nLab
category of representations

Idea

The category of all representations of some type.

If we think, fully generally, of a representation of $C$ on $D$ as nothing but a functor $C\to D$, then the representation category is just the functor category $\mathrm{Rep}(C,D)=\mathrm{Func}(C,D)$.

Notably when $G$ is a group, an ordinary linear representation is a functor $BG\to \mathrm{Vect}$ from the delooping groupoid of $G$ to Vect, and so the representation category is

$$\mathrm{Rep}(G)=\mathrm{Func}(BG,\mathrm{Vect}).$$Rep(G) = Func(\mathbf{B}G,Vect) \,.

Properties

Tannakian reconstruction

Representation categories come with forgetful functors that send a representation to the underlying object that carries the representation.

For instance for group representations the canonical inclusion $*\to BG$ induces the functor $\mathrm{Func}(BG,\mathrm{Vect})\to \mathrm{Func}(*,\mathrm{Vect})$, hence

$$F:\mathrm{Rep}(G)\to \mathrm{Vect}$$F : Rep(G) \to Vect

that sends a representation to its underlying vector space. The Tannakian reconstruction theorem says that the group $G$ may be recovered essentially as the group of automorphisms of the fiber functor $F$.

category: category