Preprint -- The RAAG Functor as a Categorical Embedding

05 Oct 2023 - Tags: my-preprints

After almost a year of sitting on my hard drive, I finally had time in August to finish revising my new preprint on Right Angled Artin Groups (Raags). And in September I had time to put it on the arxiv for people to see! Within 24 hours I had an email from somebody who had read it, and was interested in reading it closely! It’s super exciting to see that people are actually reading something I wrote, and it’s really validating to feel like the math I did last year was worth it ^_^. In this post, I’d love to give a quick informal description of what’s going on in that paper.

---

First, what’s a right angled artin group? We’ve actually talked about them before in the second ever post on this blog (!) but here’s a quick tl;dr.

For example, in case $\Gamma$ is a complete graph on $n$ vertices, $A\Gamma$ is a free abelian group of rank $n$. If $\Gamma$ has no edges, then $A \Gamma$ is a free group of rank $n$. As a quick exercise, can you convince yourself that for $\Gamma$ as below, we have $A \Gamma \cong F_2 \times F_2$ is a direct product of free groups?

Now, it’s not hard to see that the right angle artin group construction $A$ is actually a functor $A : \mathsf{Gph} \to \mathsf{Grp}$ from the category of graphs to the category of groups. After all, if $\varphi : \Gamma \to \Delta$ is a graph homomorphism¹ then we get a group homomorphism $A \varphi : A \Gamma \to A \Delta$ given by sending each generator $v \in A \Gamma$ to the generator $\varphi v \in A \Delta$.

Moreover, $A : \mathsf{Gph} \to \mathsf{Grp}$ admits a right adjoint! This makes precise the idea that the raag on $\Gamma$ is a kind of “free group” associated to $\Gamma$. Given a group $G$, we define its Commutation Graph $CG$ to be the graph² whose vertices are elements of $G$ where ${g_1, g_2} \in CG$ is an edge if and only if $g_1$ and $g_2$ commuted in $G$.

This allows us to bring much more powerful category theory to the table. In particular, it lets us use the machinery of comonadic descent to characterize the essential image of $A$. That is, to understand which groups are raags, and to understand which homomorphisms between raags are $A \varphi$ for some graph homomorphism! In fact, we show that $A$ is a categorical embedding, so that $\mathsf{Gph}$ is a (non-full) subcategory of $\mathsf{Grp}$!

Moreover, the characterization is something we can really calculate with!

---

One thing that I wish more mathematicians would talk about is the kind of meandering nature of research. When you see a paper written in defnition-theorem-proof style, it’s hard to imagine what the discovery process must have looked like. It’s almost always much messier than the final paper would lead you to believe, so I want to take a minute to talk about the history of this paper.

Why was I thinking about all this?

There’s an important open question in the theory of raags about the “nonstandard” embeddings between two raags.

Obviously if $\Gamma$ is a (full) subgraph of $\Delta$, then $A\Gamma$ is a subgroup of $A \Delta$ in a canonical way. But it turns out we can have pairs so that $A \Gamma$ embeds into $A \Delta$, but $\Gamma$ is not a subgraph of $\Delta$! Motivated by questions in geometric group theory, it’s natural to want to understand when these “nonstandard” embeddings are possible.

Now the adjunction comes into play. If we have an embedding $A \Gamma \to A \Delta$, then our adjunction gives us a map of graphs $\Gamma \to CA \Delta$. Since $CA : \mathsf{Gph} \to \mathsf{Gph}$ is a monad on the category of graphs, we (not very creatively) call this construction the Monad Graph of $\Delta$.

It would be nice to have a combinatorial characterization of when a map $\Gamma \to CA \Delta$ transposes to an embedding $A \Gamma \to A \Delta$. But such a characterization is as yet unknown³.

I actually had most of these thoughts back in 2020 when I was first thinking about raags, but they didn’t really go anywhere. Especially since I was focused on life things like passing my quals and finding an advisor. But that all changed in 2022 when I started really trying to understand stacks…

One way to understand stacks⁴ is as “categories satisfying descent”. In the same way that a sheaf on $X$ assigns a set to each open in a way that elements of the set glue along open covers, a stack on $X$ assigns a category to each open in a way that objects and arrows of the category glue along open covers!

This story is closely tied up with the story of descent theory, so I took a detour through understanding that⁵. Along the way I found out about comonadic descent which (among other things) lets you characterize the image of a left adjoint. I remembered this raags adjunction that I worked on, and knew that its image was pretty easy to understand. Maybe the adjunction was comonadic and I could use this to attack the nonstandard embedding problem!

Once I knew the right question to ask, the proof itself was surprisingly simple, and I had a rough draft within a day or two⁶.

Next came the process of writing everything up in a way that doesn’t require a ton of category theoretic background. Working out the exposition for the paper was super enlightening for me⁷. Hopefully it also makes it more accessible for people new to the subject!

---

Ok, with the history out of the way, let’s talk about

What’s in the paper

This is pretty quick to outline, since I’m assuming a category theory background of my blog readers that I wasn’t assuming of my paper readers. That said, if it ever gets too heavy, feel free to read the first few sections of the paper, since I go into much more detail there.

The main result is implied by the categorical statement that the adjunction $A \dashv C$ is Comonadic. This says that $\mathsf{Gph}$ is equivalent to the category of $AC$-coalgebras on $\mathsf{Grp}$, and moreover that the equivalence intertwice the adjunctions $A \dashv C$ (on the $\mathsf{Gph}$ side) and the co-free/forgetful adjunction (on the $\mathsf{Grp}_{AC}$ side).

That is, we have a situation as below:

Really the equivalence $A$ sends a graph $\Gamma$ to the group $A\Gamma$ with the coalgebra structure $A \eta_\Gamma : A \Gamma \to ACA \Gamma$, so that after forgetting this structure we get $A : \mathsf{Gph} \to \mathsf{Grp}$.

How do we show that this really is an equivalence? The answer is Beck’s (Co)monadicity Theorem! It says that for any adjunction $A \dashv C$ the base of that triangle is an equivalence if and only if $A$ reflects isomorphisms and preserves equalizers of “$A$-split parallel pairs”.

In the paper we use a different version of the comonadicity theorem which is easier to check, but it boils down to the same proof.

It’s “well known” in raag circles that $A$ is conservative (this can already be found in a 1987 paper of Droms), so we need to check that $A$ preserves certain equalizers. We can do this with a slightly technical argument of combinatorics on words. The key fact is that we have a good understanding of normal forms for the elements in $A\Gamma$⁸.

The last section of this paper was a small application of this machinery. We’re able to reprove a result that we can effectively recover $\Gamma$ from the isomorphism type of $G \cong A \Gamma$, as long as we’re promised that $G$ really is a raag⁹. We also show that if we have any concrete examples of groups $G$ with $AC$-coalgebra structures, we can really do all the computations that we would want to do!

---

I think I like this format of blog posts putting more emphasis on the things that were on my mind when I was working on a paper, rather than the contents of the paper itself. Maybe if I write a more detailed paper I can say some informal words about what’s in it, but I think the historical perspective might help younger mathematicians see how messy research can be, and how incidental things can all blend together into a result. I just happened to be thinking about raags a few years before I happened to learn about stacks and descent, which opened the door to a result on descent for raags. Hopefully you all also like the historical perspective ^_^.

I’m also going to try to keep these paper announcement posts a bit less polished. It’s easy to get paralysis and revise posts forever and never finish them (ask me how I know…) I want to make sure that these actually get out, so I’ll try to keep them light on the revision. That shouldn’t be hard if the main point is the history!

---

Anyways, thanks for hanging out, all. I’m super excited to have a result, and I’ll be submitting to journals in the near future. Stay warm, and we’ll talk soon ^_^.

---