Actions of monoidal categories

Action of a monoidal category

Let $(\mathcal{M}, \otimes, I_\mathcal{M}, a, l, r)$ be a monoidal category. A (left) $\mathcal{M}$-action is:

• a category $\mathcal{C}$
• a functor $\mathcal{M} \odot \mathcal{C} \rightarrow \mathcal{C}$
• a natural isomorphism $\lambda_A: I \odot A \xrightarrow{\sim} A~\forall A \in Ob(\mathcal{C})$ called the unitor
• a natural isomorphism $\alpha_{M, N, A}: (M \otimes N) \odot A \xrightarrow{\sim} M \odot (N \odot A)~\forall M, N \in Ob(\mathcal{M})~\forall A \in Ob(\mathcal{C})$ called the actor

such that the following diagrams commute with respect to the unitors and associators of $\mathcal{M}$:

Resources

ncatlab