Monoidal category action

In algebra, an action of a monoidal category ${\displaystyle (S,\otimes ,e)}$ on a category ${\displaystyle X}$ is a functor

${\displaystyle \cdot :S\times X\to X}$

such that there are natural isomorphisms ${\displaystyle s\cdot (t\cdot x)\simeq (s\otimes t)\cdot x}$ and ${\displaystyle e\cdot x\simeq x}$, which satisfy the coherence conditions analogous to those in ${\displaystyle S}$.^[1] ${\displaystyle S}$ is said to act on ${\displaystyle X}$.

Any monoidal category ${\displaystyle S}$ is a monoid object in ${\displaystyle {\mathsf {Cat}}}$ with the monoidal product being the category product. This means that ${\displaystyle X}$ equipped with an ${\displaystyle S}$-action is exactly a module over a monoid in ${\displaystyle {\mathsf {Cat}}}$.

For example, ${\displaystyle S}$ acts on itself via the monoid operation ${\displaystyle \otimes }$.

Notes

1. Weibel 2013, Ch. IV, Definition 4.7.

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