In algebra, an action of a

monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...

''S'' on a category ''X'' is a

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

\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''. If there is such an action, ''S'' is said to act on ''X''. For example, ''S'' acts on itself via the monoid operation ⊗.

References

* Monoidal categories Functors {{algebra-stub