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Stream: deprecated: mathematics

Topic: groupoids

John Baez (Dec 21 2020 at 23:47):

The new Notices of the AMS has an intro to noncommutative geometry that's actually an introduction to groupoids:

• Ernesto Lupercio, Non-Commutative geometry indomitable.

It shows how starting with the spectral lines of the hydrogen atom Heisenberg got a groupoid and then turned this into an algebra...

... except that Heisenberg didn't know about groupoids, so he sort of skipped that step and went straight to the algebra.

John Baez (Dec 21 2020 at 23:49):

This is article nicely explains how "noncommutative geometry" arises from generalizing geometry from sets to groupoids. All this is explained in Connes' book Noncommutative Geometry, but Lupercio slows down the explanation and makes it a lot easier to follow.

John Baez (Dec 21 2020 at 23:50):

And @Eric Forgy will be happy to hear that later on it explains Hochschild cohomology and cyclic cohomology.

Eric Forgy (Dec 23 2020 at 01:42):

Thanks for sharing this. It is relevant to a draft (applied) article I'm writing :raised_hands:

Matteo Capucci (he/him) (Dec 23 2020 at 14:04):

I've read it yesterday, what a fascinating subject! I didn't realize non-commutative geometry was so far-reaching. Also it's the first time I hear of the topos-theoretic bit. Wow.

Eric Forgy (Dec 23 2020 at 19:51):

This stuff is fun even for an engineer :blush:

As we mentioned here somewhere, this also ties into stochastic processes and I am pretty sure I am the first person to apply noncommutative geometry to finance :blush:

• Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance

The link to Ritz-Ryderg for the hydrogen atom is well known. This is one of my favorite references:

• Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics

Eric Forgy (Dec 23 2020 at 19:52):

One of my favorite applications, inspired by John's Azimuth project, is to climate science:

• Noncommutative Geometry and Navier-Stokes Equation

Eric Forgy (Dec 23 2020 at 19:53):

(this one is probably worth publishing :point_of_information: )

Eric Forgy (Dec 23 2020 at 19:56):

The Navier-Stokes equation can be viewed as a noncommutative version of Maxwell's equations.

Eric Forgy (Dec 23 2020 at 20:04):

The thing I love about expressing dynamical systems in terms of noncommutative geometry is that it automatically generates "discrete" versions for robust computation. Here is the original Azimuth discussion on the "Discrete Burgers Equation".

Matteo Capucci (he/him) (Dec 23 2020 at 20:55):

Just wow

Matteo Capucci (he/him) (Dec 23 2020 at 20:57):

Tonight I was wondering: how does commutativity manifest in geometry? How should I think about a non-commutative space?

Eric Forgy (Dec 23 2020 at 21:10):

It's been ages since I spent any serious cycles thinking about this stuff, but looking back at that old Azimuth discussion on the discrete Burgers equations has some interesting references to Navier-Stokes.

From the nLab:

For instance, if $Y$ is any space and $A$ a partial algebra, the resulting total complex can be seen as a generalization of Hochschild cohomology. One really interesting example (for Bruce) was where $Y$ is the interval, and $A$ is the partial algebra of forms on a Riemannian manifold M. It turns out that a certain canonical equation which pops out is precisely the Navier-Stokes equation. Gulp!

I haven't compared my stuff on Navier-Stokes to Wilson's, but I suspect it is slightly different. Sullivan-Wilson preferred maintaining graded commutativity at the expense of associativity. Urs and I preferred maintaining associativity at the expense of graded commutativity. There is probably some way to relate the two (but I still think maintaining associativity is the better way to go).

John Baez (Dec 23 2020 at 21:21):

Matteo Capucci said:

How should I think about a non-commutative space?

The best way to think of it is that it's a space where instead of a set of points you have a groupoid. We can get fancy and say "stack", but I'm trying to keep things simple and convey intuition. In fact I'll keep things so simple that I'll ignore all smooth structure, topology, etc... and work with finite sets and groupoids.

So: if your space is a mere finite set $X$, you get a commutative algebra consisting of all functions $f: X \to \mathbb{C}$, with pointwise addition and multiplication.

But if you space is a finite groupoid $G$, you get a noncommutative algebra consisting of all functions $\phi: \mathrm{Mor}(G) \to \mathbb{C}$, with pointwise addition but multiplication extending the composition of morphisms. That is, any morphism $f$ of $G$ has a characteristic function $\chi_f: \mathrm{Mor} \to \mathbb{C}$ that's 1 on this morphism $f$ and zero on all the rest. This embeds $\mathrm{Mor}(G)$ in the space of functions on $\mathrm{Mor}(G)$. There's then a unique algebra structure on this space of functions for which $\chi_f \chi_g$ is $\chi_{f \circ g}$ if $f$ and $g$ are composable, or zero otherwise.

John Baez (Dec 23 2020 at 21:22):

The point is that when your groupoid $G$ has only identity morphisms, it's really just a set, and then this fancy way to get an algebra from a finite groupoid reduces precisely to the earlier way to get an algebra from a finite set!

John Baez (Dec 23 2020 at 21:23):

But now it's possible for the algebra to be noncommutative.

John Baez (Dec 23 2020 at 21:24):

In more intuitive terms: in an ordinary space "the points just sit there", while in a groupoid "the points can move" - i.e., we have arrows between points. And one spinoff of this is that we can get noncommutativity.

John Baez (Dec 23 2020 at 21:24):

But in some ways the noncommutativity is just a spinoff of the "built-in activity", the "inherent liveliness" that points now have!

John Baez (Dec 23 2020 at 21:25):

By the way, everything so far would work just as well for a finite category.

Eric Forgy (Dec 23 2020 at 21:37):

(I wrote this before seeing John's much better explanation so I'll have a look at that too after sending this :raised_hands: )

Matteo Capucci said:

Tonight I was wondering: how does commutativity manifest in geometry? How should I think about a non-commutative space?

I'm working on a paper to explain my understanding of this as we speak :blush:

As I recently summarized here:

Given a unital $K$-algebra $A$ ("0-forms") and a product $m: A\otimes_K A\to A$, let $\tilde\Omega^1$ denote the kernel of $m.$ There is a universal derivation $\tilde d: A\to\tilde\Omega^1$ such that given an $A$-bimodule morphism $\phi: \tilde\Omega^1\to \Omega^1$, we get a unique derivation $d: A\to \Omega^1$ given by

$d := g\circ\tilde d.$

Conversely, given a derivation $d: A\to \Omega^1$, we get a unique $A$-bimodule morphism $\phi: \tilde\Omega^1\to \Omega^1$ given by

$\phi: \sum_i x_i\otimes y_i \mapsto \sum_i x_i d y_i.$

Now, if $A$ admits a basis, the above says that any "1-form" $\alpha\in\Omega^1$ can be written as a sum

$\alpha = \sum_{i,j} \alpha(i,j) e^i\otimes e^j$

where $\alpha(i,j) = 0$ if $i=j.$

Even if $A$ is commutative, we still have that 0-forms and 1-forms do not commute since

$f\, e^i\otimes_K e^j = f(i)\, e^i\otimes_K e^j$ and $e^i\otimes_K e^j\, f = f(j)\, e^i\otimes_K e^j$

or (more suggestively)

$[e^i\otimes_K e^k, f] = \left[f(j)-f(i)\right] e^i\otimes_K e^j.$

In my paper, I am relating this to directed graphs where $e^i\otimes_K e^j$ is a "dual" directed edge from node $i$ to node $j$.

With this interpretation, the universal differential calculus corresponds to a complete directed graph where there is a directed edge $(i,j)$ for every pair of nodes (except possibly loops $(i,i)$ - the math works either way). I think this has a connection to entanglement (which is the paper I also want to write up after this one).

Eric Forgy (Dec 23 2020 at 21:49):

(Btw, the connection between universal differential calculus and directed graphs is also well known and is included in the paper I referenced above: Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics. My paper agrees on the first-order calculus, but I have a new result for higher-degree forms related to "diamonds".)

Matteo Capucci (he/him) (Dec 23 2020 at 21:50):

I see! I like John's explanations of 'moving points'! I had the technical bit with me but lacked this intuition.
Has this anything to do with modeling spaces as n-groupoids? From your description, John, it seems they're different things (since non-commutative points can be richer than the fundamental groupoid of a commutative point)

Matteo Capucci (he/him) (Dec 23 2020 at 21:52):

@Eric Forgy do the chi functions used by John correspond to the basis elements $e_i$? So that the graph you talk about described the 'way points can move'?

Eric Forgy (Dec 23 2020 at 21:55):

Matteo Capucci said:

Eric Forgy do the chi functions used by John correspond to the basis elements $e_i$? So that the graph you talk about described the 'way points can move'?

Yes. I believe so. The basis $e^i$ are characteristic functions / projections / partitions of unity $e^i e^j = \delta^i e^i$ with $1 = \sum_i e^i$, i.e. the product of basis elements is 1 if it is the same "point" and 0 otherwise and the sum of all "points" is the unit element.

Eric Forgy (Dec 23 2020 at 21:58):

An arbitrary "0-form" can be expressed as

$f = \sum_i f(i) e^i$

which can be thought of as functions from nodes to $\mathbb{C}$ with $e^i(j) = \delta^i_j.$

Eric Forgy (Dec 23 2020 at 22:00):

The original paper that John referenced that started this stream defined the category I'm talking about, i.e. $\mathsf{NCAlgebras},$ where objects are algebras and morphisms are bimodules.

Eric Forgy (Dec 23 2020 at 22:01):

Translated, objects are "0-forms" and morphisms are "$p$-forms" for $p>0$.

Eric Forgy (Dec 23 2020 at 22:04):

I wouldn't describe it as "moving points" though. I think of it like points are "events" in a noncommutative spacetime and morphisms connect events (they are lightlike actually which gives a metric).

John Baez (Dec 23 2020 at 22:04):

Matteo Capucci said:

Has this anything to do with modeling spaces as n-groupoids?

Yes!

From your description, John, it seems they're different things (since non-commutative points can be richer than the fundamental groupoid of a commutative point)

That's a very interesting question. Note that I considered a groupoid without any "cohesion"- that is, the set of objects and morphisms had no topology or smooth structure.

A different thing you can do is take a traditional space with some "cohesion", like a topological space or manifold or some other sort of smooth space, and create a groupoid where the morphisms are all continuous or smooth paths in that space - up to some equivalence relation big enough to ensure that the groupoid laws hold.

So here, instead of "building in motion from the start" by starting with a groupoid, you are saying to your points "move however you want, as long as your motion is nice (continuous or smooth)".

In this second approach, you can decide whether or not you want to keep the cohesion. That is: if your original space was a topological space or manifold or some other sort of smooth space, you can get a groupoid internal to topological spaces or manifolds or some other sort of smooth spaces... or just a mere groupoid, if you prefer.

Also, there's no need to stop with a groupoid: you can get a 2-groupoid, etc.

So there are lots of variants, all connected to each other....

Eric Forgy (Dec 23 2020 at 22:06):

John, if we could sit down for a day or two, you could easily save me months of effort. This stuff comes so naturally to you :raised_hands: :blush:

John Baez (Dec 23 2020 at 22:08):

I worked on this stuff for years and wrote a bunch of papers on it. I don't want to "sit down" and talk about it for a couple of days in a row. But I'm happy to chat about it here, so everyone can hear.

Eric Forgy (Dec 23 2020 at 22:08):

I know :+1: :blush: Thank you for sharing your thoughts though :raised_hands:

John Baez (Dec 23 2020 at 22:20):

Sure, my favorite way of talking about a lot of things is in small bites, so it doesn't start feeling like "work". (Right now I'm busy writing a paper, and talking about this serves as a break....)

Matteo Capucci (he/him) (Dec 24 2020 at 11:32):

John Baez said:

Matteo Capucci said:

Has this anything to do with modeling spaces as n-groupoids?

Yes!

From your description, John, it seems they're different things (since non-commutative points can be richer than the fundamental groupoid of a commutative point)

That's a very interesting question. Note that I considered a groupoid without any "cohesion"- that is, the set of objects and morphisms had no topology or smooth structure.

A different thing you can do is take a traditional space with some "cohesion", like a topological space or manifold or some other sort of smooth space, and create a groupoid where the morphisms are all continuous or smooth paths in that space - up to some equivalence relation big enough to ensure that the groupoid laws hold.

So here, instead of "building in motion from the start" by starting with a groupoid, you are saying to your points "move however you want, as long as your motion is nice (continuous or smooth)".

In this second approach, you can decide whether or not you want to keep the cohesion. That is: if your original space was a topological space or manifold or some other sort of smooth space, you can get a groupoid internal to topological spaces or manifolds or some other sort of smooth spaces... or just a mere groupoid, if you prefer.

Also, there's no need to stop with a groupoid: you can get a 2-groupoid, etc.

So there are lots of variants, all connected to each other....

Uh--I didn't expect cohesion to enter the picture. But now I see what's the point... I think I also understand better why non-commutative geometry might be a better setting for particle physics: your points can have non-trivial symmetries now! It reminds me of this article by Urs

Chetan Vuppulury (Dec 26 2020 at 03:58):

Where could I find a reference that talks more deeply about groupoids, stacks and their relation to noncommutative geometry?

John Baez (Dec 26 2020 at 18:58):

That's a good question! I mainly just had to figure this out myself.

John Baez (Dec 26 2020 at 18:58):

But probably someone has written about it.

David Michael Roberts (Dec 28 2020 at 03:10):

Groupoids v stacks are talked about in many places, Pronk, Carchedi, Noohi, for instance. Groupoids/NC geometry is a bit harder, because the NC geometers talk their own dialect, using the single-sorted definition of groupoid, for instance, to get a C*-algebra. This goes back to Jean Renault, but you are really just looking at the operator algebra literature. Connes is probably the one who links Lie groupoids and NC geometry the most, in my experience (which is low, admittedly).

John Baez (Dec 28 2020 at 04:33):

Let's see what Connes has to say about groupoids in his book Noncommutative geometry.

He describes how instead of quotienting a space by an equivalence relation you should form the weak quotient, which is a groupoid, and then look at the groupoid algebra of that:

Equivalently one can write (in many ways) the [ill-behaved] space X as a quotient of an ordinary well behaved space by an equivalence relation. For instance the space Y of pairs (tiling, tile belonging to the tiling), is well behaved and possesses an obvious equivalence relation with quotient X. One could also use in a similar manner the group of isometries of the plane. All these constructions yield equivalence relations, or better,groupoids or pre-equivalence relations in the sense of Grothendieck. The noncommutative algebra is then the convolution algebra of the groupoid.

John Baez (Dec 28 2020 at 04:37):

He stands up for groupoids:

It is fashionable among mathematicians to despise groupoids and to consider that only groups have an authentic mathematical status, probably because of the pejorative suf-fix oid. To remove this prejudice we start Chapter I with Heisenberg’s discovery of quantum mechanics. The reader will, we hope, realise there how the experimental re-sults of spectroscopy forced Heisenberg to replace the classical frequency group of the system by the groupoid of quantum transitions. Imitating for this groupoid the construction of a group convolution algebra, Heisenberg rediscovered matrix multiplication and invented quantum mechanics.

John Baez (Dec 28 2020 at 04:40):

He promises a lot of examples in Chapter 2 and hints that smooth groupoids presenting the same stack also give Morita equivalent C*-algebras (has someone proved this?):

This is done in great detail in Chapter II with a lot of examples. The general principle is, instead of taking the quotient of a space by an equivalence relation, to retain this equivalence relation as the basic information. An important intermediate notion whichemerges is that of smooth groupoid. It plays the same role as the pre-equivalence relations of Grothendieck. The same noncommutative space can be presented by several equivalent smooth groupoids. The corresponding C∗-algebras are then strongly Morita equivalent in the sense of M. Rieffel and have consequently the same K-theory invariants (cf. Chapter II Appendix A).

John Baez (Dec 28 2020 at 04:45):

There's a fair amount about "smooth groupoids" in this book, but the word "stack" never appears. I wonder who (if anyone) has worked out the connection between stacks and Morita equivalence for the convolution C*-algebras of smooth groupoids.

David Michael Roberts (Dec 28 2020 at 06:12):

People know that 'Morita equivalent' topological/Lie groupoids give Morita equivalent C*-algebras, and that this notion of equivalence for groupoids is more or less the same as presenting the same topologial/differentiable stack.

John Baez (Dec 28 2020 at 06:13):

Okay, good. Do you know a reference for that first fact?

David Michael Roberts (Dec 28 2020 at 06:15):

Landsman, N.P. The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart. Letters in Mathematical Physics 54, 43–59 (2000). https://doi.org/10.1023/A:1007669418336

David Michael Roberts (Dec 28 2020 at 06:15):

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.

John Baez (Dec 28 2020 at 06:16):

Neat! I was just talking to Landsman about my paper on Noether's theorem. He refereed it and I misspelled his name in the references. :embarrassed:

John Baez (Dec 28 2020 at 06:18):

I should sometime go through and read everything he's done... at least skim it. I wouldn't have guessed he'd have done this!

John Baez (Dec 28 2020 at 06:19):

In some ways if what you're after is a C*-algebra, a "locally compact groupoid with Haar system" could be a bit more appropriate than a Lie groupoid. It's more analysis-flavored.

John Baez (Dec 28 2020 at 06:19):

But it's interesting to get Lie groupoids into the game.

David Michael Roberts (Dec 28 2020 at 06:22):

The more interesting problem is what to do about maps between stacks. I'm working with Rohit Holkar, a student of Ralf Meyer and Jean Renault, to understand how functorial the C*-algebra of a locally compact Hausdorff groupoid with Haar measure can be. There are really nasty measure-theoretic details, so I'm glad I have a coauthor that can deal with them!

David Michael Roberts (Dec 28 2020 at 06:23):

Lie groupoids just happen to come with a canonical Haar measure, that coming from half-densities. The same is true for étale topological groupoids, where one can use counting measure (the source and target fibres are discrete, and these are where the Haar measures are supported).

John Baez (Dec 28 2020 at 06:50):

Neat, that sounds like a cool project you're working on. When you say "nasty measure-theoretic details", what's an example?

My most nasty exposure to measure theory came when I was writing the book Infinite-Dimensional Representations of 2-Groups with Baratin, Freidel and Wise. We got into "standard Borel spaces" and "disintegration of measures" and stuff.

John Baez (Dec 28 2020 at 06:50):

Baratin did all the heavy lifting.

John Baez (Dec 28 2020 at 07:09):

Anyway, it sounds like you're doing what I was vaguely dreaming of: really nailing down the connection between noncommutative geometry and stacks.

John Baez (Dec 28 2020 at 07:10):

Can you cook up a 2-functor from a 2-category of stacks to some 2-category like C*-algebras, bimodules and bimodule homomorphisms?

David Michael Roberts (Dec 28 2020 at 07:35):

There are conditions like families of measures being quasi-invariant (whatever that means) so that convolution is respected in certain ways etc. Currently We are getting a functor from a certain wide subcategory of topological groupoids with Haar measures to the bicategory of C*-alg+bimodules, and then want to extend this to something bigger by a universal property after localisation at the ff+eso functors. I conjecture we can get it defined maps between stacks that are presented (up to possibly replacing the codomain) by maps between topological groupoids with Haar measure that are open on objects, and where there is a certain compatibility relation on the measures. This covers a lot of stuff from the literature, even though it is not fully general.

David Michael Roberts (Dec 28 2020 at 07:37):

I should say that my coauthor is not satisfied by the common assumption that everything is second countable, in order to lean on the easier versions of existence theorems for measures. He has gone back to Bourbaki's book on Integration and used it to reprove things that the operator algebra people usually only do using second countable spaces.

David Michael Roberts (Dec 28 2020 at 07:39):

And he is happy to drop Hausdorffness for local Hausdorffness, which is really pushing the envelope, as it butts up against actual topological obstructions if one goes further. So he's much more comfortable than I am with measure theory!

David Michael Roberts (Dec 28 2020 at 07:41):

Ironically, the seed for this project was planted way back when Mathai was giving me comments on my thesis introduction 11 years ago, when I cited Landsman in a lazy way for examples of bicategorical localisation.

John Baez (Dec 28 2020 at 08:00):

Interesting. I hope the introduction to your paper argues that you're figuring out the connection between stacks and noncommutative geometry. I think that message will resonate with people. The gnarly technical details are in a way less exciting to most people... except for the really macho analysts like your coauthor!

Nikolaj Kuntner (Dec 28 2020 at 13:35):

John Baez said:

He stands up for groupoids:

I've seen this kind of anthropization before :sweat_smile:
Today they are probably less neglected than as of the writing of Connes book.
Btw. I found a cool video recently
https://youtu.be/9qlqVEUgdgo

Dan Doel (Dec 28 2020 at 18:00):

So, perhaps this is too naive a question for this thread, but what makes groupoids non-commutative where groups are commutative? Is it that you can always take the product of two elements in a group, but two paths in a groupoid might not have the right endpoints to combine?

Dan Doel (Dec 28 2020 at 18:04):

It's likely I just don't know enough about how groups are used in physics to even guess the answer.

John Baez (Dec 28 2020 at 18:35):

Dan Doel said:

So, perhaps this is too naive a question for this thread, but what makes groupoids non-commutative where groups are commutative?

Nothing. We can get noncommutative algebras from groups just as easily as from groupoids!

Here's how: start with a group $G$ and take finite linear combinations of its elements. This becomes an associative algebra with a product extending multiplication in the group - the so-called group algebra. This algebra is noncommutative iff the group is noncommutative.

The stuff about getting noncommutative algebras from groupoids is just a generalization of this idea.

Dan Doel (Dec 28 2020 at 18:36):

Oh, okay.

Dan Doel (Dec 28 2020 at 18:39):

So when they talk about e.g. Heisenberg, it's not necessarily group vs. groupoid it's the particular group and groupoid.

John Baez (Dec 28 2020 at 18:39):

Your question was not "too naive", it was very satisfying. People talk about groupoids and noncommutative geometry not because they behave very differently from groups, it's because the group examples are so well-understood that the groupoids, giving new examples, seem more exciting.

John Baez (Dec 28 2020 at 18:40):

With Connes' parable about Heisenberg, the relevant groupoid is the codiscrete category on a set: the groupoid with a bunch of objects all isomorphic to each other in a unique way.

Dan Doel (Dec 28 2020 at 18:46):

I guess the set is the 'energy levels' and the equivalences are the relative energy differences or something?

Dan Doel (Dec 28 2020 at 18:48):

I guess the set could be individual states, since 'energy level' is probably quotienting states by equivalent energies.

John Baez (Dec 28 2020 at 18:59):

Yeah, the set consists of "states".

John Baez (Dec 28 2020 at 19:00):

If we start with a finite set of $n$ "states", form the codiscrete category on that, form the vector space spanned by the morphisms in this groupoid, and turn it into an algebra, we get the algebra of $n \times n$ matrices.

John Baez (Dec 28 2020 at 19:01):

This is pathetically simple, but Connes was using it as a just-so story for how Heisenberg could have invented matrix mechanics.

John Baez (Dec 28 2020 at 19:01):

Connes' actual interest lies in much more complicated groupoids, which are always at least topological groupoids.

Dan Doel (Dec 28 2020 at 19:04):

Anyhow, what it reminds me of (probably not a coincidence) is a version control system named 'darcs'. It's based on the idea of keeping track of 'patches' that combine to produce the desired state of the repository by repeated application. You can 'commute' patches past one another, but if you have $gf$ you might commute to $fg'$, because you need a different patch to 'go before' $f$ than 'go after'. It was originally written by a physicist.

And once the HoTT stuff got rolling, someone wrote a related paper where the 'patches' are paths in a type, either with a single 'point' (so, like a 'group') or relating the repository states they form (which is like a 'groupoid' I guess, but I'm not sure if it ends up being contractible, and thus not what HoTT people actually call a groupoid).

John Baez (Dec 28 2020 at 19:06):

Neat! It used to bug me that Connes' example of a codiscrete groupoid was contractible, yet it was still giving a supposedly interesting algebra, the algebra of $n \times n$ matrices.

John Baez (Dec 28 2020 at 19:06):

The solution is that this algebra, while not isomorphic to the trivial algebra $k$ (the ground field), is equivalent to it in a certain 2-category of algebras.

John Baez (Dec 28 2020 at 19:07):

At least that's a partial solution: one can still wonder why equivalent algebras can act different!

Dan Doel (Dec 28 2020 at 19:08):

The single point version would actually be a HoTT groupoid, because it has non-trivial paths. Even though it's "just" the delooping of a group.

Chetan Vuppulury (Dec 29 2020 at 12:39):

Is there a generalisation of harmonic analysis of locally compact groups (Haar measure etc) to groupoids?

John Baez (Dec 29 2020 at 18:24):

@Chetan Vuppulury - it must exist, but I don't know anything about it. I just found a paper from 1968:

• Joel Westman, Harmonic analysis on groupoids, Pacific Journal of Mathematics 27 (1968), 621-632.

It should be much more advanced by now!