# How far can I develop representation theory from category theory?

A little background: I am currently learning a bit about category theory. I studied basic representation theory last semester (definitions, Maschke's theorem and Schur lemma, character tables, ... for finite groups and vector space over $\mathbb{C}$). In order to understand the material I am reading from Tom Leinster's book (Basic Category Theory), I decided to reformulate some definitions from my lectures about representation in the language of category theory (examples below).

For example, in my notes about representation theory, we define a linear representation as a homomorphism from a group $G$ to $GL(V)$ for some vector space $V$. As a basic first example, we can see a linear representation as a functor from $G$ (seen as a category with only one element) to the category of vector spaces over a field $k$.

Then, we often define the notion of isomorphic linear representation. In my lecture notes, it was defined as a $G$-equivariant isomorphism. If I am not mistaken, we can say the two representations $F : G \rightarrow \mathbf{Vect_k}$ and $H : G \rightarrow \mathbf{Vect_k}$ are isomorphic if $F \cong G$ in the category $[G, \mathbf{Vect_k}]$ (ie, there is some invertible natural transformation between $F$ and $H$).

It is my very first time reading about category theory, so I might say some absurdities. In my current understanding of the subject, category theory is mainly about finding patterns in seemingly different objects, and therefore, does not permit to understand every little detail (because it is there to generalize concepts).

So my question is: how far can I go into rewriting representation theory in terms of category theory objects? Maybe (big question mark), Schur lemma or Maschke's theorem might be possible to rewrite, but for more advanced results, I wonder...

• I don't know much in representation theory, but I remember Frobenius reciprocity is a consequence of adjunction. – user171326 Jan 16 '17 at 16:36
• Nice question. It's curious that I've studied category theory before representation theory, which I'm gonna study this year, and so I hope someone answer your question. – Xam Jan 16 '17 at 16:44
• As an aside, one big thing you can do from this pov is that it suggests a notion of linear representation for categories, since $[\mathcal{C}, \mathbf{Vect_k}]$ makes sense for any small category. I believe representations of quivers can be seen as representations of categories. – Hurkyl Jan 16 '17 at 19:21

RE: the comments: representations of groups and quivers are both special cases of representations of categories. If $R$ is a ring, then an $R$-rep of a group is a rep of its $R$-group algebra; an $R$-rep of a quiver is a rep of its $R$-path algebra; you can view an $R$-algebra as an $R$-linear category with one object; and the category of modules over an $R$-algebra is an $R$-linear (typically abelian) category.
Everything you see in a first course in representation theory can be encoded in category theory, if you really want to. Maschke's theorem says that (if $R$ is a field of characteristic coprime to the order of $G$) the category $C$ of $R$-reps of $G$ is semisimple. There's a pairing on the objects of $C$ defined by $\langle W,V\rangle= dim\ Hom_C(W,V)$, and this is the usual pairing on characters. Frobenius reciprocity is the fact that restriction and induction are adjoint, as mentioned in the comments. The other main results have similar interpretations.
Also, a couple of words of warning: once you leave the world of finite groups, or once you allow your coefficient ring $R$ to become sufficiently "bad", things can get more difficult; you start to encounter topological issues, which might make the category theory a bit more unpleasant.