In group theory, a group can be defined abstractly as a set with a binary operation obeying certain axioms, or concretely as a bunch of permutations on some set (which doesn’t need to include all permutations, but must be closed under composition and inverse). The two views are equivalent by Cayley’s theorem, and I think the second view is more helpful, at least for beginners.
I don’t know very much about category theory, but maybe we could do something similar there? Since every small category has a faithful functor into Set, it can be defined as a bunch of sets and functions between them. It doesn’t need to include all sets or functions, but must be closed under composition and include each set’s identity function to itself.
For example, the divisibility category from the post can be seen as a category of sets like {1,...,n} and functions that are unital ring homomorphisms from Z/mZ to Z/nZ (of which there’s exactly one if n divides m, and zero otherwise). And the category of types and functions in some programming language can be seen as a category containing some sets of things-with-bottoms and monotone functions between them. So in both of these cases, going to sets leads to some nice math.
I’ve heard that the set intuition starts to break down once you study more category theory, but haven’t gotten to that point yet.
In group theory, a group can be defined abstractly as a set with a binary operation obeying certain axioms, or concretely as a bunch of permutations on some set (which doesn’t need to include all permutations, but must be closed under composition and inverse). The two views are equivalent by Cayley’s theorem, and I think the second view is more helpful, at least for beginners.
I don’t know very much about category theory, but maybe we could do something similar there? Since every small category has a faithful functor into Set, it can be defined as a bunch of sets and functions between them. It doesn’t need to include all sets or functions, but must be closed under composition and include each set’s identity function to itself.
For example, the divisibility category from the post can be seen as a category of sets like {1,...,n} and functions that are unital ring homomorphisms from Z/mZ to Z/nZ (of which there’s exactly one if n divides m, and zero otherwise). And the category of types and functions in some programming language can be seen as a category containing some sets of things-with-bottoms and monotone functions between them. So in both of these cases, going to sets leads to some nice math.
I’ve heard that the set intuition starts to break down once you study more category theory, but haven’t gotten to that point yet.