As someone who had a relatively light graduate education in algebra, the import of Yoneda’s lemma in category theory has always eluded me somewhat; the statement and proof are simple enough, but definitely have the “abstract nonsense” flavor that one often ascribes to this part of mathematics, and I struggled to connect it to the more grounded forms of intuition, such as those based on concrete examples, that I was more comfortable with. There is a popular MathOverflow post devoted to this question, with many answers that were helpful to me, but I still felt vaguely dissatisfied. However, recently when pondering the very concrete concept of a polynomial, I managed to accidentally stumble upon a special case of Yoneda’s lemma in action, which clarified this lemma conceptually for me. In the end it was a very simple observation (and would be extremely pedestrian to anyone who works in an algebraic field of mathematics), but as I found this helpful to a non-algebraist such as myself, and I thought I would share it here in case others similarly find it helpful.

In algebra we see a distinction between a polynomial form (also known as a formal polynomial), and a polynomial function, although this distinction is often elided in more concrete applications. A polynomial form in, say, one variable with integer coefficients, is a formal expression {P} of the form

\displaystyle P = a_d {\mathrm n}^d + \dots + a_1 {\mathrm n} + a_0 \ \ \ \ \ (1)

where {a_0,\dots,a_d} are coefficients in the integers, and {{\mathrm n}} is an indeterminate: a symbol that is often intended to be interpreted as an integer, real number, complex number, or element of some more general ring {R}, but is for now a purely formal object. The collection of such polynomial forms is denoted {{\bf Z}[{\mathrm n}]}, and is a commutative ring.

A polynomial form {P} can be interpreted in any ring {R} (even non-commutative ones) to create a polynomial function {P_R : R \rightarrow R}, defined by the formula

\displaystyle P_R(n) := a_d n^d + \dots + a_1 n + a_0 \ \ \ \ \ (2)

for any {n \in R}. This definition (2) looks so similar to the definition (1) that we usually abuse notation and conflate {P} with {P_R}. This conflation is supported by the identity theorem for polynomials, that asserts that if two polynomial forms {P, Q} agree at an infinite number of (say) complex numbers, thus {P_{\bf C}(z) = Q_{\bf C}(z)} for infinitely many {z}, then they agree {P=Q} as polynomial forms (i.e., their coefficients match). But this conflation is sometimes dangerous, particularly when working in finite characteristic. For instance:

  • (i) The linear forms {{\mathrm n}} and {-{\mathrm n}} are distinct as polynomial forms, but agree when interpreted in the ring {{\bf Z}/2{\bf Z}}, since {n = -n} for all {n \in {\bf Z}/2{\bf Z}}.
  • (ii) Similarly, if {p} is a prime, then the degree one form {{\mathrm n}} and the degree {p} form {{\mathrm n}^p} are distinct as polynomial forms (and in particular have distinct degrees), but agree when interpreted in the ring {{\bf Z}/p{\bf Z}}, thanks to Fermat’s little theorem.
  • (iii) The polynomial form {{\mathrm n}^2+1} has no roots when interpreted in the reals {{\bf R}}, but has roots when interpreted in the complex numbers {{\bf C}}. Similarly, the linear form {2{\mathrm n}-1} has no roots when interpreted in the integers {{\bf Z}}, but has roots when interpreted in the rationals {{\bf Q}}.

The above examples show that if one only interprets polynomial forms in a specific ring {R}, then some information about the polynomial could be lost (and some features of the polynomial, such as roots, may be “invisible” to that interpretation). But this turns out not to be the case if one considers interpretations in all rings simultaneously, as we shall now discuss.

If {R, S} are two different rings, then the polynomial functions {P_R: R \rightarrow R} and {P_S: S \rightarrow S} arising from interpreting a polynomial form {P} in these two rings are, strictly speaking, different functions. However, they are often closely related to each other. For instance, if {R} is a subring of {S}, then {P_R} agrees with the restriction of {P_S} to {R}. More generally, if there is a ring homomorphism {\phi: R \rightarrow S} from {R} to {S}, then {P_R} and {P_S} are intertwined by the relation

\displaystyle \phi \circ P_R = P_S \circ \phi, \ \ \ \ \ (3)

which basically asserts that ring homomorphism respect polynomial operations. Note that the previous observation corresponded to the case when {\phi} was an inclusion homomorphism. Another example comes from the complex conjugation automorphism {z \mapsto \overline{z}} on the complex numbers, in which case (3) asserts the identity

\displaystyle \overline{P_{\bf C}(z)} = P_{\bf C}(\overline{z})

for any polynomial function {P_{\bf C}} on the complex numbers, and any complex number {z}.

What was surprising to me (as someone who had not internalized the Yoneda lemma) was that the converse statement was true: if one had a function {F_R: R \rightarrow R} associated to every ring {R} that obeyed the intertwining relation

\displaystyle \phi \circ F_R = F_S \circ \phi \ \ \ \ \ (4)

for every ring homomorphism {\phi: R \rightarrow S}, then there was a unique polynomial form {P \in {\bf Z}[\mathrm{n}]} such that {F_R = P_R} for all rings {R}. This seemed surprising to me because the functions {F} were a priori arbitrary functions, and as an analyst I would not expect them to have polynomial structure. But the fact that (4) holds for all rings {R,S} and all homomorphisms {\phi} is in fact rather powerful. As an analyst, I am tempted to proceed by first working with the ring {{\bf C}} of complex numbers and taking advantage of the aforementioned identity theorem, but this turns out to be tricky because {{\bf C}} does not “talk” to all the other rings {R} enough, in the sense that there are not always as many ring homomorphisms from {{\bf C}} to {R} as one would like. But there is in fact a more elementary argument that takes advantage of a particularly relevant (and “talkative”) ring to the theory of polynomials, namely the ring {{\bf Z}[\mathrm{n}]} of polynomials themselves. Given any other ring {R}, and any element {n} of that ring, there is a unique ring homomorphism {\phi_{R,n}: {\bf Z}[\mathrm{n}] \rightarrow R} from {{\bf Z}[\mathrm{n}]} to {R} that maps {\mathrm{n}} to {n}, namely the evaluation map

\displaystyle \phi_{R,n} \colon a_d {\mathrm n}^d + \dots + a_1 {\mathrm n} + a_0 \mapsto a_d n^d + \dots + a_1 n + a_0

that sends a polynomial form to its evaluation at {n}. Applying (4) to this ring homomorphism, and specializing to the element {\mathrm{n}} of {{\bf Z}[\mathrm{n}]}, we conclude that

\displaystyle \phi_{R,n}( F_{{\bf Z}[\mathrm{n}]}(\mathrm{n}) ) = F_R( n )

for any ring {R} and any {n \in R}. If we then define {P \in {\bf Z}[\mathrm{n}]} to be the formal polynomial

\displaystyle P := F_{{\bf Z}[\mathrm{n}]}(\mathrm{n}),

then this identity can be rewritten as

\displaystyle F_R = P_R

and so we have indeed shown that the family {F_R} arises from a polynomial form {P}. Conversely, from the identity

\displaystyle P = P_{{\bf Z}[\mathrm{n}]}(\mathrm{n})

valid for any polynomial form {P}, we see that two polynomial forms {P,Q} can only generate the same polynomial functions {P_R, Q_R} for all rings {R} if they are identical as polynomial forms. So the polynomial form {P} associated to the family {F_R} is unique.

We have thus created an identification of form and function: polynomial forms {P} are in one-to-one correspondence with families of functions {F_R} obeying the intertwining relation (4). But this identification can be interpreted as a special case of the Yoneda lemma, as follows. There are two categories in play here: the category {\mathbf{Ring}} of rings (where the morphisms are ring homomorphisms), and the category {\mathrm{Set}} of sets (where the morphisms are arbitrary functions). There is an obvious forgetful functor {\mathrm{Forget}: \mathbf{Ring} \rightarrow \mathbf{Set}} between these two categories that takes a ring and removes all of the algebraic structure, leaving behind just the underlying set. A collection {F_R: R \rightarrow R} of functions (i.e., {\mathbf{Set}}-morphisms) for each {R} in {\mathbf{Ring}} that obeys the intertwining relation (4) is precisely the same thing as a natural transformation from the forgetful functor {\mathrm{Forget}} to itself. So we have identified formal polynomials in {{\bf Z}[\mathbf{n}]} as a set with natural endomorphisms of the forgetful functor:

\displaystyle \mathrm{Forget}({\bf Z}[\mathbf{n}]) \equiv \mathrm{Hom}( \mathrm{Forget}, \mathrm{Forget} ). \ \ \ \ \ (5)

Informally: polynomial forms are precisely those operations on rings that are respected by ring homomorphisms.

What does this have to do with Yoneda’s lemma? Well, remember that every element {n} of a ring {R} came with an evaluation homomorphism {\phi_{R,n}: {\bf Z}[\mathrm{n}] \rightarrow R}. Conversely, every homomorphism from {{\bf Z}[\mathrm{n}]} to {R} will be of the form {\phi_{R,n}} for a unique {n} – indeed, {n} will just be the image of {\mathrm{n}} under this homomorphism. So the evaluation homomorphism provides a one-to-one correspondence between elements of {R}, and ring homomorphisms in {\mathrm{Hom}({\bf Z}[\mathrm{n}], R)}. This correspondence is at the level of sets, so this gives the identification

\displaystyle \mathrm{Forget} \equiv \mathrm{Hom}({\bf Z}[\mathrm{n}], -).

Thus our identification can be written as

\displaystyle \mathrm{Forget}({\bf Z}[\mathbf{n}]) \equiv \mathrm{Hom}( \mathrm{Hom}({\bf Z}[\mathrm{n}], -), \mathrm{Forget} )

which is now clearly a special case of the Yoneda lemma

\displaystyle F(A) \equiv \mathrm{Hom}( \mathrm{Hom}(A, -), F )

that applies to any functor {F: {\mathcal C} \rightarrow \mathbf{Set}} from a (locally small) category {{\mathcal C}} and any object {A} in {{\mathcal C}}. And indeed if one inspects the standard proof of this lemma, it is essentially the same argument as the argument we used above to establish the identification (5). More generally, it seems to me that the Yoneda lemma is often used to identify “formal” objects with their “functional” interpretations, as long as one simultaneously considers interpretations across an entire category (such as the category of rings), as opposed to just a single interpretation in a single object of the category in which there may be some loss of information due to the peculiarities of that specific object. Grothendieck’s “functor of points” interpretation of a scheme, discussed in this previous blog post, is one typical example of this.