On the categorical meaning of Hausdorff and Gromov distances, I

TL;DR

This paper explores the categorical foundations of Hausdorff and Gromov distances within V-enriched categories, introducing new functorial constructions and extension theorems that deepen the theoretical understanding of these metrics.

Contribution

It introduces a categorical framework for Hausdorff and Gromov distances using V-enriched categories, including a monad structure and a general extension theorem for V-modules.

Findings

Hausdorff functor forms a monad on V-Cat with order-complete Eilenberg-Moore algebras

Gromov construction extended to any endofunctor K of V-Cat

A general extension theorem for V-modules and V-enriched categories

Abstract

Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov "distance" between V-categories X and Y we use V-modules between X and Y, rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax extension K to the category V-Mod of V-categories, with V-modules as morphisms.