## September 26, 2009

### Questions on n-Curvature

#### Posted by John Baez

Dear Urs:

I know that once you were very eager to understand Breen-Messing data in terms of $n$-transport. All of a sudden I’m very interested in applying this idea to some problems. So I’m trying to find the places where you worked out this idea most fully and explicitly — and of course I’m embarrassed that I didn’t pay more attention a couple of years ago.

Anyway, here’s a pile of questions. I bet I’m confused about some things, so some of them won’t make sense. Feel free to answer as many or as few as you want!

Take a trivial $X$-2-bundle for some Lie 2-group $X$ described in terms of a crossed module $(G,H,t,\alpha)$. By Breen-Messing data I simply mean a $Lie(G)$-valued 1-form $A$ and a $Lie(H)$-valued 2-form $B$.

In your notes you seem to say this Breen-Messing data gives well-defined $\Sigma(Inn(X))$-valued 2-transport, where $Inn(X)$ is the inner automorphism 3-group of $X$. This is already somewhat confusing to me since I normally expect $n$-transport when I have an $n$-group, but now you’re talking about $2$-transport for a $3$-group! But I guess it makes perfect sense to talk about a 3-functor

$P_2(M) \to \Sigma(Inn(X))$

from the path 2-groupoid $P_2(M)$ of a manifold $M$ to $\Sigma(Inn(X))$, which is $Inn(X)$ regarded as a 3-groupoid with one object. That’s what you mean, right? How strict is this 3-functor supposed to be? In particular, does the 3-morphism part of $\Sigma(Inn(X))$ get involved here?

Anyway, what I really want to know is some very explicit stuff:

What does a general one of these 3-functors

$P_2(M) \to \Sigma(Inn(X))$

look like in terms of differential forms? Presumably it consists of a 1-form, a 2-form and a 3-form valued in certain Lie algebras, satisfying certain equations? What are these Lie algebras in terms of the crossed module description of $X$? From your note it sounds like we have a 1-form valued in the Lie algebra of $G$ and a 2-form valued in the Lie algebra of the semidirect product of $G$ and $H$. If we’re given Breen–Messing data, what are the formulas for these Lie-algebra-valued forms? I think I can guess that.

Posted at September 26, 2009 5:29 PM UTC

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### Re: Questions on n-Curvature

Thanks, John!

This comes in at a moment where I will have to run soon. So I will postpone detailed replies.

For the moment, all I do is point out that I started developing an exposition of the general theory on my personal $n$Lab web.

This answers many of your questions. Others are scttered through my articles and notes, but I will insert them into the web eventually. (Tomorrow, if I can negotiate that.)

The entry point is here:

differential nonabelian cohomology

When you read that overview page to the end, you will see that it derives from some very general abstract nonsense the theorem that characterizes cocycles in differential nonabelian cohomology by

The upshot is that Cartan-Ehresmann $\infty$-connections are the structures introduced and studied in

Sati, Schreiber, Stasheff, $L_\infty$-algebra connections

Where much of the details that you are looking for are given. (Search the document for $inn(\mathfrak{g})$).

There we just sketched the idea of how these dg-structures come from principal $\infty$-bundles with connection. The writeup on the Lab now gives the derivation.

For the particular case that you are asking about we have a structure Lie 2-group $G$ with $\mathfrak{g}$ its Lie 2-algebra. The differential form data of a $G$-2-bundle $P\to X$ with connection is an $L_\infty$-algebra valued differential form given by a morphism of $\infty$-Lie algebroids

$\Pi^{inf}(P) \to cone(\mathfrak{g})$

with $\Pi^{inf}(P)$ the infinitesimal path $\infty$-groupoid

which is dually a dga-morphism

$\Omega^\bullet(P) \leftarrow W(\mathfrak{g}) : A$

from the Weil-algebra of the Chevalley-Eilenberg-algebra of $\mathfrak{g}$.

Unwrapping what this is we find that it consists of

- a $\mathfrak{g}_1$-valued connection 1-form $a$

- a $\mathfrak{g}_2$-valued connection 2-form $B$

- a $\mathfrak{g}_1$-valued curvature 2-form $\beta$

- a $\mathfrak{g}_2$-valued curvature 3-form $H$

with

$\beta = d_a + [a\wedge a] + \delta_* B$

$H = d_a B \,.$

This is the data that you are looking for. The equivariance/cocycle data that this has to satisfy to qualify as connection data on $P$ is the first and second $\infty$-Ehresmann condition given by the requirement that the above morphism $A$ is required to fit into a diagram

$\array{ \Omega^\bullet_{vert}(P) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& flat vertical \mathfrak{g}-valued differential form \\ \uparrow && \uparrow &&& first Ehresmann condition \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) &&& \mathfrak{g}-valued connection form on total space \\ \uparrow && \uparrow &&& second Ehresmann condition \\ \Omega^\bullet(X) &\stackrel{P(F_A)}{\leftarrow}& inv(\mathfrak{g}) &&& characteristic forms on base space }$

Posted by: Urs Schreiber on September 26, 2009 6:39 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Thanks for the reply! For starters, a couple of very basic questions.

First, when you write

$\beta = d_a + [a\wedge a] + \delta_* B$

is $d_a$ a typo for $d a$?

Second, I don’t know what $\delta_*$ means. So I’ll ask: is this whole expression simply the ‘fake curvature’ of $a$ and $B$, which we used to write as

$\beta = d a + [a,a] + dt(B) ?$

Third, when you write:

the equivariance/cocycle data that this has to satisfy to qualify as connection data on $P$ is the first and second $\infty$-Ehresmann condition…

is this something I can ignore in the case of a trivial 2-bundle? I want to restrict attention to a trivialized 2-bundle and think of all the differential forms $a, B, \alpha, H$ as living on the base space $X$ rather than up on $P$ (whatever that is).

Posted by: John Baez on September 26, 2009 11:32 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

For starters, a couple of very basic questions.

The answers are yes, yes, and yes. :-)

Just one caveat with the last one: for trivial $G$-principal 2-bundles a connection is just a globally smooth functor $\Pi_3(X) \to \mathbf{B}INN(G)$ with no further conditions and hence just the differential form data that we are talking about, but morphisms between them are not arbitrary transformations between these.

Posted by: Urs Schreiber on September 28, 2009 8:35 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Urs wrote:

The answers are yes, yes, and yes. :-)

I love it when people answer my questions that way. I just wanted to make sure I understand what’s going on here…

Just one caveat with the last one: for trivial $G$-principal 2-bundles a connection is just a globally smooth functor $\Pi_3(X) \to \mathbf{B} INN(G)$ with no further conditions and hence just the differential form data that we are talking about, but morphisms between them are not arbitrary transformations between these.

Since I don’t know what ‘arbitrary transformations’ are, I’m not sure what this means — but I do know that not all flat connections on a trivial bundle are gauge-equivalent, so hopefully you’re saying something analogous to that.

I would in fact be delighted to see, written out in the lowbrow language of group-valued functions and Lie-algebra-valued differential forms, what a morphism between two smooth functors $\Pi_3(X) \to \mathbf{B}INN(G)$ looks like. So if you ever have time to do this, or point me to someplace where you’ve done it, that would be very nice.

(I should try to work this out myself, perhaps using the ‘first and second $\infty$-Ehresmann conditions’, but I find this task a bit intimidating… and I bet I’m not the only person who feels that way.)

Posted by: John Baez on September 28, 2009 6:26 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Since I don’t know what ‘arbitrary transformations’ are,

Hey, wait, what is wrong about that? I mean transformations between 3-functors, without restrictions on them. In erudite $n$Lab-speak: (3,1)-transformations.

but I do know that not all flat connections on a trivial bundle are gauge-equivalent, so hopefully you’re saying something analogous to that.

Yes. If one forgets that this functor $\Pi_3(X) \to \mathbf{B} INN(G)$ is part of a bigger structure it might seem as if morphisms between two of these are arbitrary $(3,1)$-transformations of such 3-functors. But that would make all of them isomorphic!

I would in fact be delighted to see, written out in the lowbrow language of group-valued functions and Lie-algebra-valued differential forms, what a morphism between two smooth functors $\Pi_3(X) \to \mathbf{B} INN(G)$ looks like.

For the connection 1- and 2-forms it is the familiar transformation law from our article. The only new thing here is that the 2-form (aka “fake”) curvature doesn’t have to vanish. That is essentially the only thing that changes.

perhaps using the ‘first and second ∞-Ehresmann conditions’, but I find this task a bit intimidating… and I bet I’m not the only person who feels that way.)

Sure. The “first Ehresmann condition” simply says that the connection which locally is a 3-functor with values in $\mathbf{B}INN(G)$ has to transform on double and triple intersections by transition functions that take values in the image of the inclusion $\mathbf{B}G \hookrightarrow \mathbf{B} INN(G)$.

More precisely, let $\coprod_i U_i \to X$ be an open cover, say, let $\hat \mathcal{U}$ be the corresponding Cech 3-groupoid (following your notation here), let $\Pi_3(X,\mathcal{U})$ (following notationally your lecture here) be the 3-groupoid generated from paths in the $U_i$ and Cech jumps between the fibers,

then the first $\infty$-Ehresmann condition says that a connection on a 2-bundle that locally trivialized over $U$ is not any 3-functor

$\Pi_3(X,\mathcal{U}) \to \mathbf{B} INN(G)$

but one that fits in a diagram

$\array{ \hat \mathcal{U} &\to& \mathbf{B}G &&& underlying cocycle for 2bundle \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi_3(X,\mathcal{U}) &\to& \mathbf{B} INN(G) &&& connection and curvature }$

where the two vertical arrows are the obvious/canonical ones.

The top horizontal morphism is the cocycle for the underlying $G$-principal 2-bundle. The diagram enforces that on Chech-jumps the 3-functor on $\Pi_3$ looks like a cocycle with values in $G$, while only on paths it is allowed to do funny things in $INN(G)$.

Posted by: Urs Schreiber on September 28, 2009 8:07 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Urs wrote:

Hey, wait, what is wrong about that? I mean transformations between 3-functors, without restrictions on them. In erudite nLab-speak: (3,1)-transformations.

Okay… I usually call these ‘pseudonatural transformations’, so I was scared you might be talking about transformations where you left out the pseudonaturality condition.

You see, some people do consider transformations between 2-functors without imposing naturality or pseudonaturality! (These show up as morphisms in the internal hom for $2Cat$ that’s adjoint to the ‘White tensor product’ of 2-categories, just as natural transformations show up as morphisms in the internal hom that’s adjoint to the ‘Black tensor product’, and pseudonatural transformations show up in the internal hom that’s adjoint to the ‘Gray tensor product’)

But I couldn’t quite believe you’d be considering transformations between 3-functors that were completely arbitrary in this particular way. So I’m very relieved that you’re not.

Thanks for all the other info, too.

Posted by: John Baez on September 28, 2009 9:28 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

You see, some people do consider transformations between 2-functors without imposing naturality or pseudonaturality!

Okay. For the full story here it is important that we really are in the context of smooth $\infty$-groupids with the standard notion of $\infty$-groupoid. All 1-, 2- Gray- and whatnot structures are to be thought of as models for special cases of this. So all their notions of transformations are such that embedded in the full theory they become the right notion, e.g. the one that under realization corresponds to homotopies.

Anyway, with that out of the way, let me tell you how the second $\infty$-Ehresmann condition also has a simple consequence and leads to the desired result.

Consider a 2-cell in $\Pi_3(X, \mathcal{U})$ of the form

$\array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \,.$

The first $\infty$-Ehresmann condition said that restricted to cells like this our 3-functor $\Pi_3(X,\mathcal{U}) \to \mathbf{B} INN(G)$ looks like a 2-functor with values in $\mathbf{B}G$. This is the familiar degree 2 nonabelian cocycle.

But now consider a 2-cell in $\Pi_3(X, \mathcal{U})$ of the form

$\array{ (x,i) &\to & (x,j) \\ \downarrow^{(\gamma,i)} & \Downarrow & \downarrow^{(\gamma,j)} \\ (y,i) &\to& (y,j) }$

for $\gamma$ a path in a double overlap. Since this is not in the image of the inclusion $\hat \mathcal{U} \to \Pi_3(X,\mathcal{U})$ the first $\infty$-Ehresmann condition gives no restriction here.

But the second does: the second $\infty$-Ehresmann condition says that the curvature characteristic forms of our 2-connection do descent to globally defined differential forms on $X$.

To see what this means, the Lie $\infty$-algebraic description of the situation is valuable:

the Lie $n$-algebra sequence corresponding to our

$\mathbf{B}G \to \mathbf{B} INN(G) \to \mathbf{B}G_{ch}$

is dually

$CE(\mathfrak{g}) \leftarrow W(\mathfrak{g}) \leftarrow inv(\mathfrak{g}) \,.$

More or less by definition, the invariant polynomials $P$ in $inv(\mathfrak{g})$ are invariant in $W(\mathfrak{g})$ under action by elements in $\mathfrak{g}$. Integrating this statement we find that a curvature characteristic form $P$ evaluated on our curvature data $\beta, H$ gives a form $P(\beta,H)$ which is invariant under those gauge transformations of $\beta$ and $H$ that come from gauge transformations in the image of $\mathbf{B}G \to \mathbf{B} INN(G)$.

Because the value of our 3-functor on the above 2-cell induces such a gauge transformation, it follows that the second $\infty$-Ehresmann condition is satisfied if our 3-functor colors the 2-cells

$\array{ (x,i) &\to & (x,j) \\ \downarrow^{(\gamma,i)} & \Downarrow & \downarrow^{(\gamma,j)} \\ (y,i) &\to& (y,j) }$

with data in the image of the inclusion $\mathbf{B}G \hookrightarrow \mathbf{B} INN(G)$ instead of more generally in all of $\mathbf{B} INN(G)$.

But this then finally means that the only 2-cells in $\Pi_3(X, \mathcal{U})$ on which the functor to $\mathbf{B} INN(G)$ does not look like a functor simply to $\mathbf{B}G$ are the 2-paths

$\array{ & \nearrow && \searrow^{(\gamma_1,i)} \\ (x,i) &&\Downarrow^{(\Sigma,i)}&& (y,i) \\ & \searrow && \nearrow_{(\gamma_2,i)} } \,.$

On these the 3-functor is allowed to be more general than a 2-functor with values in $\mathbf{B}G$. This way it doesn’t have to be fake flat!

On all other types of cells on $\Pi_3(X,\mathcal{U})$, though, the two Ehresmann conditions force the 3-functor to look like the 2-functor to $\mathbf{B}G$ and hence to produce the familiar cocycle data.

Posted by: Urs Schreiber on September 29, 2009 10:21 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Unexpectedly I have another second tonight and can quickly reply to one more question:

those old notes that you point to aim to show that and how flat 3-functors

$\Pi_3(X) \to \mathbf{B} INN(G)$

from the path 3-groupoid of $X$ to the delooping of the inner automorphism 3-group $INN(G)$ for the strict 2-group $G$ encode the connection data on a $G$-principal 2-bundle.

As you note, the 3-group $INN(G)$ was described in full detail in the article with David Roberts, where we also express it as a 2-crossed module.

Two remarks on this that are not in the old notes:

a) From that 2-crossed module it is easy to obtain the corresponding Lie 3-algebra, and this is $inn(\mathfrak{g})$ from my article with Hisham Sati and Jim Stasheff.

b) Back then I didn’t have a fully developed machinery to describe and handle the smooth weak path 3-groupoid and smooth weak 3-functors $\Pi_3(X) \to \mathbf{B} INN(G)$ at the explicit algebraic level of Gray-categories. But luckily, meanwhile João Faria Martins and Roger Picken filled that gap:

in their article

they describe $P_3(X)$ (and hence my $\Pi_3(X)$ obtained by dividing out full homotopy of 3-paths) and show that a set of Lie 3-algebraic connection data given (in my language) by that morphism

$\Omega^\bullet(X) \leftarrow CE(inn(\mathfrak{g})) = W(\mathfrak{g})$

yields a smooth Gray-functor $\Pi(X) \to \mathbf{B}INN(G)$.

So their construction together with my article with David Roberts sort of completes that old set of notes.

(Where I wrote $\Sigma INN(G)$ back then I am now writing $\mathbf{B} INN(G)$ as the suspension notation $\Sigma$ for delooping should be used only for the stably abelian case).

So this gives an alternative description in “algebraically defined higher categories”, alternative to the general nonsense “geometric-style definition” that I pointed you to on the $n$Lab.

(As soon as I get the time I’ll write up the above coherently on the Lab and point you to it. Probably tomorrow.)

Posted by: Urs Schreiber on September 26, 2009 10:57 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Here are a few more basic questions.

Here you say:

those old notes that you point to aim to show that and how flat 3-functors

$\Pi_3(X) \to \mathbf{B} INN(G)$

from the path 3-groupoid of $X$ to the delooping of the inner automorphism 3-group $INN(G)$ for the strict 2-group $G$ encode the connection data on a $G$-principal 2-bundle.

First: if I leave out the word ‘flat’ in your sentence here, is it still true? I.e., does ‘flat 3-functor’ just mean ‘3-functor from $\Pi_3(X)$’? Or is it — I hope not — some special sort of 3-functor?

Second: in those old notes you instead seem to talk about some sort of functor

$P_2(X) \to \mathbf{B} INN(G)$

Are you now saying that this is correct, but just part of the full story?

I’ll read the Martins–Picken paper, too — thanks for pointing that out.

Posted by: John Baez on September 26, 2009 11:50 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

It has to be flat, because that’s how we encode the Bianchi identity. Going down a dimension, we have that a 1-connection has a curvature which is a flat 2-connection, and this flatness is precisely the usual Bianchi identity $df + [a,f] = 0$. This lower-dimensional version is I think written down somewhere, but it eludes me, and like Urs, ‘I have not time’. This last bit should be written up on the nLab, but my thesis deadline is looming, and so are the date for deliverables to stakeholders at work (ugh - weasel words)

Posted by: David Roberts on September 27, 2009 1:12 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

David wrote:

It has to be flat, because that’s how we encode the Bianchi identity.

Thanks. I sort of get the basic idea. But here’s the question that’s bugging me.

Urs wrote:

those old notes that you point to aim to show that and how flat 3-functors

$\Pi_3(X) \to \mathbf{B} INN(G)$

from the path 3-groupoid of $X$ to the delooping of the inner automorphism 3-group $INN(G)$ for the strict 2-group $G$ encode the connection data on a $G$-principal 2-bundle.

In my previous comment, I was trying to ask:

Is “flat 3-functors $\Pi_3(X) \to \mathbf{B} INN(G)$ from the path 3-groupoid of $X$ to…” just another way of saying “3-functors $\Pi_3(X) \to \mathbf{B} INN(G)$”?

I think the correct answer to this question is either “yes” or “no”. I would find that one bit of information very useful. If it’s “yes”, I understand what’s going on. If “no”, I don’t.

Posted by: John Baez on September 27, 2009 2:52 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

if I leave out the word ‘flat’ in your sentence here, is it still true?

Yes!

By definition (at least the definition that I am thinking of), the degree 3 morphisms of $\Pi_3(X)$ are homotopy-classes (rel boundary) of 3-paths in $X$ and therefor any functor out of $\Pi_3(X)$ is “flat parallel transport”.

The point is to distinguish this well from morphisms out of what we used to write $P_3(X)$, which has thin-homotopy classes of maps in degree 3.

The main insight that drives the theory of “$n$-curvature” if you wish is that these $P_n(X)$ are conceptually not really fundamental, but more of a hack. What is conceptually fundamental is $P_\infty(X) := \Pi_\infty(X) =: \Pi(X)$.

Maps out of $\Pi(X)$ are flat parallel transport. And the theory of non-flat connections and their curvature is to be thought of as the theory of obstruction classes to lifts of plain cocycles $X \to A$ to flat differential cocycles $\Pi(X) \to A$.

So the whole theory is driven by the simple-minded sounding slogan:

Curvature is the obstruction to equipping a bundle with a flat connection.

But the point is that using the perspective of twisted cohomology there is a fully formal $\infty$-categorical way to interpret “is the obstructon to” and unwinding this yields to a full theory of differential cohomology.

The whole issue with “fake curvature” came from trying to define non-flat higher connections “by hand” (by truncating something) without using obstruction theory properly.

Posted by: Urs Schreiber on September 28, 2009 8:48 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Second: in those old notes you instead seem to talk about some sort of functor

$P_2(X) \to \mathbf{B} INN(G)$

Are you now saying that this is correct, but just part of the full story?

Yes, this is correct once propertly made precise, which I didn’t fully do back then:

Namely a morphism $P_2(x) \to \mathbf{B} INN(G)$ will have a unique extension to a morphism $\Pi_3(X) \to \mathbf{B}INN(G)$:

due to the nature of $INN(G)$ there is a unique 3-cell in $\mathbf{B} INN(G)$ between any two parallel 2-cells. This 3-cell measures the failure of these two 2-cells to be equal, which is the “integrated curvature” when the 2-cells are images of 2-paths.

Anyone following this here and wondering should be alerted: yes, in fact $\mathbf{B} INN(G)$ is contractible. It is part of a model where we model an $(\infty,1)$-categorical morphism by mapping out of resolutions. So $\mathbf{B} INN(G)$ really models the point, but it is crucial to not take it equal to the point.

Posted by: Urs Schreiber on September 28, 2009 9:00 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

full homotopy as opposed to ??

Posted by: jim stasheff on September 27, 2009 2:55 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

… as opposed to thin homotopy, I guess.

Posted by: John Baez on September 27, 2009 7:33 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Jim is wondering about my sloppy terminology:

full homotopy as opposed to ??

John replies:

… as opposed to thin homotopy, I guess.

Yes, that’s what I mean. Anyone wondering what we are talking about might have a look at $n$Lab: path $n$-groupoid and links provided there.

Posted by: Urs Schreiber on September 28, 2009 9:05 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

I hope this question is not too far off topic. Fix a 2-group $G$ and consider the 2-category of principal $G$-bundles over the category of manifolds. Does this form a 2-stack? What about principal $G$-bundles with (2-)connections?

Posted by: Eugene Lerman on September 28, 2009 2:01 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Are you asking about what’s true, or what’s been proved? I only know the truth.

Posted by: John Baez on September 28, 2009 9:32 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Sorry for not being precise. I am trying to ask if proofs have been written down or are about to be written down. I am guessing from your reply that you believe that both 2-categories satisfy 2-descent. Is it in your paper with Urs? Is it in any other paper? Is a grad student working on this?

Posted by: Eugene Lerman on September 28, 2009 11:13 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Eugene wrote:

I am guessing from your reply that you believe that both 2-categories satisfy 2-descent.

Yes, I believe so.

Is it in your paper with Urs?

Certainly there’s no theorem in our paper that says this, but I suspect the necessary raw ingredients for proving such a theorem are there. I believe they might be more explicitly visible in the papers by Urs and Konrad Waldorf.

Is it in any other paper? Is a grad student working on this?

I don’t know — Urs Schreiber and Larry Breen are the people you should ask. None of my grad students are working on 2-bundles or 2-connections. We’re into other stuff these days…

Posted by: John Baez on September 29, 2009 2:15 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Thank you.

Posted by: Eugene Lerman on September 29, 2009 3:06 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

Fix a 2-group $G$ and consider the 2-category of principal $G$-bundles over the category of manifolds. Does this form a 2-stack? What about principal $G$-bundles with (2-)connections?

In my article with Konrad Waldorf, Connections on non-abelian gerbes and their holonomy we do the descent for principal 2-bundles with connection for vanishing 2-form curvature (“fake flat case”).

In fact, there with that notion of “locally smoothly trivializable transport functors” we produce even a rectified 2-stack of such bundles, because the assignment

$X \mapsto Trans(X, \mathbf{B}G)$

of the 2-category of transport functors by cnstrucion has strictly functorial pullback/restriction maps.

So what we achieve there is constructing the $\infty$-stackification of $[P_2(-), \mathbf{B}G]$ (see also this $n$Lab entry) realized within the presentation of the category of $\infty$-stacks on $Diff$ by the model structure on simplicial presheaves.

Alas, we don’t say it this way in the article, but I am claiming that this is a good way to think of the “locally trivializabel transport functor”-technology used there.

Now, as I said, the claim is that to handle 2-bundles with connection without any constraints on the lower-degree curvature components one needs to tackle this problem a bit differently:

for each given $X$ the $n$-groupoid $\bar \mathbf{H}(X,\mathbf{B}G)$ of $G$-principal bundles with connection on $X$ is the homotopy pullback

$\array{ \bar \mathbf{H}(X, \mathbf{B}G) &\to& \pi_0(\mathbf{H}(X,\mathbf{B}G_{ch})) \\ \downarrow && \downarrow^{\{P_i\}} \\ \mathbf{H}(X,\mathbf{B}G) &\to& \mathbf{H}(X, \mathbf{B}G_{ch}) }$

where

$\pi_0(\mathbf{H}(X,\mathbf{B}G_{ch})) = H^\bullet_{dR}(X,inv(\mathfrak{g}))$

is deRham cohomology with coefficients in the invariant polynomials of the Lie 2-algebra $\mathfrak{g}$, and where the morphism ${P_i}_i$ picks one representative in each cohomology class: each single $P$ is a collection of curvature characteristic forms of a 2-bundle. So The $2$-groupoid of 2-bundles with connection that have curvature characteristic class $[P]$ is the homotopy pullback

$\array{ \bar \mathbf{H}_{[P]}(X, \mathbf{B}G) &\to& {*} \\ \downarrow && \downarrow^P \\ \mathbf{H}(X,\mathbf{B}G) &\to& \mathbf{H}(X, \mathbf{B}G_{ch}) } \,.$

This makes $\bar \mathbf{H}_{[P]}(X,\mathbf{B}G)$ an instance of twisted cohomology: the non-flat differential cohomology is realized as curvature-twisted flat cohomology, if you wish.

Anyway, this is the situation for a fixed base space $X$. And this is the right answer there. For instance this gives the right notion of equivariant 2-bundles with connection. I started a brief discussion of this here, but am not done typing yet. But Eugene might like this, as this gives rise to pseudonnections and their application in equivariant cohomology.

But what i am currently not sure about is in which sense this story over a fixed $X$ is supposed to extend to an $\infty$-stack. This is a question independent of the special case of differential cohomology, and arises in all of twisted cohomology:

Let $A \to B \to C$ be a homotopy fiber sequence, then the $C$-twisted $A$-cohomology $\mathbf{H}_{tw}(X,A)$ on a given space $X$ is the homotopy pullback

$\array{ \mathbf{H}_{tw}(X,A) &\to& \pi_0 \mathbf{H}(X,C) \\ \downarrow && \downarrow^{\{c_i\}} \\ \mathbf{H}(X,B) &\to& \mathbf{H}(X,C) \,, }$

where again the $c_i$ are one choice of cocycle representative in each cohomology class.

Now, IF there were a global such choice, i.e. a morphism of $\infty$-stacks

$?!?! : \pi_0 \mathbf{H}(-,C) \to \mathbf{H}(-,C)$

then the $\infty$-stack of $C$-twisted $A$-cohomology would be the homotopy pullback of $\infty$-stacks

$\array{ \mathbf{H}_{tw}(-,A) &\to& \pi_0 \mathbf{H}(X,C) \\ \downarrow && \downarrow^{?!?!} \\ \mathbf{H}(-,B) &\to& \mathbf{H}(-,C) \,. }$

But in general such a global choice does not exist.

I am still thinking about the best way to handle this. Probably I am being dense and missing some obvious point.

Posted by: Urs Schreiber on September 29, 2009 9:20 AM | Permalink | Reply to this

### Re: Questions on n-Curvature

subscript ch denotes?

Posted by: jim stasheff on September 29, 2009 2:19 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

subscript ch denotes?

I have now started writing $A_{ch}$ for what in the case that $A$ is once deloopable I write $\mathbf{B}A_{dR}$.

The definition is at differential cohomology - nonabelian case (which needs more typing).

The letters $ch$ are supposed to remind us of Chern class. Because cohomology with coefficients in $A_{ch}$ is curvature characteristic form data for differential $A$-cocycles.

I am not sure if I have found the best notation here, probably not. Also in the abelian case $\mathbf{B}A_{dR}$ for the coefficient object for $\mathbf{B}A$-valued “deRham data” this is maybe not optimal, as it doesn’t indicate the subtle differences in the notion of deRham data in the general context here:

essentially with just a gros $(\infty,1)$-topos $\mathbf{H}$ equipped with a cosimplicial object that yields a notion of path $\infty$-groupoid there is already a notion of nonabelian deRham cohomology in the sense of trivial $\infty$-bundles with nontrivial connection:

in this generality the notion of $\infty$-bundle with connection is actually more fundamental than that of differential form, and by analogy with the ordinary case one can hence define a differential form to be a connection on a trivial $\infty$-bundle.

But a terminological conflict arises as soon as we exhibit more ambient structure, namely that of an ambient smooth $(\infty,1)$-topos. That, being modeled by simplicial synthetic spaces, now comes with a notion of differential forms in the ordinary sense.

In good cases this does coincide with the general abstract notion of “nonabelian deRham cocycles”, but this is not necessarily so, I think, and in any case not entirely trivial.

So there is a good chance that I will change my mind about terminology and notation at some point.

Posted by: Urs Schreiber on September 29, 2009 7:33 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

Urs, It will not surprise you to learn that I don’t understand $(\infty, 1)$ stacks, but I am happy to have a reference… Thanks.
Posted by: Eugene Lerman on September 29, 2009 3:38 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

It will not surprise you to learn that I don’t understand $(\infty,1)$-stacks,

By the way, there are helpful models for $\infty$-stack $(\infty,1)$-toposes that make them easier to handle.

Posted by: Urs Schreiber on September 29, 2009 7:13 PM | Permalink | Reply to this

### Re: Questions on n-Curvature

As promised, I started writing an $n$Lab entry on this. Not done yet, but have to interrupt now. It might already be useful to some extent:

principal 2-bundle with connection