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

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

This article relies largely or entirely on a single source. (May 2024)

such that there are natural isomorphisms ${\displaystyle s\cdot (t\cdot x)\simeq (s\cdot t)\cdot x}$ and ${\displaystyle e\cdot x\simeq x}$ 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 ⊗.