# nLab join-semilattice object

Contents

category theory

## Applications

#### Limits and colimits

limits and colimits

(0,1)-category

(0,1)-topos

# Contents

## Idea

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

## Definition

In a finitely complete category $C$, a join-semilattice object is a cocartesian monoidal preordered object that is also a partially ordered object.