Poset (Category)

Note that this note discusses the category where every object is a poset. See Thin Category for the categorification of a single poset.

Definition

Let $\mathrm{Poset}$ be the category of partial orders and monotone maps.

Remarks

A f-prefixpoint is when $f(x)\le x$.
A f-postfixpoint is when $f(x)\ge x$.

Note that the definitions are reversed in some cases!!

By Lambek’s lemma, a least prefixpoint is a fixed point.