> I think going down the route of trying to identify fields of maths with specific “things that can be stated in category theoretical language” is a bit wrong and bordering on hubris - sure, all group representations can be stated as some kind of functor between categories ….
I missed this point in my earlier reply, and it's too late to edit. I certainly agree that it's easy to be misled about the utility or appropriateness of a category-theoretic taxonomy, but I wasn't attempting that here. Rather, I was just looking for a way to capture the intuition I have about the different way in which representation theory looks at what are, in some sense, the same kinds of objects as those considered in abstract linear algebra.
(Come to that, I was specifically avoiding looking at group representations as functors; not that it much matters, since, again, the category theory was just a way to try to formalise intuition rather than an attempt to prove anything, but I was actually going the other way by regarding the representations as objects.)
I missed this point in my earlier reply, and it's too late to edit. I certainly agree that it's easy to be misled about the utility or appropriateness of a category-theoretic taxonomy, but I wasn't attempting that here. Rather, I was just looking for a way to capture the intuition I have about the different way in which representation theory looks at what are, in some sense, the same kinds of objects as those considered in abstract linear algebra.
(Come to that, I was specifically avoiding looking at group representations as functors; not that it much matters, since, again, the category theory was just a way to try to formalise intuition rather than an attempt to prove anything, but I was actually going the other way by regarding the representations as objects.)