On the categorical meaning of Hausdorff and Gromov distances, I |
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Authors: | Andrei Akhvlediani Maria Manuel Clementino Walter Tholen |
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Institution: | a Oxford University Computing Laboratory, Oxford OX1 3QD, United Kingdom b Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal c Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada |
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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 to the category V-Mod of V-categories, with V-modules as morphisms. |
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Keywords: | primary 18E40 secondary 18A99 |
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