Elementary examples of Yoneda embeddings

Question

Suppose you trying to sell the idea of the Yoneda embedding to perhaps a rather mixed bunch of students (so you can't presuppose too much mathematical background). Still you can say

1. Think of a group-as-a-category (one object, all the arrows isomorphisms). Then applying the Yoneda embedding theorem we get ... [a bit of chat] ... hey, Cayley's Theorem.
2. Think of a poset-as-a-category. Then applying the Yoneda embedding we get ... [a bit more chat] ... hey, the familiar result that a poset is isomorphic to a certain bunch of subsets of its objects (upper sets) ordered by inclusion.

Those are in fact the usual textbook offerings. But what third or fourth examples of such embeddings (not requiring the fully caffeinated Lemma) might work as equally accessible? Or maybe not quite as accessible but more interesting??

Answer 1

The first example has several variants. Every monoid embeds into $\mathrm{End}(S)$ for a set $S$. Every $K$-algebra embeds into $\mathrm{End}(V)$ for some $K$-vector space $V$. Every topological ring $R$ embeds into $\mathrm{End}(A)$ for some topological abelian group $A$. Every Banach algebra embeds into $\mathrm{End}(V)$ for some Banach space $V$, etc.

More generally, if $\mathcal{V}$ is a symmetric monoidal closed category, then every monoid in $\mathcal{V}$ can be considered as a one-object category enriched in $\mathcal{V}$, so that we can apply the Yoneda embedding to it. It gives us an embedding into the endomorphism monoid $\underline{\mathrm{End}}(A)$ for some object $A$.

This only covers (very) small categories, though. For large categories, some technical adjustments need to be made so that the Yoneda embedding can be formulated at all, and there are also some "philosophical" differences between small and large categories. What one usually does here are two things instead: (a) restrict to an essentiall small subcategory that is sufficient for every purpose. (b) use only the fully-faithfulness of the Yoneda embedding (which imposes no technical difficulties at all).

A typical example of (a) is the embedding $\mathbf{CRing} \to [\mathbf{CRing}^{\mathrm{op}},\mathbf{Set}]$ which is kind of important in algebraic geometry, especially in functorial algebraic geometry. But the latter functor category is ill-behaved (for example, not locally small) and hard to understand in many ways. What is often better is to consider the category of finitely presented rings (for example). Or we narrow down to the category of cofinally small presheaves (those functors $\mathbf{CRing}^{\mathrm{op}} \to \mathbf{Set}$ that can be expressed as small colimits of representable functors).

Now coming back to your question, I believe that this is the reason why there are not so many toy examples of the (full) embedding for large categories which can be explained to a mixed audience. The whole functor category is very big and very abstract.

It's quite the opposite for (b), in particular the corollary $$\hom(X,-) \cong \hom(Y,-) \implies X \cong Y$$ I have explained this probably in hundreds of posts here on MSE and elsewhere (also in my book), but already the simplest instance of the Yoneda embedding, saying

If $R,S$ are commutative rings such that there is a bijection $\alpha_A : \hom(S,A) \to \hom(R,A)$ for all commutative rings $A$ such that $\alpha_B \circ f_* = f_* \circ \alpha_A$ for all $f \in \hom(A,B)$, then there is an isomorphism $g : R \to S$ with $\alpha_A = g^*$.

helps a lot to simplify proofs in basic algebra which otherwise would require nasty calculations with elements. Here are some statements which, with the Yoneda embedding instance above and (of course) the universal properties of the involved constructions can be proven each in just one line.

• $(R/J)/(I/J) \cong R/I$ for $J \subseteq I \subseteq R$
• $R[T]/\langle fT-1 \rangle \cong R[f^{-1}]$
• $R[X][Y] \cong R[X,Y] \cong R[Y][X]$
• $R[f^{-1}][g^{-1}] \cong R[(fg)^{-1}]$
• $R[X]/\langle X \rangle \cong R$
• $R[X,Y]/\langle Y-p(X) \rangle \cong R[X]$
• ... and many more

I find this quite impressive since you can even give the above statement about rings as an exercise, without talking about categories at all, and then deduce all these "trivial" isomorphisms from it with no calculation at all.

By the way, I have interpreted your question so that it is really just about the Yoneda embedding. The full Yoneda Lemma has several other toy examples. The following one is perhaps the simplest one, but curiously it already captures the whole idea of the general case: If $G$ is a group (or just a monoid!) and $X$ is a $G$-set, then there is a bijection between the $G$-maps $G \to X$ and the elements of $X$. This is the Yoneda Lemma for one-object categories. Again, we can generalize it via enrichment. For example, if $V$ is a $K$-linear representation of a group $G$, and $R_G$ is the regular representation of $G$, then the morphisms of representations $R_G \to V$ correspond to the elements of $V$.