### Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

#### Posted by Urs Schreiber

**Update (09/14/05):** There is a disagreement about the secrecy status of the former content of this entry.

**Update (09/14/05):** There is a disagreement about the secrecy status of the former content of this entry.

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Danny has categorified this fact and found a similar result for our 2-connections.

That’s cool. Is this going to be made public anytime soon? I’d like to compare it to the locally trivialized version which I think I do understand. Does he get the fake flatness conditions?

When you define the Atiyah algebroid, are you taking the anchor map to be to $\mathrm{TM}$, or to $M$ (i.e. composing $\rho $ and $p$)?

Have you looked at the fact that the splitting

$$\mathrm{TP}/{G}_{*}\to P{\times}_{G}\mathrm{Lie}(G)=\mathrm{ad}P$$

of

$$\mathrm{ad}P\to \mathrm{TP}/{G}_{*}$$

in

$$0\to \mathrm{ad}P\to \mathrm{TP}/{G}_{*}\to \mathrm{TM}\to 0$$

also gives a connection (actually the connection form)? The s.e.s. is from “Complex analytic connections in fibre bundles” and is where your Atiyah algebroid comes from (I know you know this).

The nice thing about using this Atiyah definition of connection is it gives a “closer” link to the usual horizontal subspace/connection form defintion of connection. We just take a left or right splitting of the above s.e.s. to get the connection.

All this should lift nicely to 2-bundles, once a decent definition of 2-vector bundle is figured out (I think Danny may be working on this).

D

Posted by:
David Roberts on September 7, 2005 4:27 AM | Permalink
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are you taking the anchor map to be to $\mathrm{TM}$

Yes, as the anchor is always to $\mathrm{TM}$. By displaying the projection arrow $\mathrm{TM}\to M$ I didn’t mean anything deep.

Have you looked at the fact that […]

No, not really. But I do believe that this should be important for the following reason:

We are dealing with these two equivalent aspects of ordinary connections:

a) something that allows to *lift*, namely vectors/paths in the base space $M$ to vectors/paths in the total space $P$

b) something that defines horizontal subspaces of $\mathrm{TP}$.

Here a) is really defining a connection in terms of its holonomy. Categorifying this leads to the constraint of vanishing fake curvature.

Possibly the non-fake flat situation as obtained for nonabelian bundle gerbes with connection and curving can be understood as a categorification of b) instead of a). I believe Danny was hinting at that (but I might be wrong).

In the case of ordinary bundles b) does allow to lift paths to total space, too.

In the case of categorified bundles (gerbes) it is no longer obvious that the categorification of b) allows to *lift surfaces* from base space to total space.

I expect that adding to the categorification of b) the requirement that it should be possible to ‘lift surfaces’ will be precisely equivalent to the categorification of a) and hence of in addition requiring fake flatness.

once a decent definition of 2-vector bundle

As was mentioned here recently, as far as *tangent* 2-bundles are concerned it should be clear what one has to do:

Let $S$ be a smooth category (smooth space of objects, smooth space of morphims and all maps between them smooth maps). $S$ is a ‘2-space’. Its *tangent 2-bundle* should be the category $\mathrm{TS}$ with

(1)$$\mathrm{Obj}(\mathrm{TS})=T(\mathrm{Obj}(S))$$

(2)$$\mathrm{Mor}(\mathrm{TS})=T(\mathrm{Mor}(S))$$

(3)$${s}_{\mathrm{TS}}={\mathrm{ds}}_{S}$$

(4)$${t}_{\mathrm{TS}}=d{t}_{S}$$

(5)$${\circ}_{\mathrm{TS}}=d{\circ}_{S}\phantom{\rule{thinmathspace}{0ex}}.$$

(Here, for instance, ${s}_{\mathrm{TS}}$ denotes the source map in $\mathrm{TS}$ and ${\mathrm{ds}}_{S}$ the differential of the source map in $S$.)

This is a category internalized in the category of vector bundles and hence qualifies as a good notion of 2-vector bundle. And it exists for every smooth category $S$. This should be all that is needed for the above purposes.

—

“Let S be a smooth category (smooth space of objects, smooth space of morphims and all maps between them smooth maps). S is a 2-space;. Its tangent 2-bundle should be the category TS with […]”

— (PS - having trouble with markup and no time to fix)

What should possibly be done now is look at the “Grassmannian” of said tangent bundle and thus get a nice canonical example of a 2-bundle (poss. even a universal one, if we let “dim S” $\to \mathrm{\infty}$ (!))

D

Posted by:
David Roberts on September 13, 2005 1:40 AM | Permalink
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look at the ‘Grassmannian’ of said tangent bundle

Could you expand on that? I am afraid I don’t know what you have in mind. Are you talking of Grassmannians in the sense of spaces of $k$-dimensional subspaces of vector spaces or something else? Sorry.

if we let “dim $S$” $\to \mathrm{\infty}$

This might actually be the ‘generic’ case, in a sense. Many intersting smooth spaces of morphism will be infinite-dimensional. For instance if morphisms are paths in the space of objects.

a nice canonical example of a 2-bundle

I need to think more about this. One issue seems to be that such a tangent 2-bundle is not at all of the form of those 2-bundles which are related to gerbes, unless I am confused. I.e. it does not locally look like $S\times F$ for $F$ the typical fiber category. Or does it somehow?

—-
*
Could you expand on that? I am afraid I don’t know what you have in mind. Are you talking of Grassmannians in the sense of spaces of k-dimensional subspaces of vector spaces or something else?
*
—-

Whoops - I mean **frame bundle**. I’ve been thinking of Grassmannians and Stiefel bundles in inf dim Hilbert space recently.

Thus my “dim S” remark - I was thinking of $\mathrm{EGL}\to \mathrm{BGL}$

—-
*
One issue seems to be that such a tangent 2-bundle is not at all of the form of those 2-bundles which are related to gerbes, unless I am confused. I.e. it does not locally look like S×F for F the typical fiber category.
*
—-

That’s good - we need non-gerbe examples of 2-bundles. From the algebraic geometric side of things (a la Grothendieck, Deligne-Mumford etc) general stacks seem to me to be less “nice” than gerbes. Looking at the axioms for a gerbe:

A **gerbe** is a stack in groupoids which is

- Locally non-empty
- Locally connected

The first of these means that there is a cover such that the fibre (“set of sections”) over each element of the cover is non-empty (something like local triviality), and the second means that all the automorphism groups of objects in the fibres are isomorphic (i.e. we have a well defined fibre)

So in a sense you are correct about the nasty properties of these non-gerbe 2-bundles, but hey, I’m sure someone will find a way around this. Just remember what all these objects were invented for: gerbes for degree 2 nonabelian cohomology, 2-bundles for categorifying bundles. Both have done their job. It’s just nice they overlap a bit.

D

Posted by:
David Roberts on September 14, 2005 2:07 AM | Permalink
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That’s good - […]

I very much agree with what you said.

I might add that the different notions of ‘2-bundle’ playing a role here seem to be due to two different branches of categorification.

Roughly, whenever there is a concept $A$ one can

1) either study $A$ internalized (strictly) in $\mathrm{Cat}$

2) or study categories internalized in the category of $A$ .

There is an amazing theorem that 1) and 2) agree *if* $A$ is what is called *essentially algebraic*. But in general they need not agree.

The principal 2-bundles as defined originally by Toby Bartels are like principal bundles internalized in $\mathrm{Cat}$, i.e. a categorification of a bundle according to procedure 1).

The tangent 2-bundles that we were talking about however are categories internalized in the category of vector bundles and hence are (vector) bundles categorified according to procedure 2).

Categorification is a little bit like quantization in that it may give different and inequivalent results when applied to different but equivalent items.

So what would a 2-section of a 2-bundle in the sense 2) be? That seems to be the central question, because the relation to stacks should be that a 2-bundle has a stack of 2-sections (instead of just a sheaf of 1-sections).

I haven’t really thought hard enough, but a straightforward guess is that a 2-section of the tangent 2-bundle $\mathrm{TS}$ that we talked about must be just an ordinary section $f$ of $\mathrm{Mor}(\mathrm{TS})$.

That, however, makes it seem unlikely that these 2-sections form a stack, somehow.

Maybe one should add the (nontrivial) requirement that the image of $f$ under source and target maps $s$ and $t$ are sections of $\mathrm{Obj}(\mathrm{TS})$.

Hm…

*I haven’t really thought hard enough, but a straightforward guess is that a 2-section of the tangent 2-bundle TS that we talked about must be just an ordinary section f of Mor(TS).*

*That, however, makes it seem unlikely that these 2-sections form a stack, somehow.*

*Maybe one should add the (nontrivial) requirement that the image of f under source and target maps s and t are sections of Obj(TS). *

I would have thought that a 2-section of a 2-map $F:E\to M$ would be a functor $S:M\to E$ such that $F\circ S\Rightarrow {\mathrm{id}}_{M}$. I started working on something like this but got distracted ;) This was going to be my starting point of “2-sections of a 2-bundle form a stack”.

I’ll think about your proposed defn some more and get back.

D

Posted by:
David Roberts on September 15, 2005 1:27 AM | Permalink
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Hi David,

sorry for my slow responses. I am currently distracted by a couple of things, like preparing for my disputation (which I should be doing right now instead of chatting) and finding a place to live in Hamburg (a task that has now successfully and satisfactorily been finished).

I would have thought that a 2-section of a 2-map $F:E\to M$ would be a functor $S:M\to E$ such that $F\circ S\Rightarrow {\mathrm{id}}_{M}$.

True. In my last comment I was missing the fact that such a projection map $F$ is available here.

So at the rough level at which I have thought about this stuff, it seems that given any category $E$ internalized in the category of vector bundles, we get an internal projection functor $E\stackrel{p}{\to}B$ from it to a category $B$ internalized the category of smooth spaces as follows:

Let

(1)$${E}_{\mathrm{Obj}}\stackrel{{p}_{\mathrm{Obj}}}{\to}{B}_{\mathrm{Obj}}$$

be the vector bundle of objects of $E$ and let

(2)$${E}_{\mathrm{Mor}}\stackrel{{p}_{\mathrm{Mor}}}{\to}{B}_{\mathrm{Mor}}$$

be the vector bundle of morphisms of $E$.

The source, target and composition maps in $E$ are vector bundle morphisms which ristrict to smooth maps on the base spaces ${B}_{\mathrm{Obj}}$ and ${B}_{\mathrm{Mor}}$. Calling these restrictions ${s}_{B}$, ${t}_{B}$ and ${\circ}_{B}$, respectively, we seem to get a ‘base category’ $B$ of $E$ with

(3)$$B=\{{B}_{\mathrm{Obj}},{B}_{\mathrm{Mor}},{s}_{B},{t}_{B},{\circ}_{B}\}$$

in the (hopefully) obvious way. By simply forgetting about the fibers in the bundles involved in $E$ we should get a projection functor

(4)$$E\stackrel{p}{\to}B$$

and hence arrive at a 2-bundle in the sense 1) of my previous comment.

Ok, I didn’t realize this simple fact last time. Now with the $p$-functor available, it is clear what a 2-section should be, namely, as you say, a 2-functor

(5)$$B\stackrel{s}{\to}E$$

such that

(6)$$B\stackrel{s}{\to}E\stackrel{p}{\to}B\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\stackrel{\lambda}{\iff}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}B\stackrel{\mathrm{Id}}{\to}B\phantom{\rule{thinmathspace}{0ex}}.$$

But now, isn’t it true that such a 2-section $s$ must come (at least in the case where the above isomorphism $\lambda $ is the identity) from an ordinary section of ${E}_{\mathrm{Mor}}\stackrel{{p}_{\mathrm{Mor}}}{\to}{B}_{\mathrm{Mor}}$ ?

I guess so. But actually the example of principal 2-bundles shows that the interesting aspect of 2-sections is in their morphisms. Is there an interesting structure on the natural transformations

(7)$$B\stackrel{{s}_{1}}{\to}E\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\stackrel{\kappa}{\Rightarrow}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}B\stackrel{{s}_{2}}{\to}E$$

?

It is noteworthy that the base category $B$ is an honest category with, in general, nontrivial morphisms. This means that the ‘2-sheaf’ of 2-sections $s$ is not in an obvious way a stack. That’s because a stack is what I would call a pseudo-functor instead of a true 2-functor. Meaning that it is something from a 1-category (of ‘open sets’) to the 2-category $\mathrm{Cat}$ (or $\mathrm{Groupoids}$ if you like) instead of something whose domain is a 2-category itself.

Here it seems we need a 2-category of 2-‘open sets’ of $B$, namely something like the 2-category of ‘open subcategories’ of $B$.

I once talked more about that somewhere on the web, but haven’t really made much progress with it. But it is a general qestion that I would someday like to know the answer to:

*Why stacks instead of ‘2-sheafs’*?

Not that I would find stack theory so trivial that I would be yearning for something more intricate. But from the point of view of categorification a stack is a an inconsequent generalization of a sheaf, it seems.

Posted by:
Urs on September 18, 2005 5:20 PM | Permalink
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*Why stacks instead of ‘2-sheafs’?*

probably because John B around born yet :-D

D

Posted by:
David Roberts on September 19, 2005 7:56 AM | Permalink
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probably because John B around born yet :-D

Probably. But it bites us here. On the other hand, in special cases we can still hope to get a stack of 2-sections.

Namely if the base category $B$ has the following property.

To each open set ${U}_{i}\subset {B}_{\mathrm{Obj}}$ let

(1)$$B{\mid}_{{U}_{i}}:=\left({s}^{-1}\left({U}_{i}\right)\right)\cap \left({t}^{-1}\left({U}_{i}\right)\right)$$

be the subcategory of $B$ consisting of all morphism starting and ending in ${U}_{i}$. If all of $B$ is *generated* from

(2)$$\bigcup _{i}B{\mid}_{{U}_{i}}$$

for some covering $\{{U}_{i}{\}}_{i}$ of ${B}_{\mathrm{Obj}}$ then it does make good sense to define the category of 2-sections on ${U}_{i}$ to be that of 2-sections $B{\mid}_{{U}_{i}}\stackrel{s}{\to}E$. Hence we do get a fibered category of 2-sections. With a little luck this is a stack.

*From the algebraic geometric side of things (a la Grothendieck, Deligne-Mumford etc) general stacks seem to me to be less “nice” than gerbes.*

Also remember that stacks are closely related to moduli spaces and orbifolds, so I wouldn’t be suprised if pleasantries break down somewhere ;)

D

Posted by:
David Roberts on September 14, 2005 2:17 AM | Permalink
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* I was thinking of* $\mathrm{EGL}\to \mathrm{BGL}$

Maybe not. Maybe I was thinking of something else. *blush*

I’ll forget my own head next.

D

Posted by:
David Roberts on September 14, 2005 3:54 AM | Permalink
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Maybe not.

Don’t worry. Baking half-baked ideas is what informal discussion is meant for and good for. I enjoy it.

I think a noteworthy point of what you suggested is that by the same principle by which we construct the tangent 2-bundle to a 2-space we can also get the frame 2-bundle to a given 2-space.

Let’s see.

Denote by $\mathrm{FM}$ the ordinary (orthogonal) frame bundle of an ordinary (Riemannian) smooth space $M$.

Now let $S$ be a 2-space (a smooth category), then I guess its frame 2-bundle should be the category $\mathrm{FS}$ with

(1)$$\begin{array}{ccc}\mathrm{Obj}(\mathrm{FS})& =& O(\mathrm{Obj}(S))\\ \mathrm{Mor}(\mathrm{FS})& =& O(\mathrm{Mor}(S))\\ {s}_{\mathrm{FS}}& =& d\phantom{\rule{thinmathspace}{0ex}}{s}_{S}\\ {s}_{\mathrm{FS}}& =& d\phantom{\rule{thinmathspace}{0ex}}{t}_{S}\\ {\circ}_{\mathrm{FS}}& =& d\phantom{\rule{thinmathspace}{0ex}}{\circ}_{S}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$$

Or am I missing any subtlety?

“Update(09/14/05): There is a disagreement about the secrecy status of the former content of this entry.”

Hah, funny! So, what’s new in the world? Attempted control of blogspeak …. hilarious!

Posted by:
Kea on September 17, 2005 5:02 AM | Permalink
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Hi Kea,

Regarding what’s new in the world, I suggest you search for “blog” and “sedition” in Google News …

As to the altered post, I believe Urs has his own reason for doing what he did, so perhaps we should leave it at that.

Posted by:
Rongmin on September 17, 2005 6:56 AM | Permalink
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Read the post Pantev on Langlands, II

**Weblog:** The String Coffee Table

**Excerpt:** Pantev lectures on Langlands duality, D-branes and quantization, part II.

**Tracked:** May 11, 2006 7:50 AM

## Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Hey, this stuff is really cool!

I’ve only absorbed a fraction of it, but I’m already very happy. I’ll have to show Danny Stevenson.

You mention Atiyah’s description of connections as splittings of short exact sequences of Lie algebroids, where the splitting is only a Lie algebroid morphism if the connection is flat.

Danny has categorified this fact and found a similar result for our 2-connections.

Anyway, I’m glad you’re writing about such heavy-duty topics on this blog.