One thing I would like to understand is what limitations of set theory made it necessary to invent/discover category theory. Can someone enlighten me? What do categories let us do that we can’t do with sets?
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.
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 :)
>what limitations of set theory made it necessary to invent/discover category theory?
Category theory did not start as alternative to set theory.
But: "Category theory even leads to a different theoretical conception of set and, as such, to a possible alternative to the standard set theoretical foundation for mathematics."
>What do categories let us do that we can’t do with sets?
"At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated"
Some category theory constructions like adjoints and monads are higher level and more powerful than basic set theory constructions like power set.
"The number of mathematical constructions that can be described as adjoints is simply stunning."
I know how it relates to monoids, rather than to sets. For example, you cannot just multiply together any two matrices (like you can with monoids); they need to have the correct dimensions. So, in category theory, this corresponds to the composition of morphisms, so in this case, the objects are the number of rows/columns and the morphisms are the matrices.
