As somebody who had a comparatively gentle graduate training in algebra, the import of Yoneda’s lemma in class idea has all the time eluded me considerably; the assertion and proof are easy sufficient, however undoubtedly have the “summary nonsense” taste that one typically ascribes to this a part of arithmetic, and I struggled to attach it to the extra grounded types of instinct, corresponding to these primarily based on concrete examples, that I used to be extra comfy with. There’s a standard MathOverflow submit dedicated to this query, with many solutions that have been useful to me, however I nonetheless felt vaguely dissatisfied. Nevertheless, not too long ago when pondering the very concrete idea of a polynomial, I managed to by chance bump into a particular case of Yoneda’s lemma in motion, which clarified this lemma conceptually for me. In the long run it was a quite simple commentary (and could be extraordinarily pedestrian to anybody who works in an algebraic discipline of arithmetic), however as I discovered this beneficial to a non-algebraist corresponding to myself, and I assumed I might share it right here in case others equally discover it useful.

In algebra we see a distinction between a polynomial kind (also referred to as a formal polynomial), and a polynomial operate, though this distinction is commonly elided in additional concrete purposes. A polynomial kind in, say, one variable with integer coefficients, is a proper expression of the shape

the place are coefficients within the integers, and is an indeterminate: a logo that’s typically supposed to be interpreted as an integer, actual quantity, complicated quantity, or component of some extra common ring , however is for now a purely formal object. The gathering of such polynomial varieties is denoted , and is a commutative ring.

A polynomial kind might be interpreted in any ring (even non-commutative ones) to create a polynomial operate , outlined by the system

for any . This definition (2) appears to be like so just like the definition (1) that we often abuse notation and conflate with . This conflation is supported by the identification theorem for polynomials, that asserts that if two polynomial varieties agree at an infinite variety of (say) complicated numbers, thus for infinitely many , then they agree as polynomial varieties (i.e., their coefficients match). However this conflation is typically harmful, notably when working in finite attribute. As an example:

The above examples present that if one solely interprets polynomial varieties in a particular ring , then some details about the polynomial might be misplaced (and a few options of the polynomial, corresponding to roots, could also be “invisible” to that interpretation). However this seems to not be the case if one considers interpretations in all rings concurrently, as we will now talk about.

If are two completely different rings, then the polynomial capabilities and arising from deciphering a polynomial kind in these two rings are, strictly talking, completely different capabilities. Nevertheless, they’re typically carefully associated to one another. As an example, if is a subring of , then agrees with the restriction of to . Extra typically, if there’s a ring homomorphism from to , then and are intertwined by the relation

which mainly asserts that ring homomorphism respect polynomial operations. Observe that the earlier commentary corresponded to the case when was an inclusion homomorphism. One other instance comes from the complicated conjugation automorphism on the complicated numbers, by which case (3) asserts the identification

for any polynomial operate on the complicated numbers, and any complicated quantity .

What was stunning to me (as somebody who had not internalized the Yoneda lemma) was that the converse assertion was true: if one had a operate related to each ring that obeyed the intertwining relation

for each ring homomorphism , then there was a novel polynomial kind such that for all rings . This appeared stunning to me as a result of the capabilities have been a priori arbitrary capabilities, and as an analyst I might not anticipate them to have polynomial construction. However the truth that (4) holds for all rings and all homomorphisms is in reality somewhat highly effective. As an analyst, I’m tempted to proceed by first working with the ring of complicated numbers and profiting from the aforementioned identification theorem, however this seems to be difficult as a result of doesn’t “discuss” to all the opposite rings sufficient, within the sense that there usually are not all the time as many ring homomorphisms from to as one would really like. However there may be in reality a extra elementary argument that takes benefit of a very related (and “talkative”) ring to the speculation of polynomials, particularly the ring of polynomials themselves. Given every other ring , and any component of that ring, there’s a distinctive ring homomorphism from to that maps to , particularly the analysis map

that sends a polynomial kind to its analysis at . Making use of (4) to this ring homomorphism, and specializing to the component of , we conclude that

for any ring and any . If we then outline to be the formal polynomial

then this identification might be rewritten as

and so we’ve got certainly proven that the household arises from a polynomial kind . Conversely, from the identification

legitimate for any polynomial kind , we see that two polynomial varieties can solely generate the identical polynomial capabilities for all rings if they’re an identical as polynomial varieties. So the polynomial kind related to the household is exclusive.

We’ve got thus created an identification of kind and performance: polynomial varieties are in one-to-one correspondence with households of capabilities obeying the intertwining relation (4). However this identification might be interpreted as a particular case of the Yoneda lemma, as follows. There are two classes in play right here: the class of rings (the place the morphisms are ring homomorphisms), and the class of units (the place the morphisms are arbitrary capabilities). There may be an apparent forgetful functor between these two classes that takes a hoop and removes all the algebraic construction, abandoning simply the underlying set. A group of capabilities (i.e., -morphisms) for every in that obeys the intertwining relation (4) is exactly the identical factor as a pure transformation from the forgetful functor to itself. So we’ve got recognized formal polynomials in as a set with pure endomorphisms of the forgetful functor:

Informally: polynomial varieties are exactly these operations on rings which can be revered by ring homomorphisms.

What does this need to do with Yoneda’s lemma? Effectively, keep in mind that each component of a hoop got here with an analysis homomorphism . Conversely, each homomorphism from to will probably be of the shape for a novel – certainly, will simply be the picture of underneath this homomorphism. So the analysis homomorphism supplies a one-to-one correspondence between components of , and ring homomorphisms in . This correspondence is on the degree of units, so this provides the identification

Thus our identification might be written as

which is now clearly a particular case of the Yoneda lemma

that applies to any functor from a (domestically small) class and any object in . And certainly if one inspects the usual proof of this lemma, it’s basically the identical argument because the argument we used above to determine the identification (5). Extra typically, it appears to me that the Yoneda lemma is commonly used to establish “formal” objects with their “purposeful” interpretations, so long as one concurrently considers interpretations throughout a whole class (such because the class of rings), versus only a single interpretation in a single object of the class by which there could also be some lack of info because of the peculiarities of that particular object. Grothendieck’s “functor of factors” interpretation of a scheme, mentioned in this earlier weblog submit, is one typical instance of this.