# A (higher) categorical approach to representation theory

The representation theories of groups and Hopf algebras are very much alike. Taking the view point that both Hopf algebras and groups are Hopf monoids ("Hopf algebra objects") in their symmetric monoidal categories $Vect$ and $Set$ I wonder whether there is a general construction of a "representation" functor leading to this result. (*)

My first consideration is the following:

i) Consider a group $G$ to be a category with one object $g$ and the morphisms to be $G$.

A representation of $G$ is simply a functor $F:G\rightarrow Vect$.

For two representations $F_1,F_2:G\rightarrow Vect$ we call a natural transformation $\eta: F_1 \Rightarrow F_2$ a morphism of representations.

Therefore the representations of $G$ form a category. Infact because we know that tensor products of representations and a trivial representation exists, fullfilling the needed constraints, the representations of $G$ form a monoidal category.

Pretty much the same could be said about Hopf algebras.

Now my question is:

a) Could the category of representations of $G$ be realised as a single functor By this I mean is there a generalisation of the category $G$ (and $Vect$) that allows us to retain informations about $G$ and describes the representations with a single functor?

b) How can this theory be extended to the case of groups (and Hopf algebras). From what I understand the category of groups can be realised as a 2-category. With points as objects, groups as 1-morphisms and group homomorphisms as 2-morphisms. However the notion of a 2-category isn't quite clear to me. Could somebody help explain it to me?

And on the subject of two categories how can we model a monoidal structure in 2-categories?

(*) I should be clearer on what I mean by Hopf monoid (or a Hopf algebra object)

Suppose that $(C,\otimes,1,c)$ is a braided monoidal category. I call a object $A$ a hopf algebra object if:

1. There is a unit $u:1 \rightarrow A$
2. There is a multiplication $m:A\otimes A \rightarrow A$
3. There is a counit $q:A \rightarrow 1$
4. There is a comultiplication $\Delta:A \rightarrow A \otimes A$
5. There is a antipode $S:A\rightarrow A$

such that $(A,m,q)$ is a algebra object:

1. $m\circ (m \otimes id)=m(id\otimes m)$
2. $m\circ(u\otimes id)=id=m\circ(id \otimes u)$

and $(A,\Delta, q)$is a coalgebra

1. $(\Delta \otimes id)\circ \Delta = (id \otimes \Delta)\circ \Delta)$
2. $(id\otimes q) \circ \Delta=id=(q \otimes id )\circ \Delta$

the morphisms $\Delta$ and $q$ are algebra morphisms

1. $q \circ u=id_1$
2. $(q\otimes q)= q \circ m$
3. $\Delta \circ u= u \otimes u$
4. $m \otimes m(id \otimes c \otimes id)\circ(\Delta \otimes \Delta) =\Delta \circ m$

and the antipode satisfies

1. $m\circ (S\otimes id)\circ\Delta= u \circ q=m\circ (id \otimes S)\circ\Delta$

In the case of groups the tensor product structure is given by products of groups the comultiplication is the diagonal map, the counit the projection onto the trivial group and the antipode the inversion.

Hopf algebras are, from this viewpoint, Hopf algebra objects in the category of vector spaces.

• I am not quite sure what you mean by groups being "Hopf algebra objects" in the category of groups (so all the objects are like that?). Some of the ideas here are essentially what one does when studying algebraic groups (which are in some sense the same as commutative Hopf algebras), but I don't see any way (nor any good reason) to try to turn the entire category of representations into a single functor. – Tobias Kildetoft Feb 16 '17 at 9:44
• Also, the notion of $2$-category is probably precisely the sort of thing you will need to go any further in this sort of direction, so I would advice you to get more familiar with it (it does take a lot of getting used to). – Tobias Kildetoft Feb 16 '17 at 9:45
• I've updated my question and added a explanation of Hopf algebra objects. Hope this makes it a bit more understandable. Do you know a good introductionary book on the subject of higher categories? – SeHa Feb 16 '17 at 10:34
• I am not sure your notion of Hopf algebra object is the usual one, since I have not seen the term used before in non-additive categories. For your question about tensor categories and $2$-categories: I would invite you to try to show that a $2$-category with one object is the same as a tensor category, which is a good place to start getting an idea of what $2$-categories are like. For a reference, I am not actually sure of a good one. Mac Lane has some material on them, but not that much (there is also a bit about them in the notes arxiv.org/abs/1011.0144 by Mazorchuk) – Tobias Kildetoft Feb 16 '17 at 10:40
• You repeatedly make the mistake of saying that Hopf monoids in $\mathsf{Grp}$ are groups when you (probably) want to say that the Hopf monoids in $\mathsf{Set}$ are the groups. More concretely, the multiplication and inverse maps of a group are rarely group morphisms. As said in the nlab linked by Berci, Hopf monoids in any cartesian category are the group objects (counit and comultiplicaiton comes for free, the antipode being the inverse map): hence in $\mathsf{Grp}$ they are the abelian groups (by Eckmann-Hilton's argument). – Pece Feb 17 '17 at 15:15