Quantization and Cohomology (Week 23)

May 10, 2007

Quantization and Cohomology (Week 23)

Posted by John Baez

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This week in our seminar on Quantization and Cohomology, we tackled connections on bundles from a modern viewpoint:

Week 23 (May 7) - Principal bundles. The transport groupoid $Trans(P)$ of a principal $G$ -bundle $P$ over a smooth space $M$ . Connections as smooth functors $hol: P M \to Trans(P)$ where $P M$ is the path groupoid of $M$ . Proof that connections are described locally by smooth functors $hol: P U \to G$ where $U$ is a neighborhood in $M$ . Theorem: smooth functors $hol: P U \to G$ are in 1-1 correspondence with $Lie(G)$ -valued 1-forms on $U$ .

Last week’s notes are here; next week’s notes are here.

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To some extent we were making things up as we went along. Here’s a technical improvement that James Dolan pointed out, which didn’t make it into the notes.

We saw that not any smooth functor

$hol: P M \to Trans(P)$

gives a connection on the principal bundle $P$ over $M$ . Rather, we only want smooth functors with

$hol(x) = P_x$

for each point $x \in M$ . But, how can we say this in a nice arrow-theoretic way?

At first I thought we could to equip $P M$ and $Trans(P)$ with functors to $Disc(M)$ , the discrete category corresponding to $M$ — that is, the category with points of $M$ as objects and only identity morphisms. I wanted to express the equation $hol(x) = P_x$ by demanding that

$hol: P M \to Trans(P)$

make the resulting triangle commute.

But, this makes no sense: there are no functors from $P M$ or $Trans(P)$ to the discrete category corresponding to $M$ ! I wriggled out of this problem by equipping $P M$ and $Trans(P)$ with functors from the discrete category corresponding to $M$ , and demanding that $hol: P M \to Trans(P)$ make the resulting triangle commute.

In fancy lingo: instead of working with ‘categories over $Disc(M)$ ’, I realized I could work with ‘categories under $Disc(M)$ ’. This is what appears in the notes.

However, this feels funny — after all, we’re talking about a bundle over $M$ . James pointed out the right solution near the end of class. We should work with categories over $Codisc(M)$ , the codiscrete category corresponding to $M$ . This has the points of $M$ as objects and exactly one morphism from any object to any other.

Both $P M$ and $Trans(P)$ are categories over $Codisc(M)$ , and if we demand that

$hol: P M \to Trans(P)$

make the resulting triangle commute, we get

$hol(x) = P_x$

Moreover, $Codisc(M)$ also solves another problem for us — see the end of the notes.

So, treating path groupoids and transport groupoids as lying over $Codisc(M)$ is a good idea.

(By the way: I could explain this better if I knew how to draw nice commuting triangles in this environment!)

Posted at May 10, 2007 1:47 AM UTC

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18 Comments & 2 Trackbacks

Re: Quantization and Cohomology (Week 23)

Shouldn’t Codisc be better named Indisc? Or is that too… risqué?

Posted by: Allen Knutson on May 10, 2007 4:49 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Category theorists usually talk about ‘codiscrete’ or ‘chaotic’ categories, instead of ‘indiscrete’ ones.

But, I like the sound of the ‘risqué topology’.

Posted by: John Baez on May 12, 2007 9:48 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

I was brought up to say “indiscrete category” and “indiscrete [topological] space”, but “codiscrete” is probably a better name in both cases.

I’ve never understood the reason for “chaotic” though. What’s so chaotic about total uniformity?

Here’s hoping that risqué topology finds applications in heterotic string theory.

Posted by: Tom Leinster on May 14, 2007 3:15 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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It is well known that a $G$ -bundle on $M$ without connection is the same as an extension $Ad(P) \to Trans(P) \to codisc(M)$ (note all of these groupoids have object space $M$ .)

There is another exact sequence of groupoids with object space $M$ : $\Lambda PM \to PM \to codisc(M),$ where $\Lambda PM$ is the inertia groupoid of $PM$ - consists of only the loops.

Then what John talks about above is essentially a map from the second exact sequence to the first. The map $\Lambda PM \to Ad(P)$ materialises by universality of the kernel, and I’m talking about the strict kernel here. This last map looks like holonomy to me, and I wonder if given just $\Lambda PM \to Ad(P)$ we can find $PM \to Trans(P)$ : reconstruct the connection form from its holonomy, but using just what we have here.

Posted by: David Roberts on May 10, 2007 5:14 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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It is well known that a $G$ -bundle on $M$ without connection is the same as an extension

Yes! We have talked about this before: this is the “exponentiated Atiyah sequence” of a principal bundle. It is more familiar to many people in its differentiated version, where it becomes the ordinary Atiyah sequence of Lie algebroids over $T M$ $0 \to \mathrm{ad} P \to T P /G \to T M \to 0 \,.$

Once I summarized some cool facts related to this in $n$ -Transport and Higher Schreier Theory.

And I very much like your (David’s) obeservation that from a transport $\mathrm{tra} : P_1 X \to \mathrm{Trans}(P)$ we actually get a morphism of sequences of groupoids $\array{ \Omega_1(X) &\to& P_1(X) &\to& X \times X \\ \downarrow^{\mathrm{hol}} && \downarrow^{\mathrm{tra}} && \downarrow \\ \mathrm{Ad} P &\to& \mathrm{Trans}(P) &\to& X \times X } \,.$

Here $\Omega_1(X)$ is the skeletal groupoid which is the loop group based at $x$ over each point $x \in X$ .

(I am not sure I understand how you (David) think of this as an inertia groupoid. What’s your definition of “inertia groupoid” here?)

But I also think that one can avoid a lot of trouble by not considering $\mathrm{Trans}(P)$ in the first place. I’ll post another comment on that.

Posted by: urs on May 10, 2007 9:34 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Urs wrote:

But I also think that one can avoid a lot of trouble by not considering $Trans(P)$ in the first place. I’ll post another comment on that.

Of course I’m leading up to that! I just realized that the students would have trouble swallowing the slick approach based on smooth anafunctors before they’d seen a more old-fashioned approach to connections! I have a bunch of beginners in my class…

… so I started with the ‘old-fashioned’ definition of a connection on a fixed principal bundle $P \to M$ : namely, a smooth functor

$hol: P M \to Trans(P)$

from the path groupoid of the base manifold to the transport groupoid of the principal bundle.

The great thing about beginners is that they haven’t been brainwashed yet — they don’t know the even more old-fashioned approaches to connections.

(Some of the students do, of course. But the others were also perfectly able to follow what I was saying, with the help of lots of pictures.)

If someone is in a big rush to learn the slicker approach to connections using smooth anafunctors, start around page 3 of my third Namboodiri lecture, and then read Urs’ new paper with Konrad Waldorf. I’ll talk about it more in next week’s class.

Posted by: John Baez on May 11, 2007 3:57 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Of course I’m leading up to that! I just realized that the students would have trouble swallowing the slick approach based on smooth anafunctors

By the way, I guess I went on about that too much for anyone to stand already, but maybe here I want to emphasize this again a little:

when I think of a transport functor $\mathrm{tra} : P_1(X) \to G\mathrm{Tor}$ I am really not thinking in terms of anafunctors – not directly at least.

In particular, here $G\mathrm{Tor}$ is really supposed to be exactly that: the category of all $G$ -torsors – not its skeleton $\Sigma G$ .

I think you once indicated that in $\mathrm{tra} : P_1(X) \to G\mathrm{Tor}$ or $\mathrm{tra} : P_1(X) \to \mathrm{Vect}$ you would implicitly read $\mathrm{G}\mathrm{Tor} = \Sigma G$ and $\mathrm{Vect} = \cup_n \Sigma U(n)$ and regard the morphism above as an anafunctor.

I like that, it’s a very good thing to do, but I like even more at first not doing this, but really having the large categories $G\mathrm{Tor}$ , $\mathrm{Vect}$ , etc, around.

The thing is, that this way $\mathrm{tra}$ really is the global object which we want to consider, and not a sneaky way to talk about its local trivialization and descent data – which is what the anafunctor really is – which we obtain from the global object after choosing a smooth local structure.

The situation is this $\text{transport functor} \stackrel{\text{choose smooth local structure}}{\to} \text{descent data} \stackrel{\text{nice repackaging}}{\to} \text{anafunctor}$

Here “transport functor” is defined to be a functor which admits at least one smooth local trivialization.

And the cool thing is: one shows that if a functor admits any one smooth local trivialization at all, then this is unique up to isomorphism.

This means that we are entitled not to make a choice as long as we don’t like to, as long as we know that there is some choice.

I think that’s very convenient and rather powerful in lots of applications: whatever you would want to do to any old functor to vector spaces, you can do with your transport functor without worrying about local choices, etc. They are guaranteed to all come out right once we really need them.

The usefulness of this becomes most pronounced, to my mind, once one categorifies or once one quantizes (i.e. once one moves along the two other edges of the cube.)

In fact, that was in part the very motivation here, understanding line bundle gerbes with connection functorially.

The point is that we may think of them as being the descent data of a 2-transport $P_2(X) \to \Sigma 1d\mathrm{Vect}$ along the injection $i : \Sigma^2 \mathbb{C} \hookrightarrow \Sigma 1d\mathrm{Vect}$ .

When doing so, one finds that the “transition functions” $g$ are in fact 1-functors $g : P_1(Y) \to 1d\mathrm{Vect} \,.$ That’s how they come to us, as we turn the crank. So we would like to say that $g$ is a line bundle with connection, without having to break it down into its transition data (and hence its anafunctor version) first.

That’s already very useful for even saying what it should mean for the original 2-transport to be smooth: we say it recursively by saying that its “transition function” is a transport functor.

Similar considerations appear when quantizing a transport functor: this is most conveniently conceived in terms of the global object, without the need of making concrete choices for the local trivialization.

But of course the concrete local data obtained after making such choices is also important for many applications. And

a smooth anafunctor is a neat way of repackaging an entire object of the descent category in terms of a single functor.

The point is that there is a universal transition which every conceret choice of local transition data factors through (in the paper with Konrad this is called the “universal path pushout” and briefly discussed in the appendix).

And:

The anafunctor is the universal morphism by which the chosen local trivialization of the global transport functor factors through the universal one.

In the paper with Konrad this is briefly described in the section on anafunctors.

One could give a much more detailed discussion of the relation between anafunctors and “transport functors” (in the sense of that paper) than given there.

In particular, I think one can show the converse (and hence establish an equivalence) which shows that given any smooth anafunctor, we may extract from it an object in the category of descent data of transport functors (and hence a transport functor itself).

That didn’t make it into our paper (which is too long already), but a note on this can be obtained here.

Posted by: urs on May 11, 2007 9:50 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Very good points! Given a smooth group $G$ , is there a smooth anafunctor from $G Tor$ to $G$ which sets up an equivalence between $G$ and $G Tor$ in the 2-category

$[smooth categories, smooth anafunctors, smooth ananatural transformations] ?$

That would sort of excuse my tendency to replace $G Tor$ with $G$ when working with smooth anafunctors. I agree that working with $G Tor$ is morally superior in some contexts — e.g., the fibers of a principal bundles are really $G$ -torsors, not $G$ .

Posted by: John Baez on May 11, 2007 4:50 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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That would sort of excuse my tendency […]

Oh, it’s certainly a thing one can do, and a good thing, too!

All I think I am saying is that talking in terms of anafunctors is a way of talking about descent catgeories.

For things that form stacks, their descent categories are equivalent to the global thing – by definition of a stack – and hence one may well decide to work on the descent side of life, and re-expressing that in terms of anafunctors is certainly nifty.

But for some applications it is helpful to have the global picture available.

In particular, while I have some ideas about what 2-anafunctors should be like, this seems to be an intricate issue. Whereas on the global side it’s plain obvious.

One of the main motivations for the setup I describe is that for $n \gt 1$ there is a hierarchy of $n$ -steps of local trivializations – and the intermediate steps are important.

For instance:

- a line 2-bundle with connection is a 2-transport with values in the 2-category $\mathrm{BiMod}_{\mathrm{FinRnk}(H)}$ of bimodules for algebras of finite rank operators.

- it’s “first step local trivialization” or “local semitrivialization” is a bundle gerbe with connection.

And here I don’t mean “is equivalent to a bundle gerbe with connection”, but really is.

- it’s “second step local trivialization” is then the local trivialization of that bundle gerbe, which is finally a differential $\Sigma U(1)$ 2-cocycle, also known as a Deligne 2-cocycle. (modulo the issue that some like to have their cocycle degrees shifted by convention).

This hierarchy of local trivializations for higher $n$ is something that I am having difficulties seeing in the world of $n$ -anafunctors, at the moment.

Posted by: urs on May 11, 2007 5:23 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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My definition is: the inertia groupoid $\Lambda G$ is the subgroupoid of $G$ consisting of all the objects, and those arrows which are automorphisms. The arrow space is constructed by pulling back $(s,t):G_1 \to G_0 \times G_0$ along the diagonal $G+0 \to G_0 \times G_0$ .

Further to my question on reconstucting $P M \to Trans(P)$ from holonomy around loops, I was reminded that one can only reconstruct the connection up to gauge equivalence, but this I suppose is what one expects - uniqueness up to a unique 2-arrow.

Posted by: David Roberts on May 11, 2007 7:43 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Further to my question on reconstucting $P M \to \mathrm{Trans}(P)$ from holonomy around loops, I was reminded that one can only reconstruct the connection up to gauge equivalence, but this I suppose is what one expects - uniqueness up to a unique 2-arrow.

Yes, that’s indeed the best thing we can expect to obtain here from any kind of reconstruction.

By the way: while one can certainly reconstruct the bundle with connection (hence its transport functor) from just the holonomies – I don’t yet have a really “nice” way to do so. I mean, it’s possible in a pedestrian way, certainly, but I am not at the moment aware of a nice arrow-theoretic construction that would make this plain obvious. But I feel like there should be one. Maybe you see it. Probably something involving a choice of path from every point to the basepoint.

Posted by: urs on May 11, 2007 10:15 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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You might also mention what the smooth structure on $\mathrm{Trans}(P)$ actually is:

the space of morphisms of $\mathrm{Trans}(P)$ is in fact a manifold, namely $\mathrm{Mor}(\mathrm{Trans}(P)) = (P \times P)/G \,,$ where the action we divide out is the obvious one $g : (p_x, p_y ) \mapsto (p_x \cdot g, p_y \cdot g) \,.$

I think you discussed this on the blog before somewhere, but maybe it is worthwhile mentioning it again here.

Posted by: urs on May 10, 2007 9:44 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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Rather, we only want smooth functors with $\mathrm{tra}(x) = P_x$ for each point $x \in M$ . But, how can we say this in a nice arrow-theoretic way?

There are several ways to address this issue.

One is: don’t predefine a fixed total space of a bundle and force the functor to obey your constraints. Instead, let there be just functors $\mathrm{tra} : \mathcal{P}_1(X) \to G\mathrm{Tor}$ and then, if one really, really needs a total space (and mostly one doesn’t!), define $P_x := \mathrm{tra}(x) \,.$

The smooth structure on the functor can then be used to glue these fibers smoothly to reconstruct the bundle.

That’s particularly handy once we categorify: total spaces of higher $n$ -bundles are notoriously difficult to get one’s hands on explicitly. But we don’t need them! A principal bundle with connection is its transport functor $\mathrm{tra} : P_1(X) \to G\mathrm{Tor} \,.$

This I prove with Konrad in Parallel Transport and Functors.

Posted by: urs on May 10, 2007 9:53 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

I’m notationally or otherwise challenged.
Since hol is defined on paths, hol(x)
means on the constant path x(t) = x for all t?
And P_x is the fibre over x?
Maybe this is all in the notes?

jim

Posted by: jim stasheff on May 11, 2007 12:19 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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It’s all explained in the notes — my blog entry is just a little addendum. Given a principal bundle

P \to M

, a connection gives a functor ‘hol’ from the path groupoid of

M

points of $M$ as objects
equivalence classes of paths in $M$ as morphisms

to the transport groupoid of

P

fibers of $P$ as objects
equivariant maps between fibers as morphisms

The fun thing is what hol does to morphisms: it sends each path to its holonomy! But here’s what it does to objects:

hol(x) = P_x.

My addendum concerns the best way of understanding this requirement.

Posted by: John Baez on May 11, 2007 3:40 AM | Permalink | Reply to this

Read the post The First Edge of the Cube
Weblog: The n-Category Café
Excerpt: The notion of smooth local i-trivialization of transport n-functors for n=1.
Tracked: May 11, 2007 10:18 AM

Re: Quantization and Cohomology (Week 23)

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Way back when, we were discussing functors $P M \to Trans(P)$ as groupoids over $Codisc(M)$ . But $Codisc(M)$ is the terminal object in the category of groupoids with object space $M$ !

Does the required triangle automatically commute, or can we only say it 2-commutes and are demanding that the 2-arrow in question is the identity?

We could say the same thing about $P M$ and $Trans(P)$ as groupoids under $disc(M)$ , which is the inital object.

Posted by: David Roberts on June 6, 2007 7:04 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

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The objects in the path groupoid $P X$ are points of the space $X$ . The objects in the transport groupoid $Trans(P)$ are the fibers $P_x$ of the principal bundle $P \to X$ . We want the parallel transport functor

$hol: P X \to Trans(P)$

to send each point $x \in X$ to its own fiber $P_x$ , instead of some other fiber $P_y$ . This isn’t automatic!

To ensure that $hol(x) = P_x$ , we can require that a triangle commute, involving $hol$ together with the obvious functors from $P X$ and $Trans(P)$ to the codiscrete category on $X$ .

So, no — it’s not automatic that this triangle of smooth functors commutes.

Thus, $Codisc(M)$ cannot be the terminal object in the category of groupoids with object space $M$ . We can see this directly, too. Maps from $M$ to itself give smooth functors from $Codisc(M)$ to itself, so $Codisc(M)$ has lots of endomorphisms.

What’s true is this: $Codisc(M)$ is the terminal object in the category of groupoids with object space $M$ and functors that are the identity on objects.

Thanks for taking the trouble to test my claims. In this class we were sort of making stuff up as we went along.

Posted by: John Baez on June 7, 2007 4:41 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 23)

What’s true is this: Codisc(M) is the terminal object in the category of groupoids with object space M and functors that are the identity on objects.

Ah, of course. I was worried for a moment there.

Posted by: David Roberts on June 8, 2007 3:28 AM | Permalink | Reply to this

Read the post Curvature, the Atiyah Sequence and Inner Automorphisms
Weblog: The n-Category Café
Excerpt: On the notion of curvature 2-functor in light of morphisms from the path sequence of the base to the Atiyah sequence of the bundle.
Tracked: June 20, 2007 5:05 PM

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