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.
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 $a \in P$, $\top \leq o(a, a)$.
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 $\Omega$.
Last revised on September 22, 2022 at 18:40:57. See the history of this page for a list of all contributions to it.