The n-Category Café

My First-off thought was that distributivity is a tricky thing, because I’ve become rather used to a rather fancy sort of distributivity, where pullbacks of a colimit diagram $D$ along a map to $D$ is again a colimit diagram; the dual thing tends not to be true, and things that don’t dualize have usually taken me a lot of acclimatization.

But, then, recalled (what I think you’re aluding to) that the statement of distributivity for rings (and the ring action part of a module) is that multiplication is a $+$-homomorphism; and further that there is a good reason for this in the case of $\mathbb{Z}$ having to do with $\mathbb{Z}$ representing the underlying set functor in Abelian Groups… I’ve no hypothesis either way whether teaching this earlier will be generally helpful, except that different students learn differently, so having as many different ways to become familiar with something can’t really hurt; expecting nonspecialists to teach them all correctly, on the other hand… Generally, the trouble I find with distributivity is that students learning calculus want it to apply all the time; but the set of solutions to the equation $sin(x+y) = sin(x) + sin(y)$ is remarkably thinner than they’d like. It’s not that they don’t know of distributivity as a thing, it’s that they’ve forgotten it’s rather special.

I expect the simplest actually category-theoretic thing that grade-school students deal with is the notion of greatest common divisor; the thing I certainly didn’t learn about gcd until university Algebra is that the gcd is also the least linear combination… But to clarify, the relation “evenly divides” makes $\mathbb{Z}$ a poset in an interesting way, such that gcd is exactly $lim$ (or maybe $colim$, depending on orientation):

$gcd(a,b)$ is a thing that both divides both $a,b$ and is divisible by everything that divides both $a,b$.

The other examples I tend to hear about are actually examples of teaching a decategorified thing in grade school (natural numbers come from finite sets!), and later on motivating a categorical something by pointing out that it categorifies something you learned in grade-school.

See for instance page 8 of this and this scan of a previous NS article feature this work.