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 $\mathsf{C}\mathsf{a}\mathsf{t}$ 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 $\mathsf{C}\mathsf{a}\mathsf{t}$.

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

References

• Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. 145. American Mathematical Society. ISBN 978-0-8218-9132-2. http://www.math.rutgers.edu/~weibel/Kbook.html. 
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