they’re supposed to all be versions of that one, it’s just not always very easy to see how - joke was mainly about that. category theory calls things compact if they’re “small” in a very particular sense. algebraic compactness also has nothing to do with the topological notion, at least on the surface (abelian group that’s a direct summand of every group containing it as a pure subgroup). basically, every area of math where the topological notion makes no sense will invariably call something compact eventually, because mathematicians can’t resist.

sometimes if you squint you can see how it relates back to the topological notion but frequently it’s anything but obvious if you don’t already understand the field - which means when you’re trying to work things out for yourself, you just have to treat it like one more definition of the same word until you finally get it one day.

I think it’s easier if you have a prof who can just make the analogy clear from the start.