[agrounds]

Category theory was not a response to any limitations of set theory, but rather a collection of new abstractions, still grounded in set theory (originally anyway). The first paper introducing these abstractions was by Eilenberg and Mac Lane [1], who formalized for the first time the idea of natural functions between mathematical objects.

For a long time prior to E&M, mathematicians had used an informal notion of “natural” or “canonical” mapping, which meant something like one special mapping out of several available ones. Especially important is the idea of natural isomorphisms. Just knowing that two objects are isomorphic is often not good enough to prove results about them because you have to make a choice about which isomorphism of several you’re using, and you might have to make such an arbitrary choice about infinitely many pairs of objects all at once. Having a canonical choice solves this problem.

Prior to E&M, mathematicians couldn’t formalize this idea of canonical choice. They would hand wave about how natural their choice of isomorphism was and how this allowed them to avoid making arbitrary choices. Then E&M defined categories, functors, and natural transformations to formalize this idea of naturality. Their motivation was algebraic topology, but the abstractions they defined turned out to be extremely broadly useful across all much of mathematics.

[1] https://www.ams.org/journals/tran/1945-058-00/S0002-9947-194...

[omginternets]

Ah, this is right on the money in terms of the level of explanation I was looking for. Thank you so much for that, and thank you for the ref, which I will read during my long wait at the DMV today :)