Representation over a category

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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