# nLab semilattice object

Contents

### Context

#### Algebra

higher algebra

universal algebra

category theory

# Contents

## 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 for every morphism $a:I \to M$, $\mu \circ \Delta_M \circ a = a$, where $\Delta_M$ is the diagonal morphism of $M$.