In algebra, an action of a monoidal category ( S , ⊗ , e ) {\displaystyle (S,\otimes ,e)} on a category X {\displaystyle X} is a functor
⋅ : S × X → X {\displaystyle \cdot :S\times X\to X}
such that there are natural isomorphisms s ⋅ ( t ⋅ x ) ≃ ( s ⊗ t ) ⋅ x {\displaystyle s\cdot (t\cdot x)\simeq (s\otimes t)\cdot x} and e ⋅ x ≃ x {\displaystyle e\cdot x\simeq x} , which satisfy the coherence conditions analogous to those in S {\displaystyle S} . S {\displaystyle S} is said to act on X {\displaystyle X} . Any monoidal category S {\displaystyle S} is a monoid object in C a t {\displaystyle {\mathsf {Cat}}} with the monoidal product being the category product. This means that X {\displaystyle X} equipped with an S {\displaystyle S} -action is exactly a module over a monoid in C a t {\displaystyle {\mathsf {Cat}}} . For example, S {\displaystyle S} acts on itself via the monoid operation ⊗ {\displaystyle \otimes } .
Notes
References Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. Vol. 145. American Mathematical Society. ISBN 978-0-8218-9132-2. Module over a monoid at the nLab