nLab enriched directed loop graph

Idea

A directed loop graph may be understood as a thin (0,1)-directed pseudograph, hence as a thin directed pseudograph enriched over the cartesian monoidal poset of truth values. In generalization, one may speak of enriching directed loop graphs over other monoidal posets.

Defintion

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

Examples

• An ordinary directed loop graph is just an $\Omega$-enriched directed loop graph, with $\Omega$ the set of truth values.

• A Lawvere metric space is a $\overline{\mathbb{R}_{\geq 0}}$-enriched proset, where $\mathbb{R}_{\geq 0}$ are the non-negative 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 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.

See also

• meet-semilattice
• pseudometric space
• enriched proset
• premetric space