Monoidal category action

In algebra, an action of a monoidal category S on a category X is a functor

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

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

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

References

1. Weibel, Ch. IV, Definition 4.7.

• C. Weibel "The K-book: An introduction to algebraic K-theory"