[Paper Review] 'Hausdorff distance' via conical cocompletion

Isar Stubbe|ArXiv.org|Mar 16, 2009

Homotopy and Cohomology in Algebraic Topology13 references10 citations

TL;DR

This paper establishes that in quantaloid-enriched categories, saturated classes of weights correspond precisely to full sub-KZ-doctrines of the free cocompletion KZ-doctrine, with conical weights forming such a class. The resulting KZ-doctrine generalizes the Hausdorff distance construction to quantaloid-enriched categories, providing a categorical foundation for generalized metric distances via enriched colimits and adjointness conditions.

ABSTRACT

In the context of quantaloid-enriched categories, we explain how each saturated class of weights defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines which arise as full sub-KZ-doctrines of the free cocompletion, are characterised by two simple "fully faithfulness" conditions. Conical weights form a saturated class, and the corresponding KZ-doctrine is precisely (the generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of [Akhvlediani et al., 2009].

Motivation & Objective

• To characterize saturated classes of weights in quantaloid-enriched categories via full sub-KZ-doctrines of the free cocompletion KZ-doctrine.
• To identify necessary and sufficient conditions—expressed as two 'fully faithfulness' properties—for a KZ-doctrine to arise as a full sub-KZ-doctrine.
• To demonstrate that conical weights form a saturated class, thereby yielding a generalized Hausdorff doctrine.
• To provide a categorical framework for the Hausdorff distance construction in enriched category theory beyond metric spaces.
• To lay the groundwork for extending such constructions to Gromov distances in involutive quantaloids, though not pursued in this work.

Proposed method

• Utilizes the theory of KZ-doctrines and their full sub-KZ-doctrines to characterize cocompletion processes in quantaloid-enriched categories.
• Applies the concept of saturated classes of weights, where each class defines a unique KZ-doctrine via universal properties of enriched colimits.
• Employs adjointness conditions in the 2-category of quantaloid-enriched categories to define 'fully faithful' properties that characterize the relevant KZ-doctrines.
• Applies the general framework of conical cocompletion to show that conical weights generate a KZ-doctrine isomorphic to the Hausdorff doctrine.
• Uses the fact that distributors with left adjoint components correspond to Cauchy presheaves, and proves that the class of such presheaves is saturated.
• Leverages the universal property of free cocompletion and the structure of sup-lattices in quantaloids to derive the correspondence between saturated classes and KZ-doctrines.

Experimental results

Research questions

• RQ1 Which KZ-doctrines on quantaloid-enriched categories arise as full sub-KZ-doctrines of the free cocompletion KZ-doctrine?
• RQ2 How can saturated classes of weights in quantaloid-enriched categories be characterized via KZ-doctrines?
• RQ3 What is the categorical structure underlying the Hausdorff distance in enriched categories, and how does it generalize beyond metric spaces?
• RQ4 Can the conical weight construction be shown to define a saturated class, and what KZ-doctrine does it generate?
• RQ5 What conditions ensure that a KZ-doctrine on a quantaloid-enriched category is fully faithful in the sense of preserving essential structure?

Key findings

• Each saturated class of weights in a quantaloid-enriched category defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine.
• The KZ-doctrines that arise as full sub-KZ-doctrines of the free cocompletion are characterized by two simple 'fully faithfulness' conditions on their components.
• The class of conical weights is saturated, and the corresponding KZ-doctrine on quantaloid-enriched categories generalizes the Hausdorff doctrine previously defined for commutative quantales.
• The resulting Hausdorff KZ-doctrine sends each category to its Cauchy completion when restricted to the conical weight class, aligning with known constructions in enriched category theory.
• The framework provides a categorical foundation for the Hausdorff distance in generalized metric settings, extending Lawvere’s metric space enrichment to quantaloid-enriched categories.
• The paper establishes a general mechanism for deriving KZ-doctrines from saturated classes, with conical weights as a key example, and suggests that Gromov distances may be developed in the same framework under involutive quantaloids.