren’t Riemann surfaces with boundary. At the very least, they may have corners.

So, we can’t work with a mere category whose morphisms are Riemann surfaces with boundary, where composition of morphisms describes gluing these surfaces together.
We need some fancier algebraic structure whose operations describe ways of gluing together Riemann surfaces with corners.

An obvious candidate is a 2-category, bicategory or double category, since these structures were invented precisely to describe ways of gluing together 2d surfaces! Urs and I successfully used 2-categories for this purpose in our work on higher gauge theory.

However, in elliptic cohomology, the complex structures on our Riemann surfaces with corners must be taken into account. This, and some other issues, lead to some gnarly technical challenges.

So, when you suggested using the A-infinity operad, I got the idea of using not that but some other operad… because people do use lots of operads to describe lots of different ways of gluing together Riemann surfaces (aka “complex curves”).

But I’ve never seen people do this for Riemann surfaces with corners, so I think there’s some research left to do. But, I haven’t read the latest version of Stolz and Teichner’s paper… so maybe they have already solved the problem of formalizing “elliptic objects”.

Posted by: john baez on December 7, 2005 5:12 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

I’m still confused. Conceptually I don’t see any problem with writing down an almost 2-category with parametrized boundaries. It just seems to me that the composition (concatenation of boundaries, I assume) won’t be associative. Insetad, it will be A_oo.

Now, the technical details of doing this might be scary, but is there anything wrong with the concept?

Posted by: Aaron Bergman on December 7, 2005 11:48 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

No, nothing wrong with the basic concept.

Posted by: John Baez on April 2, 2010 6:50 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

$MathML-enabled post (click for more details).$

Hi Professor Baez,

It is good to cross paths again. I’m in your neighborhood (Santa Monica), so maybe we can get together sometime. I know my coworker (Ph.D. involved category theory as applied in computer science) has talked about trying to get some people together. You might be surprised to know how many mathematicians are working in finance these days :)

Regarding $A_{∞}$ algebras, an obvious place where they come up is in an area close to my heart: discrete geometry. Dennis Sullivan has formulated a theory of discrete geometry that is, in a sense, dual to what Urs and I did. I think it is a metatheorem that if you try to construct a theory of discrete geometry, you need to make a choice. You need your algebra to be either noncommutative or nonassociative. If you go the noncommutative route, you end up with a graded algebra that is associative, but not commutative. We didn’t show it, but I am pretty sure we could, that the failure to be commutative is similarly “up to homotopy (in the algebraic sense)”. If you go, as Sullivan did, with retaining commutativity and giving up associativity, you end up with an $A_{∞}$ algebra. The algebra we ended up with is probably closely related, maybe $C_{∞}$ (not to be confused with differentiability) :)

I think even Urs has underestimated the potential of the paper we wrote together. I think it is great to become familiar with the synthetic stuff, but in the end, I think the noncommutative route is probably more natural (and resonates more squarely with QM).

Cheers,
Eric

Posted by: Eric on December 5, 2005 7:09 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

By the way, regarding Sullivan’s work. I talked about a recent thesis of one of his students here.

Here is a direct link to the thesis

On the Algebra and Geometry of a Manifold’s Chains and Cochains

Eric

PS: If anyone has a copy of Sullivan’s preprint, “Local Constructions of Infinity Structures” and wouldn’t mind sending me a copy, I’d appreciate it. I suspect he details this in the preprint but haven’t seen it. I only know because I heard him present the material in a talk.

Posted by: Eric on December 5, 2005 7:34 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

On second thought, double categories are not related to 2-gauge theories in my sense. The crucial difference between our approaches is the group that acts on surface holonomies. In your case, it is a finite-dimensional group H (or maybe you call it G); in my case, it is a loop group LG.

It was always clear that my surface holonomy was well defined on the lattice; it is obtained by multiplying plaquette holonomies and is independent of the order that internal edge indices are contracted. However, the relation to the local continuum formulation was not completely clear, because a non-local continuum formulation was missing. The problem is that if the boundary of a lattice surface consists of n links, its holonomy is valued in the n-fold tensor product V^n, where V carries a G rep. It was unclear to me how to take the continuum limit of such a “barbed wire”.

Fortunately, some time ago I came a cross the solution in another context, in the famous history paper by Isham and Linden, I think. The important thing about V^n is that it carries a rep of the n-fold tensor product G^n. The continuum limit of G^n is the loop group LG (in higher dimensions the manifold group MG). Hence the continuum limit of V^n is the LG module LV, and the continuum surface holonomy is an element in LV.

This definition endows the surface holonomy with all the necessary properties. If we reparametrize the surface, LG and LV are reparametrized accordingly. The holonomy has the right gluing properties, which is proved in Figure 2 of math-ph/0205017. This figure was drawn for the lattice, but applies to the continuum as well. We can split the big surface into two parts (the two middle graphs), but the holonomy is the same because it is the product of the four small surfaces. The only difference is that on the lattice we contract finite-dimensional indices, whereas in the continuum we trace over paired loop group modules, i.e. we contract indices and integrate over the internal path.

Finally, we can make explicit the connection with the local continuum theory in the end of my paper. In 1-gauge theory, an infinitesimal Wilson line is completely determined by its position and direction, which is encoded in the gauge potential A_u(x). In 2-gauge theory, we need not only the position and direction on an infinitesimal Wilson surface, but also the direction s of string transport along the surface, hence the 2-gauge potential is a field A_uv(x,s) (if string length is not preserved, further data is needed). It is now clear that my expression for the field strength must be right, since it is the unique expression compatible with covariance, which is manifest in view of the global continuum formulation.

To summarize the differences (I may have misremembered or misunderstood something about your theory):

1. Your boundaries lines live in a finite-dimensional group G, and the product of two lines in G is a new line in G. My boundaries live in LV, and the join of two boundaries live in the LV associated with the combination.

2. Your surface holonomy lives in a finite-dimensional group H acting on G, mine lives in a loop group LG acting on LV.

3. You get some stringent conditions on G and H, I get no conditions on LG or LV, except that LV carries an LG rep.

4. I think you may perhaps run into problems with locality, since it is unclear where on a finite line its finite-dimensional group lives. In contrast, a loop module LV has a vector space at each point of the line, and locality is manifest.

So we are really doing different things, although both are well-defined higher analogues of 1-gauge theory.

Posted by: Thomas Larsson on December 6, 2005 10:50 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

$MathML-enabled post (click for more details).$

double categories are not related to 2-gauge theories in my sense

I think they are. You are essentially looking at 2-functors from surfaces/plaquettes to $Vect$ , regarded as a 2-category.

In your case, it is a finite-dimensional group $H$ (or maybe you call it $G$ ); in my case, it is a loop group $L G$ .

No, there is no requirement for the structure 2-group to be finite dimensional. Recall for instance the 2-group related to the String group.

But yes, for vector spaces of dimension different from 1 and from $∞$ something loopy will appear when you look at functors from surfaces to $Vect$ , like in your idea. This is where things become technical. We have talked about that before.

The case where all vector spaces (associated to edges of surfaces/plaquettes) are restricted to be 1-dimensional is that of abelian bundle gerbes. I expect that the case where all vector spaces are infinite dimensional has a relation to nonabelian $P U (∞)$ -bundle gerbes, but I haven’t worked out the details yet.

The case where the vector spaces have arbitrary dimension is technically tricky in the continuum, because here the dimension of the vector space grows with the length of an edge. There are also no weak inverses, the dimension can only grow, not shrink.

proved in Figure 2

That’s essentially the exchange law when you think of 2-functors from plaquettes to $Vect$ .

You get some stringent conditions on $G$ and $H$ , I get no conditions

These conditions (you are thinking of fake flatness) are the source/target matching condition. For 2-functors from surfaces to vector spaces this ‘fake flatness’ says that a 2-morphism from an $n$ -dimensional vector space to an $m$ -dimensional vector space must be, after a choice of basis, an $n \times m$ matrix.

Posted by: Urs on December 7, 2005 10:38 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

No, there is no requirement for the structure 2-group to be finite dimensional.

OK, my point was rather that your group can be finite-dimensional.

In my case on the lattice, a group G acts on each link. It is not strictly necessary that it is finite-dimensional, but I think about it that way. The n-fold tensor product G^n acts on the surface holonomy with n boundary links. In the limit n -> infinity, this becomes the loop group LG. So my group acting on the continuum surface holonomy is always infinite-dimensional. That’s a difference.

Actually, we can identify an n-fold tensor product with a loop group (or loop module) even if n is finite; consider loops which are piecewise constant, with n pieces. I had this idea right after I wrote my previous post, so I don’t really know what to do with it yet, but it feels potentially fruitful. Maybe one can reformulate the Yang-Baxter equation.

Posted by: Thomas Larsson on December 8, 2005 11:17 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

So my group acting on the continuum surface holonomy is always infinite-dimensional. That’s a difference.

I’d call it a special case, not a difference.

Posted by: Urs on December 8, 2005 11:53 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

No, I don’t think so. My groups are not just some arbitrary infinite-dimensional groups, but specifically loop groups (or rather path groups since there are endpoints, but I will denote them LG anyway). The important thing about path groups is that LG ⊗ LG can naturally be identified with LG itself, and the same for their modules. Namely, we can identify u(t) ⊗ v(t) with

u(2t), 0 <= t < 1/2
v(2t-1), 1/2 <= t < 1

This is what I’m using to glue adjascent strings together. It does not any special properties, since all I do is tensoring. In particular, the elements on the edges do not need to form a group.

In contrast, G ⊗ G is certainly different from G for a finite-dimensional group G. Hence, if you want that you get the same type of object with gluing edges, you must demand that the edge variables live in a group. Or perhaps something slightly weaker, but you need more than just ability to take the tensor product.

Hence, by allowing the groups líving on paths to be finite-dimensional, you are in fact imposing strong constraints on the theory.

PS. I hope ⊗ reads as tensor product - it looks like a square to me.

Posted by: Thomas Larsson on December 8, 2005 4:34 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

This was almost, but not quite, right - I only started to think about loop groups a couple of days ago. For the record, let me correct myself.

For any vector space V, let LV(p) be the space of functions from the interval [0,p[ -> V, and let LV be the union of LV(p) for all p. LV is naturally a semigroup, with the product defining a map LV(p) x LV(q) –> LV(p+q) by

u*v(t) = u(t), 0 <= t < p
= v(t-p), p <= t < p+q

This product is evidently associative, so LV is a semigroup. No assumptions are required about V for this.

This is really a very natural object from a 2-gauge point of view. If the boundary of a surface splits into an “in” side of length p, and an “out” side of length q, the surface holonomy is an operator LV(p) –> LV(q).

I am somewhat confused why I used tensor products in my previous post, though; the loop product * is the Cartesian product, isn’t it?

Posted by: Thomas Larsson on December 9, 2005 7:27 AM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

Thomas writes:

[…] double categories are not related to 2-gauge theories in my sense […]

Urs responds:

I think they are. You are essentially looking at 2-functors from surfaces/plaquettes to Vect, regarded as a 2-category.

Good point! Actually a bicategory….

To expand on this slightly: Vect is a monoidal category with its usual tensor product. A monoidal category is a bicategory with one object. So, Larsson’s notion of connection is mapping

0) points of a lattice to the one object in Vect regarded as a bicategory

1) paths in the lattice to vector spaces (which are morphisms in Vect regarded as a bicategory)

2) paths-of-paths in the lattice to linear operators (which are 2-morphisms in Vect regarded as a bicategory)

Part 0) contains no information since there’s only one way to do it.

Composition of paths corresponds to tensoring vector spaces - and the interesting thing (which Urs notes) is that tensoring vector spaces is not usually invertible.

So, Thomas is giving an example of some kind of “2-connection” where the holonomies take values in a monoidal category that’s not a 2-group!

On sci.physics.research, Toby Bartels and Miguel Carrion studied gauge theory where instead of a gauge group one had a “gauge monoid”. I’ve wondered about that for a long time - wondered if that sort of thing was ever actually interesting. Now we’re seeing a categorified version which seems pretty natural and interesting!

Posted by: john baez on December 7, 2005 5:40 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

1) paths in the lattice to vector spaces (which are morphisms in Vect regarded as a bicategory)

I don’t think that it is meaningful to regard my paths as morphisms, since the don’t really have an orientation. My understanding is that in every kind of categorical approach, the edges always come with little arrows, and the surfaces with fatter arrows. This is because the edges are morphisms, which act on points, and the surfaces are higher morphisms. In contrast, my edges don’t have any arrows on them, only the surfaces. My edges don’t act on points; in fact, the don’t act at all. They just sit there, as passive pieces of a string, until they are acted upon by the surface holonomy.

But then again, I don’t really have a feeling for category theory, so maybe I’m wrong.

Posted by: Thomas Larsson on December 8, 2005 3:08 PM | Permalink | Reply to this

Re: 2-Connections on 2-Bundles

$MathML-enabled post (click for more details).$

I dont think that it is meaningful to regard my paths as morphisms, since the dont really have an orientation. My understanding is that in every kind of categorical approach, the edges always come with little arrows, and the surfaces with fatter arrows.

This is not an issue if you think of the paths as morphisms in a (strict) 2-groupoid - they are all invertible and so orientation takes a back seat.

My edges don’t act on points; in fact, the dont act at all. They just sit there, as passive pieces of a string, until they are acted upon by the surface holonomy.

In that case take the fundamental 2-groupoid $Π_{2} (L)$ for $L$ your lattice, and this is the `domain’ of the surface holonomy 2-functor

$Hol : Π_{2} (L) \to G$

for $G$ the group, 2-group or whatever the holonomy is valued in.

Posted by: David Roberts on December 9, 2005 4:11 AM | Permalink | Reply to this

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December 2, 2005