I would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc.
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, but this is just convenient compact language for describing the definition. Any deep theorems that follow can almost certainly not be deduced from the categorical description, and are many times inhibited or obscured by it.
> 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 would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc.
That's a very good point about other kinds of 'representers' —though I think it doesn't change the central point that, in some sense, linear algebra seems to focus on "spaces as objects", whereas representation theory, loosely, focuses on "morphisms as objects".
I agree that most people will mean when they refer to representations to refer to linear actions, but every so often one does encounter references to, say, actions on sets as 'permutation representations' (although that also has a, closely related, linear-action meaning). I just meant to acknowledge that occasional extra inclusiveness.
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, but this is just convenient compact language for describing the definition. Any deep theorems that follow can almost certainly not be deduced from the categorical description, and are many times inhibited or obscured by it.