nLab
join

this entry is about the notion of colimits in posets. For the notion of join of simplicial sets and join of quasi-categories see there.

Definition

If $x$ and $y$ are elements of a poset, then their join, or supremum, is an element $x\vee y$ of the poset such that:

• $x\le x\vee y$ and $y\le x\vee y$;
• if $x\le a$ and $y\le a$, then $x\vee y\le a$. Such a join may not exist; if it does, then it is unique.

In a proset, a join may be defined similarly, but it need not be unique. (However, it is still unique up the natural equivalence in the proset.)

The above definition is for the join of two elements of a poset, but it can easily be generalised to any number of elements. It may be more common to use ‘join’ for a join of finitely many elements and ‘supremum’ for a join of (possibly) infinitely many elements, but they are the same concept. The join may also be called the maximum if it equals one of the original elements.

A poset that has all finite joins is a join-semilattice. A poset that has all suprema is a suplattice.

A join of subsets or subobjects is called a union.

Special cases

A join of zero elements is a bottom element. Any

element $a$ is a join of that one element.

Properties

As a poset is a special kind of category, a join is simply a coproduct in that category.

Related concepts

• join

• meet