nLab Pos

Definition

$Pos$ (or $Poset$ or $Posets$) is the category whose objects are posets and whose morphisms are monotone (weakly increasing) maps.

Since posets can be identified with (0,1)-categories, $Pos$ can be identified with the full subcategory of Cat spanned by these; thus it might also be called $(0,1)Cat$.

Properties

The hom-sets of $Pos$ themselves have the structure of posets, because $Pos$ is a cartesian closed category. This means $Pos$ is a 2-poset (aka $(1,2)$-category) or locally posetal 2-category. If Set is the primordial example of a category and Cat is the primordial example of a 2-category, then $Pos$ is the primordial example of a $2$-poset.

The category $Pos$ is a locally presentable category. This implies it is complete and cocomplete. This can also be seen by viewing $Pos$ as a reflective subcategory of PreOrd, which is topological over $Set$ and therefore cocomplete.

We may illustrate the latter point of view by showing how to construct coequalizers in $Pos$. Concretely, to form the coequalizer of a pair of poset maps $f, g: X \rightrightarrows Y$, one first takes their coequalizer $q: Y \to Z$ in $Set$, and endows $Z$ with the smallest reflexive transitive relation that makes $q$ order-preserving. This makes $Z$ a preorder; it is the coequalizer of $f, g$ in $PreOrd$. Then one passes to the poset reflection of this preorder (i.e., identifying $z$ and $z'$ in $Z$ if $z \leq z'$ and $z' \leq z$) to form the coequalizer in $Pos$.

Related categories

• Set

• Grpd, ∞Grpd

• Cat, (∞,1)Cat

• (∞,n)Cat

category: category