Codiscrete Preorder

definition example

From every Set $X$ we may construct its codiscrete preorder $(X,\le )$ by equipping it with the total relation $X\times X\subseteq X\times X$: for all $x,y$ we have $x\le y$ (and hence $y\le x$). Its poset reflection is the one-element poset.

Sources: 7 Sketches Example 1.33; Kittenlab Lecture 5.

The construction is a functor $\mathbf{Set}\to \mathbf{Preord}$, right adjoint to the underlying-set functor (compare the Discrete Preorder, which is the left adjoint). As a category, a codiscrete preorder is the Codiscrete Category (an “indiscrete” Groupoid in which every hom-set is a singleton), and it is a Dagger Preorder.