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May 4, 2007

The First Edge of the Cube

Posted by Urs Schreiber

One knows one is getting to the heart of the matter when the definitions in terms of which one conceives the objects under consideration categorify effortlessly.

Take the ordinary definition of a smooth fibre bundle with connection. Write it out in full detail. Stare at that page for a while and try to categorify everything in sight.

Then, when sufficiently frustrated – to increase the effect – notice that the connection on the bundle gives rise to a parallel transport functor from paths in base space, 𝒫 1(X)\mathcal{P}_1(X), to the category of fibers.

For a principal bundle this is tra:𝒫 1(X)GTor \mathrm{tra} : \mathcal{P}_1(X) \to G\mathrm{Tor} for a vector bundle this is tra:𝒫 1(X)Vect. \mathrm{tra} : \mathcal{P}_1(X) \to \mathrm{Vect} \,. This are nothing but representations of the path groupoid.

Now categorify this. This is so trivial that trying to avoid it is like trying to avoid to breathe. Here is a parallel nn-transport on a principal nn-bundle with connection: tra:𝒫 n(X)G nTor. \mathrm{tra} : \mathcal{P}_n(X) \to G_n\mathrm{Tor} \,. Here is a parallel transport on an nn-vector bundle with connection tra:𝒫 n(X)nVect. \mathrm{tra} : \mathcal{P}_n(X) \to n\mathrm{Vect} \,. Here is a parallel transport on an nn-bundle whose fibers are objects in the nn-category TT of your choice: tra:𝒫 n(X)T. \mathrm{tra} : \mathcal{P}_n(X) \to T \,. We are just looking at representations of the path nn-groupoid of a space XX.

Not all of these nn-bundles with connection will be useful in applications. Most will be badly pathological. The useful ones will be locally trivial with respect to a predefined local structure.

Again, we want to say this in a way that it’s a breeze to categorify. This means we draw arrows.

A little contemplation reveals that a local structure on an nn-functor is a square

(1)𝒫 n(Y) π 𝒫 n(X) triv t tra T i T \begin{aligned} \mathcal{P}_n(Y) & \overset{\quad \pi\quad}{\to} && \mathcal{P}_n(X) \\ \mathrm{triv}\downarrow & \sim\Downarrow t && \downarrow \mathrm{tra} \\ T^\prime & \underset{\quad i\quad}{\to} && T \end{aligned}

where π\pi is epi and ii is mono.

If we can complete the nn-functor to such a square, it makes sense to say that (π\pi-)locally the functor has ii-structure.

All that remains is to choose the ii that describes your application.

In fact, it was the observation that plenty of intricate-looking structures that people consider in the literature on gerbes and other higher gauge theory (notice that the term is catching on slowly but steadily) all follow from nothing but a square as above, for various choices of ii.

In particular, every such square naturally produces an object in a category of descent data, just by itself. All that is required is playing Lego with these squares: two of them paste together to give a transition function. Six to give a triangle. Twelve a tetrahedron – these shapes are the differential cocycle representing the differential cohomology class of the nn-transport.

And a morphism of squares gives a morphism of differential cocycles in the obvious way, so we have an nn-functor Ex π:Triv π n(i)Desc π n(i) \mathrm{Ex}_\pi : \mathrm{Triv}^n_\pi(i) \to \mathrm{Desc}_\pi^n(i) which extracts from an nn-transport with π\pi-local ii-trivialization tt this cocycle information.

All this is god-given. The first point where one actually has to do work comes when we ask if Ex π\mathrm{Ex}_\pi may be inverted.

Notice that two, six, twelve, etc. are even numbers. So the descent data Desc π n(i)\mathrm{Desc}_\pi^n(i) can only know about pairs of squares (1). Can we take the square root?

If we can, and in examples we can, we get a weak inverse nn-functor Rec π:Desc π n(i)Triv π n(i). \mathrm{Rec}_\pi : \mathrm{Desc}_\pi^n(i) \to \mathrm{Triv}^n_\pi(i) \,. which reconstructs an nn-bundle with connection from its differential cocycle data.

Thus showing that locally ii-trivializable nn-functors form an nn-stack, as they should.

One aspect of the power of this formulation of differential cohomology is that I haven’t even had to mention the word differentiable yet. Nor the word continuous.

While nice, this is a tad more general than what one would be willing to deal with on a normal day: we will want to be able to say that an nn-transport functor is not just locally trivializable – but also that its descent data has extra structure, like topological, or smooth structure.

For satisfying this demand, essentially no additional work is required beyond translating this demand into arrow-theory. It will take care of itself then.

As John Baez teaches (and as his students have written up here and here – the lore of stuff, structure and property) extra structure (on sets, here) is a faithful functor f:CSet. f : C \to \mathrm{Set} \,. Extra structure on our nn-categories requires that CC admits enough pullbacks.

So if we want continuous transport, we choose C=TopC = \mathrm{Top}. If we want smooth transport, we choose C=S C = S^\infty, the category of Chen-smooth spaces.

Given that, realize the local domain 𝒫 n(X)\mathcal{P}_n(X) and the local codomain T T^\prime internal to CC. (For instance, let T =GrT^\prime = \mathrm{Gr} be a Lie groupoid, a groupoid internal to smooth spaces. This will enforce that ii-trivial transport functors locally take values in objects and morphisms of that Lie groupoid.) Then denote by Desc π n(i) C {\mathrm{Desc}^n_\pi(i)}^C the nn-category of descent data as before, now with everything internal to CC. Define Triv π n(i) C {\mathrm{Triv}_\pi^n(i)}^C to be the pre-image of that under Ex π\mathrm{Ex}_\pi: this are those nn-transport functors whose local can be lifted through f:CSetf : C \to \mathrm{Set} CC, i.e. those whose descent data has CC-structure. Check that this restricts to an equivalence Ex π C:Triv π n(i) CDesc π n(i) C \mathrm{Ex}_\pi^C : {\mathrm{Triv}_\pi^n(i)}^C \to {\mathrm{Desc}^n_\pi(i)}^C and thus obtain a notion of nn-transport nn-functor equipped with tt a local-CC ii-trivialization. Such a gadget is what really deserves to be called “transport” (as opposed to being an arbitrary functor). And if so, we have the urge to forget the choice of local trivialization tt and call the image of the forgetful morphism Triv π n(i) CFunct n(𝒫 n(X),T) {\mathrm{Triv}_\pi^n(i)}^C \to \mathrm{Funct}_n(\mathcal{P}_n(X),T) the nn-category of transport functors (locally ii-trivializable with CC-structure) Trans n(i) CFunct n(𝒫 n(X),T). {\mathrm{Trans}^n(i)}^C \subset \mathrm{Funct}_n(\mathcal{P}_n(X),T) \,.

While this, again, is all given to us, the second point where we may want to do real work is in checking if an nn-functor in Tra n(i) C{\mathrm{Tra}^n(i)}^C admits a unique (up to equivalence) ii-trivialization with CC-structure.

If so, we arrive at one of the main goals of the entire exercise: an equivalence Trans n(i) CDesc π n(i) C {\mathrm{Trans}^n(i)}^C \simeq {\mathrm{Desc}^n_\pi(i)}^C which tells us that given any old nn-functor tra:𝒫 n(X)T \mathrm{tra} : \mathcal{P}_n(X) \to T all we need is the guarantee that there is some locally-CC ii-trivialization of it – but we don’t need to carry it around.

All you ever want to do with such an nn-functor, like taking its sections e:tra otra e : \mathrm{tra}_o \to \mathrm{tra} or forming its curvature curv(tra):Π n+1(X)T n+1 \mathrm{curv}(\mathrm{tra}) : \Pi_{n+1}(X) \to T_{n+1} (which corresponds to things like finding gerbe modules or quantizing the nn-particle coupled to this ) you’ll do as for plain ordinary nn-functors 𝒫 n(X)T\mathcal{P}_n(X)\to T.


This is, all in all, what the first edge of the cube is about: local trivialization of nn-functors.

And for n=1n=1 it has now finally been written up:

K. Waldorf & U. S.
Parallel Transport and Functors
arXiv:0705.0452

Some relation to other work:

For n=2n=2, and i:ΣGIdΣG 2i : \Sigma G \stackrel{\mathrm{Id}}{\to} \Sigma G_2 the identity of a strict 2-group, and C=S C = S^\infty, the descent category Desc π 2(i) C {\mathrm{Desc}_\pi^2(i)}^C which turns out to be equivalent to differential G 2G_2-cocycles subject to the constraint of fake flatness, is what I worked out with John Baez.

(The generalization to non-fake-flat G 2G_2-cocycles will actually involve 3-transport.)

A couple of things at that time were not as cleanly worked out as they eventually should be. This includes the relation between smooth 1-functors 𝒫 1(X)ΣG\mathcal{P}_1(X) \to \Sigma G and Lie(G)\mathrm{Lie}(G)-valued differential forms. In fact, one finds an equivalence of the category of these smooth 1-functors with the category of global differential GG-1-cocycles: Funct(𝒫 1(X)) Z¯ id X 1(G). \mathrm{Funct}(\mathcal{P}_1(X))^\infty \simeq \bar Z^1_{\mathrm{id}_X}(G) \,.

The category of local such cocycles Z¯ π 1(G) \bar Z^1_\pi(G) is in fact canonically equivalent to that of suitable smooth anafunctors. A full proof of this equivalence didn’t make it into the paper.

These constructions all involve crucially the category of paths in the transition groupoid.

In fact, much of the structure one encounters here can be traced back to the fact that the functor 𝒫 1:S Cat S \mathcal{P}_1 : S^\infty \to \mathrm{Cat}_{S^\infty} which sends a space to its path groupoid preserves the square Y [2] Y Y X \array{ Y^{[2]} &\to& Y \\ \downarrow && \downarrow \\ Y &\to& X } as a pullback square, but not as a pushout square. Instead, the pushout after applying 𝒫 1\mathcal{P}_1 is that category of paths in the transition groupoid.

This discussion didn’t make it into the paper.

Another equivalence which didn’t make it into this paper is that with the notion of smooth functors used by Stolz and Teichner.

All this is obtained by choosing suitable ii in the general theory of locally trivializable 1-transport.

Curiously, it does make good sense to consider i=Idi = \mathrm{Id}. For instance, we might start with a principal transport tra:𝒫 1(X)GTor \mathrm{tra} : \mathcal{P}_1(X) \to G\mathrm{Tor} and then postcompose this with the morphism f:GTorΣG f : G\mathrm{Tor} \to \Sigma G which is – at the level of categories internal to Set\mathrm{Set} an equivalence that identifies the groupoid with its vertex group – here by choosing for each GG-torsor in the world an isomorphism to GG itself.

This is of course not smooth. In fact ftraf \circ \mathrm{tra} will be badly pathological in general.

Such arbitrary functors from paths to the group are called generalized connections in parts of the literature. Applying our theorem to such pathological functors we find: either a generalized connection comes from a smooth connection by applying an ff as above, or it doesn’t. But if it does, then this smooth bundle with connection that it comes from is unique up to equivalence.

We give a condition to check this explicitly: a generalized connection can be equipped with a smooth local structure if and only if all its “Wilson lines” are smooth.


Update: It turns out that some discussion of this paper is taking place in the thread Quantization and Cohomology (Week 23).

Posted at May 4, 2007 5:10 PM UTC

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12 Comments & 24 Trackbacks

Re: The First Edge of the Cube

Very neat. One day I hope to understand it.

Do you think it is possible for someone who barely knows enough category theory to state what a natural transformation is could ever have a chance of understanding “arrow theory”. It sounds like what was in the background of my thinking for ages, but I wasn’t smart enough to enunciate it.

For example, could you apply the “arrow theory” to what we did as an example? Or is that overkill?

Posted by: Eric on May 4, 2007 10:31 PM | Permalink | Reply to this

“arrow theory”

have a chance of understanding “arrow theory”

Yes. Thanks for asking.

This term is slightly ideosyncratic, but other people use it, too. And I have grown fond of it ever since I learned this way of thinking from John Baez and from the early days of Toby Bartels’ thesis.

Sometime people will not say “arrow theory” maybe, but, “internal formulation” or something.

Anyway, the idea is simple: whatever concept you want to describe, you “have found its arrow theory” (as I said, this is somewhat ideosyncratic) when you have formulated it “without words” but using just diagrams.

So instead of saying “we have this and that satisfying these and those relations”, you say “consider the folling diagram internal to some category CC”.

Often it is completely obvious which diagram you have to draw for something, and finding the diagram itself is not the big deal. Even then, nevertheless drawing the diagram may help seeing further structures.

I could start listing general-interest examples, but right now I will simply use those that appear in the above entry:

my continuous emphasis of the “magic square” 𝒫 n(Y) π 𝒫 n(X) triv t tra T i T \begin{aligned} \mathcal{P}_n(Y) & \overset{\quad \pi\quad}{\to} && \mathcal{P}_n(X) \\ \mathrm{triv}\downarrow & \sim\Downarrow t && \downarrow \mathrm{tra} \\ T^\prime & \underset{\quad i\quad}{\to} && T \end{aligned} has made some people I have talked to raise an eyebrow. Especially when only confronted with the n=1n=1-case, they will complain that all this diagram expresses is that we have a transformation t:π *trai *triv. t : \pi^* \mathrm{tra} \to i_* \mathrm{triv} \,. (In fact, that’s why you only see this equation in the introduction of our paper, not the diagram.)

Some people may feel that I am being in a silly way overly sophisticated by drawing a simple equation which just says “such and such a natural transformation exists” in terms of a diagram.

But this is not so.

There is more in the diagram (in the “arrow theory”) than the corresponding equation does convey.

It’s the diagram-Lego: these squares can be glued (“pasted”) to each other in various ways (obvious ways when you draw them as such) and each such way of gluing them among each other gives another diagram that encodes a phenomenon canonically associated with the former.

For instance concepts like “reduction of the structure group” of a bundle with connection, or “refinemnet of a cover” and thiungs like that all come from playing the obvious Lego of squares – and this way these concepts all immediately appear at the right position in the rest of the theory.

That may still appear little gain for n=1n=1 (but only if you knew these concepts before!), even though here already it may be helpful when one generalizes “horizontally”: how much of the general theory of fiber bundles with connection carries over to principal groupoid bundles? Using arrow theory, this doesn’t even require new thinking.

Of course the full power of this “arrow theory” appears when one moves up to n>1n\gt 1. And that’s what we are preparing for here.

(Further discussion of this aspect, too, has not made it into the paper, like so many other things.)

Most crucially, as I at least vaguely indicated in the above entry, the squares above may be glued as equation (2.21). This pasting diagram is the transition “function” corresponding to the chosen local trivialization.

Much of what many people in “higher gauge theory” ever consider are, I claim, structures built from pairs of glued squares this way, for n=2n=2.

If you know how these structures look likes as pastings of squares, many things becomes obvious: the notion of morphisms between triangles fromed by them for instance.

If you don’t see these squares (because they are “nothing but a weird way of writing dow a transformation”) it may be much harder to navigate your theory.

For instance bundle gerbes: people first wrote down the wrong notion of morphisms between them. The right notion of morphism, which is obvious once you have the arrow theory, has been found only indirectly, because one noticed that the former morphisms didin’t gives the isomorphismsm classes that one was hoping to get. The right notion is now known as a “stable isomorphism” of bundle gerbes. Unfortunately.

(The “stable” here refers to the fact that before equating two thing one would naively want to equate, one first tensors both of them hither and wither. These kind of “twists”, which appear frequently, are almost always shadows of arrow-theory squares in the backround: the square is like a morphism from its right edge to its left edge, but the top and bottom edge have to enter the picture, too. That’s the twist.)

Another important aspect is the formulation of curvature of transport. What if we have the hull of the magic square above – but cannot fill it?

This is, when either read from left to right or from right to left, the general “lifting problem” or “descent problem”, respectively, in nn-bundle theory.

How does one solve it? By fitting these squares into larger diagrams (p. 6 here) and letting the arrow theory take care of the rest. It’s not human invention, it happens by itself. Which means: it answers lots of questions for how to conceive nn-connections on nn-bundles. In general.

Hm, it dawns on me that I did not really answer your question properly. I will vevertheless post this now and maybe say more later.

Posted by: urs on May 5, 2007 10:59 AM | Permalink | Reply to this

Re: “arrow theory”

What would an “arrow theory” of classical electromagnetic theory, i.e. Maxwell’s equations, look like? And would such an arrow theory formulation help with “discretizing” it so you could implement numerical algorithms?

If my interest is in finding “natural” discretizations of otherwise well understood continuum theorems, would I gain much by thinking arrow theoretically?

When I was searching for natural discretizations, I thought hard about what was important in the continuum theory and settled on a few things:

- Exterior derivative d
- Wedge product /\
- Hodge star *

I’ve seen you mention some difficulty finding an arrow theory description of wedge product and the difficulties reminded me of things I struggled with although at a much more mundane level.

Am I just thinking of totally irrelevant stuff?

Posted by: Eric on May 12, 2007 4:49 AM | Permalink | Reply to this

Re: “arrow theory”

What would an “arrow theory” of classical electromagnetic theory, i.e. Maxwell’s equations, look like?

The idea of “arrow theory” is to separate, – as much as possible – concepts from implementation (by distinguishing diagrams from their internalization).

So if you have a given setup, you first want to identitfy the concepts it involves, while making as few assumptions on the concrete implementation as possible.

Electromagnetism is a theory of connections on bundles. More precisely, it is the study of a certain functional on the space of all connections on all line bundles.

So in order to formulate that “arrow-theoretically”, we first want to find the “walking connection” – the pure idea of a connection on a line bundle.

Given that, one can then think about implementing that concept of connection in the setup one is interested in.

Two interesting implementations would be:

a) one that reproduces the ordinary notion of connection on smooth bundles – such that we are reassured that we did something sensible when extracting the abstract notion of connection

b) one that applies to discretized versions of electromagnetism.

I believe that when regarding line bundles with connections as functors tra:P 11DVect \mathrm{tra} : P_1 \to 1D\mathrm{Vect} with a certain regularity property, then one can make progress here.

As discussed above, by choosing the domain category P 1P_1 to be the smooth path groupoid in a smooth space XX, one does recover connections on smooth line bundles.

But one can use another domain P 1P_1 instead. For instance one can take P 1:=P 1(E,V) P_1 := P_1(E,V) to be the groupoid freely generated from some fixed graph (E,V)(E,V). So objects of P 1P_1 would be exactly the vertices in EE of the graph, and morphisms would be all concatenations of edges in VV of the graph, together with their inverses.

Moreover, I think that at least when the graph is nice in some way, for instance in that it is a diamond graph, one can also form higher path groupois P n:=P n(E,V) P_n := P_n(E,V) this way, which express the idea of discretized nn-paths between (n1)(n-1)-paths etc.

You, see, the idea here would be to entirely encode the discretization (or not) in the domain nn-category of our transport functor.

Then, by just turning the general crank, the “arrow-theory” should spit out for us the answer to “What is the covariant exterior derivative in this setup?” and things like that.

And I think it does. I think that applying the general concept of nn-curvature to this setup, we do reobtain the lattice covariant exterior derivative acting on discrete differential forms.

Strangely, what I don’t see yet from the general point of view (“from the arrow-theoretic viewpoint”) is how to formulate the wedge product. In order to do that, at the moment I have to make a construction which explicitly uses information encoded in the choice of P 1P_1.

The covariant exterior derivative, on the other hand, I can define without knowing anything about which particular choice for P 1P_1 you made.

That’s arrow-theory.

Posted by: urs on May 14, 2007 3:55 PM | Permalink | Reply to this

Re: “arrow theory”

Thanks Urs for taking the time to respond to my questions from the peanut gallery :)

That is very interesting! And I think I understood most of it. Again, we come back to the wedge product. I think this issue is probably THE fundamental issue in order to naturally “discretize” continuum theories.

If my gut is any indication (it has guided us to a certain extent in the past), I think that getting the wedge product straightened out will have implications far beyond electromagnetism and discretizations.

This is a case where I think that the continuum has thrown a monkey wrench into understanding what is really going on.

Thanks again,
Eric

Posted by: Eric on May 14, 2007 5:04 PM | Permalink | Reply to this

Re: “arrow theory”

Hi Eric,

could be, could well be.

But it could also be that there is a nice way to formulate the wedge product of an nn-functor with an mm-functor in a nice way (something universal, something pull-push, probably) and I just don’t see it yet.

I mean, it took me quite a while to arrive at the full “arrow-theory of nn-curvature” (and hence of the exterior differential).

Quite a while. First I thought I’d need curvators. Actually, that went in the right direction, but without recognizing the true pattern. Then I recognized that taking inner automorphisms INN()\mathrm{INN}(\cdot) achives the “shift in degree” that I had tried to invent curvators for. (And I am still learning more about that inn()\mathrm{inn}(\cdot)-construction). But only quite a time later did I understand the universal construction that gives rise to inn()\mathrm{inn}(\cdot)).

And there are still aspects of this which I don’t understand yet.

But from curvators to the current state, the answer has continually become more elegant and more powerful, I think. With hindsight it now looks rather simple, almost tautalogous.

So, that’s why I am hoping that, even though the arrow-theory of the wedge product is currently mysterious to me, there might one day emerge a simple and natural answer, such that we will ask ourselves why it took so long to find it.

Posted by: urs on May 14, 2007 5:38 PM | Permalink | Reply to this

Re: “arrow theory”

So, that’s why I am hoping that, even though the arrow-theory of the wedge product is currently mysterious to me, there might one day emerge a simple and natural answer, such that we will ask ourselves why it took so long to find it.

Hi Urs,

I hope nothing I said makes you think I have other ideas than this very concept. I think that finding what you are looking for will involve wedge product falling out naturally. However, I also think the converse could be a helpful way to think of it, i.e. I think that by thinking about the wedge product, you will be guided to the natural formulation you are looking for. Either way, I do not have any doubt that wedge product will appear quite obvious in the rearview mirror :)

Good luck! It’s still fun watching the progress from the sidelines :)

Eric

Posted by: Eric on May 14, 2007 6:27 PM | Permalink | Reply to this

Re: “arrow theory”

Has there been any progress here defining an “arrow theory” of wedge product?

Posted by: Eric on December 15, 2008 10:38 PM | Permalink | Reply to this

Re: “arrow theory”

Has there been any progress here defining an “arrow theory” of wedge product?

Yes. I think the idea I sketched elsewhere on the blog (forget where, should post it to the nnLab where it doesn’t get lost) should go through.

So the general issue is: how do we define cup products in differential nonabelian cohomology.

Consider a line bundle as a B\mathbf{B}\mathbb{Z}-cocycle, i.e. as an ana-2-functor XB 2. X \to \mathbf{B}^2 \mathbb{Z} \,.

This comes from cover

XYgB 2. X \stackrel{\simeq}{\leftarrow} Y \stackrel{g}{\to} \mathbf{B}^2 \mathbb{Z} \,.

Suppose you have another such line bundle on XX

XYgB 2. X \stackrel{\simeq}{\leftarrow} Y' \stackrel{g'}{\to} \mathbf{B}^2 \mathbb{Z} \,.

Then, since XXX \otimes X (Gray tensor product) is again XX (since XX is a 0-category, being a space), it should be true that

XYY X \stackrel{\simeq}{\leftarrow} Y \otimes Y'

is still a (hyper)cover of XX. On this we canonically get a B 2B 2\mathbf{B}^2\mathbb{Z} \otimes \mathbf{B}^2\mathbb{Z}-cocycle

XYYggB 2B 2. X \stackrel{\simeq}{\leftarrow} Y \otimes Y' \stackrel{g \otimes g'}{\to} \mathbf{B}^2\mathbb{Z} \otimes \mathbf{B}^2\mathbb{Z} \,.

But using the group structure on \mathbb{Z} there is a natural morphism

B 2B 2B 4 \mathbf{B}^2\mathbb{Z} \otimes \mathbf{B}^2\mathbb{Z} \to \mathbf{B}^4 \mathbb{Z}

postcomposing with this we finally obtain a B 3\mathbf{B}^3 \mathbb{Z}-cocycle, i.e. a line 3-bundle aka 2-gerbe canonically associated to our two line bundles

XYYggB 2B 2B 4. X \stackrel{\simeq}{\leftarrow} Y \otimes Y' \stackrel{g \otimes g'}{\to} \mathbf{B}^2\mathbb{Z} \otimes \mathbf{B}^2\mathbb{Z} \to \mathbf{B}^4 \mathbb{Z} \,.

The claim is that this is indeed the right refinement on the cup product

:H 2(X,)×H 2(X,)H 4(X,). \cup : H^2(X,\mathbb{Z}) \times H^2(X,\mathbb{Z}) \to H^4(X, \mathbb{Z}) \,.

But I still need to write this out and check details.

Now, the point of this exercise is that, using the idea of transport functors and differential nonabelian cohomology, once we know how to do something for nn-bundles in this functorial way, we know how to do it for nn-bundles with connection. In particular then for trivial line bundles with connection. Which is the same thing as differential forms.

In particular, replace in the above B\mathbf{B}\mathbb{Z} with ()(\mathbb{Z} \to \mathbb{R}) which is a weak resolution of U(1)U(1) ()U(1)(\mathbb{Z} \to \mathbb{R}) \stackrel{\simeq}{\to} U(1). And replace XX by P ω(X)P_\omega(X), its fundamental path ω\omega-groupoid.

Then, I think, the analogous construction as above goes through. And now its describes the cup product refined to differential integral cohomology. So in particular we get the wedge product of two curvature 2-forms of two line bundles to the curvature 4-form of their cup product line 3-bundle.

Posted by: Urs Schreiber on December 16, 2008 9:36 AM | Permalink | Reply to this

Re: “arrow theory”

Ok. I didn’t understand a word of that :)

Assuming, with effort, I can understand that. Does it give the wedge product of forms OR cup product on cohomology?

We know that the de Rham map is an algebra map in cohomology, but not on the nose at the cochain level.

I know I probably didn’t say it correctly and it was probably as clear as mud, but I’m suggesting that perhaps “wedge” is not the right product to formulate arrow theoretically. Instead we should use cup on cochains to deduce a product on forms. Unless a miracle happens, I don’t think the product deduced from cup is the same as wedge product. I could be wrong.

Again, I think it helps to look at de Rham and Whitney maps.

With the wedge product, we have

R(αβ)R(α)R(β).R(\alpha\wedge\beta) \approx R(\alpha)\smile R(\beta).

This approximation becomes exact when you pass to cohomology.

On the other hand, if we begin with cochains and define a new product on forms, we have a full algebra map

W(a)˜W(b)=W(ab)W(a)\tilde\wedge W(b) = W(a\smile b)

even at the cochain level and this passes through to cohomology since dd commutes with WW.

Explicitly,

dW(ab) =Wd(ab) =W(dab+(1) |a|adb) =d(Wa)˜W(b)+(1) |a|W(a)˜dW(b). \begin{aligned} dW(a\smile b) &= W d(a\smile b) \\ &= W (da\smile b + (-1)^{|a|} a\smile db) \\ &= d(Wa)\tilde\wedge W(b) + (-1)^{|a|} W(a)\tilde\wedge dW(b). \end{aligned}

You can say, I think, I am suggesting something a little bit radical that was born from arrow theory.

Posted by: Eric on December 16, 2008 8:18 PM | Permalink | Reply to this

Re: The First Edge of the Cube

“differential cocylce” should be
“differential cocycle”

“Chen” or “Chern”?

Either way, defined how?

Posted by: Jonathan Vos Post on May 5, 2007 3:17 AM | Permalink | Reply to this

Re: The First Edge of the Cube

“differential cocylce” should be “differential cocycle”

Thanks! Fixed.

“Chen” or “Chern”?

Either way, defined how?

It’s really “Chen”. A guy who spent his life’s work with thinking about parallel transport, loop spaces and path spaces. Maybe more famous (currently) for his theory of “iterated integrals” which capture and generalize the idea of the “Dyson formula” (as physicists call it) for the “path ordered exponential” Pexp( γA)P \mathrm{exp}(\int_\gamma A).

But here I am referring to his notion of “smooth space”. We had various discussions of that here at Café. See for instance John’s lecture Quantization and Cohomology (Week 20).

Posted by: urs on May 5, 2007 11:05 AM | Permalink | Reply to this
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