Metric axioms: a structural study

TL;DR

This paper explores the structure of weight functions on a set, analyzing how metric axioms shape their properties using order theory and category theory, leading to lattice-embeddability results.

Contribution

It provides a novel order-theoret and categorical analysis of collections of weight structures satisfying metric axioms, including lattice-embeddability theorems.

Findings

Order-theoretic characterization of metric structures

Categorical connections via adjunctions

Lattice-embeddability theorems for weight structures

Abstract

For a fixed set $X$, an arbitrary \textit{weight structure} $d \in [0,\infty]^{X \times X}$ can be interpreted as a distance assignment between pairs of points on $X$. Restrictions (i.e. \textit{metric axioms}) on the behaviour of any such $d$ naturally arise, such as separation, triangle inequality and symmetry. We present an order-theoretic investigation of various collections of weight structures, as naturally occurring subsets of $[0,\infty]^{X \times X}$ satisfying certain metric axioms. Furthermore, we exploit the categorical notion of adjunctions when investigating connections between the above collections of weight structures. As a corollary, we present several lattice-embeddability theorems on a well-known collection of weight structures on $X$.