For me category theory is more concrete. The basic reason is simple: categories provide type information. If all you want to do is grind out some calculations to pass an exam, then yeah, memorize the ring axioms and have at it. Anything more than that, and you need direction, or context, Ie. types, categories. Once you de-categorify that's when things become mysterious and confusing, and the impressive calculations abound. But it's just because you forgot the concrete thing that you are calculating.
It's all a bit paradoxical, and I'm not explaining it very well. (And I have no clue about the Haskell stuff.)
I fail to see how a category theory perspective is going to help with "grinding out" any interesting theorems about commutative rings, such as:
- Every commutative ring has a maximal ideal.
- The quotient of a commutative ring by a maximal ideal is a field.
- The quotient of a commutative ring by a prime ideal is an integral domain.
- The classification of finitely generated modules over a principal ideal domain.
There are some universal properties which define what it means to be a quotient, which can help you work with them. But for example, the only "categorical" definition of prime ideals that I know is based on that third theorem.
I think it is easy to forget just how much underlying theory you need to know before you can check that the categorical analogues of definitions are even the same as the original ones.
It's all a bit paradoxical, and I'm not explaining it very well. (And I have no clue about the Haskell stuff.)