Robbie Lyman

Topologies on Free Products

Wed, 12 Aug 2026 00:30:00 +0000

In a previous post, I introduced a set of axioms for identity neighborhoods in topological groups. In general, many of the familiar categorical constructions for topological groups appear to me to be extremely poorly understood so far. Some of this appears to me to be intrinsic to the subject, but plenty of it does not.

For example, in this post I want to describe a collection of group topologies on the free product $G * H$ of topological groups $G$ and $H$ , and determine which one makes the free product into the categorical coproduct in the category of topological groups.

Free Products, A Refresher

If $S$ is a set (called an “alphabet”), the “free monoid” $S^*$ over $S$ , also sometimes called the collection of “words” or “strings” in $S$ is the collection of finite sequences $s_1s_2\ldots s_n$ , where each $s_i \in S$ . Concatenation of strings is an associative, typically noncommutative binary operation on $S^*$ with a two-sided identity, namely the empty string. (The operation, its associativity and identity are what make $S^*$ deserve the appellation “monoid”.)

In general, $S^*$ is not a group, because there is no inverse to concatenation. However, when $G$ and $H$ are groups, there is an equivalence relation on $(G \cup H)^*$ whose quotient is a group. If $\omega_1$ and $\omega_2$ are strings with equivalence classes $[\omega_1]$ and $[\omega_2]$ respectively, the group operation is

$[\omega_1][\omega_2] = [\omega_1\omega_2].$

In other words, concatenation of strings descends to become the group operation.

Here is the equivalence relation: if $g_1 g_2 = g_3$ in $G$ or $h_1 h_2 = h_3$ in $H$ , we set

$\omega_1 g_1 g_2 \omega_2 \sim \omega_1 g_3 \omega_2 \quad\text{and}\quad \omega_1 h_1 h_2 \omega_2 \sim \omega_1 h_3 \omega_2,$

where $\omega_1$ and $\omega_2$ are arbitrary strings. Also if $1_G$ and $1_H$ are the identity elements of $G$ and $H$ respectively, we also set

$\omega_1 1_G \omega_2 \sim \omega_1\omega_2 \quad\text{and}\quad \omega_1 1_H \omega_2 \sim \omega_1\omega_2.$

Two strings are equivalent if one can be transformed into the other by a finite collection of moves.

The quotient of $(G \cup H)^*$ by the equivalence relation $\sim$ generated by these four operations is a group $G * H$ , which we call the free product of $G$ and $H$ , where the inverse of a word $s_1\ldots s_n$ is represented by $s_n^{-1}\ldots s_1^{-1}$ , where the inversion of each letter happens in the group $G$ or $H$ as the case may be.

This equivalence relation is extremely nice from a theory of computation standpoint, by the way. Every word can be represented uniquely by a word which cannot be shortened by any of these four operations, and such words can be recognized by a finite state automaton with four states: START, G, H and REJECT. Only the REJECT state is not an accept state, and the start state is START. From START, every nonidentity element of $G$ takes you to G, and every nonidentity element of $H$ takes you to H. At G nonidentity elements of $H$ take you to H and conversely at H, nonidentity elements of $G$ take you to G. All other possibilities take you to REJECT.

Provided you know how to compute in $G$ and $H$ , you can adapt the above automaton to help you do computations in $G * H$ . A “greedy algorithm”, for example, which works by seeing where in the word we move to REJECT and performing corresponding one of our four basic operations which reduces word length and trying again will run in more or less linear time.

Free products of nontrivial groups $G$ and $H$ are always infinite, nonabelian (i.e. the group operation is noncommutative) and contain canonical copies of both $G$ and $H$ as (non-normal) subgroups.

The free product satisfies a “universal property”: If $K$ is a group and $G \to K$ and $H \to K$ are homomorphisms, there exists a unique homomorphism $G * H \to K$ such that the restriction of this homomorphism to the canonical copy of $G$ or $H$ respectively gives back the homomorphisms we started with.

The Bass-Serre Tree

The abstract group $G * H$ acts on a tree called the Bass-Serre tree with one orbit of edges and two orbits of vertices. The edges of the tree correspond to elements of the group, and vertices of the tree correspond to cosets of the canonical copies of $G$ and $H$ . The edge $g$ connects the vertices $g G$ and $g H$ . The action of $G * H$ is multiplication in the labels. The stabilizer of the vertex $g H$ , for example, is the conjugate $g H g^{-1}$ of the subgroup $H$ in $G * H$ .

It is perhaps not immediate to see that the graph so described is a tree; this is part of a very beautiful subject called Bass-Serre theory and named for Jean Pierre Serre, who wrote a book Trees on the subject, and Hyman Bass who was his grad student at the time, contributed to the theory from the beginning and continued to develop the subject for some time after.

At the time of this writing, Serre is 99 years old and posted a math paper to the arXiv within the past two years.

Some Topologies

Update, 16 August 2026: Just a note to say that some of this section is not correct. In particular, none of the proposed families of identity neighborhoods below appear to be closed under products. The problem being that while it is true that $(gUg^{-1})^2 = gU^2g^{-1}$ , the actual products of elements of the families below must contain elements which are not conjugate into any vertex stabilizer. I noticed this while thinking about the neighboring case of free topological groups. If I find a fix, it will appear in a later post.

In the previous post alluded to earlier, we found a set of axioms characterizing group topologies in terms of their identity neighborhoods. In short, identity neighborhoods are closed under finite intersections, inversion, multiplication, and conjugation.

Supposing $G$ and $H$ are topological groups, we are interested in group topologies on $G * H$ such that the canonical inclusions of $G$ and $H$ are embeddings. To this end, we must allow any identity neighborhood in $G$ or $H$ to be at least contained in an identity neighborhood of $G * H$ . The challenges, however, are finite intersections between identity neighborhoods in $G$ with those in $H$ , as well as ensuring that conjugates of identity neighborhoods are identity neighborhoods.

Thinking about it this morning, the problem feels very similar to the question of topologizing Cartesian products of spaces. I have three families of sets, $\mathcal{B}$ , $\mathcal{C}$ and $\mathcal{P}$ , each of which satisfies the axioms and induces a different group topology on $G * H$ .

First, a little notation: let $T$ be the Bass-Serre tree from above and $v$ a vertex, and let $\mathop{\mathrm{Stab}}(v)$ be its stabilizer. As we saw above, $v$ corresponds to a coset of the form $g G$ or $g H$ , and we may choose a unique preferred representative of this coset by requiring that the reduced word representing $g$ either ends with an letter in $H$ in the case of $gG$ or with a letter in $G$ in the case of $gH$ . Call this element $g_v$ . In all cases below, let $U_v$ denote an identity neighborhood in $\mathop{\mathrm{Stab}}(v)$ .

$\mathcal{P} = \left\{ \bigcup_{v \in T} U_v : U_v = \mathop{\mathrm{Stab}}(v) \text{ for all but finitely many } v \right\}$

I called this $\mathcal{P}$ because its definition is reminiscent to me of the product topology on a Cartesian product.

$\mathcal{C} = \left\{ \bigcup_{v \in T} U_v : U_v = g_v U_G g_v^{-1} \text { or } g_v U_H g_v^{-1} \text{ for fixed } U_H \subset H \text{ and } U_G \subset G \right\}$

The identity neighborhoods feel a bit like “cylinders”, since you fix once and for all $U_H$ and $U_G$ and just conjugate them around, hence why I chose the name $\mathcal{C}$ .

$\mathcal{B} = \left\{ \bigcup_{v \in T} U_v \right\}$

Beyond being open, there are no constraints here; one picks a separate identity neighborhood for each $v$ . This topology felt most similar to the box topology on a Cartesian product, hence the name $\mathcal{B}$ .

A little elementary set theory should convince you that each element of $\mathcal{P}$ contains an element of $\mathcal{C}$ , and that each element of $\mathcal{C}$ is an element of $\mathcal{B}$ , so of the three resulting topologies, $\mathcal{B}$ is the finest, meaning it has the most open sets, while $\mathcal{P}$ is the coarsest, having the fewest open sets.

Since the intersection of $\mathop{\mathrm{Stab}}(v)$ with $\mathop{\mathrm{Stab}}(w)$ is trivial (i.e. just the identity) when $v \ne w$ , you can work term by term to see that if $U$ and $V$ belong to one of the collections above, then so do $U^{-1}$ , $g U g^{-1}$ for any $g$ and $U \cap V$ . Likewise, for each $U$ there is a $W$ such that $W^2 \subset U$ . Therefore each of the collections $\mathcal{P}$ , $\mathcal{C}$ and $\mathcal{B}$ produces a group topology on $G * H$ .

For each of these topologies, the canonical inclusions of $G$ and $H$ are embeddings. Also for each of these topologies, one can show that if $\Phi\colon G \to K$ and $\Psi\colon H \to K$ are continuous homomorphisms of topological groups, the induced homomorphism $G * H \to K$ is continuous.

The idea I have for this is to begin by supposing that $g \in G$ and $h \in H$ , and considering an identity neighborhood $\Phi(g)\Psi(h)W$ in $K$ . By continuity of multiplication in $K$ , there are identity neighborhoods $V_g$ and $V_h$ in $K$ such that $\Phi(g)V_g \Psi(h)V_h \subset \Phi(g)\Psi(h)W$ . By continuity of $\Phi$ and $\Psi$ , there are identity neighborhoods $U_g \subset G$ and $U_h \subset H$ such that $\Phi(g U_g)\Psi(g U_h) \subset \Phi(g) V_g \Psi(h)V_h$ . We can then use $U_g$ and $U_h$ to define elements of the desired family of identity neighborhoods. Some careful induction should complete the proof.

Differentiating the topologies

Exercise. The canonical homomorphism $G * H \to G \times H$ , where the latter has the product topology, is continuous for the $\mathcal{C}$ -topology (and hence the $\mathcal{B}$ -topology) but in general not for the $\mathcal{P}$ -topology.

The point, you should convince yourself in doing the exercise, is that the preimage of the basic identity neighborhood $U \times V$ contains the element of $\mathcal{C}$ determined by $U$ and $V$ , but contains no element of $\mathcal{P}$ .

It’s fairly obvious that the $\mathcal{B}$ -topology is in general genuinely finer than the $\mathcal{C}$ -topology, but I’m not aware of a universal construction that lets you readily see this. That is, I don’t know of a topological group $K$ not isomorphic to $G * H$ equipped with homomorphisms from the factors for which the abstract homomorphism $G * H \to K$ is continuous for the $\mathcal{B}$ -topology but not for the $\mathcal{C}$ -topology. If you can think of one, email me!, I’d love to hear about it.

Proposition. The $\mathcal{B}$ -topology is the finest group topology on $G * H$ for which the inclusion maps $G \to G * H$ and $H \to G * H$ are continuous. It is therefore the coproduct of $G$ and $H$ in the category of topological groups.

Proof. Suppose that $\tau$ is a group topology on $G * H$ as in the statement. We show that an arbitrary identity neighborhood in $\tau$ contains a $\mathcal{B}$ -identity neighborhood. This will show that $\tau$ -open sets are $\mathcal{B}$ -open, completing the proof.

If $U$ is a $\tau$ -identity neighborhood, we know that $U_G = U \cap G$ and $U_H = U \cap H$ are identity neighborhoods, as is $gUg^{-1}$ for any $g$ . More generally, if $g = g_v$ for $v \in T$ , notice that $g_v^{-1} U g_v \cap U \ni 1$ is an identity neighborhood, so we conclude that $U \supset g_v U_v g_v^{-1}$ for some identity neighborhood $U_v \subset \mathop{\mathrm{Stab}}(v)$ . This constructs an element of $\mathcal{B}$ contained in $U$ . $\QED$

Who's afraid of a non sofic group

Sun, 02 Aug 2026 21:00:00 +0000

This is a post mostly about AI, I guess, and to complain about a bad definition which somehow managed to stay relevant in group theory despite having very few known consequences for twenty-five years or so.

Dynamical Systems, Surjunctivity

Suppose $X$ is a space and $\rho \colon G \times X \to X$ is an action of $G$ on $X$ by homeomorphisms. A continuous map $\Phi \colon X \to X$ is $G$ -equivariant if for each $g \in G$ and $x \in X$ , we have $\Phi(\rho(g,x)) = \rho(g,\Phi(x))$ .

For example, $G$ might be the integers under addition, and $X = \mathbb{R}$ , where the action is $\rho(k,x) = x + k$ . In this setting, you can check that the maps $\Phi_1, \Phi_2\colon \mathbb{R} \to \mathbb{R}$ defined as

$\Phi_1(x) = \pi x \qquad \Phi_2(x) = x + \pi$

have the property that $\Phi_1$ is not equivariant, but $\Phi_2$ is.

Suppose $S$ is some finite set (typically called an “alphabet”). Let $S^G$ denote all the functions $\omega \colon G \to S$ . This is also the product of copies of $S$ indexed by $G$ . If we give $S$ the discrete topology and $S^G$ the product topology, then by Tychonoff’s theorem, since $S$ is compact (it’s finite and discrete), so too is the (typically infinite) space $S^G$ . Any group $G$ always admits an action on this space by the rule $\rho(g,\omega)(h) = \omega(g h)$ .

Walter Gottschalk called a group $G$ surjunctive if every equivariant continuous map $S^G \to S^G$ which is injective is automatically surjective.

Surjunctivity is sort of like finiteness: just by counting, you can see that any function from a finite set to itself which is injective is also surjective. On the other hand, it is much more mysterious: To my knowledge, there is still no known example of a non-surjunctive group.

There are actually lots of properties in mathematics which similarly evade our grasp. To name a couple off the top of my head: We do not know whether every finitely presented group with one end is semistable at infinity. We do not know whether every word hyperbolic group is residually finite. We do not know whether every finitely generated purely pseudo-Anosov subgroup of the mapping class group is also convex-cocompact. We do not know whether every Artin group is torsion free. We do not know whether every mapping class group (of a surface group of finite type) is linear.

Sofic groups

A finitely generated group is sofic (forgive me for handwaving) if it has a Cayley graph with respect to a finite generating set which can be approximated to any desired specificity by finite graphs.

Explicitly, such a Cayley graph $\Gamma$ has edges labeled by the generating set. We seek for each $r > 0$ and $\epsilon > 0$ a finite graph $(V,E)$ with edges similarly labeled such that there is a subset $V_0 \subset V$ of size at least $(1 - \epsilon)|V|$ such that the $r$ -neighborhood of each vertex $v_0 \in V_0$ is isomorphic (respecting labels) to the $r$ -neighborhood of any vertex in $\Gamma$ .

This is, I must say, a bad definition. The reason it is the way that it is is best explained by introducing ultraproducts, which are gorgeous things that I can’t waste your time with today.

The reason for the definition (perhaps), is twofold: on the one hand, one can prove that sofic groups are surjunctive. It was Gromov who did this first, in 1999. The second reason, which is quite pretty, but sort of difficult to explain quickly, is that two important but obviously limited families of groups are easily seen to be surjunctive: the amenable groups and the residually finite groups.

A group $G$ is amenable if every continuous action on a compact space $X$ admits an invariant probability measure. The “volume” (Lebesgue) measure on a unit ball is an example of a probability measure. A math-pilled reader may be aware of the Banach-Tarski paradox which doubles the volume of a 3-dimensional ball. Amenability was introduced by John von Neumann to name the phenomenon allowing (or disallowing) similar paradoxes to occur in various settings.

A group $G$ is residually finite if whenever $g \in G$ is nontrivial, there is a homomorphism $\varphi\colon G \to H$ with $H$ finite so that $\varphi(g)$ is still nontrivial.

These two properties are ubiquitous and useful in group theory, but it is not difficult to come up with non-examples. If it were the case that every group is sofic, it would follow that every group is surjunctive. Weiss, naming the term “sofic”, writes

It is not likely that all groups are sofic — but I don’t know of any definite example of a non sofic group. A concrete case that I haven’t been able to resolve is the universal Burnside group on a finite set of generators.

How to feel about non-sofic groups

Recently, OpenAI announced an example of a non-sofic group. OpenAI calls this a “central open question in group theory”. Respectfully, and I’m so sorry Alex Lubotsky, I do have to disagree. Many, if not most papers about sofic groups cannot be bothered to really define them. Similar problems exist for open questions in K-theory with group theoretic interest, so this is not a problem unique to sofic groups, but it’s hard to defend a claim that such a question is central if its definition remains largely inaccessible.

I was heartened to read the Leiden declaration on AI and Mathematics recently. The declaration is clear-headed, and does a good job of steering clear of both hype and Luddism. OpenAI doesn’t really know what to do about this declaration, since the paper is very clearly not written by somebody who can understand the model’s output. This is a shame. They write

We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.

I do have to ask, though: what ideas?

In terms of the existence of a non-sofic group, like most research-level results I’ve seen so far from LLMs, this one resolves the easier and less sexy (forgive me) direction of a question. A proof that all finitely generated groups are sofic would necessarily be rather difficult, since the stated hypotheses give you nearly nothing to work with. OpenAI’s disproof just needs one non-sofic group.

I skimmed the non-sofic groups chapter of the OpenAI paper fairly quickly. The claimed group is apparently a group of units of a certain algebra. The algebra is simple to write down, but the model neglects to give a presentation of the claimed non-sofic group in a way that caught my eye. The method of (dis)-proof seemed more or less straightforward, but I wasn’t able to follow it mostly because I had other things to do with my day.

I think the deep idea here is this: we love our conjectures too much to give them the Riemann hypothesis treatment. The Riemann hypothesis, asking about the pattern of zeros of the zeta function, has been tested extensively for more or less every reasonably computable value (at least, such is my non-expert understanding). Reading the AI disproofs we’ve seen so far, it appears to me that many well-known conjectures have not seriously been subject to the same sort of treatment.

I know a little bit about how this goes! Here’s a little story.

Over the lockdown era, one of my projects was about ends of the groups $\mathrm{Out}(A_1 * \cdots * A_n * F_k)$ , where the $A_i$ are all finite and $F_k$ is free of finite rank $k$ . I was following methods of Karen Vogtmann, with input from a paper of Collins and Zieschang with the delightful title Rescuing the Whitehead method for free products.

Vogtmann’s ideas got me pretty far: relying on her work I was able to reduce the problem to three cases. One case showed up in her work as well, but two were new, and it was these two that really had me stumped for a while. Since I wanted my proof to apply to all the relevant groups above, I kept wanting to choose some extra bit of information that was not guaranteed to exist.

Finally, one weekend in the fall after spending yet another summer more or less stumped, I decided to sit down and spend the weekend attempting to prove the opposite of my desired statement. To my great surprise, I found a beautiful little pattern waiting for me that showed that the very smallest case I was worried about did indeed not work out. That example, together with some supporting material, is my paper When is the Outer Space of a free product CAT(0)?. After a bit more hemming and hawing, I did manage to get the rest of the cases of my initial theorem to work out; that project is One-endedness of outer automorphism groups of free products.

More recently, having not learned my lesson, an LLM was able to set me straight on a joint project where we became enamored of a possibility which would be pretty and difficult to prove.

Living without a moat

I’ll close by offering some thoughts about AI in general. In software companies, apparently, there is this concept of a “moat”. The moat is what keeps your lunch from being stolen by another company willing to undercut you on cost. Sometimes your moat is a facet of your software, other times maybe it is a costly form of regulatory approval you’ve undergone.

Many people share their worries with me about AI as it impacts their work as mathematicians, musicians or programmers. A common thread I hear in all of these fears is a sort of identification of their worth as a person (at least, in an economic sense) being tied to skills that they are suddenly feeling like no longer provide them with much of a moat. Here are some things to ponder in that regard.

You never had a moat. I have not met anyone who is inarguably the single best person in their field. Even among exceptionally talented people, many of whom I haven’t met, many are either neck-and-neck with their “competition”, or have a niche where they’re better which doesn’t extend to the whole field. It’s really easy to tie your self-work to the worth of the work that you do: I do it too, all the time. It hurts every time for me too, so I’m trying not to.
You do have the work, and your team. If we accept the above as true, in our extremely connected era, perhaps the real reason that pure disinterested economics has not pushed all of us out of our jobs is that the job was always about more than the skillset. For example, my work as a university professor is not just about research or giving lectures on calculus. How I am doing at my job is also about connections I am able to build with and for my students, and support I’m able to give my colleagues.
The reason for the work is not volume of production. Many jobs which have aspects that are automatable are still alive and kicking. For example, although most clothing on earth is produced in bulk by machines and invisibilized workers typically far from where I live in the US, many, if not most of the people working in fashion design know how to sew, and this skill serves them some purpose even if their work does wind up mass-produced.
First time? If we’ve discovered that some of the purpose of the work that people did without LLMs has benefits not directly correlated to the output of the work (e.g. coding to figure out a problem space, even if you throw the code away before writing a final solution), we may have found some aspects of the work which are more like art or literature or craft. Many people continue to learn and build their skills in fields like this long after the market has “killed” it as a profession, and some do continue to have it be their main source of income. Once AI could produce coherent videos, why did the film industry not implode? I’m sure I don’t have a full answer, but it does appear that our culture values the skills involved in making a movie enough to pay a number of people to do that work full time.
“Theorem credits” was always the wrong metric. Similarly, number of users on a given product, for example, is well-known to not correlate with the quality of the product. We have a whole new word about that, “enshittification”. If you haven’t read Bill Thurston’s On Proof and Progress in Mathematics recently, you might find it comforting to read again.

Now, if you’ll forgive me, I have much more interesting stuff, which was even written by humans, to go read.

Topologies, Groups, Graphs

Thu, 30 Jul 2026 14:00:00 +0000

This is a post about topology and about groups. While theoretically it is possible and in scope for me to introduce topology from the bottom up, I suspect that you won’t get anything out of the story that I want to tell if it really is your first time seeing the definition of a topology, so I won’t. However, if you know topology and groups separately, this is a perfect place to meet topological groups, since I’m going to talk only about pretty elementary things.

Topological Groups

A topological group is a group $G$ which is at the same time a topological space so that multiplication $(g,h) \mapsto g h$ and inversion $g \mapsto g^{-1}$ are continuous as maps $G \times G \to G$ and $G \to G$ respectively.

An identity neighborhood is a an open subset of a topological group containing the identity, $1$ . Identity neighborhoods satisfy the following axioms.

If $U, V$ are identity neighborhoods, so is $U \cap V$ .
If $U$ is an identity neighborhood, so is $g U g^{-1}$ and $U^{-1}$ for every $g \in G$ .
If $U$ is an identity neighborhood, there is another identity neighborhood $V$ such that $V^2 \subset U$ .

To clarify notation, $g U g^{-1} = \{ g u g^{-1} : u \in U \}$ , $U^{-1} = \{ u^{-1} : u \in U \}$ , and $V^2 = \{ u v : u, v \in U \}$ . The second bullet point follows from continuity of the map $x \mapsto (g,x) \mapsto g x \mapsto (g x, g^{-1}) \mapsto g x g^{-1}$ and inversion, respectively, and the third bullet point follows from continuity of multiplication.

A slight weakening of these axioms suffice to characterize group topologies.

Proposition 1. Suppose that $\mathcal{B}$ is a collection of subsets of a group $G$ satisfying the following axioms.
We have $1 \in U$ for all $U \in \mathcal{B}$ .
For each $U, V \in \mathcal{B}$ , there exists $W \in \mathcal{B}$ satisfying $W \subset U \cap V$ .
For each $U \in \mathcal{B}$ , there exists $V \in \mathcal{B}$ satisfying $V^2 \subset U$ .
For each $U \in \mathcal{B}$ , there exists $V \in \mathcal{B}$ satisfying $V \subset U^{-1}$ .
For each $U \in \mathcal{B}$ and $g \in G$ , there exists $V \in \mathcal{B}$ satisfying $V \subset g U g^{-1}$ .
Then there is a group topology on $G$ where a subset $S$ is open if and only if for each $g \in S$ there exists $U \in \mathcal{B}$ such that $g U \subset S$ .

Proof. The claimed topology really is a topology: the defining property of open sets is clearly closed under arbitrary unions. If $g \in g U \subset S$ and $g \in g V \subset T$ , then by assumption there exists $W \subset U \cap V$ so that $g W \subset S \cap T$ , so finite intersections of open sets are open.

Suppose $g h \in S$ where $S$ is open. To show that multiplication is continuous, we show that there are sets $U, V \in \mathcal{B}$ so that $g U h V \subset S$ . Indeed, since $S$ is open, it contains $g h W$ for $W \in \mathcal{B}$ , and we choose $U, V \in \mathcal{B}$ so that $V^2 \subset W$ and $U \subset h V h^{-1}$ . We compute:

$g U h V \subset g h V h^{-1} h V = g h V^2 \subset g h W$

as required.

To show that inversion is continuous, given $U \in \mathcal{B}$ , take $V \subset g U g^{-1}$ and $W \subset V^{-1}$ . Then

$g^{-1}W \subset g^{-1} V^{-1} \subset g^{-1} g U^{-1} g^{-1} = U^{-1} g^{-1} = {(g U)}^{-1}.$

Thus this topology makes $G$ a topological group. $\QED$

Groups acting on graphs

Let $G$ be a group acting on a connected, simplicial graph $\Gamma$ by graph automorphisms without inversions in edges. Simplicial means that between two vertices of $\Gamma$ , there is at most one edge, and that no edge forms a loop. Without inversions in eges means that if an edge of $\Gamma$ is fixed setwise by a group element, it is fixed pointwise. A consequence of this latter condition is that the stabilizers of an edge $e$ incident to the vertices $v$ and $w$ is naturally a subgroup of the stabilizers of both $v$ and $w$ . Another is that the quotient map $\Gamma \to G \backslash \Gamma$ is a graph map.

Proposition 2. Suppose $G$ acts on a connected, simplicial graph $\Gamma$ by graph automorphisms without inversions in edges, and that for each vertex $v \in \Gamma$ , the stabilizer $G_v$ is equipped with a group topology such that the following conditions hold.
For each $g \in G$ , conjugation by $g$ induces a homeomorphism $G_v \to g G_v g^{-1} = G_{g.v}$ .
If $e$ is an edge of $\Gamma$ incident to vertices $v$ and $w$ , the inclusion of the stabilizer $G_e$ into $G_v$ and $G_w$ yields the same topology on $G_e$ , such that both inclusions are open embeddings.
Then the family $\mathcal{B} = \{ U \subset G_v : U \text{ is an identity neighborhood } \}$ satisfies the hypotheses of Proposition 1. The induced topology on $G$ is the finest group topology on $G$ such that each inclusion $G_v \to G$ is an open embedding. In this topology the group $G$ acts continuously on $\Gamma$ .

Proof. We verify the hypotheses of Proposition 1. The first, that each $U \in \mathcal{B}$ contains the identity, is clear, as are the third and fourth, since each $G_v$ is a topological group. The final hypothesis, about conjugation, holds by definition. Thus we need only verify that $U \cap V$ contains an element of $\mathcal{B}$ . Supposing that $U \subset G_u$ and $V \subset G_v$ for vertices $u, v \in \Gamma$ , we will show this by induction on the distance between $u$ and $v$ . More precisely, given any geodesic edge path from $u$ to $v$ , we show that the subset of $U \cap V$ which additionally stabilizes this path pointwise belongs to $\mathcal{B}$ , and is in fact an identity neighborhood in both $G_u$ and $G_v$ .

The base case where the distance is $0$ (i.e. $u = v$ ) holds because $G_v$ is a topological group. When $u$ and $v$ are connected by an edge $e$ , the intersection $U \cap V$ is contained in the edge stabilizer $G_e$ .

Suppose now that the result holds for all pairs $U$ and $V$ at distance at most $n$ , we prove the statement for vertices at distance $n + 1$ . Let $e_0 \ldots e_n$ be a geodesic edge path from $u$ to $v$ , and let $w$ be the initial vertex of $e_n$ .

By induction, the subset of the intersection $U \cap G_w$ which stabilizes the (possibly trivial) path $e_0 \ldots e_{n-1}$ is an identity neighborhood in both $G_u$ and $G_w$ . Call this subset $W$ .

Since $G_{e_n}$ inherits the same topology as an open subgroup of each of $G_v$ and $G_w$ , we see that $V \cap W \subset G_{e_n}$ is an open identity neighborhood in both $G_v$ and $G_w$ . The subset of $U \cap V$ stabilizing $e_0 \ldots e_n$ is clearly contained in $V \cap W$ , and the reverse is also true: if $g \in V \cap W$ , then by assumption $g \in U \cap V$ stabilizes the path $e_0 \ldots e_{n-1}$ and also $v$ , whence since $\Gamma$ is simplicial, it stabilizes the whole path.

Finally, to see that this topology $\tau$ is the finest group topology on $G$ such that each inclusion $G_v \to G$ is an open embedding, observe that any group topology $\tau'$ satisfying this latter hypothesis actually makes each $U \in \mathcal{B}$ into an identity neighborhood, so the identity map $(G,\tau') \to (G,\tau)$ is continuous, as required. $\QED$ .

Proposition 2 has an immediate consequence for certain (2-categorical) colimits in the category of topological groups.

Corollary 3. Suppose $\mathcal{G}$ is a connected graph of topological groups such that each edge-to-vertex group homomorphism is an open embedding. The fundamental group of the graph of groups $\pi_1(\mathcal{G})$ is canonically a topological group. $\QED$

One limitation of Corollary 3 is that it is not strong enough to topologize some of the most familiar tree-like constructions on groups. For example, I’ll close by leaving you to ponder the following.

Exercise 4. Suppose $G_1,\ldots,G_n$ are groups. The only topology on their free product $G_1 * \cdots * G_n$ which makes the action on the natural Bass-Serre tree associated to the free product continuous is the discrete topology. $\QED$

Category Theory Day One

Tue, 21 Jul 2026 17:00:00 +0000

This is a repost of a post I wrote in February 2020 that got lost over the years with my tendency to reinvent myself.

Category Theory Day One

My office-mate asked me to give a “second lecture” in category theory in Eminar, the Tufts graduate student seminar. I gave a first lecture last semester, but I thought it might be nice to have something to point to for people who missed it, in order to stick to my assignment and begin at the Yoneda lemma. So, here is yet another rendition of the first hour of a category theory class.

The treatment is brisk. To the interested reader, I would highly recommend Emily Riehl’s Category Theory in Context, which is the main inspiration here and has the advantage of being available on the author’s website. I also recommend Saunders Mac Lane’s Category Theory for the Working Mathematician and Francis Borceux’s Handbook of Categorical Algebra.

Functors

If $C$ and $D$ are categories, a functor $F \colon C \to D$ is an assignment

$\begin{CD} a @.{\mapsto} @.Fa \\ @VfVV @. @VVFfV \\ b @.{\mapsto} @.Fb. \end{CD}$

I.e. for each object $a$ of $C$ , there is an object $Fa$ in $D$ , and for each arrow $f\colon a \to b$ in $C$ , there is an arrow $Ff\colon Fa \to Fb$ . Additionally, $F$ is required to satisfy

$F1_a = 1_{Fa}$

for each object $a$ in $C$ , and $F$ must respect composition in the sense that

$F(gf) = Fg Ff.$

Functors are sometimes qualified with the adjective covariant. The reason being that every category $C$ has an opposite category $C^{\mathrm{op}}$ , which is obtained by formally reversing the direction of all the arrows. Thus if $C$ has arrows

$\begin{CD} a @>f>> b @>g>> c, \end{CD}$

then $C^{\mathrm{op}}$ has arrows

$\begin{CD} a @<f^{\mathrm{op}}<< b @<g^{\mathrm{op}}<< c. \end{CD}$

Any statement about objects and arrows in has a dual statement about objects and arrows in $C^{\mathrm{op}}$ , which is obtained by reversing the direction of all the arrows. If the proof of the original statement is categorical, dualizing the proof yields a proof of the dual statement for $C^{\mathrm{op}}$ .

The definition of a functor admits dualization in $C$ or in $D$ . The import is essentially the same as long as only one dualization is performed: the result is a contravariant functor $G\colon C^{\mathrm{op}} \to D$ :

$\begin{CD} a @.\mapsto @.Ga \\ @VfVV @. @AAGfA \\ b @.\mapsto @.Gb. \end{CD}$

Functors abound in mathematics. Many are so simple that we hardly think about them: for instance the “forgetful” functor from the category of groups and homomorphisms to sets that sends a group $G$ to its set of elements, and a group homomorphism to its action on underlying sets.

If $C$ is a locally small category and $c$ is an object, there are two particularly important functors $C \to{\operatorname{Set}}$ . Write $C(c,b)$ and $C(b,c)$ for the set of arrows in $C$ with domain and codomain $c$ , respectively. If $f\colon b\to d$ is an arrow in $C$ , there are functions

$f_*\colon C(c,b) \to C(c,d) \text{ and }f^*\colon C(d,c) \to C(b,c)$

defined as $f_*(g) = fg$ and $f^*(h) = hf$ , respectively. The assignments

$\begin{CD} b @.\mapsto @.C(c,b) @.\qquad\qquad @.b @.\mapsto @.C(b,c) \\ @VfVV @.\mapsto @VVf_*V @.\qquad\qquad @VfVV @.\mapsto @AAf*A \\ d @.\mapsto @.(c,d) @.\qquad\qquad @.d @.\mapsto @. C(d,c) \end{CD}$

define covariant and contravariant functors $C(c,-)\colon C \to {\operatorname{Set}}$ and
$C(-,c)\colon C^{\mathrm{op}}\to{\operatorname{Set}}$ , respectively.

Natural Transformations

Just as we have defined functors between categories, it is useful to consider functor categories, whose elements are functors $C\to D$ . To do this, we need a notion of a morphism between functors. It turns out that the right notion is a natural transformation. If $F,G\colon C \to D$ are functors, a natural transformation

$\eta\colon F\Rightarrow G$

is a collection of maps $\eta_c\colon Fc \to Gc$ , one for each object $c$ of $C$ such that for every arrow $f\colon c \to d$ , the following diagram commutes

$\begin{CD} Fc @>\eta_c>> Gc \\ @VFfVV @VVGfV \\ Fd @>\eta_d>> Gd. \end{CD}$

Any directed path in a diagram of objects and arrows within a category like the above yields an arrow from the initial “vertex” of the path to the terminal one by reading off the labelled arrows. A diagram commutes if any two directed paths with the same endpoints yield equal arrows. Thus, to say that the above diagram commutes asserts the equality

$Gf\eta_c = \eta_d Ff.$

The standard example of a natural transformation is the following. Let $k$ be a field, and consider the category of $k$ -vector spaces and linear transformations. There is a contravariant functor that sends a vector space $V$ to its dual $V^*$ , namely the vector space of linear transformations $V\to k$ . A linear map $f\colon V \to W$ is sent to a map $f^*\colon W^*\to V^*$ which is defined as follows. If $\lambda\colon W \to k$ is an element of $W^*$ , $f^*(\lambda)$ is the linear map

$\lambda f\colon V\to k.$

Thus there is a covariant functor sending a vector space $V$ to its “double dual” $V^{**}=(V^*)^*$ . There is a natural transformation $ev$ from the identity functor to the double dual functor. At a given vector space $V$ , the map $ev_V$ sends a vector $v$ to the linear map $V^*\to k$ defined by “evaluation at $v$ ,” namely

$\lambda \mapsto \lambda(v),\quad \lambda \in V^*.$

Naturality of $ev$ asserts that for $f\colon V \to W$ a linear map, the following diagram commutes

$\begin{CD} V @>ev_V>> V^{**} \\ @VfVV @VVf^{**}V \\ W @>ev_W>> W^{**}. \end{CD}$

To show this, consider a vector $v \in V$ . The element $f^{**}(ev_V(v)) \in W^{**}$ is the linear map

$ev_V(v)\circ f^*\colon W^*\to k,$

so its action on $\lambda \in W*$ is

$ev_V(v)(f^*(\lambda)) = ev_V(v)(\lambda f) = \lambda (f(v)) = ev_W(f(v))(\lambda),$

demonstrating naturality.

The map $ev_V$ is an isomorphism when $V$ is finite dimensional. Thus we say that when restricted to the subcategory of finite dimensional vector spaces, $ev$ is a natural isomorphism.

The Yoneda Lemma

Perhaps the fundamental result in category theory is the following:

Theorem. (The Yoneda Lemma)
Let $C$ be a locally small category and $F\colon C \to {\operatorname{Set}}$ a functor. For each object $c$ of $C$ , there is a bijection

$\{\alpha\colon C(c,-)\Rightarrow F\} \longleftrightarrow \{x \in Fc\}$

implemented by sending a natural transformation $\alpha$ to the element $\alpha_c(1_c)$ . This bijection is natural in $F$ and $c$ .

The proof is straightforward, once one figures out what needs proving. The utility of the Yoneda lemma is likely far from obvious when one first meets it, but it turns out to be a major tool in category theory.

Nuh-uh! Infinity plus one!

Sat, 23 May 2026 21:00:00 +0000

This is a blog post about ordinal numbers, ordinal arithmetic, and a brief advertisement for a result I proved with George Domat, and Hannah Hoganson. The paper is now available on the arXiv.

Nuh-uh! Infinity plus one!

I’m sure you’ve probably spent enough time with children or were once a child to remember a variant of the line above. Somehow you and a friend are competing to name bigger and bigger numbers. Maybe your friend thinks they’ve won because they know about infinity and say “Infinity!” and you have to top that, so you say “Nuh-uh! Infinity plus one!” At a vibes level, ordinal numbers in mathematics are a way to actually take that idea seriously.

Two definitions

A totally ordered set is a set $S$ with an order $>$ ; given any two distinct elements $x$ and $y$ of the set, one is always bigger, maybe $x$ is bigger, so we say $x > y$ . The order is transitive as usual.

Some total orders go on forever in one or both directions. For example, the integers $\mathbb{Z}$ . Given an integer like 16, there’s always an element smaller than it, and an element bigger than it. A set is well-ordered if it only goes on forever as you get bigger. That is, a set is well-ordered if any nonempty subset has a smallest element.

So the integers are not well-ordered, but the natural numbers $\mathbb{N} = \{0,1,2,\ldots\}$ are well-ordered.

An ordinal is the order-isomorphism class of a well-ordered set.

The von Neumann definition of the ordinals provides a convenient (although initially very confusing because the quickest way to say it gets circular fast) way to represent ordinals:

We start with $\mathbf{0} = \varnothing$ . The empty set is the zero ordinal. Next $\mathbf{1} = \{\mathbf{0}\} = \{\varnothing\}$ . The set containing the empty set is the first successor ordinal. In general, given an ordinal $\alpha$ , there is an ordinal $\alpha + 1$ which, as a set, is the collection of all ordinals up to and including $\alpha$ in the order that they appear. So for example $\mathbf{2} = \mathbf{1} + 1 = \{\mathbf{0}, \mathbf{1}\}$ .

In this definition, each ordinal is a prefix of all ordinals which are bigger than it. Given an increasing sequence of ordinals $\{\alpha_n\}_{n \in \mathbb{N}}$ , their union is a well-ordered set which we call $\lim_{n\to\infty}\alpha_n$ . The first example is $\mathbb{N} = \omega = \lim_{n\to\infty} \mathbf{n}$ . The notation $\omega$ is standard when referring to the set of natural numbers as an ordinal, so we’ll use it throughout this post.

Every ordinal turns out to either be $\mathbf{0}$ or a successor ordinal $\alpha + 1$ , or a limit ordinal $\lambda = \lim_{n \to \infty} \alpha_n$ , where $\alpha$ and the $\alpha_n$ are previously constructed ordinals. This is the basis for a proof technique called transfinite induction.

Notice, this definition makes it totally sensible to say $\omega + 1 = \omega \cup \{\omega\} = \mathbb{N} \cup \{\omega\}$ ! Here is a picture representing $\omega + 1$ .

Ordinal arithmetic

Given two totally ordered sets $X$ and $Y$ , you can produce a new one $X + Y$ by sticking them “end-to-end” so that every element of $Y$ is bigger than every element of $X$ . When these sets $X$ and $Y$ are well-ordered, so too is $X + Y$ . This is ordinal addition.

Ordinal addition is not commutative: you should convince yourself that $\mathbf{1} + \omega = \omega$ but that $\omega + \mathbf{1}$ is different from $\omega$ by virtue of having a largest element.

There is also ordinal multiplication. Given two totally ordered sets $X$ and $Y$ , their cartesian product $X \times Y$ is totally ordered by saying that $(x,y) > (x',y')$ when either $y > y'$ or $y = y'$ and $x > x'$ . This order is called reverse lexicographic order, because the “second letter” in the “word” is more important, which is the opposite from when you alphabetize a bookshelf.

Again, when $X$ and $Y$ are totally ordered, the product with this reverse lexicographic order is totally ordered, and its order-isomorphism class is the product $X\cdot Y$ of these ordinals. This product is also not commutative: you can convince yourself that $\mathbf{2} \cdot \omega = \omega$ , but it turns out that $\omega \cdot \mathbf{2} = \omega + \omega$ is different from $\omega$ .

One final operation I want to mention is ordinal exponentiation. Here I’ll rely a little more on von Neumann’s terminology. A function $f \colon \alpha \to \beta$ where $\alpha$ and $\beta$ are ordinals is finitely supported if for all but finitely many elements $x \in \alpha$ , we have $f(x) = 0 \in \beta$ .

Two finitely supported functions can be compared: take $x$ to be the largest element for which $f(x) \ne g(x)$ , and say that $f > g$ if $f(x) > g(x)$ . This makes the collection of finitely supported functions into a well-ordered set, which we denote $\beta^\alpha$ . Exponentiation with base $\omega$ has the nice property that $\omega^{\alpha + 1} = \omega^\alpha\cdot\omega$ .

Picturing ordinals

What we’ve talked about so far can help us visualize ordinals. For example

$\omega^2 = \omega\cdot\omega = \lim_{n\to\infty}\omega\cdot n = \lim_{n\to\infty}\underbrace{\omega + \omega + \cdots + \omega}_{n\text{ times}}.$

So to picture $\omega^2$ , the rule for ordinal addition says that we should stitch together infinitely many copies of $\mathbb{N} = \omega$ end-to-end.

Actually, we can be even braver! Now that we can picture $\omega^2$ , we can stitch together those to build $\omega^3$ and so on. The observation that if $\alpha > \beta$ , then $\beta + \alpha = \alpha$ tells us that

$\omega^\omega = \lim_{n\to\infty} \omega^n = \mathbf{1} + \omega + \omega^2 + \omega^3 + \cdots.$

So we can picture this ordinal by lining up a copy of $\omega^n$ for each finite $n$ in increasing order!

Okay I’ll finish this section with a challenge straight from your kid self: using the same strategies as before, we could picture $\omega^{\omega^\omega}$ and then $\omega^{\omega^{\omega^\omega}}$ (so four $\omega$ s, and then 5, and so on) and line up all of those ordinals to picture their limit, which is called $\epsilon_0$ . The ordinal $\epsilon_0$ is a tower of repeated exponentiations with base $\omega$ and has the weird property that $\epsilon_0^\omega = \epsilon_0$ .

It turns out that there may (or may not be!) many many many countable ordinals on beyond $\epsilon_0$ !

Classifying homeomorphisms of ordinals

In a 2025 paper with myself, Branman, Domat and Hoganson, we studied the group of homeomorphisms of countable ordinals equipped with the order topology. The reason for studying homeomorphism, which is weaker than order isomorphism, is that by definition ordinals have no order automorphisms, but they do have many homeomorphisms.

For example, the homeomorphism group of a finite ordinal $\mathbf{n}$ is the symmetric group $S_n$ . The homeomorphism group of $\omega$ or of $\omega + 1$ is the group $S_\omega$ of bijections of a countable set.

When $\alpha$ is a countable, compact ordinal, the group $\mathop{\mathrm{Homeo}}(\alpha)$ is a Polish topological group with the compact–open topology.

Such (nonempty) ordinals are classified up to homeomorphism by two pieces of data: a countable ordinal $\alpha$ and a natural number $n \ge 1$ . In the notation above, the ordinal is homeomorphic (although not necessarily equal) to $\omega^\alpha \cdot n + 1$ .

Using a framework introduced by Christian Rosendal, we showed in that previous paper that the following properties hold.

Theorem (Branman–Domat–Hoganson–Lyman, 2025). The groups $G(\alpha,n) = \mathop{\mathrm{Homeo}}(\omega^\alpha \cdot n + 1)$ have a well-defined coarse structure. Moreover, they fall into three camps:
If $n = 1$ , the group $G(\alpha,n)$ is coarsely bounded, so coarsely equivalent to a point.
If $n > 1$ and $\alpha$ is a successor ordinal $\alpha = \beta + 1$ , the group $G(\alpha,n)$ is generated by a coarsely bounded set, and has a well-defined, unbounded quasi-isometry type.
If $n > 1$ and $\alpha$ is a limit ordinal, the group $G(\alpha,n)$ is not generated by any coarsely bounded set.

In a forthcoming paper we will complete the classification of homeomorphism groups of countable, compact ordinals by proving the following result.

Theorem (Branman–Domat–Hoganson–Lyman 2026). The groups $G(\alpha,n) = \mathop{\mathrm{Homeo}}(\omega^\alpha \cdot n + 1)$ fall into exactly three coarse equivalence classes as described above.

That is, we show that all groups in any of the three camps above are coarsely equivalent to each other and not to ones from the other camps. This is obviously already known for the first camp by our previous work. For the other two camps, the result is to our knowledge one of the first of its kind in the setting of non-locally compact Polish groups.

Subgroupoids of Groups

Wed, 25 Feb 2026 17:00:00 +0000

There’s an annoying philistinism in some areas of mathematics, including my own area of geometric group theory, of pooh-poohing category theory.

It’s easy to see why: if you browse the nLab or the Stacks project or talk to a PL theory person who knows too much Haskell, you mostly get what sounds like meaningless gobbledigook—excuse me, I meant abstract nonsense. This isn’t to bash the work that went into these projects, which are valuable references once you learn how to read them. The problem is that many people don’t discover why they might want to.

Ostensibly, the aim of a field like geometric group theory, low-dimensional topology or analysis, is concerned with answering concrete questions about fairly specific objects. Such and such family of solitons, or Artin groups or knots, or what have you. Often a strong geometric intuition or a willingness to get one’s hands dirty with the simplest nontrivial example is a key part of the work. Knowledge of category theory often does not help here. Knowing that an HNN extension is an example of a co-inserter is very different from being able to comfortably work with its Bass–Serre tree. Everyone in mathematics has met an approximation of the bright senior undergrad or first-year grad student who appears to have swallowed Category Theory for the Working Mathematician whole. This student might use abstract nonsense as a deflection, a way of avoiding actually getting their hands dirty learning a field.

That being said, there really is a time for theory-building, and geometric group theory for the most part falls on its face at this step. It’s great for definitions to be difficult so that theorems can be easy, for instance, but some definitions are still too difficult. To me, this signals a lack of time spent trying to fit tools into a bigger picture.

Anyway, that’s enough polemics from me for today. The purpose of this post is to introduce a concept familiar to folks working in category theory that suffers from a lack of attention from group theorists: the notion of a subgroupoid of a group. In an effort to make it relevant to a geometric group theorist, I’ll close by reformulating the definitions of height and width of subgroups as properties of a subgroupoid.

Background

I’m going to fly through this because you probably have seen this stuff before, and if you haven’t, you should do more work understanding it than this post is gonna provide.

A groupoid is a small category where every arrow is invertible.

If that’s too fast for you, you have two sets $V \mathsf{G}$ of objects and $E \mathsf{G}$ of arrows. I’ll use function notation for arrows: $f\colon x \to y$ saying that $f \in E \mathsf{G}$ is an arrow with source object $\iota(f) = x \in V\mathsf{G}$ and target object $\tau(f) = y \in V\mathsf{G}$ . Arrows $f\colon x \to y$ and $g \colon y \to z$ can compose like functions as $g \circ f = g f \colon x \to z$ , and each object $x$ has an identity arrow $1_x \colon x \to x$ . Composition is associative and identity arrows are (two-sided) identities for composition. Finally if $f\colon x \to y$ is an arrow, there is an arrow $f^{-1}\colon y \to x$ such that $f \circ f^{-1} = 1_y$ and $f^{-1} \circ f = 1_x$ .

You might have heard “a group is a groupoid with one object.” This is true: if $G$ is a group, let $G$ itself denote the object and each element $g \in G$ becomes an arrow $g \colon G \to G$ . Composition is the group operation and inversion is inversion.

Morphisms of groupoids are functors: $F \colon \mathsf{H} \to \mathsf{G}$ sends, for example $f \colon x \to y$ in $\mathsf{H}$ to $F f \colon F x \to F y$ in $\mathsf{G}$ , so there are assignments of objects to objects and arrows to arrows. Composition and identities must be respected.

A natural transformation, natural isomorphism, or my emerging favorite terminology, a conjugacy $\eta$ between functors $F, K \colon \mathsf{H} \to \mathsf{G}$ is an assignment, for each object $x \in V \mathsf{H}$ , of an arrow $\eta_x \colon F x \to K x$ in $\mathsf{G}$ such that for each arrow $f \colon x \to y$ in $\mathsf{H}$ , the equation

$K f \circ \eta_x = \eta_y \circ F f$

is true in $\mathsf{G}$ . The notation is $\eta \colon F \Rightarrow K$ .

I like the word conjugacy because in the case where $H$ and $G$ are groups, a functor is a homomorphism and a conjugacy is simply a choice of an element $\eta \in G$ with the property that $K(h) = \eta F(h) \eta^{-1}$ for all elements $h \in H$ .

An equivalence of groupoids $\mathsf{G}$ and $\mathsf{H}$ is a pair of functors $F \colon \mathsf{G} \to \mathsf{H}$ and $K \colon \mathsf{H} \to \mathsf{G}$ and conjugacies $\eta \colon 1_{\mathsf{G}} \Rightarrow K F$ and $\epsilon \colon F K \Rightarrow 1_{\mathsf{H}}$ .

Assuming the Axiom of Choice, a functor $F$ is an equivalence if and only if it is full (surjective on arrows $Fx \to Fy$ ), faithful (injective on arrows $x \to y$ ) and essentially surjective (every object $y$ in the target groupoid is connected to an object $Fx$ by an arrow $Fx \to y$ ).

Group Systems are Groupoids

A groupoid is connected if between every pair of objects $x$ and $y$ , there is an arrow $x \to y$ . A groupoid is contractible if it is connected and the aforementioned arrow is unique. Every nonempty connected groupoid $\mathsf{H}$ is equivalent to the group $\mathsf{H}(v,v)$ of arrows $v \to v$ for some (and hence any) object $v \in V\mathsf{H}$ . A nonempty contractible groupoid is equivalent to the trivial group.

If $v \in V\mathsf{G}$ is an object, its connected component is the set of objects $w$ in $\mathsf{G}$ such that there is an arrow $v \to w$ . Suppose $\{ [v_i] : i \in I \}$ is the set of connected components of $\mathsf{G}$ . We can construct a spanning groupoid from this set by choosing for each $w \in [v_i]$ a unique arrow $s_w \colon v_i \to w$ , with the stipulation that the arrow $s_{v_i} \colon v_i \to v_i$ is $1_{v_i}$ . Completing this by addding in identities and compositions, we get a contractible groupoid $\mathsf{S}$ with vertex set $V\mathsf{G}$ . With all of this data, one can show that each arrow $f \colon v \to w$ factors uniquely as $f = s_w f_i s_v^{-1}$ , where $s_v \colon v_i \to v$ and $s_w \colon v_i \to w$ are in $\mathsf{S}$ and $f_i \colon v_i \to v_i$ is an element of the group $G_i = \mathsf{G}(v_i, v_i)$ .

Every groupoid, then, is equivalent to what J.P. Mutanguha denotes a group system: an indexed disjoint union of groups. Explicitly, if $\{ [v_i] : i \in I \}$ is the set of connected components of $\mathsf{G}$ , then $\mathsf{G}$ is equivalent to the disjoint union of groups $\bigsqcup_{i \in I} G_i$ , where $G_i = \mathsf{G}(v_i,v_i)$ .

Subgroupoids of Groups

A subgroupoid $\mathsf{H}$ of a group $G$ is a groupoid $\mathsf{H}$ together with a functor $F \colon \mathsf{H} \to G$ with trivial kernel, meaning that if $F f = 1_G$ , then $f$ is an identity arrow.

Let’s analyze what this means. By our discussion in the previous section, we should consider the groups $H_i = \mathsf{H}(v_i,v_i)$ where $\{ [v_i] : i \in I \}$ is the set of connected components of $\mathsf{H}$ . On each such group $F$ restricts to an injective homomorphism, so each group $H_i$ is a subgroup of $G$ . Each arrow $s_v \colon v_i \to v$ maps to an element $g_v \in G$ and the stipulation that no arrow (necessarily) of the form $s_w h_i s_v^{-1}$ maps to the identity says that if $v$ and $w$ are distinct elements of $[v_i]$ , the cosets $g_v H_i$ are distinct in $G$ .

A connected subgroupoid $\mathsf{H}$ of a group $G$ therefore corresponds to a set of cosets of a given subgroup $H \le G$ . The groupoid $\mathsf{H}$ therefore has at most $G / H$ many objects I’ll say that a component $\mathsf{H}_i$ of a subgroupoid $\mathsf{H}$ is total if it has $G / H_i$ for its set of objects, and that $\mathsf{H}$ is total if each component is. (I don’t love “total”. Maybe another name could be “wide”, which generally has a slightly different meaning; if you have a better name, please let me know.)

Height and Width

Suppose $G$ is a group. Typically $G$ should be finitely generated, perhaps even finitely presented, but the following definition makes some amount of sense for a group $G$ .

First, we say the height of an infinite subgroup $H \le G$ is $n$ if there exist distinct cosets $\{ g_1H, \ldots, g_nH \}$ such that the intersection $\bigcap_{i=1}^n g_iHg_i^{-1}$ is infinite and $n$ is maximum possible.

Next, we say that the width of an infinite subgroup $H \le G$ is $n$ if there are distinct cosets as above so that pairwise we have that $g_iHg_i^{-1} \cap g_jHg_j^{-1}$ and again $n$ is maximum possible.

Finite width implies finite height.

This is phrased both simply and cleanly in the language of subgroupoids: If $\mathsf{H}$ is a total subgroupoid of $G$ , the inclusion witnesses “vertex” or “isotropy” groups $\mathsf{H}(v,v)$ as subgroups of $G$ . A component $\mathsf{H}_i$ of $\mathsf{H}$ has width (or respectively height) $n$ if $H_i = \mathsf{H}(v_i,v_i)$ is infinite and there is a collection of $n$ objects in $\mathsf{H}_i$ (which are, remember, subgroups of $G$ conjugate to $H_i$ by conjugators from distinct cosets) with pairwise (or respectively total) infinite intersection and $n$ is chosen maximum possible. A subgroupoid $\mathsf{H}$ therefore has width (or height) $n$ if $n$ is the maximum width (or height) among its components.

So, is coding dead yet or what? My students want to know

Fri, 13 Feb 2026 19:30:00 +0000

CEOs of AI startups have been claiming that software engineering is about 6 months away from total obsolescence more or less continuously since maybe 2023, with Anthropic’s CEO the most recent to say so about three weeks ago. As in Aesop’s fable of the boy who cried wolf, I think there are diminishing returns to making these claims again and again. Even if or as LLM capabilities improve and the job of software engineering changes, the doomsday speak is tiresome.

To be sure, LLM models have increased dramatically in their ability to code, from laughably unable to do anything sensible in the pre-ChatGPT era to new “agentic” workflows starting to see widespread adoption. If you pattern-match, some of the stuff currently squarely in the “laughably unable to do anything sensible” camp, like Gas Town, may stand to see significant improvement over a timeline best measured in years, not months.

But: let’s cut the crying wolf routine. I don’t want to read any more blog posts about how coding is dead.

I would love to read posts about how to focus on product engineering if the writing code step can be either mostly or completely outsourced to LLM agents. What have you learned about what makes that work successful or difficult? Are those skills complementary to being able to reason about code or orthogonal to it? Unfortunately, every post I read is either predicting the end of the world—at least, jobs-wise—suffers from a worrying lack of a grip on reality, or is obviously just trying to sell me something.

Here’s why it matters to me: I’m a relative outsider—my employment work is at a university, not a tech company. Most of my coding work goes into maintaining and building open source projects in Zig. Since I’m interested as much in building my own knowledge as I am about the quality of the code I produce, I’m more likely to implement a classical algorithm, (for example constructing a Delaunay triangulation of a point cloud) than to build a CRUD app with or without AI assistance.

However, as a teacher in a department of math and computer science, I know that my students are entering a very different technical landscape than I did graduating college. I’m sure that they need engineering skills that look a little different than being able to write depth-first search correctly, and I would love to help convey those skills to them so they can succeed.

None of the three kinds of LLM booster writing I diagnose above is helpful to me in that goal. In fact, little of it is good writing.

Folks in the first camp, appear to want to step to, e.g., Cory Doctorow. The point of these posts is to be right about the future. That’s certainly interesting, but parrotting the kind of hype performances that AI CEOs put on for the benefit of their shareholders is not particularly incisive insight, if I may be so bold.

Folks in the Steve Yegge camp, whether they’re right or not, are basically impossible to learn from. Steve’s article is compelling the way a train wreck is. Imagine reading programming classics like Crafting Interpreters or Seven Languages in Seven Weeks but having to worry at the same time about the wellness of the author. It’s hard to build credibility if you type like you’re halfway through a crazed 48-hour coding bender.

Then there’s the selling. Content works great as advertising! I’ve been on Instagram, I know how it is. That’s not to say that it can’t be educationally valuable at the same time, but this kind of stuff is ultimately aimed to entertain, rather than to teach. This kind of writing is all over the internet, of course, too, so it’s well-within ChatGPT’s training data to generate more.

Now, I’m not saying you have to write a book—although please don’t let me stop you! But, if you’re learning a lot using AI and are interested to be part of the public conversation as it shapes the future of coding, here’s my suggestion to you:

Write what you’ve learned about your work, now that you’re not nose-to-the-terminal in code all the time. As you do so, write in the kind of measured, informative tone of voice you remember from writing that has inspired you. You don’t need to sound like a textbook, but you do need to build trust, not provoke a reaction. Above all, do not cry wolf. Do not pass Go, do not collect $200.

Where to Put my Little Yops?

Sat, 13 Dec 2025 23:00:00 +0000

Caring by Way of Tying Your Hands

I got in on Twitter not quite on the ground floor, but pretty close. (It’s kind of fun being in your 30s, because you can almost truly say “back in the day” unironically, but the luster of that ability hasn’t gone away.) This was back in the day, like 2009. You could text 40404 or whatever it was and send and receive tweets. In fact, in those days, I think the 140 character limit was designed so that the rest of the message’s information (its “header”, if you will) could fit in the remaining 20 characters an SMS message gave you.

I don’t think I had anything truly interesting to say, nor, really, did anybody I followed. There was a woman I knew tangentially through some long-defunct website; I think at the time she must have been in her 20s or 30s, living in England, possibly a PhD student, and a self-published author of fantasy stories that I couldn’t tell you a thing about now (except that “Icahn” was a proper name in her world; when I see the signs for Icahn Stadium out the window of an airplane near LaGuardia, I think about her). A couple years later, early in college, when I tweeted sheepishly that my strategy of “stay up much too late, work very last-minute” appeared to have gone off without too much of a hitch, as far as my finals grades were concerned, I remember her saying words to the effect of “hey, maybe that’s what works for you”.

Anyway, by the end of college, I had a sneaking little worry that I was really thinking mostly in tweets. Like, developing a take that wouldn’t fit in 140 characters sometimes felt difficult. In grad school in 2017, reflecting that looking at Twitter really was not helping me feel any better about anything going on in the world, I deleted my account. I’m not sure that I ever actually really needed to worry about the way it was shaping my thinking, but I think it’s fair to say I can think at length now.

Where do you Put your little Yops?

Similar stories happened with other forms of (essentially) blogging, like Tumblr and previous iterations of this blog. I hopped back onto Twitter during the height of the pandemic, and then left quietly in 2022 as I started to feel like I was seeing a bit too much of the uglier sides of the communinity I ran with. I’m sure many people are huge fans of X and Instagram lately, right?

There are lots of ways to choose your own social media adventure. I think lately the things I’m interested in are surprisingly “mailing list”-shaped. Unlike a blog, you can just reply to an email. Unlike a blog, the contents of an email are semi-impermanent by default. Unlike a blog (the Recurse Center community’s excellent “Blaggregator” project notwithstanding), newsletters arrive surrounded by their friends: other emails. A mailing list also allows for a little more community creation, although less so than something like a Discord server.

I want to share my little yops. Apparently it matters to me at least a little the company that the yops keep, and I want a certain amount of ownership over the yops.

Anyway so apparently I have a home for yops now.

Think of it like a trade offer: you give me one (1) email address, and I return to you many (eventually) fun (hopefully) things to read in your inbox.

What are your Yops like?

Some of them, surely a bit like this: a post that winds up telling me what it is by the writing of it. I like having these sorts of posts about culture and cultural artifacts (uhh, art pieces of varying kinds), about writing itself—I’m sure you’ll get the spirit quick. Another vibe I like is “the checkin”. A third, in a similar vein, is an idea I read on the WaniKani forums: you need to have adventures every day. I’m sure I don’t need to have a full little essay every day, but more frequently would be very welcome.

So, the first one is this post, essentially. A bit more and a bit less.

Reinit / How to Draw a Triangle in Three Years

Wed, 12 Nov 2025 00:00:00 +0000

Reinit

Hi again. This is Robbie. For most of a decade, I used to go by Rylee. I do math, I make music, I write code, I’m learning Japanese. I like graphic design, writing, cooking, some other stuff too. Lately I’ve been discovering that running and the gym are actually fun and good things, which would have blown my sixteen-year-old self’s mind.

This is a blog. Let’s go.

How to Draw a Triangle in Three Years

I am finally owning up to thinking that it would be fun to make a game. So I’m working on that, instead of, like, making a Lua environment ostensibly for music or wrapping Cocoa for easy access from Zig or any number of ways one could aim at making a game and miss perhaps deliberately.

Part of the discomfort that might cause one to miss, I think, is that making a game relies pretty fundamentally on other people’s code. Paraphrasing Carl Sagan, if you want to make a DAW from scratch, you must first invent the Zig programming language. Which is to say, software has the same problem as mathematics, and alas it is taking me about as long to recognize and resolve that problem.

Like in mathematics, in programming you can go all the way to the big bang if you’d like. You could write assembly. Or an assembler. Or an operating system. Or a C compiler. Or your own programming language. You could even write your own instruction set. Peeking one layer deeper into the abstraction is always at least theoretically possible.

For me, this leads directly to not-invented-here syndrome. Do not pass “go”, do not collect 10,000 hours of making-things time. If I don’t know how it works, how can I use it?

Unfortunately, being not-Andrew Kelley, I fear that if I attempt the same sort of from-first-principles exploration, I won’t even have demos of a game to show for it. (In fact, you don’t see Andrew Kelley reinventing Linux, for example, so picking on him obliquely is obviously unfair.)

Part of resolving the problem for me with math was intensifying the need to actually do something. I told my advisor I wanted to graduate in a year, and then did what I needed to do to do that: get out of my head and prove a theorem.

Another part of it, and this is sort of what I’m leaning on with programming, is the realization that “understanding,” here, is not an objective quantity that can be possessed seperate from experience using the thing. Putting the pieces together is an important part of graduating from armchair enthusiast to expert.

So, here is how to draw a triangle.

The First Year

In the first year or so, you should not know that you want to draw a triangle. You should be interested primarily in learning a programming language sufficiently powerful to let you shoot yourself in the foot (that is to say, link against C). I’m resisting giving Rust my customary noogie here.

As a result of your programming efforts, maybe you’ve discovered the edges of an ability to draw pixels to the user’s screen.

By the way, you should be running on macOS. After all, we’re using Vulkan, so anything else would be easy mode.

The Second Year

In the second year, you should learn about graphics pipelines. Enough to follow the Vulkan tutorial, but not enough to, like, read their “Getting Started” documentation. Since you’re coding along, you should be able to verify that your program type-checks but then doesn’t work at all. Ideally it stalls out at the “have GLFW load Vulkan” step, so that you can believe that your code works, but that something is just broken about GLFW or Vulkan on macOS, but not really solve it.

Then you should learn about Metal. It’s actually pretty helpful to do, since Metal and Vulkan agree to an embarrassing (for Apple? for not cooperating? for Khronos? for not working harder to get them onboard? answering these questions is above your pay grade) degree about what a graphics card does and how to talk to it. Maybe you can draw a triangle with Metal!

The Third Year

In the third year, you should ideally burn out for most of it. Give everything a break and reconnect with what’s actually interesting to you and what you are doing in order to please other people.

Really find yourself, tho. Like, learn jj. Delete and remake your website. Switch to fish. Lose track of the little notes apps and things that you used to keep yourself on track and then find them months later as a sad Tigger, a sorry Tigger, an “Oh Rabbit, am I glad to see you,” Tigger.

Then, clear-eyed and rarin to go, here are the steps.

Download Zig master

This can be done by clicking, or by being a script kiddie. In the interests of embarrassing myself, here’s a script-kiddie approach.

mkdir dwnld && cd dwnld
curl https://ziglang.org/download/index.json -o zig_idx.json
cat zig_idx.json | \
    jq '"url = \(.master.["aarch64-macos"].tarball)"' -r | \
    curl -K- -o zig.tar.xz
cat zig_idx.json | \
    jq '"\(.master.["aarch64-macos"].shasum)  zig.tar.xz"' -r | \
    shasum -a 256 -c -

These commands download the JSON blob from the zig website into a new directory named dwnld, then grab and download the master tarball for aarch64-macos and check its sha256 checksum by using jq to parse the JSON. You should probably not continue if the final command above doesn’t tell you that the sum was OK, but probably you shouldn’t even use this approach in the first place.

mkdir zig && cd zig
tar -xf ../zig.tar.xz --strip-components=1
cd ..
rm -rf ~/bin/doc ~/bin/lib
mv zig/* ~/bin
xattr -d com.apple.quarantine ~/bin/zig
cd ..
rm -rf dwnld

The first pair of commands extracts the tarball into a new directory named zig. Status quo is that the tarball packages everything under a folder named something like zig-macos-aarch64-[VERSION-NAME-CRUFT], which is fine as far as it goes, but not very conducive to script kiddie powers, hence the --strip-components=1 flag on the tar command. As you can see, I put the zig compiler in ~/bin, which I had to add to my PATH. The xattr line avoids the “macOS won’t let you run this program bc you downloaded it” dance.

Install GLFW and Vulkan

For GLFW, you can just do

brew install glfw

Probably a similar invocation could work for Vulkan, but you might as well go to their website and download and install the SDK.

Get started with Vulkan-Zig

Go ahead and grab the repo

jj git clone git@github.com:Snektron/vulkan-zig
cd vulkan-zig

The repo should build fine with

zig build --build-file examples/build.zig

but probably it does not, complaining that GLFW cannot be found. The fix is pretty easy:

--- a/examples/build.zig
+++ b/examples/build.zig
@@ -22,7 +22,7 @@
         .use_llvm = true,
     });
     b.installArtifact(triangle_exe);
-    triangle_exe.linkSystemLibrary("glfw");
+    triangle_exe.linkSystemLibrary("glfw3");

     const registry_path: std.Build.LazyPath = 
        if (maybe_override_registry) |override_registry|
         .{ .cwd_relative = override_registry }

After this, the repo should build, but it should fail when you try to run it, complaining that Vulkan cannot be found. To fix this, you need to make it so that dlopen can find Vulkan. This is much less scary than it sounds: for a development build it suffices to add a path to your DYLD_LIBRARY_PATH variable. For me, since DYLD_LIBRARY_PATH is empty, that looks like this:

set -gx DYLD_LIBRARY_PATH "/Users/robbie/VulkanSDK/1.4.328.1/macOS/lib"

Then you can run the program and see a beautiful Vulkan triangle! It only took you three years! Maybe the rest will be much quicker ;)

Robbie Lyman

Topologies on Free Products

Free Products, A Refresher

The Bass-Serre Tree

Some Topologies

Differentiating the topologies

Who's afraid of a non sofic group

Dynamical Systems, Surjunctivity

Sofic groups

How to feel about non-sofic groups

Living without a moat

Topologies, Groups, Graphs

Topological Groups

Groups acting on graphs

Category Theory Day One

Category Theory Day One

Categories

Functors

Natural Transformations

The Yoneda Lemma

Nuh-uh! Infinity plus one!

Nuh-uh! Infinity plus one!

Two definitions

Ordinal arithmetic

Picturing ordinals

Classifying homeomorphisms of ordinals

Subgroupoids of Groups

Background

Group Systems are Groupoids

Subgroupoids of Groups

Height and Width

So, is coding dead yet or what? My students want to know

Where to Put my Little Yops?

Caring by Way of Tying Your Hands

Where do you Put your little Yops?

What are your Yops like?

Reinit / How to Draw a Triangle in Three Years

Reinit

How to Draw a Triangle in Three Years

The First Year

The Second Year

The Third Year

Download Zig master

Install GLFW and Vulkan

Get started with Vulkan-Zig