Monoidal category action

Template:One source 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 $\otimes$.

References

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