The String Coffee Table

Did Ross Street do that ages ago?

Could be, don’t know. With a little luck John sees this and gives you a better reply.

All I know is that double categories were first identified by Ehresmann. He was the one who came up with the idea of internalization, where you externalize any concept by defining it in terms of abstract diagrams, and then internalize it in some category by thinking of all these diagrams as diagrams in that category. As I said, when you define a category itself by abstract diagrams[1] and then internalize these diagrams in $\mathbf{Cat}$, you get the concept of a double category. When you repeat this process you get $n$-fold categories, or whatever they are called.

The picture that arises from your description of a double category seems to be essentially an n-diamond.

I see what you mean. On the other hand, a 2-morphism in a 2-category is in a sense closer to the concept of a 2-diamond, actually. Just draw the source and target morphisms of the bigon not as round arcs, but as segments of straight lines with a rectangular kink in the middle (it’s just notation, after all!).

I have some diagrams where I compute the surface holonomy of a torus in an orbifold using such ‘rectangular’ bigons and it does look precisely like the 2-diamond graph that you are probably thinking of. I cannot make these public yet, but as soon as I can I will show you.

if we augment the gauge theory of our paper with group algebra value 2-forms

Yes, true, that should work and should be interesting.

I am beginning, slowly but surely, to delve into synthetic constructions. Right now I am in the process of describing 2-connections on nonabelian bundle gerbes by ‘synthetically differentiated’ 2-functors. That involves essentially the step that you mentioned, where the graph in question is that of ‘infinitesimal neighbours’.

As soon as I have something presentable, I’ll show you the details.

using techniques from higher gauge theory to model the yield curve

Yes, I remember. I never understood if in that context one really wants to integrate along a path in the space of yield curves. Does one? Is there really a quantity associated to a fixed yield curve and one wants to know how this quantity changes as the yield curves is deformed?

If yes, then that might be a surface transport. Is it?

[1]

In case anyone wonders: by defining a category by abstract diagrams I mean the following:

a category $C$ is a tuple $(\mathrm{Obj}(C),\mathrm{Mor}(C))$ of abstract objects together with arrows

(the identity morphism-assignment)

(source and target)

(composition of morphisms)

and diagrams like

(associativity)

as well as diagrams expressing the property of the identity morphisms. When one thinks of this in $\mathbf{Set}$ it is an ordinary category. When one thinks of this in $\mathbf{Cat}$ one gets a double category.

It sounds like double categories would be the right setting for 2-gauge theories in my sense.

Yes, maybe. You would probably want to take the little plaquettes that you are considering and then take the double category generated by these under composition. (Sort of the 2-analog of the path category of a graph.)

I might not fully recall the details of the construction that you have proposed. I guess you want a single object, 1-morphisms to be vector spaces, composition of 1-morphisms to be the tensor product of vector spaces and 2-morphisms to be linear maps between these vector spaces.

That should give a decent double category.

In fact, I think when you pass to the 2-category equivalent to this double category you end up with the monoidal category $\mathbf{Vect}$ of vector spaces, but regarded as a 2-category with a single object. (Where the tensor product, as above, corresponds to horizontal composition.)

I have a pre-pre-print showing that 2-functors from surfaces to $\mathbf{Vect}$-regarded-as-a-2-category-with-a-single-object yield bundle gerbes with connective structure, when ‘pre-trivialized’.

So what is a thin structure, and why do we want it?

[…] a thin structure is equivalent to a connection pair.

A thin structure on a double category $D$ in this context is a morphism of double categories (a ‘double functor’) from the double category of commuting squares in the edge category of $D$ to $D$ itself.

In plain english this means that whenever four 1-edges of a double category form a commuting square, then there is at least one 2-morphism (square/plaquette) ‘filling’ this square, i.e. with these 1-morphisms as boundary.

This is a very natural requirement. It essentially says that all ‘identity 2-morphisms’ do exist.

Brown and Mosa re-express this condition in terms of connection pairs, but I think the word ‘connection’ here is not related to the same word as it appears in ‘bundle with connection’.

You explained the problem with 2-categories or bicategories; one problem with double categories is that composition of morphisms must be strictly associative, which forces one to use unparametrized edges, which (I think) makes gluing complex structures along these edges tricky.

Any reason you can’t generalize to A-infinity structures?

Re: 2-Connections on 2-Bundles

Re: 2-Connections on 2-Bundles

Hi Professor Baez,

It is good to cross paths again. I’m in your neighborhood (Santa Monica), so maybe we can get together sometime. I know my coworker (Ph.D. involved category theory as applied in computer science) has talked about trying to get some people together. You might be surprised to know how many mathematicians are working in finance these days :)

Regarding $A_\infty$ algebras, an obvious place where they come up is in an area close to my heart: discrete geometry. Dennis Sullivan has formulated a theory of discrete geometry that is, in a sense, dual to what Urs and I did. I think it is a metatheorem that if you try to construct a theory of discrete geometry, you need to make a choice. You need your algebra to be either noncommutative or nonassociative. If you go the noncommutative route, you end up with a graded algebra that is associative, but not commutative. We didn’t show it, but I am pretty sure we could, that the failure to be commutative is similarly “up to homotopy (in the algebraic sense)”. If you go, as Sullivan did, with retaining commutativity and giving up associativity, you end up with an $A_\infty$ algebra. The algebra we ended up with is probably closely related, maybe $C_\infty$ (not to be confused with differentiability) :)

I think even Urs has underestimated the potential of the paper we wrote together. I think it is great to become familiar with the synthetic stuff, but in the end, I think the noncommutative route is probably more natural (and resonates more squarely with QM).

Cheers,
Eric

Re: 2-Connections on 2-Bundles

So my group acting on the continuum surface holonomy is always infinite-dimensional. That’s a difference.

I’d call it a special case, not a difference.

Re: 2-Connections on 2-Bundles

I dont think that it is meaningful to regard my paths as morphisms, since the dont really have an orientation. My understanding is that in every kind of categorical approach, the edges always come with little arrows, and the surfaces with fatter arrows.

This is not an issue if you think of the paths as morphisms in a (strict) 2-groupoid - they are all invertible and so orientation takes a back seat.

My edges don’t act on points; in fact, the dont act at all. They just sit there, as passive pieces of a string, until they are acted upon by the surface holonomy.

In that case take the fundamental 2-groupoid $\Pi_2(L)$ for $L$ your lattice, and this is the `domain’ of the surface holonomy 2-functor

$$Hol:\Pi_2(L) \rightarrow G$$

for $G$ the group, 2-group or whatever the holonomy is valued in.