# Representation over a category

## Definition

Suppose $G$ is a group and $\mathcal{C}$ is a category. A representation of $G$ over $\mathcal{C}$ is the following data $(A,\rho)$: an object $A \in \operatorname{Ob} \mathcal{C}$, and a homomorphism of groups $\rho:G \to \operatorname{Aut}(A)$.

We are generally interested in studying representations up to equivalence where two representations $(\rho_1,A_1)$ and $(\rho_2,A_2)$ of $G$ are termed equivalent if there exists an isomorphism $\alpha:A_1 \to A_2$ such that, for all $g \in G$:

$\alpha \circ \rho_1(g) = \rho_2(g) \circ \alpha$

## Particular cases

Category What a representation over that category is called
category of sets group action or permutation representation
category of vector spaces over a field $K$ linear representations over the field $K$