On the categorical meaning of Hausdorff and Gromov distances, I

Journal article

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.

Authors: Akhvlediani, A; Clementino, M; Tholen, W

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Topology and its Applications
• Volume: 157
• Issue: 8
• Pages: 1275–1295
• Publication date: 2010-06-01
• Edition: Publisher's version
• DOI: 10.1016/j.topol.2009.06.018
• ISSN: 0166-8641
• Language: English
• Keywords: <em>V</em>-category; Lawvere metric space; Gromov metric; <em>V</em>-module; Hausdorff metric
• Subjects: Analytic Topology or Topology