On the categorical meaning of Hausdorff and Gromov distances, I

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.