The n-Category Café

All the typos you caught are really typos! I’ll fix them. Thanks.

I actually love it when people point out typos and other mistakes in This Week’s Finds — I like to get these things fixed.

Another thing that’s not clear: I remember discussion about “groupoids” being the thing to use for studying “semi-symmetries”. IIRC, a semi-symmetry was something like the finite pattern “||||||||” being almost symmetric - in the sense of mapping almost onto itself - under translation by one bar-separation, with the issue being that for the finite pattern the end bar doesn’t have anything to map onto. This was related to the fact that you could have a tranlsation morphism (?) T which was defined for all the bars except the final one, and likewise for the inverse T.

Is this a separate concept that also goes under the name groupoid, a different application of the same mathematical gadget “groupoid” or an aspect to the story about spans and groupoids? I’ve noodled about with the framework in the TWF and as far as I can see your morphisms in the definition of a groupoid could fit with the partial symmetry, but then I can’t see how with your F and G functors you don’t end up with “full” symmetries. (Maybe it’s that in your concrete example of incidence geometry all symmetry-type things are full symmetries?)

Whilst trying to see if I understood the concepts properly, I tried to come up with another concrete groupoidfication example. I think if you work with numbers you can define “not being coprime” as the relationship and your witness being one of the non-unity common factors. Then, say, multiplying everything by a value X again gives you a valid relation, ie, it’s a symmetry. But maybe dividing by a value X is a semi-symmetry: when your witness is divisible by X your statement remains valid, but in some cases you witness and all the other possible witnesses won’t be divisible by X, so it’s not a “full symmetry”. Is this on the right lines?

Alan Weinstein’s Groupoids: Unifying Internal and External Symmetry gives a good little introduction to groupoids. You can see that the two Wikipedia definitions Thomas mentions are equivalent. (The other sense of ‘groupoid’ - merely a set with a binary operation - is clearly different.)

A situation similar to dave tweed’s finite number of bars is also treated. This is a finite patch of tiles, whose symmetries are considered in terms of a transformation groupoid (page 746).

Thomas Larsson wrote:
I am confused about the term groupoid.

The term was originally used by Oystein Ore back in the 1930s to refer to a set with a binary operation. That usage is still quite common; its MSC classification number is 20N02. What is now called just “groupoid” used to be called a “Brandt groupoid”. You can find the noncategorical definition in, for instance, R.H. Bruck’s classic Survey of Binary Systems.

I’m not sure when exactly it happened, but eventually these just came to be called “groupoids”. That might be due to the influence of Bourbaki. The replacement term “magma” for sets with a binary operation is certainly a Bourbakism (Bourbaki-ism?).

Folks who work a lot in quasigroups, loops, and the like (MSC 20N05) still tend to use “groupoid” in its classical sense. I work in that area, but have coauthored a paper with one of the main proponents (Weinstein) of the modern sense of the term, so I switched to “magma”. I still get annoyed referee reports insisting I switch back to “groupoid”.

It sounds like you’re talking about two ways to define a groupoid: as a set with partially defined multiplication and totally defined inverses satisfying some axioms, and as a category where morphisms have inverses. As the Wikipedia explains, these definitions are equivalent if one gets the axioms right in the first approach.

There’s also an unrelated, archaic definition of groupoid: ‘a set equipped with an arbitrary binary operation’. I urge everyone to avoid this confusingly different usage. The preferred term for such an entity is ‘magma’, which nicely captures the feel of this raw, primeval concept.

Then would a degroupoidification form a kind of three-dimensional matrix?

Perhaps ‘tensor’ would have been better.

David wrote:

Perhaps ‘tensor’ would have been better.

You got it. Jim, Todd and I use the awkward term ‘multispan’ to mean an object $S$ equipped with $n$ different morphisms

$$f:S\to X_i$$

By the process of degroupoidification, multispans of groupoids give rank-$n$ tensors. These turn out to be very useful.

The curious thing is that normally, rank-$n$ tensors are covariant in some slots and contravariant in others. When we groupoidify, the distinction between covariant and contravariant disappears.

I’ll let you guess why.

But you’re making me give away some of the fun to come…

Regarding my last paragraph, what should we make of efforts to introduce probabilistic ideas in topics such as algebraic topology. E.g., Betti numbers of random manifolds looks at the expectations of Betti numbers. It begins

In various fields of applications, such as topological robotics, configuration spaces of mechanical systems depend on a large number of parameters, which typically are only partially known and often can be considered as random variables. Since these parameters determine the topology of the configuration space, the latter can be viewed in such a case as a random topological space or a random manifold. To control such a system one has to understand geometry, topology and control theory of random manifolds.

One of the most natural notion to investigate is the mathematical expectation of the Betti numbers of random manifolds. Clearly, these average Betti numbers encode valuable information for engineering applications; for instance they provide an average lower bound for the number of critical points of a Morse function (i.e. observable) on such manifolds.

If measures can be placed on Grassmanians, as in Klain and Rota’s wonderful lectures, might there be scope in the groupoidified version?

I saw this linked from Bad Astronomy in one of the comments. Am I horribly off base to assume it’s likely from a spherical harmonic with roughly hexagonal symmetry? It really does look a lot like some of their graphs that I’ve seen.

I’ve been on a solar binge since reading Week 248, and Dan P’s telescope comment. If any cafe goers are interested the New York amateur astronomy soc will have some H-alpha telescopes set up in Central Park next Saturday (weather permitting).

The Sun is indeed quite having its day in itself. There was an interesting BBC article today too. (And some nice sound samples to download)

Maybe you didn’t read all the way to the end of week248?

If we are going to demand composition of arrows, can anyone explain this puzzle in Sloane’s? It is asbout semigroups, not groupoids, so there is more structure, not less, but still puzzling.

A071536 Extensions to a semigroup of the (categorical) composition of arrows in the complete directed graph on n labeled nodes.

1, 1, 3, 3, 147

OFFSET

0,3

COMMENT

Terms obtained, especially the 147, lend some support to the peculiar conjecture that the composition in any category can be extended to a total, associative operation, that is to a semigroup (not, of course, a monoid).

EXAMPLE

For n=2, arrows 0:0->0, 1:1->1, f:0->1, g:1->0, the self-dual solution (commutes with reversing arrows) has multiplication table (rows and columns indexed by 0, 1, f, g in order) with rows: 0 f f 0; g 1 1 g; 0 f f 0; g 1 1 g.

KEYWORD

hard,nonn,nice

AUTHOR

F. Lockwood Morris (lockwood(AT)ecs.syr.edu), May 29 2002

This relation to Hecke operators should be pretty interesting. I once grew fond of the fact that lots of “duality” transformations in the literature are all examples of 2-linear maps i.e. (morphisms of 2-vector space) obtained by having a vector bundle on a span (also, now, known as a bi-brane) and performing pull-push through this correspondence: I once listed some examples in Fourier-Mukai, T-Duality and other linear 2-Maps. The Hecke transformation in geometric Langlands is among these, at least in its “classical limit”.

And in fact that makes good sense and fits very nicely into the very big picture, parts of which is the picture drawn by Kapustin/Witten, which becomes, quite clear once you match it to the general theory of 2-vectorial duality in 2-dimensional QFT, as hinted at in Duality and defects in rational conformal field theory.