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On the categorical meaning of Hausdorff and Gromov distances, I

Hausdor and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdor 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 de...