On the categorical meaning of Hausdorff and Gromov distances, I

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

2009-01-06

arxiv.org/abs/0901.0618v1

Mathematics

Category Theory

27 pages

Scientific paper

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.

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