On the Categorical Meaning of Hausdorff

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 K̃ to the category V-Mod of V-categories, with V-modules as morphisms.