Should the elementary concepts of category theory be taught in pre-college courses? Could that be a benefit for the general population?

I appear to have been asked to answer, so I'll give it a try. A disclaimer, though: while I can perhaps claim some expertise on category theory, I am not an expert on mathematics pedagogy.

Depending on how the question is intended, I end up with a couple of different answers, so I'll try to address several different options. (Also: I’m presuming that by `elemental' you mean `elementary'. I'm not terribly familiar with Spanish, but it appears to be a common Spanish-English false friend. )

1) Should an introductory category theory course be added to the pre-college mathematics curriculum?

My answer here is a pretty firm "no." For better or for worse, much of the power of the categorical framework derives from the way in which it unifies constructions from seemingly disparate branches of mathematics. Moreover, nearly all of the interesting examples in category theory (and, in particular, the examples that provide the most intuition) come from branches of mathematics which are typically taught in college. Algebra (i.e. groups, rings, fields, etc.), topology, and geometry (of the manifold sort--not Euclidean geometry) are all fairly essential background to make sense of category theory.

As a small example: One of the nicer basic applications of adjunctions is to provide a definition for what a "free" object[1] is. Given a category[math] \mathbf{C}[/math] and a forgetful functor [math]F:\mathbf{C}\to\mathbf{Set}[/math], the free functor [math]\mathbf{Set}\to\mathbf{C}[/math] is a left adjoint [math]G[/math] of [math]F[/math]. The image of a set [math]X[/math] under [math]G[/math] is then a free object in [math]\mathbf{C}[/math]. This framework covers free groups, free vector spaces, and even discrete topological spaces. If a student is not familiar with these examples of free constructions, then it will be very difficult for them to build a good intuition for the more general case.

2) Should some elementary constructions from category theory be used in the existing pre-college mathematics curriculum (without introducing category theory as a subject more generally)?

Here my answer is still "no," but a rather more qualified "no" than before. This answer hinges on what is meant by "pre-college mathematics." Some constructions from category theory—in particular, the practice of defining objects via so-called universal properties[2]— have found their way into introductory college mathematics courses (particularly in algebra, but also in topology and elsewhere). For courses of this sort, I think there are great advantages to a slightly categorical approach--though, of course, too much "abstract nonsense" still runs the risk of depriving beginning students of valuable intuition.

Here's the thing, though: when I think of pre-college mathematics, I typically think of calculus and its various prerequisites. At least in the US, where I grew up, this seems to be the standard paradigm for mathematics education. While I'm very much in favor of trying to change this (and of introducing a broader spectrum of mathematics in high school), as things currently stand, it looks somewhat unlikely. Given that that's the case, there are simply no subjects taught at the pre-college level where the application of categorical constructions is pedagogically useful or appropriate. It would likely serve only to confuse students and muddle the curriculum.

3) Should intuitions and ideas from category theory influence the way pre-college mathematics is taught?

Here my answer is a somewhat qualified "yes." While categorical constructions and definitions are usually rather "high-tech", some of the insights gained from category theory are decidedly not. At it's core, category theory rests on the principle that mathematical objects are characterized by the kinds of maps between them: Sets are characterized by maps of sets, groups are characterized by homomorphisms, topological spaces by continuous maps, and so on. This kind of insight has implications, not only for the nitty-gritty technical details, but also for the way we think about mathematics.

One of my favorite examples (and one I think I've alluded to on Quora before) is an alternative axiomatization of set theory, called the Lawvere[3] axioms. There is a wonderful exposition for a general mathematical audience written by Tom Leinster and posted on the Arxiv[4]. As Leinster notes, these axioms are not an application of category theory. Rather, they are a new way of thinking about axiomatic set theory influenced by categorical ways of thinking. Because of this influence, the Lawvere axioms offer a different perspective on set theory--in terms of maps between sets--that I believe could be of great pedagogical utility.

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If I may be permitted one further distortion of your question: I think that, more generally, learning category theory--or any branch of mathematics, for that matter--could be of great benefit to members of the general public. Math--in all its many forms--involves grand aesthetics, new ways of thinking, and deep connections to other realms of human thought. I realize reading back over my answer, that it could be taken as an attempt to dissuade non-mathematicians from attempting to learn category theory. It is not, and I hope no one takes it as such.

Footnotes

[1] Free object - Wikipedia

[2] Universal property - Wikipedia

[3] William Lawvere - Wikipedia

[4] [1212.6543] Rethinking set theory

I’ve been struggling with this question for some time. One thing is painfully obvious: the way we currently teach math is a disaster. We are preparing kids to be 19th century accountants. Memorizing the multiplication table or perfecting long division in the era when almost every child has access to a powerful computer in their cell phone makes no sense. We teach kids how to follow a set of rigid instructions without making any errors: that’s exactly what computers are good at. Humans are bad at following instructions, and we make mistakes. We shouldn’t be competing with computers in this rigged game.

Some of these kids grow up to be computer programmers and they carry this baggage of bad math to their jobs. I’ve seen this over and over when teaching functional programming. Instead of forcing computers to work with human abstractions, most programmers try to compete with them at following rigid instructions. It puzzled me why this style of imperative programming seems more natural to most programmers until I realized that we’ve been brainwashed into it from early childhood. The damage has already been done and it’s very hard to overcome early training.

Category theory is very visual: it’s all about drawing arrows between objects. The problem is, in order to teach category theory we have to come up with good examples. Traditionally, the examples have all been taken from other, usually very advanced, branches of mathematics, so a natural assumption was that you had to first master math before you can learn category theory. This is not true, and I was pretty successful in teaching advanced category theory to programmers using mostly examples from programming. So one possibility would be to teach kids elements of category theory in parallel with programming, which is another very important skill in the 21st century.

But many concepts can be taught using simpler examples. For instance, a great example of a monoid is given by string concatenation. You don’t need to know math or programming to understand that “Mary” plus “Ann” is “MaryAnn”. You can teach kids what the initial and terminal objects are in a poset by letting them form a row in order of height or age (which can also, as a bonus, teach them a few sorting algorithms).

I realize that it’s a gargantuan task to rewrite the whole math curriculum and retrain a whole generation of math teachers, but I don’t see how we can prepare kids to be successful in the 21st century using a 19th century curriculum.

I'm gonna have to go with a “no” on this, as much as I think it'd be nice if more people learned category theory.

The thing is, while category theory is all about “universality”, which makes it ostensibly a nice thing for anybody to study, it's still very much rooted in abstract mathematics, and most people simply have no practical need to go to the level of abstraction involved.

Perhaps in 500 years, when math is refined enough that kids learn Grothendieck style results in elementary school.