A Cute Application of the Yoneda Lemma

17 Mar 2025

Every few weeks recently I’ve been putting a new fun problem on one of the whiteboards in the first year office. These are often inspired by something I saw on MSE, and I’m usually choosing problems that force you to understand something fundamental really well. Then I usually have a “challenge” bonus problem that uses the same ideas, but requires you to do some independent research to finish the puzzle! I’ve just decided what the next problem is going to be, and I’m going to post about it here because I think it’s too cute to not share more widely! This is also going to serve as a nice excuse to talk about some important reasons to care about yoneda and representability that I don’t think we often tell beginners!

Before we start, though, some quick life updates: I’m still waiting to hear back positively from postdocs, but there’s still plenty of departments where I’d be very happy. I’m trying to not stress out until the end of april. What’s more stressful is my actual dissertation – because we had to move everything up a year, I didn’t have as much time as I expected to solve my main problem, and while I solved some very simple thing that’s needed along the way… It’s really not a very good thesis, haha. So I’m trying really hard to finish the whole problem in the time I have, and we’ll see how I manage. My mental health has been pretty hit or miss with the *gestures at the state of the world*, but I’m doing what I can where I can to help, and that’s been good for what would otherwise be an overwhelming sense of helplessness. I know everyone is feeling some amount of this, especially in the queer and trans community, and I encourage everyone to find something actionable to do on a local level – it helps the community, and it really does help keep the feelings manageable.

BUT, enough about all that! I haven’t been writing much because I’ve been so focused on research, and I’ve really missed posting here. The quicker I am the more likely I’ll find time to do it again, so let’s get to the fun part!

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I’m going to assume that most people reading my blog have at least heard of the Yoneda Lemma. But it’s possible that some people reading this are confused as to why you might care about it. Like, what does it buy you? How does it help prove theorems? One answer is that it’s foundational for thinking about the functor of points approach to lots of subjects, which tells you how to study things like stacks, etc. Said in a different way, it’s an extremely useful perspective for studying certain moduli problems – you can see my old blog post about this here if you’re interested.

But that’s all quite high powered. Is there a more concrete reason to care about yoneda? The answer here is inspired by one of Terry Tao’s old blog posts – the yoneda lemma tells you how to understand polymorphic transformations between two constructions (which are a priori complicated) by just studying regular old functions between representing objects!

For one version of this, say you want to understand cohomology operations. That is, say you want to understand when there’s a map on cohomology $H^p(X,G_1)\to H^q(X,G_2)$ which is “polymorphic in $X$” or “uniform in $X$” in the sense that the same map works no matter which $X$ you choose¹. Just as in analysis where uniformity allows you to prove lots of ~bonus results~, requiring uniformity here gives us access to many additional theorems – such as the yoneda lemma. Here this “uniformity” is usually called naturality of the transformation, and the precise relationship to polymorphism will be well understood by anyone who learned category theory via functional programming², and is well explained in some of Bartosz Milewski’s old blog posts here and here.

Now we know that $H^p(-,G)$ is represented by a space called $K(G,p)$, so asking for one of these “cohomology operations” is asking for a natural transformation

$$H^p(-,G_1)\Rightarrow H^q(-,G_2).$$

Which, by representability, is a natural transformation

$$[-,K(G_1,p)]\Rightarrow [-,K(G_2,q)]$$

but now we finally get to use yoneda! These natural transformations are exactly the same thing as ordinary continuous functions (up to homotopy)

$$[K(G_1,p),K(G_2,q)]!$$

This means if we can understand all continuous functions between these fixed spaces, that’s enough to understand all cohomology operations $H^p(X,G_1)\to H^q(X,G_2)$ for every space $X$ simultaneously! The ability to work with the representing objects directly is a huge reduction in complexity, and one of the main reasons people work with objects like “stacks” and “spectra” is because they want more things to be representable so that they can play exactly this game³.

That’s well and good…. But surely that can’t be the “down to earth” example, right? Cohomology operations are interesting, and might eventually feel down to earth, but objectively most people would laugh at that idea. This, finally, brings us to the problem I’ll be writing on the first year’s whiteboard later today:

I’m about to spoil both of these problems, so if you want to think about them (and I really encourage you to think about them! At least the first problem!) now is your last chance to be unbiased by my explanation…

Ok, so. Inspired by the rest of this post you’ll want to think about how to represent the constructions above. For the first problem, where we want set functions $R^n\to R$, that means we want to represent⁴

• the functor sending a ring $R$ to the underlying set of $R^n$
• the functor sending a ring $R$ to its underlying set

The second one isn’t so hard to do – a moment’s thought shows that ring homs $\mathbb{Z}[x_1]\to R$ are in bijection with elements of $R$ (we just have to choose where to send $x_1$). Once you’ve figured this out it’s not such a big jump to see that ring homs $\mathbb{Z}[x_1,\dots ,x_n]\to R$ are in bijection with elements of $R^n$ (we just have to choose where to send each of the $x_i$ separately).

So classifying “polymorphic” transformations $R^n\to R$ is classifying natural transformations

$$[\mathbb{Z}[x_1,\dots ,x_n],-]\Rightarrow [\mathbb{Z}[y],-]$$

which, by yoneda, is asking to classifying good old fashioned ring homs

$$[\mathbb{Z}[y],\mathbb{Z}[x_1,\dots ,x_n]]$$

and, as before, we just need to decide where $y$ goes! So this is in bijection with the underlying set of $\mathbb{Z}[x_1,\dots ,x_n]$. That is, with integer-coefficient polynomials in $n$ variables.

It’s not hard to see that any such polynomial gives you such a natural map! Indeed, for a concrete example take $p=x_1^2+2x_1{x}_2-3$. Then we get a function $R^2\to R$ sending $(r_1,r_2)\mapsto r_1^2+2r_1{r}_2-3$, where an integer like $-3$ is interpreted in $R$ as $0_R-1_R-1_R-1_R$.

If one thought about this problem without knowing category theory, I think it would be pretty easy to conjecture that integer polynomials are the only polymorphic maps $R^n\to R$… But I’m not sure how you would prove it without basically rediscovering this special case of yoneda!

Now… what about the challenge problem? We want to classify “polymorphic” maps

$$\{{\mathfrak{S}}_3\text{-torsion collections in }R\}\to \{C_2\text{-torsion collections in }R\}$$

Again we have a natural⁵ conjecture, since if we have a group homomorphism ${\mathfrak{S}}_3\to R^{\times }$ we get some obvious group homomorphisms $C_2\to R^{\times }$ too (namely the images of $(12)$, $(23)$ and $(13)$, the 2-torsion elements in ${\mathfrak{S}}_3$)… Could this be all of them?

Well, a “$G$-torsion collection” in $R$ is a group homomoprhism $G\to R^{\times }$, and these are in bijection with ring maps $\mathbb{Z}G\to R$! So the same idea as before shows that classifying natural transformations

$$\{{\mathfrak{S}}_3\text{-torsion collections in }R\}\to \{C_2\text{-torsion collections in }R\}$$

is the same as classifying natural transformations

$$[\mathbb{Z}{\mathfrak{S}}_3,-]\Rightarrow [\mathbb{Z}{C}_2,-]$$

which, by yoneda, means classifying ring maps

$$[\mathbb{Z}{C}_2,\mathbb{Z}{\mathfrak{S}}_3]$$

and thus means classifying $C_2$-torsion collections in $\mathbb{Z}{\mathfrak{S}}_3$. So we want to understand the group of units in $\mathbb{Z}{\mathfrak{S}}_3$, and in particular we want to classify the $2$-torsion elements in this group!

Now is where the independent research comes in. You might naively guess that the group of units in $\mathbb{Z}G$ is just $G$ again… But you would be wrong! Of course if $g\in G$ then both $+g$ and $-g$ are units in $\mathbb{Z}G$, but even this isn’t enough! Understanding the units in group rings turns out to be a subtle and interesting problem, and I think it’s extra cute that this little puzzle helps people learn about this surprisingly rich subject⁶.

We want to understand the group of units in $\mathbb{Z}{\mathfrak{S}}_3$, and searching for things related to this will quickly turn up the paper The Group of Units of the Integral Group Ring $\mathbb{Z}{S}_3$ by Hughes and Pearson, which certainly seems related!

This paper is only 6 pages long, and is a very polite read⁷. Part (2) of their Theorem 1 is exactly what we’re looking for:

(2) … there are $2$ conjugacy classes of elements of order $2$, with generic elements $(12)$ and $t=(12)+3(13)-3(23)-3(123)+3(132)$ respectively.

What does this mean for us? Their Part (1) to this same theorem gives an isomorphism of $(\mathbb{Z}{\mathfrak{S}}_3{)}^{\times }$ with a certain $2\times 2$ matrix group, so it’s easy to compute conjugation. So knowing this and knowing both conjugacy representatives we have a classification of all order 2 elements in $\mathbb{Z}{\mathfrak{S}}_3$, and by extension (crucially using yoneda and representability!) a classification of all natural transformations

$$\{{\mathfrak{S}}_3\text{-torsion collections in }R\}\to \{C_2\text{-torsion collections in }R\}$$

Indeed from this we’ve learned that our naive conjecture was wrong! This element $t$ is not one of the 2-torsion elements we were expecting, and it gives us a new and unexpected natural transformation!

If $f:{\mathfrak{S}}_3\to R^{\times }$ is a ${\mathfrak{S}}_3$-torsion collection in $R$, then of course we have $C_2$-torsion elements $f(12)$, $f(23)$ and $f(13)$. But it turns out we also have a secret, special $C_2$-torsion element

$$f(t)=f(12)+3f(13)-3f(23)-3f(123)+3f(132)$$

Or, said another way, if $x,y\in R^{\times }$ with $x^2=1$, $y^3=1$, and $xyx=y^2$ determine a ${\mathfrak{S}}_3$-torsion collection in $R$, then

$$x+3yx{y}^{-1}-3y^{-1}xy-3y+3y^{-1}$$

is a (probably surprising!) $C_2$-torsion element in $R$.

Of course, as with the last problem, I’m not sure how you would even begin to approach this classification problem without rediscovering the relevant version of the yoneda lemma!

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Thanks again for hanging out, everyone! I’m working on a lot of really fun stuff and I can’t wait to have time to talk about it. I think I want a quick outro today – I’ve had the sniffles all weekend, probably because I didn’t dress well before hanging out in the cold with one of my best friends who visited me last week! We made creme brulee for the first time (since I recently bought a blowtorch), and it was surprisingly easy and turned out delicious! I already make a lot of lava cakes when I want a decadent and fancy dessert, but I’ll have to add creme brulee into the rotation!

Talk soon, everyone! Stay warm 💖

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