Monoidal category action

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

[math]\displaystyle{ \cdot: S \times X \to X }[/math]

such that there are natural isomorphisms [math]\displaystyle{ s \cdot (t \cdot x) \simeq (s \cdot t)\cdot x }[/math] and [math]\displaystyle{ e \cdot x \simeq x }[/math] 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

• 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.