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April 25, 2007

n-Curvature

Posted by Urs Schreiber

We have learned that parallel nn-transport in an nn-bundle with connection over a base space XX is an nn-functor tra:𝒫 n(X)T \mathrm{tra} : \mathcal{P}_n(X) \to T from the nn-path nn-groupoid of XX to some nn-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 nn-functor tra\mathrm{tra} is canonically associated a curvature (n+1)(n+1)-functor curv tra:Π n+1(X)T n+1. \mathrm{curv}_{\mathrm{tra}} : \Pi_{n+1}(X) \to T_{n+1}\,. The functor tra\mathrm{tra} is flat precisely if curv tra\mathrm{curv}_{\mathrm{tra}} is trivial on all (n+1)(n+1)-morphisms.


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

3) Parallel sections ee of the nn-bundle with connection associated with tra\mathrm{tra} are equivalent to morphisms from the trivial nn-transport into tra\mathrm{tra}: e:tra 0tra. e : \mathrm{tra}_0 \to \mathrm{tra} \,.

4) General sections ee together with their covariant derivative e\nabla e are equivalent to morphisms from the trivial curvature (n+1)(n+1)-transport into the curvature (n+1)(n+1)-transport (e,e):curv 0curv tra. (e,\nabla e) : \mathrm{curv}_0 \to \mathrm{curv}_{\mathrm{tra}} \,.

What is the curvature associated with a transport, really?

Whatever it is, it should vanish if our transport factors through the fundamental nn-groupoid Π n(X)\Pi_n(X) of XX. As opposed to 𝒫 n(X)\mathcal{P}_n(X), whose nn-morphisms are thin homotopy classes of nn-paths, the nn-morphisms of Π n\Pi_n are ordinary homotopy classes of nn-paths.

Hence we have a canonical projection π:𝒫 n(X)Π n(X) \pi : \mathcal{P}_n(X) \to \Pi_n(X) that sends any nn-path to its homotopy class.

Definition. We say that tra\mathrm{tra} is flat precisely if there is an nn-functor f:Π n(X)T f : \Pi_n(X) \to T such that 𝒫 n(X) π Π n(X) tra f T = T. \array{ \mathcal{P}_n(X) &\stackrel{\pi}{\to}& \Pi_n(X) \\ {}^{\mathrm{tra}}\downarrow\;\;\, &\Downarrow^\sim& \;\;\;\downarrow^f \\ T &=& T } \,.

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

Proposition. Given any nn-transport tra:𝒫 n(X)T\mathrm{tra} : \mathcal{P}_n(X) \to T, we canonically obtain an (n+1)(n+1)-category T n+1 T_{n+1} and an (n+1)(n+1)-functor curv tra:Π n+1(X)T n+1 \mathrm{curv}_\mathrm{tra} : \Pi_{n+1}(X) \to T_{n+1} such that curv tra\mathrm{curv}_{\mathrm{tra}} is trivial on (n+1)(n+1)-morphisms (sends every (n+1)(n+1)-morphism to an identity (n+1)(n+1)-morphism) if and only if tra\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:

Curvature of Transport

Posted at April 25, 2007 7:38 PM UTC

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6 Comments & 13 Trackbacks

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 nn-thing P n(X) tra T \array{ P_n(X) \\ \downarrow^{\mathrm{tra}} \\ T } on P n(X)P_n(X) and want to push it down to Π n(X)\Pi_n(X) by completing a square to the right P n(X) Π n(X) tra f T = T. \array{ P_n(X) &\to& \Pi_n(X) \\ {}^{\mathrm{tra}}\downarrow \;\; &\Downarrow^\sim& \;\; \downarrow^f \\ T &=& T } \,.

The obstruction is an (n+1)(n+1)-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(1)HG U(1) \to H \to G of groups, and given a principal GG-transport P 1(X) tra GTor \array{ \P_1(X) \\ \;\;\downarrow^{\mathrm{tra}} \\ G\mathrm{Tor} } we want to know if we can lift by completing a square to the left P 1(X) = P 1(X) f tra HTor GTor. \array{ \P_1(X) &=& \P_1(X) \\ {}^f \downarrow\;\; &\Downarrow^\sim& \;\;\downarrow^{\mathrm{tra}} \\ H\mathrm{Tor} &\to& G\mathrm{Tor} \,. }

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 aa and bb in XX such that aba b is defined If EXE\to X were a fibration, i.e. with path lifting but not unique, then even with initial starting points the lift of aba b need not be the composite of lift aa lift bb. 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:tra otra e:\mathrm{tra}_o \to \mathrm{tra} is defined for every oo so really we have ObjT×P nXT\mathrm{Obj} T \times P_n X \to T ??

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

- what’s TT'? (mid page)

- =d+A\nabla = d + A??

- whoops, what’s dd?

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

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

For any nn-functor f:Π nXTf: \Pi_n X \to T fπtraf\circ \pi - \mathrm{tra} is the obstruction which should be just a representativie of something like a cohomology class as ff varies then your construction of T n+1T_{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 XTX\to T, which is just a set, although XX 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(X,π)=H^n(X, \pi) = set of homotopy classes XK(π,n)X \to K(\pi,n) which is isomorphic to set of homotopy classes XΩK(π,n+1)X \to \Omega K(\pi,n+1) which is isomorphic to set of homotopy classes ΣXK(π,n+1)\Sigma X \to K(\pi,n+1) where here Σ\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 (x 1,x 2)(x_1,x_2) with both in the same connected component so this is a subthing of the usual (x 1,x 2)(x 2,x 3)(x 1,x 3)(x_1,x_2) (x_2,x_3) \to (x_1,x_3) ??

p. 7 - why e f(x)e^f(x) just to get F 1=dfF_1 = 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+1T_{n+1} is precisely analogous to the Koszul-Tate construction.

Oh, that’s interesting! I’ll have to think about this. Do you think this can be made precise?

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 tra:𝒫 n(X)T \mathrm{tra} : \mathcal{P}_n(X) \to T I can already say what flat sections are, namely morphisms into this e:tra otra. e : \mathrm{tra}_o \to \mathrm{tra} \,. 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 ee above is nothing but the “integrated” version of the usual differential condition e=0\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 (e,e):curv ocurv tra. (e,\nabla e) : \mathrm{curv}_o \to \mathrm{curv}_{\mathrm{tra}} \,. Indeed, as a morphism of (n+1)(n+1)-functors, this transformation is an nn-functor on 𝒫 n(X)\mathcal{P}_n(X) itself. As such, if smooth, it comes from pp-form data (which encodes the differential description of the “integrated” nn-functor). This pp-form data is the covariant derivative of the section (by my definition for n>1n\gt 1, and by inspection for n=1n=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:tra otrae : \mathrm{tra}_o \to \mathrm{tra} is defined for every oo, so we really have Obj(T)×𝒫 n(X)T\mathrm{Obj}(T) \times \mathcal{P}_n(X) \to T?

Hm, interesting point. I’ll think about this.

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

If the vector bundle is rank nn, one might be tempted to set o= no = \mathbb{C}^n. The a morphism e:tra otrae : \mathrm{tra}_o \to \mathrm{tra} is not just one flat section, but nn of them. On the other hand, it then makes sense to ask if ee is an isomorphism, in which case it would trivialize the vector bundle.

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

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

- what’s TT'

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

- =d+A\nabla = d + A

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

Defn 1 - if there is an nn-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 nn-categories and everything else strict.

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

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

But from my first entry on curvature here at the nn-café I know that the kind of 2-connection that Breen-Messing considered is really a flat 3-transport with values in INN(G 2)\mathrm{INN}(G_2) – and the latter is not strict but has a nontrivial isomorphism in the exchange law. I also know that for G 2=String k(G)G_2 = \mathrm{String}_k(G) 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 XTX \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 (x 1,x 2)(x_1,x_2) with both in the same connected component so this is a subthing of the usual (x 1,x 2)(x 2,x 3)(x 1,x 3)(x_1,x_2)(x_2,x_3) \to (x_1,x_3) ??

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

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

p. 7 - why e f(x)e^{f(x)} just to get F 1=dfF_1 = 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 Π n(X)\Pi_n(X) as having as objects separate points (not disjoint unions of them), and having as pp-morphisms pp-surfaces cobounding their source and target (p1)(p-1)-morphisms. That impliees that everything here is pp-disk-shaped.

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

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 (n+1)(n+1)-curvature functor from a given nn-transport that I descibe above may be regarded as a generalization of the exterior derivative:

- for nn-functors with values in Σ nU(1)\Sigma^n U(1) it reduces to the ordinary exterior derivative (under the equivalence of these nn-functors with degree-nn differential forms)

- for nn-functors with values in other Lie nn-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 (n+2)(n+2)-functor which acts trivially on (n+2)(n+2)-morphisms: the (n+1n+1)-curvature itself is always flat.

Then the natural question is: if we start with a flat nn-transport: under which conditions may we regard this as the curvature of some (n1)(n-1)-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 GG-gerbes with connection is actually not that of a 2-transport, but that of a 3-transport with values (locally) in the 3-group INN(AUT(G))\mathrm{INN}(\mathrm{AUT}(G)).

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 INN(AUT(G))\mathrm{INN}(\mathrm{AUT}(G)), 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 3T_3 of some 2-category TT (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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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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