details
.: download
Author(s)
Andrei Akhvlediani; Maria Manuel Clementino; Walter Tholen;
Title 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 Vcategory X, provides the powerset of X with a suitable Vcategory structure, is part of a monad on VCat whose EilenbergMoore algebras are ordercomplete. The Gromov construction may be pursued for any endofunctor K of VCat. In order to define the Gromov "distance" between Vcategories X and Y we use Vmodules between X and Y, rather than Vcategory 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 VMod of Vcategories, with Vmodules as morphisms.
Preprint series Prépublicações do Departamento de Matemática da Universidade de Coimbra
Issue 0901
Year 2009
