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Stream: community: events

Topic: This Week's Finds seminar

John Baez (Sep 16 2022 at 08:25):

I'm giving eleven lectures on topics from This Week's Finds in Mathematical Physics, one every Thursday from September 22nd to December 1st. The talks will be at 3 pm UK time and they will be here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

John Baez (Sep 16 2022 at 08:29):

If you're Edinburgh, come by! They'll be in Room 6206 of the James Clerk Maxwell Building, home of the Department of Mathematics. There will be tea in the common room on the 5th floor at 2:45, and I hope some of us go back there after my lecture so we can discuss the topics in more detail.

John Baez (Sep 16 2022 at 08:29):

Also, we can discuss the topics here!

John Baez (Sep 16 2022 at 08:32):

The first lecture will be on Young diagrams and representations of the classical groups:

Young diagrams

Young diagrams are combinatorial structures that show up in a myriad of applications. Among other things, they classify conjugacy classes in the symmetric groups $S_n$, irreducible representations of $S_n$, irreducible representations of the groups $\mathrm{SL}(n)$ over any field of characteristic zero, and irreducible unitary representations of the groups $\mathrm{SU}(n)$.

Dylan Braithwaite (Sep 16 2022 at 09:54):

Can you give an idea of what level of background you'll be assuming in representation theory/algebra?

John Baez (Sep 16 2022 at 10:22):

I'm trying to keep things simple, or at least semisimple. So if you've taken an algebra course that talks about groups, rings and fields you should be fine, at least if you ask enough questions.

For example, in my lecture notes about Young diagrams and representation theory I start by defining a representation of a group (or more generally a monoid, since I have a few tricks up my sleeve). But I may now and then assume you can catch on to things, or else ask questions. For example, I may forget to define the direct sum of representations... based on the assumption that "people who understand algebra know that you can take direct sums of things".

John Baez (Sep 16 2022 at 10:23):

I'll put my lecture notes here soon, and that should help you gauge what's going on, and start asking questions.

Dylan Braithwaite (Sep 16 2022 at 10:35):

Thanks! Sounds great

Matteo Capucci (he/him) (Sep 16 2022 at 13:24):

John Baez said:

I'm trying to keep things simple, or at least semisimple.

:laughing:

John Baez (Sep 16 2022 at 16:16):

Here are lecture notes for the first one or two seminars:

• Young diagrams and classical groups.

John Baez (Sep 16 2022 at 16:18):

I probably won't cover all this material in the seminar, just some portion of it. The important stuff is the stuff up to and including the classification of irreducible representations of the monoid $\mathrm{End}(\mathbb{C}^n)$.

David Egolf (Sep 16 2022 at 16:32):

On the topic of direct sums of representations, a question occurred to me.

Let $G$ be a group, viewed as a category with a single object $*$, where every morphism is an isomorphism.
Let $Vect$ be the category of vector spaces and linear maps between them (over some fixed field $k$).
Then consider the category $Rep(G)$ where objects are functors from $G$ to $Vect$, and morphisms are natural transformations between such functors. I think the objects of $Rep(G)$ are what we call "group representations". Now I want to talk about direct sums of representations in this setting.

Assume that $\rho, \pi: G \to Vect$ are two group representations, with $\rho(*) = V_1$ and $\pi(*) = V_2$. Then I believe $(\rho \oplus \pi)(*) = V_1 \oplus V_2$ and $(\rho \oplus \pi)(g)(v_1, v_2) = (\rho(g)(v_1), \pi(g)(v_2))$, for any morphism $g \in G$ and any $(v_1, v_2) \in V_1 \oplus V_2$. Hopefully this is what we actually mean by the direct sum of two group representations; I'm still a bit shaky on this.

My question is: does the coproduct of two group representations $\rho$ and $\pi$ exist in $Rep(G)$? If so, is it the same thing as $\rho \oplus \pi$ described above? I would try to work this out myself, and maybe I still will, but I don't have the energy at the moment, and I thought it might be an interesting question for this topic.

John Baez (Sep 16 2022 at 16:39):

David Egolf said:

Assume that $\rho, \pi: G \to Vect$ are two group representations, with $\rho(*) = V_1$ and $\pi(*) = V_2$. Then I believe $(\rho \oplus \pi)(*) = V_1 \oplus V_2$ and $(\rho \oplus \pi)(g)(v_1, v_2) = (\rho(g)(v_1), \pi(g)(v_2))$, for any morphism $g \in G$ and any $(v_1, v_2) \in V_1 \oplus V_2$. Hopefully this is what we actually mean by the direct sum of two group representations; I'm still a bit shaky on this.

You're exactly right. That's the direct sum of two group representations.

John Baez (Sep 16 2022 at 16:43):

David Egolf said:

My question is: does the coproduct of two group representations $\rho$ and $\pi$ exist in $\mathsf{Rep}(G)$? If so, is it the same thing as $\rho \oplus \pi$ described above? I would try to work this out myself, and maybe I still will, but I don't have the energy at the moment, and I thought it might be an interesting question for this topic.

Yes, the direct sum is the coproduct! You can cook up dream up morphisms $\rho \to \rho \oplus \pi$ and $\pi \to \rho \oplus \pi$ and check that they have the universal property of the coproduct.

Actually all this is vastly easier if you first check that the direct sum of vector spaces is the coproduct in $\mathsf{Vect}$. Then the other structure sort of goes along for the ride. Trying to check everything all at once is bit heavy.

John Baez (Sep 16 2022 at 16:44):

Another cool thing to check is that the direct sum of two vector spaces is also their product in $\mathsf{Vect}$! And this is true for group representations, too: the direct sum of two representations is both a product and coproduct.

John Baez (Sep 16 2022 at 16:47):

I would say that one of the great charms of linear algebra is that the product of two objects is also the coproduct. This gives rise to the theory of matrices.

David Egolf (Sep 16 2022 at 16:56):

Awesome, thanks! I will have to try checking out the universal property at some point when I'm feeling more energetic.

I was wondering what the product of two group representations was. It's interesting that it shares this nice property with $Vect$, where products and coproducts are the same. That makes me wonder, for example, if monoid actions also have this nice property.

I did not know that the theory of matrices was related to this connection between products and coproducts! That is also very interesting. In the engineering work I've done, we like to use matrices (describing linear maps between vector spaces) any time we can. The programming language I use the most - MATLAB - is actually named after matrices. So we really like them! Maybe we could also learn to love group representations somehow, if they also have this nice coproduct/product property.

John Baez (Sep 16 2022 at 17:04):

David Egolf said:

Awesome, thanks! I will have to try checking out the universal property at some point when I'm feeling more energetic.

I was wondering what the product of two group representations was. It's interesting that it shares this nice property with $Vect$, where products and coproducts are the same. That makes me wonder, for example, if monoid actions also have this nice property.

Monoid actions on what? Monoid actions on vector spaces do indeed have this property, yes! That's probably what you meant. But I would just call those representations of monoids. If you just walked up to me on the street and said "monoid actions", I'd think you meant monoid actions on sets... or maybe monoid actions on objects of an arbitrary category, if you looked exceptionally erudite.

David Egolf (Sep 16 2022 at 17:07):

Ah, yes, I meant monoid actions on vector spaces. I'm still learning the terminology.

John Baez (Sep 16 2022 at 17:09):

Yeah, it's fine - some of the conventions are rather hard to pick up except by actually talking to people and having them look funny at you when you say certain things.

John Baez (Sep 16 2022 at 17:10):

For example "action" and "representation" and "module" are overlapping concepts when you generalize them enough, but still people tend to use them in different contexts.

Chris Grossack (they/them) (Sep 16 2022 at 19:56):

Will the seminar be recorded? I could be up at 7 PST every thursday, but it would be more convenient (if less engaging) to watch on youtube

John Baez (Sep 16 2022 at 20:15):

Yes, they will eventually show up on YouTube.

John Baez (Sep 16 2022 at 20:15):

I think people on Zoom can ask questions, though I'm not sure how it'll work.

David Egolf (Sep 21 2022 at 17:31):

Taking a look at the notes, I found myself confused by this part:
notes image

To my understanding, we are trying to establish an isomorphism between $\mathbb{C}[G]$ and an algebra of matrices of the following form, with complex-valued entries:
matrix form

Let $f: A \to \mathbb{C}[G]$ be the isomorphism we seek, where $A$ is the algebra of matrices of the above form. To figure out this isomorphism, the notes talk about first finding $f(p_1)$, $f(p_2)$, and $f(p_3)$, where the $p_i$ are shown in the first image above.

The notes then (I think) claim that figuring out the rest of the isomorphism is easy given these $f(p_i)$. However, the explanation of this claim is confusing me. The notes state that $\mathbb{C}[G]$ is a direct sum of "blocks", and then writes: $\{p_i a p_i | a \in \mathbb{C}[g]\}$. I'm finding this statement confusing.

For one thing, the $p_i$ are in $A$, while the $a$ here are in $\mathbb{C}[G]$. So, I'm not sure how we can multiply these things. Perhaps what is meant is instead of $p_i a p_i$, rather $f(p_i) a f(p_i)$?

I'm also a little confused by the statement "$\mathbb{C}[G]$ is the direct sum of blocks". Let $s_i = \{f(p_i) a f(p_i) | a \in \mathbb{C}[g]\}$. Is the idea that each $s_i$ forms a algebra, and so we can take a direct sum of these to get something isomorphic to $\mathbb{C}[G]$?

Maybe I'm figuring this out. Consider matrices of the form $p_i M p_i$, where $M$ is any square complex-valued matrix of the right size. Then I am guessing that $p_i M p_i$ are the matrices only having nonzero entries in a single block. We can write any matrix in $A$ as a sum of matrices in $p_i M p_i$ as $i$ varies. Then the idea is to apply $f$ to all this, to write any element of $\mathbb{C}[G]$ as a sum of elements of the images of the sets $p_i M p_i$, maybe?

I'm not sure I was able to pose a clear question... but related thoughts would be welcome.

David Egolf (Sep 21 2022 at 18:00):

Also, what does it mean for something to "symmetrize", in particular over row permutations of a Young diagram?
Oh, maybe I get it.
Find the permutations $\pi_i \in S_n$ that permute elements while keeping them in the same row of a Young diagram. Then the "symmetrization" of these is the sum of all such permutations in $\mathbb{C}[S_n]$, divided by the number of these permutations.

dusko (Sep 22 2022 at 11:39):

John Baez said:

I'm giving eleven lectures on topics from This Week's Finds in Mathematical Physics, one every Thursday from September 22nd to December 1st. The talks will be at 3 pm UK time and they will be here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

i was totally hoping to attend the young diagrams one, but it will be 2am here when it starts and the 6am meeting i thought was on friday is just in 4 hours, and i teach tomorrow. i mean today. how about you push the record button on zoom and let us download the mov from somewhere?

Dylan Braithwaite (Sep 22 2022 at 11:41):

I think John said on twitter that they'll be uploaded to YouTube at least

David Egolf (Sep 22 2022 at 15:11):

I was glad to be able to watch the first talk! Thanks for putting this on. It's awesome how people from around the world can join in.

David Egolf (Sep 23 2022 at 01:11):

Is there a notion of Young diagrams for monoids? A monoid element acting on a set isn't always going to create a bunch of cycles of the elements of the set, but maybe we could use the "connected components" of the set induced by applying it as a substitute?

Beppe Metere (Sep 23 2022 at 06:47):

I was wondering the same question. Indeed I would like to understand at least in the case of monoid of endomaps of a finite set. This should play a role as the symmetric group does for group representation. A difference with the group case is that, while for symmetric group you can describe permutations as disjoint cycles, in the case of endomaps these cycles are decorated with trees, and these are the connected components you are talking about.

John Baez (Sep 28 2022 at 09:12):

David Egolf said:

Also, what does it mean for something to "symmetrize", in particular over row permutations of a Young diagram?
Oh, maybe I get it.
Find the permutations $\pi_i \in S_n$ that permute elements while keeping them in the same row of a Young diagram. Then the "symmetrization" of these is the sum of all such permutations in $\mathbb{C}[S_n]$, divided by the number of these permutations.

Exactly. I will add some formulas that say this sort of thing.

John Baez (Sep 28 2022 at 09:13):

For anyone who couldn't show up to my first talk but wants to see it, it's on YouTube now:

Lecture 1: Young diagrams and classical groups

John Baez (Sep 28 2022 at 09:14):

You can read lecture notes here:

• Young diagrams and classical groups.

John Baez (Sep 28 2022 at 09:15):

These notes will be good for the first couple of lectures, and contain more than I'll have time to actually talk about!

John Baez (Sep 28 2022 at 09:16):

Tomorrow (Thursday September 29th) I'll talk about how to actually get irreps of $S_n$ and also some classical groups from Young diagrams.

John Baez (Sep 28 2022 at 10:39):

Beppe Metere said:

I was wondering the same question. Indeed I would like to understand at least in the case of monoid of endomaps of a finite set. This should play a role as the symmetric group does for group representation. A difference with the group case is that, while for symmetric group you can describe permutations as disjoint cycles, in the case of endomaps these cycles are decorated with trees, and these are the connected components you are talking about.

Sorry to take so long to respond - I've been busy writing lecture notes for my talks!

Since I brought monoid representations into this course on Young diagrams and classical groups, classifying representations of the monoid of endomaps of a finite set seems like a really interesting question!

I brought in monoid representations for a purely practical reason! Namely, before classifying representations of the general linear group consisting of all invertible linear maps $f: \mathbb{C}^n \to \mathbb{C}^n$, its good to classify representations of the full linear monoid of all linear maps $f: \mathbb{C}^n \to \mathbb{C}^n$. This is simpler!

John Baez (Sep 28 2022 at 10:41):

But to do this we have to start by classifying representations of the symmetric group consisting of all invertible maps $f : \{1,\dots, n\} \to \{1,\dots, n\}$.

John Baez (Sep 28 2022 at 10:41):

So maybe it's also good to study representations of the monoid of all maps $f : \{1,\dots, n\} \to \{1,\dots, n\}$.

John Baez (Sep 28 2022 at 10:43):

I have never seen those classified! But someone must have done it, like Benjamin Steinberg, who has a book Representation Theory of Finite Monoids (not free here, alas).

John Baez (Sep 28 2022 at 10:46):

I see this:

• C. Hindlycke, Irreducible representations of finite monoids.

develops a general theory and then classifies irreps of the monoid of partial injections $f : \{1,\dots, n\} \to \{1,\dots, n\}$. I believe these are partially defined functions that are 1-1. Not what I want now. :neutral:

John Baez (Sep 28 2022 at 10:48):

But it makes me realize that all 16 of my favorite categories with finite sets as objects should give monoids whose representations are worth studying! Another one would be the monoid all relations from a finite set to itself!

John Baez (Sep 28 2022 at 10:49):

Does each of these have its own mutant theory of Young diagrams?

:question: :question: :question:

Morgan Rogers (he/him) (Sep 28 2022 at 11:41):

I'm curious about this too! Back when I was learning representation theory, monoids seemed scary with a lot more added complications. Now that I've spent years working on monoids, I can't help but wonder what happens when I replace "group" with "monoid" in a given context!

Matteo Capucci (he/him) (Sep 28 2022 at 12:52):

Here's a cute observation: while automorphisms of finite sets break into cycles, endomorphisms break into generic graphs.

Matteo Capucci (he/him) (Sep 28 2022 at 12:54):

Vertices are given by elements of the set and an arrow $i \to j$ means $f(i)=j$

Daniel Geisler (Sep 28 2022 at 12:55):

Young diagrams enumerate the integer partitions while the Faà di Bruno's formula uses the enumeration of the integer partitions to determine the terms for the derivatives of composite functions. So when I see that a system is represented by Young diagrams I ask myself if some form of differentiation and functional composition are present.

Beppe Metere (Sep 28 2022 at 13:00):

@Matteo Capucci (he/him) No, they don' break into generic graphs, but into cyclic permutations of trees.
Schermata-2022-09-28-alle-14.59.53.png

Beppe Metere (Sep 28 2022 at 13:01):

The pictures is from "Introduction to the Theory of Species of Structures", by Bergeron, Labelle, Leroux.

Beppe Metere (Sep 28 2022 at 13:04):

Then, as I wrote in my previous message, in order to classify finite set endomaps, you first count the partitions of the finite set, and then, for each part you count how many cyclic trees permutations can be obtained... In fact is a "composition" of three species: sets (E), cyclic permutations (C) and rooted trees (T). If we use P for permutations, we have PT=(EC)T=E(CT).

John Baez (Sep 28 2022 at 15:15):

Yes, everything Beppe says is correct, and it's very nice to read about this in Bergeron, Labelle and Leroux.

John Baez (Sep 28 2022 at 15:16):

Also, Tom Leinster has a nice study of these things here:

• Tom Leinster, The eventual image, The n-Category Cafe.

John Baez (Sep 28 2022 at 15:19):

For a small sample of the delights in this blog article:

Suppose $f: X \to X$ is a function from a finite set to itself. Then the collection of functions

$f, f^2, f^3, \dots$

has exactly one idempotent in it: i.e. a map $g = f^n$ such that $g^2 = g$. This maps $X$ to its "eventual image", i.e. the intersection of the images of the maps $f, f^2, f^3, ...$.

Matteo Capucci (he/him) (Sep 28 2022 at 18:20):

Beppe Metere said:

Matteo Capucci (he/him) No, they don' break into generic graphs, but into cyclic permutations of trees.
Schermata-2022-09-28-alle-14.59.53.png

Ha, they're less generic than I thought. My first guess was indeed trees but then cycles ruined it. Beautiful!

Egbert Rijke (Sep 30 2022 at 11:26):

Hi everyone! I had to ask John where to find the zulip discussion of the seminar, but I made it:)

My idea to classify endomorphisms up to isomorphism and to connect to the seminar is as follows: The endomorphisms on an n-element set are classified by young diagrams where you stick finite rooted trees in the boxes, so that the total number of nodes is n.

Edit: the trees have to be rooted.

Morgan Rogers (he/him) (Sep 30 2022 at 11:27):

Seems like there's a consensus on that :joy:

Egbert Rijke (Sep 30 2022 at 11:28):

Ah good, I better read up on the discussion here:)

Egbert Rijke (Sep 30 2022 at 14:40):

So I guess we can also classify arbitrary relations on an n-element sets: They are classified by Young diagrams where we put a (nonempty) connected digraph in each box, with at most one edge between two vertices and loops are allowed. If you allow for multiple edges then you classify endospans. Not sure if this is useful at all, but at least it is entertaining to see a feint image of Young diagrams here as well.

John Baez (Oct 01 2022 at 16:15):

I like this stuff a lot - but what would really thrill me is anything interesting about the classification of representations of the monoid of all endomaps of a finite set... especially if it could be done using 'mutant Young diagrams' of some sort.

John Baez (Oct 01 2022 at 16:17):

Here's the second lecture:

Lecture 2: Young diagrams and classical groups

John Baez (Oct 01 2022 at 16:17):

Here I explain how to classify irreducible representations of the symmetric groups Sₙ using Young diagrams. Then I introduce some 'classical groups' - famous groups whose irreducible representations can also be classified using Young diagrams.

John Baez (Oct 01 2022 at 16:18):

I got some good questions from the Zoom audience this time, both during the talk and at the end.

John Baez (Oct 04 2022 at 18:56):

Here are the lecture notes for my talks about Dynkin diagrams!

• Coxeter and Dynkin diagrams.

John Baez (Oct 04 2022 at 18:57):

This Thursday (October 6) I will finish up talking about Young diagrams. Then I'll move on to Dynkin diagrams.

Simon Burton (Oct 07 2022 at 07:27):

The free 2-rig on one generator, could we call this the walking Young diagram category?

Simon Burton (Oct 07 2022 at 07:38):

I was wondering how the Schur-Weyl duality works for symplectic groups, and if we would replace the group algebra $\mathbb{C}[S_n]$ with the group algebra of the Weyl group for Sp(n). But it seems this is the wrong guess, instead you use something called the Brauer algebra. This is just like $\mathbb{C}[S_n]$, which you can view as linear combinations of "string diagrams" for permutations, but the Brauer algebra also has cups and caps in these string diagrams. Viewed this way, the "walking Brauer-Young diagram category" (??) would be a 2-rig with duals, something like that, so we can interpret these Brauer algebra string diagrams freely...

John Baez (Oct 09 2022 at 12:06):

Simon Burton said:

The free 2-rig on one generator, could we call this the walking Young diagram category?

I usually call it $\mathsf{Schur}$ or "the category of Schur functors": its objects are formal finite direct sums of Young diagrams.

If I were trying to be technically correct but also cute, and we'd already agreed we were working in the doctrine of 2-rigs, I could call $\mathsf{Schur}$ the "walking object". This means that if you have any 2-rig $\mathsf{R}$ and any object $r \in \mathsf{R}$, there's a unique map from $\mathsf{Schur}$ to $\mathsf{R}$ sending the 1-box Young diagram to $r$.

John Baez (Oct 09 2022 at 12:08):

This is like how Jim Dolan would call the free ring on one element the "walking element", if we knew already that we were working in the theory of rings.

John Baez (Oct 09 2022 at 12:10):

Simon Burton said:

I was wondering how the Schur-Weyl duality works for symplectic groups, and if we would replace the group algebra $\mathbb{C}[S_n]$ with the group algebra of the Weyl group for Sp(n). But it seems this is the wrong guess, instead you use something called the Brauer algebra. This is just like $\mathbb{C}[S_n]$, which you can view as linear combinations of "string diagrams" for permutations, but the Brauer algebra also has cups and caps in these string diagrams. Viewed this way, the "walking Brauer-Young diagram category" (??) would be a 2-rig with duals, something like that, so we can interpret these Brauer algebra string diagrams freely...

I am quite fuzzy on Brauer algebras and how they're connected to the group algebra of the Wel group of Sp(n). But this is a fun subject!

I know that the category of finite-dimensional representations of Sp(n) should be the "free 2-rig on a symplectic object $x$ of dimension $n$.

John Baez (Oct 09 2022 at 12:13):

Let me explain. We can talk about an object $x$ in a 2-rig being dualizable. You probably know this means there is an object $x^\ast$ equipped with morphisms called the unit

$\iota: I \to x \otimes x^\ast$

and counit

$\epsilon : x^\ast \otimes x \to I$

obeying the zig-zag axioms.

John Baez (Oct 09 2022 at 12:19):

Since a 2-rig is symmetric monoidal you can cook up a specific isomorphism

$x^{\ast \ast} \cong x$

and I'll use this to ignore the distinction between an object and its double dual.

John Baez (Oct 09 2022 at 12:23):

This helps us describe a "360 twist" map

$x \cong x \otimes I \xrightarrow{1 \otimes \iota} x \otimes x \otimes x^\ast \xrightarrow{S_{x,x} \otimes 1} x \otimes x \otimes x^\ast \xrightarrow{1 \otimes \epsilon} x \otimes I \cong x$

John Baez (Oct 09 2022 at 12:26):

:warning: Here in $1 \otimes \epsilon$ I'm treating $x \cong x^{\ast \ast}$ as the dual of $x^\ast$ - so this $\epsilon$ is the counit for $x^\ast$, not our original $\epsilon$! :warning:

John Baez (Oct 09 2022 at 12:26):

You probably know all this stuff - it's easier to explain with string diagrams.

John Baez (Oct 09 2022 at 12:28):

Now, my point is this: if this 360 twist map is the identity we can call our dualizable object $x$ an orthogonal object, and if it's minus the identity we can call it a symplectic object.

John Baez (Oct 09 2022 at 12:29):

Now here's the point:

Theorem. The category of finite-dimensional representations of O(n) is the free 2-rig on an orthogonal object of dimension n, while the category of finite-dimensional representations of Sp(n) is the free 2-rig on a symplectic object of dimension n.

John Baez (Oct 09 2022 at 12:34):

Well, I'm using another thing I didn't explain, which is how (and when!) we can define the "dimension" of an object in a 2-rig. Briefly, we can define it to be the trace of the identity morphism on that object. This will usually be an endomorphism of the unit object $I$, but when our 2-rig is connected the algebra of endomorphisms of $I$ is just $\mathbb{C}$ - or whatever field we're using here.

John Baez (Oct 09 2022 at 12:35):

That's the definition of a connected 2-rig, and I should stick 'connected' into my theorem somewhere.

David Michael Roberts (Oct 09 2022 at 23:44):

A vague question: is 'connected' here at all related to how one can consider Galois categories, which are essentially representation categories for fundamental groups of pointed connected toposes, as opposed to rep cats for fundamental groupoids for more arbitary toposes?

dusko (Oct 10 2022 at 08:32):

John Baez said:

I like this stuff a lot - but what would really thrill me is anything interesting about the classification of representations of the monoid of all endomaps of a finite set... especially if it could be done using 'mutant Young diagrams' of some sort.

sorry that i am running behind, but it took me a while to watch it all. two comments:

1) young diagrams are sometimes used in cryptanalysis, and there some people programmed irreducible decompositions of inverse semigroups, which seem to correspond to a sort of "mutant young diagrams". a message is a function $n\to A$, where $A$ is the alphabet of some sort. so it induces a partition on $n$. hence one young diagram, which can be viewed as corresponding to the substitution cipher if we only remember the parts. (if the alphabet is known, then the sizes of the parts are proportional to the frequencies of the symbols. some people pad and dualize the partition...) a transposition cipher is a permutation of the positions $n$. hence the second young diagram. so the space of ST-boxes is the space of young diagrams. (pity that i don't understand how is this space related to the space of representations. i never managed to read fulton's book, though i had it on loan for years.) inverse semigroups enter the scene because encryption is usually randomized, to avoid that the attacker enciphers the messages of interest and sits and waits to see them. so the encryption-decryption pairs are not inverses but retractions. inverse semigroups have a very nice decomposition theory, but it would be even nicer if someone worked out proper young tableaux.

2) bob pare worked out the structure of endofunctors on the category of finite sets, and the induced cardinal functions are simply all weighted sums of stirling numbers of the second kind, summed over the numbers of parts. along the yoneda this turns out to determine representations in terms of weighted sums of surjections. maybe the representations of endofunctions over a finite set decompose in a similar way? he found that the pushouts of surjections between finite sets are absolute (as are the coequalizers of reflexive relations!).

Simon Burton (Oct 10 2022 at 10:09):

John Baez said:

Now, my point is this: if this 360 twist map is the identity we can call our dualizable object $x$ an orthogonal object, and if it's minus the identity we can call it a symplectic object.

Wow, I'm just bowled over by this "high-brow" monoidal category theory so naturally yields all the classical representation theory... still trying to pick myself up off the floor here..

John Baez (Oct 10 2022 at 10:59):

David Michael Roberts said:

A vague question: is 'connected' here at all related to how one can consider Galois categories, which are essentially representation categories for fundamental groups of pointed connected toposes, as opposed to rep cats for fundamental groupoids for more arbitary toposes?

Yes! For current purposes @Joe Moeller, @Todd Trimble and I say a 2-rig is a symmetric monoidal Cauchy-complete category enriched over $\mathrm{Vect}_k$ for some field $k$. I say it's "connected" if the endomorphism ring of the unit object is just $k$.

Then the category of representations of a group is connected, while the category of representations of a groupoid is not. That's because in the groupoid case, the unit object is the direct sum of representations that are 1-dimensional on one connected component and 0-dimensional on the rest!

John Baez (Oct 10 2022 at 11:04):

Simon Burton said:

John Baez said:

Now, my point is this: if this 360 twist map is the identity we can call our dualizable object $x$ an orthogonal object, and if it's minus the identity we can call it a symplectic object.

Wow, I'm just bowled over by this "high-brow" monoidal category theory so naturally yields all the classical representation theory... still trying to pick myself up off the floor here..

Thanks! I developed this stuff in HDA2: 2-Hilbert spaces, but there I was studying unitary representations on Hilbert spaces so I used a somewhat different framework, more heavy on ideas from quantum mechanics. But for finite-dimensional representations of compact Lie groups it doesn't matter much whether we consider unitary representations or not, so it's easy to transplant these ideas to a simpler context - namely, the theory of 2-rigs.

John Baez (Oct 10 2022 at 11:14):

dusko said:

1) young diagrams are sometimes used in cryptanalysis, and there some people programmed irreducible decompositions of inverse semigroups, which seem to correspond to a sort of "mutant young diagrams". a message is a function $n\to A$, where $A$ is the alphabet of some sort. so it induces a partition on $n$. hence one young diagram, which can be viewed as corresponding to the substitution cipher if we only remember the parts. (if the alphabet is known, then the sizes of the parts are proportional to the frequencies of the symbols. some people pad and dualize the partition...) a transposition cipher is a permutation of the positions $n$. hence the second young diagram. so the space of ST-boxes is the space of young diagrams.

Wow. That's interesting. I don't understand it because I don't know what's an inverse semigroup, substitution cipher, transposition cipher, or ST-box. But actually I can look up the first one and look up or maybe even just guess the latter three.

John Baez (Oct 10 2022 at 11:15):

(pity that i don't understand how is this space related to the space of representations. i never managed to read fulton's book, though i had it on loan for years.)

My first three lectures do the easy stuff in Fulton's book.

2) bob pare worked out the structure of endofunctors on the category of finite sets, and the induced cardinal functions are simply all weighted sums of stirling numbers of the second kind, summed over the numbers of parts.

I should learn about this, because I've just been running around telling people about (algebraic) endofunctors on the category FinVect of finite-dimensional vector spaces: any one of these can be described as a direct sum of endofunctors coming from Young diagrams - the functors called $\tilde{Y}$ in my third talk. (I start describing them at 10:45.)

Here "algebraic" rules out weird endofunctors coming from, e.g., the Galois group of the field we're implicitly using here. FinVect is enriched over algebraic varieties, and an endofunctor on FinVect is algebraic if it's enriched over algebraic varieties. Simply put, the maps from hom-spaces to hom-spaces can be written out using polynomials!

dusko (Oct 10 2022 at 11:19):

Simon Burton said:

John Baez said:

Now, my point is this: if this 360 twist map is the identity we can call our dualizable object $x$ an orthogonal object, and if it's minus the identity we can call it a symplectic object.

Wow, I'm just bowled over by this "high-brow" monoidal category theory so naturally yields all the classical representation theory... still trying to pick myself up off the floor here..

while john is explaining this in terms of 2-rigs very very nicely, reconstructing the group out of the monoidal structure of its representations was the goal of tanaka duality, which goes back to the 70s. and it is as an instance of functorial semantics. eg the paper Matrices, Relations and Group Representations by aurelio carboni tells the story that way. categories of representations do have a tendency to be universal constructions. eg the category of representations of a category is the free category with colimits.
sorry that i am interferring, but precisely because what john is presenting is so fundamental, it wouldbe a misunderstanding to view it as "high-brow".

dusko (Oct 10 2022 at 11:29):

John Baez said:

Wow. That's interesting. I don't understand it because I don't know what's an inverse semigroup, substitution cipher, transposition cipher, or ST-box. But actually I can look up the first one and look up or maybe even just guess the latter three.

given a text $n\to A$, a substitution cipher is a permutation on $A$, a transposition cipher is a permutation on $n$, and ST-boxes are compositions of substitutions and transpositions with something in-between to prevent them from commuting (like sigmoids etc in neural nets). an inverse semigroup satisfies $xyx=x$. i may be oversimplifying, but even adding what i forgot might not take more than 5 min of search.

John Baez (Oct 10 2022 at 11:32):

Okay, that's good enough! The stuff about ciphers makes sense and it's enough to make me have lots of pleasant thoughts about how the set of functions $f: m \to n$ is acted on by both $S_m$ and $S_n$, giving stuff that can be studied with Young diagrams.

John Baez (Oct 10 2022 at 18:56):

Here is the third talk:

Lecture 3: Young diagrams and classical groups

John Baez (Oct 10 2022 at 18:57):

Unfortunately the recording starts a bit after my talk. But I'm just going through the diagram of four "classical groups", which you can see in my lecture notes on Young diagrams. There are a few other suboptimal things about the video - we'll try to do better next time.

John Baez (Oct 10 2022 at 18:58):

On this Thursday I'll explain Coxeter groups and Coxeter diagrams, and here are the notes:

• Coxeter and Dynkin diagrams.

Simon Burton (Oct 11 2022 at 09:46):

dusko said:

sorry that i am interferring, but precisely because what john is presenting is so fundamental, it wouldbe a misunderstanding to view it as "high-brow".

Clearly my eyebrows have a long way to go.

John Baez (Oct 14 2022 at 22:12):

Here's the fourth talk. New subject now, no prerequisites!

• Lecture 4: Coxeter and Dynkin diagrams.
  

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I begin by explaining how Coxeter groups classify finite reflection groups.

John Baez (Oct 24 2022 at 21:43):

Here's the fifth talk:

• Lecture 5: Coxeter and Dynkin diagrams.
  

John Baez (Oct 24 2022 at 21:44):

Here I explain how Dynkin diagrams classify root lattices.

John Baez (Oct 28 2022 at 20:30):

• Lecture 6: Coxeter and Dynkin diagrams.

John Baez (Oct 28 2022 at 20:33):

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I use Dynkin diagrams to classify compact Lie groups---and especially compact semisimple Lie groups.

I made 3 mistakes, caught in real time by audience members. First, while it's true that a compact Lie algebra is 'semisimple' iff for every $latex z$ there exist $x,y$ such that $[x,y] = z$, and I was mainly talking about compact Lie algebras, this is not a valid definition of semisimplicity for arbitrary Lie algebras. A better definition of semisimplicity is to require the 'Killing form' to be nondegenerate:

• Wikipedia, Killing form.

The Killing form for is something you can define for any Lie algebra, and for a compact semisimple Lie algebra, the negative of the Killing form is what I called the 'god-given inner product'.

Second, the Weyl group is not the normalizer of the maximal torus; it's the normalizer mod the centralizer. I fixed this during the talk:

• Wikipedia, The Weyl group of a connected compact Lie group.

Third, the root lattice associated to the Lie algebra of a compact semisimple Lie group $G$ is actually the dual lattice of the kernel of the exponential map $\exp: \mathrm{Lie}(T) \to T$, where $T$ is the maximal torus---and even this is only true when $G$ has trivial center. But every compact semisimple Lie group $G$ is a covering space of a unique compact semisimple Lie group with trivial center. What I called the root lattice is actually what Adams calls the 'integer lattice', and I recommend his treatment of these issues:

• J. F. Adams, Lectures on Lie Groups, University Of Chicago Press, 1983.

Ieva Cepaite (Oct 31 2022 at 11:20):

Hi, what's the topic of the seminar this week? I might finally have the time to pop down to Edinburgh for the afternoon :)

John Baez (Nov 10 2022 at 09:58):

Wow, I didn't see your message until now @Ieva Cepaite. Sorry!

Today I'll be talking about E8, quaternions and octonions.

John Baez (Nov 10 2022 at 09:58):

And a warning to everyone: today, November 10th, the seminar will be in a different building.

John Baez (Nov 10 2022 at 10:00):

Today we're in Lecture Theatre 1 of the Daniel Rutherford Building. This is a few minutes' walk from the usual James Clerk Maxwell Building, just along from the Darwin Building.

John Baez (Nov 10 2022 at 10:00):

Next week we'll be back in the usual place.

John Baez (Nov 10 2022 at 10:02):

• Lecture 7: Coxeter and Dynkin diagrams.
  

John Baez (Nov 10 2022 at 10:04):

In Lecture 7, I went through all the connected Dynkin diagrams and described all the Lie groups they correspond to. All of them give symmetries of geometries connected to the real numbers, complex numbers, quaternions and/or octonions.

Matteo Capucci (he/him) (Nov 10 2022 at 14:08):

Did the lecture start yet? @Tom Leinster

Tom Leinster (Nov 10 2022 at 14:57):

Starting in 8 mins

John Baez (Nov 10 2022 at 14:59):

It always starts at 3 pm, dude. :upside_down:

Matteo Capucci (he/him) (Nov 10 2022 at 22:19):

:face_palm: sorry folks, I was too impatient!

John Baez (Nov 11 2022 at 16:50):

• Lecture 7: E8 and the octonions.
  

Here I explain the E8 root lattice and how it gives rise to the octooctonionic projective plane, a 128-dimensional manifold on which the compact Lie group called E8 acts as symmetries. I also discuss how some special root lattices give various notions of 'integer' for the real numbers, complex numbers, quaternions and octonions.

Ieva Cepaite (Nov 13 2022 at 14:19):

Ah no worries @John Baez ! The lecture tends to be at the same time as my group meetings so I’ve found it hard to make it but I’m very keen. This week is 100% sure though unless something completely unexpected happens :)

John Baez (Nov 13 2022 at 16:03):

If you come by, try to drag some other folks over. We can go to a pub afterwards and talk, which will be a bit less scary for both of us if you have some friends around.

John Baez (Nov 14 2022 at 11:52):

On this Thursday (November 17) I will show how to construct the quaternions and octonions using the dot product and cross product of vectors. For the quaternions this is well-known (but good), while for the octonions it's much less well known and requires a slight twist!

John Baez (Nov 14 2022 at 11:52):

On next Thursday (November 24) universities across the UK will be on strike, and I will not give a talk.

John Baez (Sep 11 2023 at 10:10):

It’s almost time for more talks - this time about category theory! As before, I’ll be doing these on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building, home of the School of Mathematics at the University of Edinburgh.

The first will be on Thursday September 21st and the last on Thursday November 30th. I’ll skip October 19th… and any days there are strikes.

John Baez (Sep 11 2023 at 10:12):

Tom Leinster and I are planning to

1) make the talks hybrid on Zoom so that you can join online:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of the famous lemma in category theory.

John Baez (Sep 11 2023 at 10:13):

2) make lecture notes available on my website.

3) record them and make them publicly available on my YouTube channel.

4) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s right here!

John Baez (Sep 11 2023 at 10:17):

I'll start with a lecture or two on this:

Categorification and the periodic table of $n$-categories. Categorification is a not-completely-systematic process of taking known math and replacing sets by categories, functions by functors, and equations by natural isomorphisms. Often categorifying simple results in math leads to deeper, more interesting results. When we iterate categorification we're pushed into higher categories. Higher categories exhibit striking patterns visible in the the "periodic table" of $n$-categories, such as the Stabilization Hypothesis, Tangle Hypothesis and Cobordism Hypothesis. I'll concentrate on sketching the basic ideas, since the evidence is much easier to explain than the rigorous proofs. This will be an elementary introduction to higher categories.

John Baez (Sep 13 2023 at 08:33):

There was a typo in that announcement, which I've fixed now: the first talk is on Thursday September 21st.

Simon Burton (Sep 26 2023 at 16:59):

Here is the video from the first lecture: https://www.youtube.com/watch?v=ZVecriTCBLU . It's very nice to hear John give a breezy introduction to higher categories and am looking forward to seeing where this goes..

David Egolf (Sep 26 2023 at 22:38):

I enjoyed watching the first 10 minutes! Maybe I'll watch the rest too, but I wanted to first ask a question that occurred to me in response to the basic idea sketched at the start of the lecture.

When categorifying, I believe we follow this pattern:

• sets $\to$ categories
• functions $\to$ functors
• equations $\to$ natural isomorphisms

I think the intuition is roughly that, just as natural isomorphisms go between functors, in some sense "equations go between functions". On a related note, two naturally isomorphic functors decategorify to the same function, so if $d$ denotes decategorification, then for two naturally isomorphic functors $F,G: C \to D$ we have $d(F) = d(G)$. Here $d(F)$ is the function induced by $F$ that sends isomorphism classes of objects in $C$ to isomorphism classes of objects in $D$, and $d(G)$ is defined similarly. To see that $d(F) = d(G)$, note that since $F(c) \cong G(c)$ for all $c$, we have $d(F)[c] = [F(c)] = [G(c)] = d(G)[c]$. So, in summary, a natural isomorphism $\alpha: F \to G$ induces the equation $d(F) = d(G)$.

David Egolf (Sep 26 2023 at 22:38):

I was wondering how to expand on the rough intuition that "equations go between functions". In particular, I was wondering if there are other relationships between functions that might categorify to natural transformations that are not necessarily natural isomorphisms. As an initial guess, I was wondering if one could try to place some kind of ordering on functions with the same source and target, such that if $f \leq g$ and $g \leq f$ we'd have $f=g$. Then if $f \leq g$ but we don't have $g \leq f$, we'd hope that the inequality $f \leq g$ "between" $f$ and $g$ would categorify to a natural transformation that is not a natural isomorphism.

However, I think the specific idea I had in mind for doing this actually works better if we start with the category $\mathsf{Rel}$ of sets and relations between them. "Categories for Quantum Theory: An Introduction" has this nice picture illustrating what a relation from $A$ to $B$ can look like:
a relation

Notice in particular that a given element can "explode" outwards to multiple elements, in the sense that we can have $(a,b)$ and $(a,b')$ for $b \neq b'$ in our subset of $A \times B$ that describes our relation $R$. I will use the notation $R(a)$ to denote the subset of $B$ consisting of all the elements $b \in B$ related to $a$, so $R(a) = \{b \in B | (a,b) \in R\}$. Then we might try introducing an order of some kind on relations, by saying that for relations $R,R': A \to B$ we have $R \leq R'$ exactly when $R(a) \subseteq R'(a)$ for all $a \in A$.

Well, we now have some kind of additional structure between our morphisms in our starting (not-yet-categorified) setup, but to do this we have drifted from $\mathsf{Set}$ to $\mathsf{Rel}$. I am wondering if it is possible to still "categorify" some objects and morphisms in $\mathsf{Rel}$ (in some sense), in a way roughly analogous to how we can categorify objects and morphisms in $\mathsf{Set}$.

To conclude, I'm now wondering if we can be inspired by this pattern:

• sets $\to$ categories
• functions $\to$ functors
• equations $\to$ natural isomorphisms

...to obtain a related pattern:

• sets $\to$ ?
• relations $\to$ ?
• inequalities between relations (in the sense above) $\to$ ?

David Egolf (Sep 26 2023 at 22:46):

Perhaps a related (and maybe easier?) question is: Can we decategorify enriched categories to obtain something fancier than sets and functions between them?

Kevin Arlin (Sep 26 2023 at 22:47):

Yes, very cool thoughts!

An intuition I find valuable here is that a category is a "proof-relevant" set. That is, in a set all we can know is that two things are equal (for some reason or other.) In a category, the set of morphisms can sometimes usefully be thought of as the set of ways or reasons two objects are related.

Again, then, given a relation $R$ between sets $A$ and $B$, we're allowed to know $aRb,$ that is, the fact that two elements are related. Thus if we categorify $A$ and $B$ to categories, we might try to categorify the relation $R$ to some sort of structure that allows us to produce a set of ways two objects in a category can be related.

Possibly I should pause here and see if you want to think about that further on your own before I reveal any words!

Cole Comfort (Sep 26 2023 at 22:48):

David Egolf

Sets $\to$ categories
Relations $\to$ [[profunctors]]
Inclusion of relations $\to$ natural transformations between profunctors

For a more elementary categorification, consider the category of Bool-enriched profunctors. It is almost Rel, except now for every 0-cell, there is a opposite 0-cell. This is different than Rel, where the the opposite of a set is the set itself!

Morally, I would argue that profunctors are more of a categorification of matrices over the natural numbers, which is equivalently [[spans]] of finite sets.

The picture you linked for the graph of a function has a formal interpretation. The category of finite sets and functions is a monoidal category under the disjoint union. It is equivalent to the strict monoidal category generated by a commutative monoid on a single object. So the picture for the graph of a function are the string diagrams for this monoidal category. So here is a fun excercise: find the generators and relations for the monoidal categories of finite sets and relations and finite sets and spans under the disjoint union.

edit: @Kevin Arlin I didn't see your comment when I started writing my comment :upside_down:

Cole Comfort (Sep 26 2023 at 23:02):

For a harder exercise, can you do the same for some subcategory of Bool-enriched categories and similarly for Bool-enriched profunctors.

now that I am reading the comments above me I realize I am spoiling John's lecture series, so maybe try this exercise when you get to lecture 60.

John Baez (Sep 27 2023 at 11:49):

Go ahead and "spoil" it, Cole! Anything that helps alert people that there's something worthwhile in those lectures is fine with me!

David Egolf (Sep 27 2023 at 16:39):

Kevin Arlin said:

An intuition I find valuable here is that a category is a "proof-relevant" set. That is, in a set all we can know is that two things are equal (for some reason or other.) In a category, the set of morphisms can sometimes usefully be thought of as the set of ways or reasons two objects are related.

Again, then, given a relation $R$ between sets $A$ and $B$, we're allowed to know $aRb,$ that is, the fact that two elements are related. Thus if we categorify $A$ and $B$ to categories, we might try to categorify the relation $R$ to some sort of structure that allows us to produce a set of ways two objects in a category can be related.

That's an interesting way to think about this! If I'm following, the idea is that in our starting setup (either a set or a relation) we have a "truth-value" relationship between every appropriate pair of elements:

• in a set $S$, for any two elements $a,a' \in S$, either $a=a'$ is true, or it is false
• in a relation $R$ between sets $A$ and $B$, for an element $a \in A$ and an element $b \in B$, either $a \sim_R b$ is true, or it is false

(I'm guessing this relates to the idea that a set can be viewed as a "0-category" and a truth value can be viewed as a "(-1)-category").

Then, when we categorify a set, our new objects are related by a set of morphisms, not just a truth value. By analogy, when we categorify a relation $R$ between $A$ and $B$, we'd expect the categorified elements of $A$ and $B$ to be objects that are related in a set of ways.

However, if $A$ gets categorified to a category $A'$, and $B$ gets categorified to a category $B'$, then we would like to have a set (of "morphisms") that relates an object $a \in A'$ and an object $b \in B'$. This is a bit different than categorifying a set, because now we want to relate objects in different categories. I'm not sure how to create "morphisms" that go between objects in different categories. Maybe we could make a bigger category where we set $A'$ and $B'$ "side by side" (I think this is $A \coprod B$), and then introduce a set of morphisms from a categorified version of an element $a$ (an object in $A'$) to a categorified version of an element $b$ (an object in $B'$), if $a \sim_R b$.

Kevin Arlin (Sep 27 2023 at 16:41):

The bigger category is a good idea and that can work, yes! You could also let the "morphisms" from objects of $A'$ to objects of $B'$ be an ontologically novel thing: you have categories $A'$ and $B'$, and now introduce a notion of "heteromorphism" [as they're sometimes called] from objects of $A'$ to objects of $B'$, then try to think what kinds of axioms you need to make the notion of heteromorphism play well with $A'$ and $B'$'s category structures.

David Egolf (Sep 27 2023 at 16:55):

I think I've run across heteromorphisms once before, but I hadn't realized they have anything to do with relations!

One challenge, I think, for the "bigger category" approach I mention above (where we set $A'$ and $B'$ side by side and add in some morphisms) is that this single category would just decategorify to a single set, unless we changed our decategorification procedure. (And we want to decategorify to obtain two sets and a relation between them.)

Kevin Arlin (Sep 27 2023 at 16:56):

Yes, you'd need to decategorify in a way that remembers that your big category has a "left side", a "right side", and some "stuff" in the middle.

David Egolf (Sep 27 2023 at 17:09):

Thanks, @Cole Comfort for telling me that relations categorify to profunctors. That really caught me by surprise! "Profunctor" is a word that has mostly just been intimidating for me - I don't really know what it means yet. But now I have a little bit of intuition for trying to understand profunctors, going forward.

To understand the rest of your answer, I probably need to first do some practice with the concept of profunctor (and also with string diagrams). But I'll make note of this for later!

Thanks also for highlighting that these lectures cover profunctors. That looks like a great resource.

John Baez (Sep 27 2023 at 20:31):

Profunctors shouldn't be scary. My lectures (and Fong and Spivak's book) discuss them in detail, but here's the one-sentence explanation:

A relation from a set $X$ to a set $Y$ is a function $f: X \times Y \to 2$ where $2$ is the set of truth values, and similarly a profunctor from a category $\mathsf{X}$ to a category $\mathsf{Y}$ is a functor $f: \mathsf{X} \times \mathsf{Y}^{\mathrm{op}} \to \mathsf{Set}$.

John Baez (Sep 27 2023 at 20:36):

So a profunctor is a categorified relation, and the only big surprise is the "op", which turns out to be very useful: it was invisible in the case of relations, since the "op" of a set it itself.

Bruno Gavranović (Oct 02 2023 at 21:31):

Is there going to be a seminar this Thursday?

John Baez (Oct 03 2023 at 12:32):

There certainly will! There's no strike this week, and no known strikes in future weeks. As I announced in the first talk, I will be gone to Cambridge on Thursday October 19th, so no talk then. But I'll be speaking October 5th and 12th.

John Baez (Oct 03 2023 at 12:33):

This Thursday I'm going to talk about the Periodic Table of n-categories.

Bruno Gavranović (Oct 03 2023 at 16:23):

Fabulous. See you there!

John Baez (Oct 03 2023 at 18:07):

Great - see you! Try to drag @Jules Hedges and @Jade Master and others along (if they're willing to be dragged).

By the way, my wife Lisa will be out of town so I will be more-than-usually happy to go to a pub and/or dinner afterward.

David Egolf (Oct 05 2023 at 17:56):

I just finished watching "This Week's Finds 11: n-categories" (the one from two weeks ago), and I really enjoyed it. Thanks for recording and posting these.

One thing I would note - it is very difficult to make out what is written on the chalkboard. It looks like this for me:
chalkboard

Thankfully, you do a great job at speaking aloud what you are writing, so I am still able to follow. I'm looking forward to future seminars!

John Baez (Oct 06 2023 at 08:13):

Thanks! Yes, Tom Leinster recognized that the image quality was bad but was unable to do anything about it at the time. (The camera he'd been using last year stopped working after a firmware update, so he was winging it with another one.)

He used a different camera for yesterday's seminar, the second in this year's series. And he thinks it's better! (I haven't seen the video yet myself.)

John Baez (Oct 08 2023 at 12:46):

Here's my second lecture:

• The periodic table of n-categories.
  

John Baez (Oct 08 2023 at 12:46):

And here's a description for a broad audience including non-category-theorists:

What happens in math if there are no equations? All you have are things, processes that turn one thing into another, meta-processes that turn one process to another, and so on... forever!

If this is too scary you can truncate it at the nth level. Then you're dealing with an 'n-category'. This has things (called 'objects'), processes (called 'morphisms'), meta-processes (called '2-morphisms') and so on up to n-morphisms. You can use equations... but only between n-morphisms.

In this talk I explain the periodic table of n-categories - a fundamental structure that emerges when you think about this stuff.

I put a lot of work into making it fun and easy to follow... and I think it worked!

(Alas, the video quality is still not great, but it's better than last week's lecture where I introduced n-categories. The volume is low so you have to really crank up your speaker... and the only way I have to boost the volume of a video also makes the file a lot bigger.)

John Baez (Oct 10 2023 at 12:17):

On Thursday October 12 at 3 pm I'll talk about this - as always, it's available on Zoom or later on YouTube:

Topology and the Periodic Table

The "periodic table of n-categories" is a chart showing n-categories that are trivial at the bottom k levels. This helps organize our understanding of higher categories and their relation to topology and physics. There are many ways to hop around the periodic table, and I will explain a few of the main ones: suspension, looping, delooping, and decategorification. I will also explain how free k-tuply monoidal categories let us enhance familiar structures such as the natural numbers and integers and obtain such things as braid groups and the homotopy groups of spheres. With luck we'll have time to use these ideas to think about the isomorphisms $\pi_3(S^2) \cong \mathbb{Z}$ and $\pi_4(S^3) \cong \mathbb{Z}/2$.

I will not be speaking on Thursday October 17th or the 26th (when there's a celebration of the 30th anniversary of the ICMS, with Graeme Segal speaking), but I will resume on Thursday November 2nd, on a new topic.

David Egolf (Oct 13 2023 at 02:20):

John Baez said:

Here's my second lecture:

• The periodic table of n-categories.

Thanks for sharing another enjoyable lecture! I love the idea of replacing some equality between two things with a process that describes how to change one thing into the other, in a reversible way. It reminds me of something you told me before (I think): it's a lot more useful to know of a specific isomorphism $f: a \to b$ than to just know that $a$ and $b$ are isomorphic (so their isomorphism classes are equal). Keeping around this extra information seems potentially quite useful and interesting!

After watching the lecture, a question did occur to me: Can we apply this perspective of replacing equality with a reversible process to contexts involving a quotient group? For example, we can think of the elements of the group $\mathbb{Z}/4\mathbb{Z}$ as equivalence classes of integers. Then we have, for example, that $[0] = \{\dots, -8, -4, 0, 4, 8, \dots\}$ is an element of $\mathbb{Z}/4\mathbb{Z}$. So, we started with a bunch of distinct things (the integers) then we placed an equivalence relationship on them, and finally we work with the resulting equivalence classes. Could we instead remember the distinct integers, and just keep track of specific ways of changing integers into one another? For example, instead of saying that $[-4]$ and $[0]$ are equal, we could say that $-4$ and $0$ are not equal, but that we have a "process" +4 that changes $-4$ to $0$, which is reversible. I'm wondering if we can get some kind of category with 2-morphisms in this way:

• there is a single object $*$
• the 1-morphisms are integers (composed by addition)
• the 2-morphisms go between integers with a difference that is a multiple of 4, and they describe how to change the source integer to the target integer by adding (or subtracting) 4 a certain number of times (and these 2-morphisms are composed by addition)

It intuitively seems like this resulting category (if it is a category) should contain all the information in $\mathbb{Z}/4\mathbb{Z}$, plus some more.

John Baez (Oct 13 2023 at 09:36):

I'm glad you liked my lecture, @David Egolf!

After watching the lecture, a question did occur to me: Can we apply this perspective of replacing equality with a reversible process to contexts involving a quotient group?

Yes, this is a famous and important construction! Given a group $G$ acting on a set $X$, there's a category $X//G$ called the weak quotient that has elements of $X$ as objects and a morphism $g: x \to x'$ whenever $gx = x'$.

John Baez (Oct 13 2023 at 09:38):

You can figure out how to compose morphisms, check this gives a category, and check that it's a groupoid since any morphism $g: x \to x'$ has an inverse $g^{-1} : x' \to x$.

John Baez (Oct 13 2023 at 09:40):

I've spent a lot of time advertising the virtues of this construction, which is also called the action groupoid.

John Baez (Oct 13 2023 at 09:40):

In particular there's a way to assign a cardinality to any finite groupoid, which is a nonnegative rational number, and then we get

$|X//G| = |X|/|G|$

John Baez (Oct 13 2023 at 09:41):

For example if you have a 2-element group acting on a 3-element set, the weak quotient is a groupoid with cardinality $3/2$.

David Egolf (Oct 13 2023 at 15:59):

John Baez said:

After watching the lecture, a question did occur to me: Can we apply this perspective of replacing equality with a reversible process to contexts involving a quotient group?

Yes, this is a famous and important construction! Given a group $G$ acting on a set $X$, there's a category $X//G$ called the weak quotient that has elements of $X$ as objects and a morphism $g: x \to x'$ whenever $gx = x'$.

Very cool! It's also great to know this is called the "weak quotient" or "action groupoid" - knowing the name makes it easier to learn more about it.

From your description, it looks to me like the action groupoid is similar to but a bit different from the construction I sketched above. To compare the two (using my example with integers modulo 4), I'll let $X = \mathbb{Z}$ (the underlying set of the additive group of integers) and let $G = 4 \mathbb{Z}$, the additive group of integer multiples of 4. Then let $4 \mathbb{Z}$ act on $\mathbb{Z}$ by $(g,x) \mapsto g+x$. Then in $X//G$ the objects are integers, and we have a morphism $g: x \to x'$ whenever $x'-x$ is a multiple of four corresponding to $g$.

This is similar to the construction I sketched above. In my construction, the 1-morphisms are the integers and we have a 2-morphism $g \in 4\Z$ from $x$ to $x'$ whenever $x'-x=g$. So, this construction has similar data to that of the action groupoid, but I've "shifted everything up by level" by introducing a boring bottom level (a single object). I'm remembering here the discussion on $k$-tuply monoidal $n$-categories in the lecture, which roughly have $k$ "boring" layers of objects, 1-morphisms, ..., and (k-1)-morphisms before we get to the interesting morphisms. As a resulting of shifting everything up, the objects of $X//G$ I think corresponds to the 1-morphisms of my construction. So it's sort of like $X//G$ is inside my construction, but in my construction I also have a way to compose the objects of $X//G$. This leads me to wonder if people study action groupoids that are also monoidal categories.

John Baez (Oct 13 2023 at 16:29):

I'm sorry, I didn't pay enough attention to your whole story. So we're starting with the action of the subgroup $4\mathbb{Z}$ on the group $\mathbb{Z}$ where $a \in 4\mathbb{Z}$ acts on $x \in \mathbb{Z}$ to give $a + x$.

John Baez (Oct 13 2023 at 16:30):

(Of course $4\mathbb{Z}$ is isomorphic to $\mathbb{Z}$ but it's less confusing to give it a different name here.)

John Baez (Oct 13 2023 at 16:31):

If we treat $\mathbb{Z}$ as a mere set acted on by $4\mathbb{Z}$ then the weak quotient $\mathbb{Z}//4 \mathbb{Z}$ is a groupoid as I explained.

John Baez (Oct 13 2023 at 16:34):

But you want to take into account the fact that $\mathbb{Z}$ is a group, and maybe use this operation to make the groupoid $\mathbb{Z}//4 \mathbb{Z}$ into a monoidal category.

John Baez (Oct 13 2023 at 16:37):

That sounds doable. The objects are elements of $\mathbb{Z}$, and we tensor them using addition. A morphism $a: x \to a + x$ can be tensored with another morphism $a' : x' \to a' + x'$ to give a morphism

$a + a' : x + x' \to a + a' + x + x' = (a + x) + (a' + x')$.

John Baez (Oct 13 2023 at 16:38):

Note I used the fact that $\mathbb{Z}$ is abelian there.

John Baez (Oct 13 2023 at 16:39):

Note that the monoidal category we're getting is a "groupal groupoid": every morphism has an inverse, but also every object has an inverse (for the tensor product, which is addition here).

John Baez (Oct 13 2023 at 16:40):

Almost nobody says "groupal groupoids": Hoàng Xuân Sính studied them in her thesis and called them "Gr-categories", and later I called them "2-groups".

John Baez (Oct 13 2023 at 16:41):

So what you seem to be saying is that if we have an abelian group $A$ and a subgroup $B$, which acts as translations in the obvious way, the weak quotient $A//B$ is a 2-group.

John Baez (Oct 13 2023 at 16:42):

In fact it will be a very "abelian" sort of 2-group, which Sính called a "strict Picard category".

John Baez (Oct 13 2023 at 16:43):

A strict Picard category is an abelian group object in the category of groupoids, or equivalently a groupoid object in the category of abelian groups.

John Baez (Oct 13 2023 at 16:44):

Are you happy yet? :face_with_spiral_eyes:

John Baez (Oct 13 2023 at 16:44):

I'm not happy yet because I haven't seen the one-sentence argument for why $A//B$ will always be a strict Picard category.

John Baez (Oct 13 2023 at 16:45):

The thing that bugs me is that $B$ is not acting on $A$ as group homomorphisms.

John Baez (Oct 13 2023 at 16:47):

Anyway, your idea is a nice one and I think there should be some very slick way to generalize it and show that whenever $B$ is a subgroup of an abelian group $A$, the weak quotient $A//B$ is a monoidal category of a very nice sort, namely a strict Picard category.

David Egolf (Oct 14 2023 at 17:11):

John Baez said:

That sounds doable. The objects are elements of $\mathbb{Z}$, and we tensor them using addition. A morphism $a: x \to a + x$ can be tensored with another morphism $a' : x' \to a' + x'$ to give a morphism

$a + a' : x + x' \to a + a' + x + x' = (a + x) + (a' + x')$.

Ah, I see now that I had forgotten that we have to define two ways to compose the morphisms in a monoidal category. (I suppose these are the vertical and horizontal ways, viewing the morphisms as 2-morphisms in a bicategory with one object). Interestingly, in this particular case, it looks like the tensoring composition for morphisms is forced to be as you describe, once we've set how to tensor our objects together. (I think this is roughly because $\Z$ is (1) commutative and (2) a group).

John Baez said:

A strict Picard category is an abelian group object in the category of groupoids, or equivalently a groupoid object in the category of abelian groups.

I assume the "strict" part of "strict Picard category" refers to the fact that our tensor composition $\otimes$ is actually associative and unital (I hope "unital" is the right word here...), not just up to isomorphism.

However, "abelian group object in the category of groupoids" or "groupoid object in the category of abelian groups" is a bit outside my understanding at the moment! When we talk about a $X$ object in a category $C$, it seems like this usually involves a lot of diagrams in $C$ relating to $X$ that need to commute, and that seems a bit intimidating to check. To talk about an abelian group in the category of groupoids, I guess we would need to specify a groupoid $G$, a identity element map to it from the terminal groupoid, a composition map from $G \times G$ to $G$, a "inverse" map from $G$ to $G$, a diagram (in the category of groupoids) describing the associativity of composition, a diagram describing the commutativity of composition, a diagram describing the unitality of composition, and a diagram describing how inverses work.

Setting all this up and checking that the needed diagrams commute sounds tough! But maybe it's not too bad in practice? (I haven't tried yet!)

John Baez said:

I'm not happy yet because I haven't seen the one-sentence argument for why $A//B$ will always be a strict Picard category.

It sounds like sometimes (in some related settings) there is a clever way to avoid setting up and manually checking all the diagrams I describe above? That sounds useful!

John Baez (Oct 14 2023 at 20:27):

This is unusual (e.g. it's certainly not true in the category of vector spaces with $\otimes$ or sets with $\times$).

John Baez (Oct 14 2023 at 20:30):

In fact in your example we can take the braiding to be the identity for all pairs of objects. (You might expect this since addition of integers is commutative, but it still takes a bit of thought since the braiding needs to obey some equations.) But I believe any strict Gr-category is equivalent to one where the braiding is the identity for all pairs of objects.

John Baez (Oct 14 2023 at 20:35):

David Egolf said:

However, "abelian group object in the category of groupoids" or "groupoid object in the category of abelian groups" is a bit outside my understanding at the moment! When we talk about a $X$ object in a category $C$, it seems like this usually involves a lot of diagrams in $C$ relating to $X$ that need to commute, and that seems a bit intimidating to check.

It sounds like it's not outside your understanding at all. It sounds like you understand it well enough to want to avoid checking those diagrams. :upside_down:

They're really not hard.

John Baez (Oct 14 2023 at 20:37):

To talk about an abelian group in the category of groupoids, I guess we would need to specify a groupoid $G$, a identity element map to it from the terminal groupoid, a composition map from $G \times G$ to $G$, a "inverse" map from $G$ to $G$, a diagram (in the category of groupoids) describing the associativity of composition, a diagram describing the commutativity of composition, a diagram describing the unitality of composition, and a diagram describing how inverses work.

Right, so it's not at all outside your understanding.

Setting all this up and checking that the needed diagrams commute sounds tough! But maybe it's not too bad in practice? (I haven't tried yet!)

They're all really easy, because your groupoid is built from the integers (and the integers times 4). All the maps you listed are built using the corresponding maps involving integers: e.g. your map $G \times G \to G$ will be built using addition in $\mathbb{Z}$, the "inverse" map from $G$ to $G$ will be built using negatives in $\mathbb{Z}$, and the map from the terminal groupoid will be built using $0 \in \mathbb{Z}$. And all the laws you need to check will follow from the corresponding laws for integers!

John Baez (Oct 14 2023 at 20:37):

I'm not happy yet because I haven't seen the one-sentence argument for why $A//B$ will always be a strict Picard category.

It sounds like sometimes (in some related settings) there is a clever way to avoid setting up and manually checking all the diagrams I describe above? That sounds useful!

Yes, you shouldn't really have to do a bunch of manual labor in this game.

John Baez (Oct 14 2023 at 20:39):

In this kind of math, manual labor can often be avoided by thinking more abstractly. That's not always less work, but it's less tedious.

John Baez (Oct 14 2023 at 20:41):

I think I sort of see how to prove $A//B$ is a strict Gr-category more elegantly now, but I think I've punished you (and myself) enough for now.

John Baez (Oct 16 2023 at 13:05):

Here is my third lecture:

• Topology and the periodic table
  

John Baez (Oct 16 2023 at 13:06):

The "periodic table of n-categories" is a chart showing n-categories that are trivial at the bottom k levels. This helps organize our understanding of higher categories and their relation to topology and physics. There are many ways to hop around the periodic table, and I explain 4 of the main ones: looping, delooping, forgetting monoidal structure and stabilization. I also explain how free k-tuply monoidal categories let us enhance familiar structures such as the natural numbers and integers and obtain such things as braid groups and the homotopy groups of spheres. At the end I explain how to use these ideas to understand the isomorphism π₃(S²)≅ℤ.

John Baez (Oct 30 2023 at 09:47):

My next This Week's Finds lecture will be this week on Thursday November 2nd at the usual time, 3 pm UK time, in the usual place, the James Clerk Maxwell Building room 6206. You can also join here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

John Baez (Oct 30 2023 at 09:48):

Here's what I'll be talking about:

The 3-strand braid group

The 3-strand braid group has striking connections to the group SL(2,Z) of invertible 2x2 matrices with integer entries, the Lorentz group from special relativity, modular forms (famous in number theory), and the trefoil knot. They all fit together in a neat package, which I explain here.

John Baez (Oct 30 2023 at 09:50):

So, I'm taking a little break from the theme of categorification, but still with braided monoidal categories lurking beneath the surface.

John Baez (Nov 06 2023 at 10:05):

Here's my talk on the 3-strand braid group:

https://www.youtube.com/watch?v=MnS4hduP5xg

John Baez (Nov 06 2023 at 10:12):

Starting this Thursday, November 9th, my talks will turn to combinatorics:

Combinatorics and categorification

The theory of generating functions is a simple and fun but powerful tool in enumerative combinatorics, which I will explain in the next few lectures. Digging into it, we shall see that it rests on some ideas from "categorification": the more or less systematic replacement of sets by categories. One is "groupoid cardinality": just as finite sets have cardinalities that are natural numbers, finite groupoids have cardinalities that are nonnegative rational numbers! Another is Joyal's theory of species. A species is a type of structure that can be put on finite sets, of the sort we count in enumerative combinatorics. Just as polynomials in one variable form the free ring on one generator, the category of species is the free "2-rig" on one generator, a 2-rig being a categorified analogue of a rig. I will explain these ideas with a minimum of prerequisites.

My talks will be very loosely based on this paper:

• John Baez and James Dolan, From finite sets to Feynman diagrams.

And here's some more reading material - free books:

• François Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species of Structures.

• Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics

• Herbert S. Wilf, Generatingfunctionology.

Dominic Barraclough (Nov 08 2023 at 22:25):

Re Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics
the pdf at http://algo.inria.fr/flajolet/Publications/book.pdf seems to be corrupted
I found a good version at https://ac.cs.princeton.edu/home/AC.pdf

Chris Grossack (they/them) (Nov 09 2023 at 01:21):

Also, for people who like lectures to supplement their reading, there are detailed videos with examples going chapter by chapter through Flajolet and Sedgewick's book. See the links here

John Baez (Nov 11 2023 at 10:42):

Thanks! I forgot to announce the readings in my talk on Thursday....

John Baez (Nov 11 2023 at 10:43):

Dominic Barraclough said:

Re Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics
the pdf at http://algo.inria.fr/flajolet/Publications/book.pdf seems to be corrupted.

Hmm, I've used this pdf a lot and haven't noticed a problem. What problem are you seeing?

I found a good version at https://ac.cs.princeton.edu/home/AC.pdf

Thanks!

John Baez (Nov 16 2023 at 13:15):

.

In about two hours - 3 pm UK time November 16th 2023 - I'll talk about combinatorial species and their generating functions. And I'll use these ideas to count binary trees with 𝑛 nodes! We get these famous numbers:

1, 1, 2, 5, ...

This is fun because we go all the way from category theory to a very concrete problem!

My talk will be in Edinburgh, in the James Clerk Maxwell Building room 6206. You can also join here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

Jules Hedges (Nov 16 2023 at 14:07):

Oh, I totally forgot it was thursday today

John Baez (Nov 16 2023 at 14:52):

:cry:

John Baez (Nov 16 2023 at 14:53):

Btw, sometime we should talk about categorical cybernetics and agent-based models. It would be cool if there were enough overlap that I could get you pulled into my Fields Institute climate change program.

Jules Hedges (Nov 16 2023 at 15:26):

Ok! I'd be extremely happy if that works out, I was trying and failing to get a link from compositional game theory to climate econ for a long time... let's talk in december when I have 5 minutes to think

John Baez (Nov 16 2023 at 16:25):

Okay!

George Zipperlen (Nov 16 2023 at 20:06):

In This Week's Finds seminar, 16-Nov-2023, Dr. Baez mentioned some references for combinatorial species. Is this the web page? Seminar - Fall 2019 -- Combinatorics

Books mentioned are:

François Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species of Structures.

Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics.

Herbert S. Wilf, Generatingfunctionology.

John Baez (Nov 16 2023 at 22:08):

That seminar is one of two courses I've given on the subject. But in my talk today I was pointing people to the videos and reading materials for the seminar I'm giving now:

• This Week's Finds seminar.

At the end you'll see links to those same free books you just mentioned, and a paper of mine.

John Baez (Nov 16 2023 at 22:09):

But you can also get them here:

• John Baez and James Dolan, From finite sets to Feynman diagrams.

• François Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species of Structures.

• Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics

• Herbert S. Wilf, Generatingfunctionology.

Dominic Barraclough (Nov 17 2023 at 18:20):

Hmm, I've used this pdf a lot and haven't noticed a problem. What problem are you seeing?

The pdf would render but most of the characters were gibberish - with hindsight I probably should have retried the download as the pdf from the link now seems fine. In my defense, these days I so rarely see corruption during a download, I implicitly expect a download to go without a hitch.

John Baez (Nov 18 2023 at 09:50):

Okay, thanks. Then I'll have to decide which link seems more likely to survive a long time.

John Baez (Nov 18 2023 at 09:52):

I think I'll pick my original link since it's on the author's website.

John Baez (Nov 18 2023 at 09:54):

Here are the next two lectures:

John Baez (Nov 18 2023 at 09:55):

Combinatorics, groupoid cardinality and species

https://youtu.be/yLtgs7Fz8aw

The theory of generating functions is a simple and fun but powerful tool in enumerative combinatorics, which I will explain in the next few lectures. Digging into it, we shall see that it rests on some ideas from 'categorification': the more or less systematic replacement of sets by categories. One is 'groupoid cardinality': just as finite sets have cardinalities that are natural numbers, finite groupoids have cardinalities that are nonnegative rational numbers! Another is Joyal's theory of 'species'. A species is a type of structure that can be put on finite sets, of the sort we count in enumerative combinatorics.

In the rest of my lectures I'll continue talking about combinatorics and categorification, loosely following this paper:

John Baez and James Dolan, From finite sets to Feynman diagrams, http://arxiv.org/abs/math.QA/0004133

And here's some more reading material — free books:

François Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species of Structures, http://bergeron.math.uqam.ca/wp-content/uploads/2013/11/book.pdf

Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics, http://algo.inria.fr/flajolet/Publications/book.pdf

Herbert S. Wilf, Generatingfunctionology, https://www.math.upenn.edu/~wilf/gfology2.pdf

John Baez (Nov 18 2023 at 09:59):

Species and their generating functions

https://youtu.be/EAM3K4rzLTQ

The theory of species and their generating functions is a powerful tool in enumerative combinatorics, and here we use it to count trees. A 'species' is a type of structure that can be put on finite sets: technically, it is a functor from the groupoid of finite sets and bijections to the category of sets. Any species has a 'generating function', which is a formal power series whose coefficients count how many ways we can put that structure on an $n$-element set for each $n$. We show that some operations on species correspond to operations on their generating functions, and use this to count the binary planar rooted trees with $n$ nodes.

George Zipperlen (Nov 22 2023 at 09:11):

John Baez said:

That seminar is one of two courses I've given on the subject. But in my talk today I was pointing people to the videos and reading materials for the seminar I'm giving now:

• This Week's Finds seminar.

At the end you'll see links to those same free books you just mentioned, and a paper of mine.

Thank you, next time I'll check the TWF seminar page first.

Also, thank you for making this series publically available, in live stream / video, as well qw your wonderful expository essays over the years. I followed sci.math on Usenet ages ago, and am now getting back into math after a career in software engineering.

John Baez (Nov 22 2023 at 12:13):

Great, thanks - it's exactly people like you who I'm trying to reach!

John Baez (Nov 22 2023 at 12:14):

I sometimes miss those usenet days.

John Baez (Nov 23 2023 at 13:34):

A 'derangement' is a permutation of a set where no element gets mapped to itself. There are 24 permutations of the set {A,B,C,D}, but only 9 derangements - shown in blue below:

Today I'll explain how to count derangements! And I'll prove something interesting about them using category theory.

My talk will be at 3 pm UK time, November 23rd. I'm giving it in room 6206 in the James Clerk Maxwell Building of the University of Edinburgh. You can also join us on Zoom:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

This is my second to last talk on categories and combinatorics - and also the second to last talk in this series.

Todd Trimble (Nov 23 2023 at 17:02):

A fun fact I didn't know about until a few months ago: the number of derangements $D(n)$ on an $n$-element set satisfies the following recurrence: $D(1) = 0$, $D(2) = 1$, and

$D(n+1) = n(D(n) + D(n-1))$

John Baez (Nov 23 2023 at 20:09):

Yes! There's a nice proof of that using species and the thing I proved today:

$\mathrm{Perm} \cong \mathrm{Exp} \cdot \mathrm{D}$

where

• $\mathrm{Perm}$ is the species "being a finite set equipped with a permutation"

• $\mathrm{Exp}$ is the species "being a finite set"

• $\mathrm{D}$ is the species "being a finite set equipped with a derangement".

Anyone curious can read it here:

• Let's get deranged!

Todd Trimble (Nov 23 2023 at 21:26):

Yes, in fact the example of derangements and the derivation of the species isomorphism is given in the original paper on species by Joyal!

I didn't happen to know the recurrence until recently, but it's not hard to see directly, by taking a derangement on $\{1, \ldots, n+1\}$, and considering the two cases according to whether or not $n+1$ belongs to a 2-cycle.

John Baez (Nov 23 2023 at 23:29):

Oh, actually the recurrence in my course notes (linked above) is different: it's

$D(n+1) = (n+1) D(n) + (-1)^{n+1}$

It's kind of amazing to me that both these recurrences are true!

John Baez (Nov 23 2023 at 23:30):

I got "my" recurrence by differentiating

$\mathsf{Perm} \cong \mathsf{Exp} \cdot \mathsf{D}$

John Baez (Nov 23 2023 at 23:31):

but none of it was original to me; I learned it in some book.

John Baez (Nov 24 2023 at 13:36):

Here is the video of my penultimate This Week's Finds lecture:

Counting derangements

https://youtu.be/19XDNVWUFOs

A 'derangement' is a permutation where no element is mapped to itself: the cover picture lists the 24 permutations of the 4-element set {A,B,C,D}, and the 9 derangements are shown in blue. Here I show how to count derangements and find the generating function for derangements. A 'species' is a type of structure that can be put on finite sets: technically, it is a functor from the groupoid of finite sets and bijections to the category of sets. Any species has a 'generating function', which is a formal power series whose coefficients count how many ways we can put that structure on an n-element set for each n.

John Baez (Nov 24 2023 at 13:39):

.....

And here's the title and abstract of the last one, which is on Thursday November 30th at 3 pm UK time at the usual zoom link:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

John Baez (Nov 24 2023 at 13:41):

Categorifying the Quantum Harmonic Oscillator

Classically, light in a mirrored box can be described as a collection of harmonic oscillators, one for each vibrational mode of the light. Planck ‘quantized’ the electromagnetic field by assuming that energy of each oscillator could only take on discrete, evenly spaced values. Later Einstein took this seriously, and realized that light comes in discrete energy packets called 'quanta'. Surprisingly, when we categorify the mathematics describing this situation we are led to the theory of species - one of the basic tools of combinatorics. The commutation relations between annihilation and creation operators, and the inner product on the Hilbert space of a quantum harmonic oscillator, then receive a natural interpretation in terms of structures on finite sets.

John Baez (Dec 01 2023 at 13:14):

Here is the video of my final This Week's Finds talk:

Categorifying the Quantum Harmonic Oscillator

Classically, light in a mirrored box can be described as a collection of harmonic oscillators, one for each vibrational mode of the light. Planck ‘quantized’ the electromagnetic field by assuming that energy of each oscillator could only take on discrete, evenly spaced values. Later Einstein took this seriously, and realized that light comes in discrete energy packets called 'quanta'. Surprisingly, when we categorify the mathematics describing this situation we are led to the theory of species - one of the basic tools of combinatorics. The commutation relations between annihilation and creation operators, and the inner product on the Hilbert space of a quantum harmonic oscillator, then receive a natural interpretation in terms of structures on finite sets.

https://www.youtube.com/watch?v=pvVm3L92pdc&list=PLuAO-1XXEh0a4UCA-iOqPilVmiqyXTkdJ