nLab idempotent monoid object

This article is about internalizing the algebraic notion of idempotent monoid in general enough categories. For the category theoretic notion of idempotent monoid, see idempotent monoid in a monoidal category.

Idea

The notion of an idempotent monoid object is the generalization of the algebraic notion of idempotent monoid as one passes from the ambient category of sets into more general ambient categories with suitable properties.

Definition

In a monoidal category with diagonals $(C, \otimes, I, \Delta)$, an idempotent monoid object is a monoid object $(M, \mu, \eta)$ such that $\mu \circ \Delta_M = \mathrm{id}_M$, where $\Delta_M$ is the diagonal morphism of $M$ and $\mathrm{id}_M$ is the identity morphism of $M$.

See also

• monoid object

• semilattice object

• idempotent monoid