## July 5, 2005

### 2-Holonomy in Terms of 2-Torsors

#### Posted by Urs Schreiber

One thing that is sort of obvious, but which has not been written out in detail yet, is that the global 2-holonomy 2-functor defined in terms of local trivializations as discussed here more intrinsically is a 2-functor

(1)$\mathrm{hol}:{𝒫}_{2}\left(M\right)\to {G}_{2}-2\mathrm{Tor}$

from 2-paths (surfaces) in the base manifold $M$ to the 2-category of ${G}_{2}$-2-torsors.

A discussion of this can be found here:

$p$-Functors from $p$-Paths to $p$-Torsors
http://www-stud.uni-essen.de/~sb0264/Transport.pdf

as well as in section 12.4 of this.

Speaking of which, I would also like to mention here that no less than three of the early-morning talks at the Streetfest will be about aspects of categorified gauge theory.

Michael Kapranov will talk about Noncommutative Fourier-Transforms, Chen’s iterated integrals and Higher-Dimensional Holonomy. In the abstract he says:

[…] Then we explain how to extend this correspondence to represent higher-dimensional membranes by elements of a certain differential graded algebra $A$. This is related to the concept of holonomy of gerbes that attracted a lot of attention recently. We will also give the interpretation of higher gerbe holonomy in terms of Chen’s iterated integrals of forms of higher degree with coefficients in Lie-algebraic analogs of crossed modules and crossed complexes. If one views higher holonomy as a “pasting integral” then 2-dimensional associativities translate into vanishing of some brackets in the structure dg-Lie algebra which is automatic in the crossed module case but has to be imposed in general.

I am eager to learn more about the details he has in mind. This must be closely related to what I have been thinking about lately.

Then, Ezra Getzler will talk about Lie theory for ${L}_{\infty }$-algebras.

The Deligne groupoid is a homotopy functor from nilpotent differential graded Lie algebras concentrated in positive degree to groupoids. We generalize this to a functor from nilpotent ${L}_{\infty }$-algebras concentrated in degrees $\left(-n,\infty \right)$ to $n$-groupoids, or rather, to their nerves. On restriction to abelian differential graded Lie algebras, i.e. chain complexes, our functor is the Dold–Kan functor. The construction uses methods from rational homotopy theory, and gives a generalization of the Campbell-Hausdorff formula.

This is presumeably based on

E. Getzler
Lie theory for nilpotent ${L}_{\infty }$-algebras
math.AT/0404003

As far as I understand, this is about the problem of integrating weak (semistrict, really) Lie $p$-algebras to weak Lie $p$-groups. You might recall that I had recently made some elemetary comments on that here.

Posted at July 5, 2005 10:19 AM UTC

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### Streetfest

Hi Urs

Yes, Mathematical Physics is one of the focuses of the Streetfest. Ross Street himself is publishing in physics journals now, with for instance

“Frobenius monads and pseudomonoids” J. Math. Phys. 45, 10 (2004) 3930

There is also a workshop following the Streetfest, with some exciting talks. For example, Tim Porter will speak about “Homotopy Quantum Field Theories with background a 2-type, formal maps and gerbes” and Robin Cockett will talk about “Differential and Smooth Categories”.

Posted by: Kea on July 5, 2005 11:45 PM | Permalink | Reply to this

### Re: Streetfest

Mathematical Physics is one of the focuses of the Streetfest

Right, but it is still interesting that in particular ‘categorified gauge theory’ seems to be getting a lot of attention.

For example, Tim Porter will speak about ‘Homotopy Quantum Field Theories with background a 2-type, formal maps and gerbes’

I wish I better understood what this means. A while ago I had read about HQFTs, but I am afraid I would have to look at the details again. What is the ‘background’ here? An Eilenberg-MacLane space? So is it simply the target space of the HQFT?

Has anyone heard of any genuine physics context where HQFTs arise? Like, might there be some twisting or something of the fundamental string such that it becomes something described by a 2D HQFT?

Robin Cockett will talk about ‘Differential and Smooth Categories’.

The abstract sounds interesting, but I am unfamiliar with many of the technical terms. So what’s the upshot? Say you hand me a ‘differential category’ in Cockett’s sense. What could I do with it?

Cocket writes

The coKleisli category of a differentiable category is a smooth category.

I know what a smooth category is. But what is a coKleisli category? What is a Kleisli category?

Posted by: Urs on July 6, 2005 9:48 AM | Permalink | Reply to this

### Re: Streetfest

I wish I better understood what this means. A while ago I had read about HQFTs, but I am afraid I would have to look at the details again. What is the ‘background’ here? An Eilenberg-MacLane space? So is it simply the target space of the HQFT?

A HQFT according to Turaev is a TFT where the objects and morphisms in the cobordism category comes with homotopy classes of maps into some fixed space (not necessarily an Eilenberg-MacLane space). I guess therefore that the “background” is this space.

Has anyone heard of any genuine physics context where HQFTs arise? Like, might there be some twisting or something of the fundamental string such that it becomes something described by a 2D HQFT?

Well, in the example of an Eilenberg-Maclane space K(\pi,1) where \pi is a finite group (which is the main example of Turaev), one gets a TFT for manifolds with principal \pi-bundles. In the 3D case this is very close to how the orbifold construction is carried out in CFT. I guess the 2D case would correspond to the TFT version of orbifolding. Moore and Segal have considered this case in their work on open-closed 2D TFT, but I haven’t read that in detail yet.

Jens

Posted by: Jens on July 6, 2005 11:23 AM | Permalink | Reply to this

### HQFTs

Hi Jens,

sorry that I didn’t come by your office again yesterday to say good-bye. But I’ll try to come next Thursday to your talk, even though it is only the second of two parts. This stuff sounds interesting.

So, let’s see. Say we have a $K\left(\pi ,1\right)$ target space and a 2D HQFT. Assuming we have foliated our surface, then a homotopy class of maps from it into $K\left(\pi ,1\right)$ assigns an element of $\pi$ to every leaf.

Hm, in which sense is that

a TFT for manifolds with principal $\pi$-bundles

?

Posted by: Urs Schreiber on July 8, 2005 6:50 PM | Permalink | Reply to this

### Re: 2-Holonomy in Terms of 2-Torsors

Urs,

have you thought about looking at “bundles of groups” (a la Moerdijk)? Really it’s a groupoid

$\begin{array}{rl}m& :{G}_{1}{×}_{{G}_{0}}{G}_{1}\to {G}_{1}\\ s,t& :{G}_{1}\to {G}_{0}\\ u& :{G}_{0}\to {G}_{1}\\ i& :{G}_{1}\to {G}_{1}\end{array}$

such that $s=t$. This is like a smooth version of a sheaf of groups, and is a more general structure “group” type object. Clearly thinking of a gerbe as a stack of groupoids pulls this concept in, and the relevance to 2-bundles is obvious to anyone following this blog.

The concept “bundle of groups” seems to me to be easily categorifiable (has anyone used that tense before?) as so should fit in without too much bother

————————

As I mentioned to Urs, I’m having to miss the workshop - is anyone going that will make decent notes (and is willing to let me have a copy)? All the stuff I want to know is going on there.

Aside from the ones mentioned by Kea, I really wanted to see Kock’s and Hermida’s talks(and others of course)

Can anyone help me in my dilemma?

David

Posted by: David Roberts on July 6, 2005 6:25 AM | Permalink | Reply to this

### Re: 2-Holonomy in Terms of 2-Torsors

have you thought about looking at ‘bundles of groups’ (a la Moerdijk)?

Hm, not sure. I am currently in an internet cafe and cannot access the page behind the link that you gave. I’ll try again from my office later today.

is anyone going that will make decent notes (and is willing to let me have a copy)? All the stuff I want to know is going on there.

Yes, it would be great if we could have some live-blogging from the Streetfest. If anyone feels like becoming a member of the string coffee table so that she or he can post entries, we can contact Jacques Distler about it.

Posted by: Urs on July 6, 2005 10:11 AM | Permalink | Reply to this

### Re: 2-Holonomy in Terms of 2-Torsors

Hi Urs

Interesting idea. I don’t see how I can volunteer. I am not and have never been a student or researcher of String Theory (except for 3 or 4 lovely lectures given by David Gross in Australia in 1994).

Posted by: Kea on July 6, 2005 11:45 PM | Permalink | Reply to this

### to string or not to string

I am not and have never been a student or researcher of String Theory

As for me, as long as you have something interesting to say about physics or mathematical physics, or maybe just math, I’d be interested. Right now the SCT is not threatened by being flooded by off-topic posts. If that should ever change we might want to talk about restricting the topics a little, but right now I do not see the need.

In particular, if you really take a lot of notes at the Steetfest and want to share them they should be very welcome here. While not explicitly about strings, maybe, there is a wealth of interesting relationships between the topics discussed in Sydney and string theory.

Posted by: Urs Schreiber on July 8, 2005 6:23 PM | Permalink | Reply to this

### bundles of groups

Urs,

Actually I access the online version of that paper by Moerdijk through my university’s library - perhaps you could look at this one instead - “Lecture 4” which is on bundle gerbes (really just a treatement of gerbes in a smooth setting).

Yrs,

D.

Posted by: David Roberts on July 7, 2005 3:23 AM | Permalink | Reply to this

### Re: bundles of groups

Ah, ok, now we are on the same page.

Sure, a smooth discrete groupoid is nothing but a copy of a Lie group $G$ at every point of some manifold $M$.

Similarly, a smooth discrete $p$-groupoid is nothing but a copy of a Lie $p$-group at every point of some manifold.

On the other hand, a principal $G$-bundle over $M$ has a $G$-torsor over every point of $M$. But locally it does look like a bundle of groups.

Similarly, a principal ${G}_{p}$-$p$-bundle $E$ over a categorically trivial base space really has a ${G}_{p}$-$p$-torsor ${E}_{x}$ associated to every point $x\in M$ and locally, over contractible patches $U\subset M$, it looks just like $M×{G}_{p}$, which we can regard as a smooth discrete $p$-groupoid.

What is kind of interesting is that the description of $p$-bundles with $p$-connection and $p$-holonomy that I am talking about also involves morphisms between groupoids, but apparently in a somewhat different way than in the bundle gerbe context. I am inclined to say in a more transparent way, but I might be biased. ;-)

So actually it seems that a principal ${G}_{p}$-$p$-bundle $E$ with $p$-connection and $p$-holonomy over a base manifold $M$ is nothing but a smooth (I am glossing over a technicality here) $p$-functor

(1)$\mathrm{hol}:{𝒫}_{p}\left(M\right)\to {G}_{p}-p\mathrm{Tor}$

that assigns $n$-morphisms in the category of $p$-torsors of the structure $p$-group ${G}_{p}$ to n-dimensional hypermvolumes inside $M$ in the $p$-path $p$-groupoid ${𝒫}_{p}\left(M\right)$.

This is the most banal thing that one could say about $p$-holonomy, and yet it says everything there is to say. This is the content of the notes mentioned above.

I should mention that John Baez was telling me all along that this must be true. Still, it is kind of interesting to see how it works out and for instance gives rise to the cocycle description of a (fake-flat) nonabelian gerbe with connection and curving.

Anyway, my point is that ${𝒫}_{p}\left(M\right)$ (the $p$-category whose objects are points in $M$, whose morphisms are paths in $M$, whose 2-morphisms are surfaces in $M$, and so on) as well as ${G}_{p}-p\mathrm{Tor}$ are both $p$-groupoids and hence a principal $p$-bundle with $p$-connection is nothing but a $p$-groupoid morphism.

This says nothing but that when you compute $p$-holonomy along some $p$-dimensional surface which retraces itself, you have done nothing - up to equivalence. This, again, is a most banal statement, really.

Locally, as I show, all this is described by simplicial maps of the sort shown in the third figure of the previous entry. This says that locally this $p$-groupoid morphism is given by $p$-functors from $p$-paths to ${G}_{p}$ itself, which are glued on $\left(n+1\right)$-fold overlaps by $p$-functor $n$-morphisms.

I find it interesting that this description in terms of local $p$-functors ,glued together on overlaps, easily generalizes away from groupoids. Without much ado I can write down the same simplicial map for much more general $p$-functors that do not need to take values in a $p$-group.

For instance they might take values in a $p$-groupoid. (Recall that I am now talking about the local functors which previously took values in a $p$-group only.) I show in section 13 of my thesis that the differential version of such a gadget is related to the algebroid Yang-Mills theory that have been considered by Thomas Strobl.

I don’t know if this situation still gives rise to a nonabelian gerbe. The Breen & Messing definition of nonabelian gerbes is very general.

But the target of these local functors does not have to be a $p$-groupoid either and still the whole thing makes sense and is certainly no longer described by any gerbe. I wonder if there are any interesting examples of such more general ‘$p$-bundles with $p$-connection and $p$-holonomy’.

One thing that might go wrong, though, is the independence of the local description of the choice of good covering. For the $p$-group case this independence follows easily from the equivalent intrinsic description in terms of $\mathrm{hol}:{𝒫}_{p}\left(M\right)\to {G}_{p}-p\mathrm{Tor}$. But this no longer works when we do away with weak invertibily.

All right, I am rambling on here and should stop at some point.

Posted by: Urs Schreiber on July 8, 2005 6:10 PM | Permalink | Reply to this

### Re: bundles of groups

So when you say you have a functor from the modified Cech p-path groupoid (I think that’s what you called it) to the p-torsor (I like to think of them as principal transitive ${G}_{p}$-spaces and save torosr for just principal ${G}_{p}$-spaces), or say the content of the following paragraph

>Anyway, my point is that ???p(M) (the p->category whose objects are points in M, >whose morphisms are paths in M, whose 2->morphisms are surfaces in M, and so on) >as well as G p-pTor are both p-groupoids >and hence a principal p-bundle with p->connection is nothing but a p-groupoid >morphism.

You’re just looking at something on the truncated homotopy type, aren’t you? A lot of higer information can propbably be thrown away - unless you’re a glutton fro punishment and want to work in omega-holonomies ;)

When you say

>I wonder if there are any interesting >examples of such more general ‘p-bundles >with p-connection and p-holonomy’.

you probably want some stack theory.

Apologies for the rotten typographical quality of the post, I’m at a public library.

Got to run,

D.

Posted by: David Roberts on July 9, 2005 2:05 AM | Permalink | Reply to this

### Re: bundles of groups

you say you have a functor from the modified Cech $p$-path groupoid

More precisely, I either have

- a functor from the Čech-extended $p$-path $p$-groupoid to the structure $p$-group ${G}_{p}$

or

- a functor from the ordinary path groupoid to the $p$-category of ${G}_{p}$-$p$-torsors

(which are both smooth in an appropriate sense), and I am claiming that gauge equivalence classes of the former are in 1-1 correspondence to the latter. I think I should be able to prove that the respective $p$-categories of such $p$-functors are equivalent.

I like to think of them as principal transitive ${G}_{p}$-spaces and save torsor for just principal ${G}_{p}$-spaces

Hm, ok. I am following Baez et al.’s nomencalture here. I don’t know which one is more standard.

You’re just looking at something on the truncated homotopy type, aren’t you?

Thanks for the question - I don’t know! :-)

If you could briefly tell me about what the truncated homotoy type is, that’d be great.

you probably want some stack theory.

True, that’s plausible. But there would be higher connections on these stacks. Is there any notion of ‘stack with connection’ for the case that the stack is not a stack in groupoids?

Posted by: Urs on July 9, 2005 2:27 PM | Permalink | Reply to this

### Re: bundles of groups

Urs,

if you could briefly tell me about what the truncated homotoy type is, that’d be great.

Given a topological space $X$, one can form the generalisation of the fundamental groupoid ${\Pi }_{1}\left(X\right)$, i.e. ${\Pi }_{n}\left(X\right)$, as in

Z. Tamsamani, Sur des notions de n-categorie et n-groupoide non strictes via des ensembles multi-simpliciaux, K-Theory, Volume 16, Issue 1 (1999), pp 51 - 99

much like your definition of Cech p-groupoid (but probably not with thin homotopy - and please note I haven’t read the paper). We also have a geometric realisation functor going the other way. It (truncated homotopy type) is basicallly taking a space and killing of all homotopy groups ${\pi }_{k}\left(X\right)$ for $k\ge n+1$.

But there would be higher connections on these stacks. Is there any notion of ‘stack with connection’ for the case that the stack is not a stack in groupoids?

not sure, but Grothendieck defines a stack to _be_ a stack-in-groupoids, which for most cases is what we want anyway. Wouldn’t we want to look at stacks which are also torsors (my defn), since general stacks are sort of like looking at general sheaves which come from something merely like a submersion, or an etale map or something admitting local sections (since, there is, I think, a theorem which states “all” sheaves (conditions, anyone?) come from sections of some sort of bundle.) This isn’t so useful. What would be good is a theorem which showed some sort of equivalence between the 2-cat of stacks (actually a 2-stack) and the 2-cat of (generalised) 2-bundles. I say generalised as maybe we need a notion of 2-Groth. topology? Getting too speculative there…

D.

Posted by: David Roberts on July 10, 2005 10:23 AM | Permalink | Reply to this

### Re: bundles of groups

Given a topological space $X$, one can form the generalisation of the fundamental groupoid ${\Pi }_{1}\left(X\right)$, i.e. ${\Pi }_{n}\left(X\right)$

OK, I should emphasize that when I am talking about paths and $p$-paths in the present context I do not mean homotopy equivalence classes of paths, but really parameterized paths (or thin homotopy classes, which means dividing out (only) by reparameterizations and by self-retracing paths).

So, when I say that a $p$-bundle with $p$-connection and $p$-holonomy is a $p$-functor from $p$-paths to something, I am not restricting to flat $p$-connections, which can only see the fundental $p$-path $p$-groupoid ${\Pi }_{p}\left(X\right)$.

Grothendieck defines a stack to be a stack-in-groupoids

I haven’t read Grothendiek in the original. In the Moerdijk article that we talked about before a stack is just any fibered category with descent data. A (non-empty and transitive) stack in groupoids is then a gerbe.

What would be good is a theorem which showed some sort of equivalence between the 2-cat of stacks (actually a 2-stack) and the 2-cat of (generalised) 2-bundles.

Right. Toby Bartels is currently working that out, defining and showing the equivalence of the 2-category of nonabelian gerbes and that of $\mathrm{Aut}\left(G\right)$-2-bundles on categorically trivial base spaces. We already know that the cocycle description of both is the same, so this equivalence should be true.

But I agree that it would be interesting to have a more general statement. That’s what I alluded to above.

Posted by: Urs on July 11, 2005 10:30 AM | Permalink | Reply to this

### Re: bundles of groups

OK, I should emphasize that when I am talking about paths and p-paths in the present context I do not mean homotopy equivalence classes of paths, but really parameterized paths (or thin homotopy classes, which means dividing out (only) by reparameterizations and by self-retracing paths).

So, when I say that a p-bundle with p-connection and p-holonomy is a p-functor from p-paths to something, I am not restricting to flat p-connections, which can only see the fundental p-path p-groupoid Ð p(X).

OK - I see what you mean, but what I think I meant is that all those p-paths dont care about the higher-connectedness of the space (whatever it is).Although thinking about it, attaching cells (or however one chooses to kill homotopy groups) probably mangles all those nicely parameterised p-paths of yours.

Right. Toby Bartels is currently working that out, defining and showing the equivalence of the 2-category of nonabelian gerbes and that of Aut(G)-2-bundles on categorically trivial base spaces. We already know that the cocycle description of both is the same, so this equivalence should be true.

I knew that, and it’s sort of obvious that it should be true, but I checked that theorem about sheaves/bundles, and it’s true that every sheaf of sets comes from a (topological) bundle, but the total space is usually not Hausdorff (actually this is the way the associated sheaf functor is defined). What I was wondering is that can we do the same for stacks?

Really got to run, I’m going to miss the loop space talk

Posted by: David Roberts on July 11, 2005 11:59 PM | Permalink | Reply to this

### Re: bundles of groups

it’s true that every sheaf of sets comes from a (topological) bundle

Cool! Where can I find that theorem?

What I was wondering is that can we do the same for stacks?

Right, yes, good question. So there is the interesting

Conjecture: Every stack can be realized as a a stack of 2-sections of a topological 2-bundle.

That would be nice, if true.

what I think I meant is that all those $p$-paths dont care about the higher-connectedness of the space

All right, now please bear with me, I might still be missing the point you are making. Let’s maybe concentrate on the simplest example, an ordinary principal $G$-bundle $E\to M$ with connection. This is a functor

(1)$\mathrm{hol}:{𝒫}_{1}\left(M\right)\to G-\mathrm{Tor}$

which assignes morphisms of $G$-torsors to thin homotopy equivalence classes of parameterized paths in base space $M$.

Now, you seem to be saying that this functor is insensitive to the higher connectedness of $M$. Is that so? While it sort of sounds right, I could readily agree on it only in the case that the functor is supposed to represent a flat connection, otherwise I am not sure yet what the statement really is.

Posted by: Urs Schreiber on July 12, 2005 11:31 AM | Permalink | Reply to this

### Re: 2-Holonomy in Terms of 2-Torsors

Hi David

Well, I’m going to everything and I plan to write furiously and take everything in that I can. Whether I have enough of an intersection with you to be able to satisfy your curiosity, I’m not sure, but I’ll certainly try.

Posted by: Kea on July 6, 2005 11:25 PM | Permalink | Reply to this

### who are you?

Kea,

so who are you and where are you based? From your posts it seems you’re in Australia - Macquarie/Sydney category group?

Posted by: David Roberts on July 7, 2005 3:50 AM | Permalink | Reply to this

### liveblogging

I plan to write furiously and take everything in that I can

Sounds great! I am looking forward to it.

Posted by: Urs Schreiber on July 8, 2005 6:26 PM | Permalink | Reply to this

### coffee table contributor (was: Re: 2-Holonomy in Terms of 2-Torsors)

So how do I join this fine electronic establishment to post entries about Streetfest?

I should warn that although I did my honours thesis on T-duality (it is available online, but if you want to read it that bad, you can find it!), I’ve left the world of physics - and apologies Kea, but that is including string theory - this time ;)

yrs,

D.

Posted by: David Roberts on July 8, 2005 4:18 AM | Permalink | Reply to this

### Re: coffee table contributor

So how do I join this fine electronic establishment to post entries

You, or anyone else who would like to join and become a member of the String Coffee Table, should send an email to Jacques Distler, who has set up and is administrating this blog.

Posted by: Urs Schreiber on July 8, 2005 6:16 PM | Permalink | Reply to this

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