action

Definition

An action of an object $X$ on an object $Y$ of a category $\mathcal{C}$ is a homomorphism

$$\mathcal{A}:X⟶{\text{Aut}}_{\mathcal{C}}(Y)$$

Note, this requires $X$ to have some sort of compatible structure with ${\text{Aut}}_{\mathcal{C}}(Y)$. There is generally a lot of structure from ${\text{Aut}}_{\mathcal{C}}(Y)$, so this is not impossible to find interesting examples.

Examples of actions

• group action
• R-module