nLab enriched preorder

Contents

Context

$(0,1)$ -Category theory

Idea
Defintion
Examples
See also

Idea

Recall (here) that a preorder may be understood as a thin (0,1)-category, hence as a thin category enriched over the cartesian monoidal proset of truth values. In generalization, one may speak of enriching preorders over other monoidal posets.

Defintion

Let $(M, \leq, \wedge, \top)$ be a monoidal poset. A $M$ -enriched proset or proset enriched over/in $M$ is a set $P$ with a binary function $o:P \times P \to M$ such that

for every $a \in P$ , $b \in P$ , and $c \in P$ , $o(a, b) \wedge o(b, c) \leq o(a, c)$
for every $a \in P$ , $\top \leq o(a, a)$ .

Examples

An ordinary poset is just an $\Omega$ -enriched poset, with $\Omega$ the set of truth values.
A Lawvere metric space is a $\overline{\mathbb{R}_{\geq 0}}$ -enriched proset, where $\overline{\mathbb{R}_{\geq 0}}$ are the non-negative extended Dedekind real numbers.
A quasipseudometric space is a $\mathbb{R}_{\geq 0}$ -enriched proset, where $\mathbb{R}_{\geq 0}$ are the non-negative Dedekind real numbers.
A quasimetric space is a $\mathbb{R}_{\geq 0}$ -enriched poset, where $\mathbb{R}_{\geq 0}$ are the non-negative Dedekind real numbers.
A pseudometric space is a $\mathbb{R}_{\geq 0}$ -enriched symmetric proset, where $\mathbb{R}_{\geq 0}$ are the non-negative Dedekind real numbers.
A metric space is a $\mathbb{R}_{\geq 0}$ -enriched set, where $\mathbb{R}_{\geq 0}$ are the non-negative Dedekind real numbers.
A set with an irreflexive comparison, such as an apartness relation or a linear order, is an $\Omega^\op$ -enriched poset, where the co-Heyting algebra $\Omega^\op$ is the opposite poset of the set of truth values $\Omega$ .