nLab semilattice object

Idea

The notion of a semilattice object is the generalization of that of semilattice as one passes from the ambient category of sets into more general ambient categories with suitable properties.

Definition

In a symmetric monoidal category with diagonals $(C, \otimes, I, \Delta)$, a semilattice object is a commutative 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

• commutative monoid object

• idempotent monoid object

• join-semilattice object

• meet-semilattice object

• lattice object

• semilattice