nLab
MSet

If $M$ is a monoid, then $M\mathrm{Set}$ or $M‑\mathrm{Set}$ is the category of $M$-sets. That is, an object of this category is a set $X$ equipped with an action of $M$ on $X$, and a morphism is an equivariant function?.

Don’t confuse this with the category of multisets.

• See also Understanding M-Set

category: category