Category theory and micro-epiphanies

Once in a while something big suddenly makes sense. Far more often, things make sense a little at a time.

I was talking with someone a few days ago, and we both said that we never had an epiphany when category theory suddenly made sense. Instead, we both said we had a sequence of micro-epiphanies.

Here's one micro-epiphany I had along the way: Category theory seems much more existential than other areas of math, but it's not. Or not much.

By "existential" I mean definitions and theorems that state something exists. For every set of objects that do this, there exists an object that does that. Instead of applying a series of operations, such as when you solve an algebraic equation or differentiate a function, you're always having to argue that one thing exists because another thing exists. I found this frustrating until I realized that category theory isn't that different from other areas where I find existential arguments so intuitive that I hardly notice them.

Analysis and topology are filled with existential theorems. For every I there exists a I " For every open cover there exists a subcover " For every point there exists an open set " However, the existential arguments are strongly motivated by geometric intuition. It takes more work, at least for me, to motivate the existential arguments in category theory. But it was helpful for me to realize that category theory isn't that different from other areas of math I find more intuitive.