Monoidal category action

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

$\cdot :S\times X\to X$

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

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

For example, S acts on itself via the monoid operation ⊗.

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