The goal of this post is to unpack a remark in [1]:
… we can say this in fancier terms. Fix a field k …. We say that an elliptic curve E defined over k is that functor which …
Well that is fancy. But what does it mean?
Looking for objects
A functor is a pair of functions [2]. At the base level, a functor takes objects in one category to objects in another category. The quote above continues
… that functor which associates fields K containing k to an algebraic set of the form …
The key word here is “associates.” That must be our function between objects. Our functor maps fields containing k to a certain kind of algebraic set. So the domain of our functor must be fields containing the field k, or fields more generally if you take “containing k” to apply on both sides for all fields k.
Looking for morphisms
Categories are more than objects, and functors are more than morphisms.
A category consists of objects and morphisms between those objects. So our category of fields must also contain some sort of morphisms between fields, presumably field homomorphisms. And our category of algebraic sets must have some kind of morphisms between algebraic sets that preserve the algebraic structure.
The functor between fields and algebraic sets must map fields to algebraic sets, and maps between fields to maps between algebraic sets, in a structure-preserving way.
Categorification
The algebraic sets are what we normally think of as elliptic curves, but the author is saying we can think of this functor as an elliptic curve. This is an example of categorification: taking something that doesn’t initially involve category theory, then building a categorical scaffold around it.
Why do this? In order to accentuate structure that is implicit before the introduction of category theory. In our case, that implicit structure has to do with fields.
The most concrete way to think of an elliptic curve is over a single field, but we can look at the same curve over larger fields as well. We might start, for example, by thinking of a curve over the rational numbers ℚ, then extending our perspective to include the real numbers ℝ, and then extending it again to include the complex numbers ℂ.
The categorification of elliptic curves emphasizes that things behave well when we go from thinking of an elliptic curve as being over a field k to thinking of it as a field over an extension field K that contains k.
A functor between fields (and their morphisms) and algebraic sets (of a certain form, along with their morphisms) has to act in a “structure-preserving way” as we said above. This means that morphisms between fields carry over to morphisms between these special algebraic sets in a way that has all the properties one might reasonably expect.
Coda
This post has been deliberately short on details, in part because the line in [1] that it is expanding on is short on details. But we’ve seen how you might tease out a passing comment that something is a functor. You know the right questions to ask if you want to look into this further: what exactly are the morphisms in both categories, and does functorality tell us about elliptic curves as we change fields?
There are many ways to categorify something, some more useful than others. Useful categorifications express some structure; some fact that was a theorem before categorification becomes a corollary of the new structure after categorification. There are other ways to categorify elliptic curves, each with their own advantages and disadvantages.
Related posts
- Category theory without categories
- Category theory make a little easier for programmers
- Applied category theory
[1] Edray Herber Goins. The Ubiquity of Elliptic Curves. Notices of the American Mathematical Society. February 2019.
[2] Strictly speaking, a pair of “morphisms” that may or may not be functions.
I’m not sure if it makes any difference here, especially as this is, as you say, deliberately short on details. But my mind kept getting caught on the ideas of “field homomorphism(s)”. It’s probably because I’m used to looking at structures where you have kernels and a first homomorphism theorem, but you don’t have that with fields (kernels are ring ideals, which are trivial for fields). So “homomorphisms” in this case are actually just embeddings, i.e. extensions.
Maybe all the readers already know this, and the phrase “… over a single field, but we can look at the same curve over larger fields as well” indicates that they’re implicitly assuming that. Maybe this is also an indicator that when thinking categorically one shouldn’t get too bogged down in specifics, but “bogged down in specifics” is rather how I tend to do math. :-)