April 25, 2007

n-Curvature

Posted by Urs Schreiber

We have learned that parallel $n$-transport in an $n$-bundle with connection over a base space $X$ is an $n$-functor $\mathrm{tra}:{𝒫}_{n}\left(X\right)\to T$ from the $n$-path $n$-groupoid of $X$ to some $n$-category of fibers.

With every notion of connection we expect to obtain notions of

1) curvature;
2) Bianchi identity;
3) parallel sections;
4) covariant derivative.

Here we describe the functorial incarnation of these concepts. We find

1) To every transport $n$-functor $\mathrm{tra}$ is canonically associated a curvature $\left(n+1\right)$-functor ${\mathrm{curv}}_{\mathrm{tra}}:{\Pi }_{n+1}\left(X\right)\to {T}_{n+1}\phantom{\rule{thinmathspace}{0ex}}.$ The functor $\mathrm{tra}$ is flat precisely if ${\mathrm{curv}}_{\mathrm{tra}}$ is trivial on all $\left(n+1\right)$-morphisms.

2) The curvature $\left(n+1\right)$-functor, regarded as an $\left(n+1\right)$-transport itself, is always flat.

3) Parallel sections $e$ of the $n$-bundle with connection associated with $\mathrm{tra}$ are equivalent to morphisms from the trivial $n$-transport into $\mathrm{tra}$: $e:{\mathrm{tra}}_{0}\to \mathrm{tra}\phantom{\rule{thinmathspace}{0ex}}.$

4) General sections $e$ together with their covariant derivative $\nabla e$ are equivalent to morphisms from the trivial curvature $\left(n+1\right)$-transport into the curvature $\left(n+1\right)$-transport $\left(e,\nabla e\right):{\mathrm{curv}}_{0}\to {\mathrm{curv}}_{\mathrm{tra}}\phantom{\rule{thinmathspace}{0ex}}.$

What is the curvature associated with a transport, really?

Whatever it is, it should vanish if our transport factors through the fundamental $n$-groupoid ${\Pi }_{n}\left(X\right)$ of $X$. As opposed to ${𝒫}_{n}\left(X\right)$, whose $n$-morphisms are thin homotopy classes of $n$-paths, the $n$-morphisms of ${\Pi }_{n}$ are ordinary homotopy classes of $n$-paths.

Hence we have a canonical projection $\pi :{𝒫}_{n}\left(X\right)\to {\Pi }_{n}\left(X\right)$ that sends any $n$-path to its homotopy class.

Definition. We say that $\mathrm{tra}$ is flat precisely if there is an $n$-functor $f:{\Pi }_{n}\left(X\right)\to T$ such that $\begin{array}{ccc}{𝒫}_{n}\left(X\right)& \stackrel{\pi }{\to }& {\Pi }_{n}\left(X\right)\\ {}^{\mathrm{tra}}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}& {⇓}^{\sim }& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{f}\\ T& =& T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

Whatever the curvature of an $n$-functor is, it should be an obstruction for this construction.

Proposition. Given any $n$-transport $\mathrm{tra}:{𝒫}_{n}\left(X\right)\to T$, we canonically obtain an $\left(n+1\right)$-category ${T}_{n+1}$ and an $\left(n+1\right)$-functor ${\mathrm{curv}}_{\mathrm{tra}}:{\Pi }_{n+1}\left(X\right)\to {T}_{n+1}$ such that ${\mathrm{curv}}_{\mathrm{tra}}$ is trivial on $\left(n+1\right)$-morphisms (sends every $\left(n+1\right)$-morphism to an identity $\left(n+1\right)$-morphism) if and only if $\mathrm{tra}$ is flat.

When I wrote my first entry on this topic I think I had the right idea. But I did it by hand. This time I want to let the Dao do it by its own means:

Posted at April 25, 2007 7:38 PM UTC

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Lifting gerbes?

Here is a problem whose solution I do not understand yet.

The way I have discussed curvature above, it is the obstruction to a descent problem:

we have some $n$-thing $\begin{array}{c}{P}_{n}\left(X\right)\\ {↓}^{\mathrm{tra}}\\ T\end{array}$ on ${P}_{n}\left(X\right)$ and want to push it down to ${\Pi }_{n}\left(X\right)$ by completing a square to the right $\begin{array}{ccc}{P}_{n}\left(X\right)& \to & {\Pi }_{n}\left(X\right)\\ {}^{\mathrm{tra}}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& {⇓}^{\sim }& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{f}\\ T& =& T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

The obstruction is an $\left(n+1\right)$-thing.

This curiously smells like it should be one aspect of a general mechanism of which lifting gerbes are another.

For lifting gerbes, the problem is essentially “the same but opposite”:

given an extension $U\left(1\right)\to H\to G$ of groups, and given a principal $G$-transport $\begin{array}{c}{P}_{1}\left(X\right)\\ \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{\mathrm{tra}}\\ G\mathrm{Tor}\end{array}$ we want to know if we can lift by completing a square to the left $\begin{array}{ccc}{P}_{1}\left(X\right)& =& {P}_{1}\left(X\right)\\ {}^{f}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& {⇓}^{\sim }& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{\mathrm{tra}}\\ H\mathrm{Tor}& \to & G\mathrm{Tor}\phantom{\rule{thinmathspace}{0ex}}.\end{array}$

We know from pedestrian reasoning that this lift of 1-things is obstructed by a 2-thing: the lifting gerbe.

Therefore I expect that there is, as there was for curvature, a canonical arrow-theoretic construction that reads in the above extension problem and canonically spits out the parallel transport 2-functor of the lifting gerbe.

The entire problem here looks like that for curvature, with arrows reversed. Therefore I thought it would be simple to see how it works. But I still don’t see it.

Posted by: urs on April 26, 2007 5:12 PM | Permalink | Reply to this

Re: n-Curvature

Does the definition of covariant derivative come before parallel sections? Even though parallel transport comes first?

Here’s one way to look at curvature from a homtopy point of view, though I think it is also implicit in your version:

consider paths $a$ and $b$ in $X$ such that $ab$ is defined If $E\to X$ were a fibration, i.e. with path lifting but not unique, then even with initial starting points the lift of $ab$ need not be the composite of lift $a$ lift $b$. The difference could give the 2-transport?

p.2 - The diagram with the diagonal double arrow says the diagram commutes up to the double arrow (I’ll think of it as a homotopy) but is something stronger implied by labeling it with $\sim$ ??

p.3 - $e:{\mathrm{tra}}_{o}\to \mathrm{tra}$ is defined for every $o$ so really we have $\mathrm{Obj}T×{P}_{n}X\to T$ ??

Concerning flat sectins of principal bundles: there aren’t any sections at all unless $P$ is equivalent to $X×G$ so are flat section with respect to a particular equivalence the ones that correspond to $X×g$ for some $g\in G$?

- what’s $T\prime$? (mid page)

- $\nabla =d+A$??

- whoops, what’s $d$?

Defn 1 - if there is an $n$-functor and a transformation (the diagonal one in the diagram) ??

Yes, your construction of ${T}_{n+1}$ is precisely analogous to the Koszul-Tate construction.

For any $n$-functor $f:{\Pi }_{n}X\to T$ $f\circ \pi -\mathrm{tra}$ is the obstruction which should be just a representativie of something like a cohomology class as $f$ varies then your construction of ${T}_{n+1}$ kills the class – though I haven’t a clue yet as to what this ‘cohomology’ is

- except for one sentence in line 6 I see no need for weak cats

- If I’m right, a 0-transport is just a function $X\to T$, which is just a set, although $X$ is a topological space

cf. Bott’s remarks in the intro to Bott and Tu, especially in re: constant on each connected component (not necessariuly path component)

N.B. ordianary cohomology of nice spaces ${H}^{n}\left(X,\pi \right)=$ set of homotopy classes $X\to K\left(\pi ,n\right)$ which is isomorphic to set of homotopy classes $X\to \Omega K\left(\pi ,n+1\right)$ which is isomorphic to set of homotopy classes $\Sigma X\to K\left(\pi ,n+1\right)$ where here $\Sigma$ is suspension

p.6 - I’m tempted to draw these diagrams to look more cubical either in the usual 3D perspective or the square within a square

But first I must master your pullback. It consists of pairs $\left({x}_{1},{x}_{2}\right)$ with both in the same connected component so this is a subthing of the usual $\left({x}_{1},{x}_{2}\right)\left({x}_{2},{x}_{3}\right)\to \left({x}_{1},{x}_{3}\right)$ ??

p. 7 - why ${e}^{f}\left(x\right)$ just to get ${F}_{1}=\mathrm{df}$?

homotopy classes of surfaces - so you really want cobordisms i.e. allow the surfaces to have handles as opposed to just disks?

Posted by: Jim Stasheff on April 30, 2007 7:07 AM | Permalink | Reply to this

Re: n-Curvature

Yes, your construction of ${T}_{n+1}$ is precisely analogous to the Koszul-Tate construction.

Does the definition of covariant derivative come before parallel sections? Even though parallel transport comes first?

Right, so from the point of view of parallel transport functors, the logical order of some of these constructions is different from what one is ordinarily used to.

So given just the parallel transport $\mathrm{tra}:{𝒫}_{n}\left(X\right)\to T$ I can already say what flat sections are, namely morphisms into this $e:{\mathrm{tra}}_{o}\to \mathrm{tra}\phantom{\rule{thinmathspace}{0ex}}.$ I haven’t even introduced the covariant derivative at this point yet. But when one looks at what these constructions mean in detail, one sees that the existence of the morphism $e$ above is nothing but the “integrated” version of the usual differential condition $\nabla e=0$.

This relation is then made explicit and precise by the realization that arbitrary sections come from parallel sections of the curvature transport $\left(e,\nabla e\right):{\mathrm{curv}}_{o}\to {\mathrm{curv}}_{\mathrm{tra}}\phantom{\rule{thinmathspace}{0ex}}.$ Indeed, as a morphism of $\left(n+1\right)$-functors, this transformation is an $n$-functor on ${𝒫}_{n}\left(X\right)$ itself. As such, if smooth, it comes from $p$-form data (which encodes the differential description of the “integrated” $n$-functor). This $p$-form data is the covariant derivative of the section (by my definition for $n>1$, and by inspection for $n=1$).

p.2 - The diagram with the diagonal double arrow says the diagram commutes up to the double arrow (I’ll think of it as a homotopy) but is something stronger implied by labeling it with ∼ $\sim$??

If you think homotopy anyway, then the $\sim$ doesn’t add anything. That symbol is supposed to indicate that the 2-arrow here is an equivalence, hence that it may be inverted up to higher stuff.

p.3 - $e:{\mathrm{tra}}_{o}\to \mathrm{tra}$ is defined for every $o$, so we really have $\mathrm{Obj}\left(T\right)×{𝒫}_{n}\left(X\right)\to T$?

Right now, I’d fix one $o\in \mathrm{Obj}\left(T\right)$ once and for all and then consider flat sections with respect to this given $o$. For principal 1-bundles and choosing $o=G$ one obtains the ordinary notion. Same for vector bundles and $o=ℂ$.

If the vector bundle is rank $n$, one might be tempted to set $o={ℂ}^{n}$. The a morphism $e:{\mathrm{tra}}_{o}\to \mathrm{tra}$ is not just one flat section, but $n$ of them. On the other hand, it then makes sense to ask if $e$ is an isomorphism, in which case it would trivialize the vector bundle.

So, different purposes are served by looking at flat sections for different $o$. I am not sure yet that I fully understand what this is trying to tell me. Somehow having to choose such an $o$ is a nuisance.

So maybe you have good point here, and I should be looking at all $o$ at once. I’ll think about this.

- what’s $T\prime$

Oh, a typo. Usually this denotes my “catgeory of typical fibers”, but here it’s a typo. I’ll correct that.

- $\nabla =d+A$

Well, that’s at least the idea. $\nabla$ denotes the covariant derivative associated with a connection. Locally and with $A$ acting suitably, it does look like $d+A$, yes. And $d$ is the just the ordinary exterior derivative.

Defn 1 - if there is an $n$-functor and a transformation (the diagonal one in the diagram) ??

Yes, that’s what I mean.

- except for one sentence in line 6 I see no need for weak cats

Right, you could do all this just for strict $n$-categories and everything else strict.

But for many applications that won’t be interesting enough. So we would want to have a notion of $n$-curvature that applies also to the weak case.

Right now I don’t address this in detail, really. I spell out the cases $n=0,1,2$ where I consider all categories to be strict, all $n$-functors to be strict and all 2-transformations to be pseudo.

But from my first entry on curvature here at the $n$-café I know that the kind of 2-connection that Breen-Messing considered is really a flat 3-transport with values in $\mathrm{INN}\left({G}_{2}\right)$ – and the latter is not strict but has a nontrivial isomorphism in the exchange law. I also know that for ${G}_{2}={\mathrm{String}}_{k}\left(G\right)$ this is intimately related to Chern-Simons transport.

And this means that I am interested in non-strict parallel transport. :-)

- If I’m right, a 0-transport is just a function $X\to T$,

Yes!

p.6 - I’m tempted to draw these diagrams to look more cubical either in the usual 3D perspective or the square within a square

Yes, that’s probably a good idea. There are a couple of ways one could defomr these diagrams without changing them, but possibly making them look more suggestive. I wouldn’t be surprised if it turned out that these diagrams are a special case of some general construction. My hope is that somebody will see them here and recognize them as what they really are.

But first I must master your pullback. It consists of pairs $\left({x}_{1},{x}_{2}\right)$ with both in the same connected component so this is a subthing of the usual $\left({x}_{1},{x}_{2}\right)\left({x}_{2},{x}_{3}\right)\to \left({x}_{1},{x}_{3}\right)$ ??

Yes, exactly. The groupoid here that appears for curvature of 0-transport is not the full pair groupoid of $X$, but the disjoint union of the pair groupoids of the connected components of $X$.

(And, by the way, is different from the fundamental groupoid of $X$, except when $X$ is simply connected.)

p. 7 - why ${e}^{f\left(x\right)}$ just to get ${F}_{1}=\mathrm{df}$?

Yes. First I didn’t have that. Then I realized that if I don’t assume the 0-transport to be non-vanishing in this example, I’d need to be more careful with the corresponding 1-curvature. So for simplicity I then just assumed that it is non-vanishing. Just to get the main point across.

homotopy classes of surfaces - so you really want cobordisms i.e. allow the surfaces to have handles as opposed to just disks?

Well, currently I am still thinking of ${\Pi }_{n}\left(X\right)$ as having as objects separate points (not disjoint unions of them), and having as $p$-morphisms $p$-surfaces cobounding their source and target $\left(p-1\right)$-morphisms. That impliees that everything here is $p$-disk-shaped.

But I believe this is not all that essential. A similar discussion of $n$-curvature should go through if I allow my domain $n$-category to be more like extended $n$-cobordisms in $X$.

But in that case the codomain must be able to mimick the required tensor product structures. So then we can no longer talk about principal transport anymore, but are restricted to vector transport.

Posted by: urs on April 30, 2007 7:49 AM | Permalink | Reply to this

Poincaré Lemma

Another open question is: “What happens to the Poincaré lemma”?

The construction of an $\left(n+1\right)$-curvature functor from a given $n$-transport that I descibe above may be regarded as a generalization of the exterior derivative:

- for $n$-functors with values in ${\Sigma }^{n}U\left(1\right)$ it reduces to the ordinary exterior derivative (under the equivalence of these $n$-functors with degree-$n$ differential forms)

- for $n$-functors with values in other Lie $n$-groups it reduces to the appropriate notion of covariant exterior derivative

As I mentioned, the Bianchi identity is built in, so “taking that generalized exterior derivative” twice always produces an $\left(n+2\right)$-functor which acts trivially on $\left(n+2\right)$-morphisms: the ($n+1$)-curvature itself is always flat.

Then the natural question is: if we start with a flat $n$-transport: under which conditions may we regard this as the curvature of some $\left(n-1\right)$-transport?

I think we have one interesting example that demonstrates the relevance of this question:

if you belive my computation here, the Breen-Messing differential cocycle data for nonabelian $G$-gerbes with connection is actually not that of a 2-transport, but that of a 3-transport with values (locally) in the 3-group $\mathrm{INN}\left(\mathrm{AUT}\left(G\right)\right)$.

Since this 3-group is codiscrete at top level, this transport is – as a 3-transport – necessarily flat.

This would be perfectly consistent with the idea that this data should really describe parallel transport in a 1-gerbe, namely 2-transport, if we could regard this flat 3-transport as the curvature of some 2-transport.

But I am pretty sure that we cannot: the 3-group $\mathrm{INN}\left(\mathrm{AUT}\left(G\right)\right)$, regarded as a 3-groupoid with a single object, is not strict – the exchange law holds only up to 3-isomorphism – and hence, I think, impossible to realize as the ${T}_{3}$ of some 2-category $T$ (in my above notation).

This would mean that there is no 2-transport which integrates the Breen-Messing connection data, but that this data is genuinely a (flat, though), 3-transport – unless of course the “fake curvature” vanishes.

Posted by: urs on April 30, 2007 2:31 PM | Permalink | Reply to this
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Excerpt: Anders Kock gives a synthetic differential description of parallel n-transport using strict n-fold categories.
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Re: n-Curvature

I have a question that may be born of ignorance: so in these notes, you define the curvature of an n-transport using a pullback.

You seem to be assuming that these pullbacks must exist, but it’s not clear to me why.

Posted by: Creighton Hogg on August 14, 2007 4:39 AM | Permalink | Reply to this

Re: n-Curvature

You seem to be assuming that these pullbacks must exist

That’s true. One would need to think about under which conditions they actually do exist.

My point in these notes was that in the cases we care about, where these functors go from paths to group elements, for instance, the pullback happens to exist.

On the other hand, notice that the curvature of a functor the way I define it in Arrow-theoretic differential theory always exists, for every functor whatsoever.

And in those “cases of interest”, the result happens to coincide (up to some canonical identifications) with the result I get here using a pullback construction.

So, now I tend to think of the “arrow-theoretic exterior differential” of a functor as the general concept, and as the description in terms of pullbacks as a useful alternative point of view for certain applications.

Posted by: Urs Schreiber on August 14, 2007 7:24 AM | Permalink | Reply to this
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