The n-Category Café

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.)

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 $INN(G)$.

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.

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.

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)

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.

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.

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.

… as opposed to thin homotopy, I guess.

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.

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?

subscript ch denotes?