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:
- There is a unit $u:1 \rightarrow A$
- There is a multiplication $m:A\otimes A \rightarrow A$
- There is a counit $q:A \rightarrow 1$
- There is a comultiplication $\Delta:A \rightarrow A \otimes A$
- There is a antipode $S:A\rightarrow A$
such that $(A,m,q)$ is a algebra object:
- $m\circ (m \otimes id)=m(id\otimes m)$
- $m\circ(u\otimes id)=id=m\circ(id \otimes u)$
and $(A,\Delta, q)$is a coalgebra
- $(\Delta \otimes id)\circ \Delta = (id \otimes \Delta)\circ \Delta)$
- $(id\otimes q) \circ \Delta=id=(q \otimes id )\circ \Delta$
the morphisms $\Delta$ and $q$ are algebra morphisms
- $q \circ u=id_1$
- $(q\otimes q)= q \circ m$
- $\Delta \circ u= u \otimes u$
- $m \otimes m(id \otimes c \otimes id)\circ(\Delta \otimes \Delta) =\Delta \circ m $
and the antipode satisfies
- $m\circ (S\otimes id)\circ\Delta= u \circ q=m\circ (id \otimes S)\circ\Delta$
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.