On the categorical meaning of Hausdorff and Gromov distances, I
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 ˜K to the category V-Mod of V-categories, with V-modules
as morphisms.
