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Author(s)
Andrei Akhvlediani; Maria Manuel Clementino; Walter Tholen;

Title
On the categorical meaning of Hausdorff and Gromov distances, I

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.

Preprint series
Pré-publicações do Departamento de Matemática da Universidade de Coimbra

Issue
09-01

Year
2009

 
     
 

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