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 $K$