Address: Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail: achirvas@buffalo.edu
Abstract: We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally $\aleph _1$-presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-$\aleph _0$-generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include the automatic completeness of a colimit of a diagram of bi-Lipschitz morphisms between complete metric spaces and a characterization of those pairs (metric space, unital $C^*$-algebra) that have a tensor product in the CMet-enriched category of unital $C^*$-algebras.
AMSclassification: primary 54E50; secondary 54E40, 51F30, 18A30, 18C35, 18D20, 18D15, 46L05, 46L09.
Keywords: complete metric space, path metric, intrinsic metric, gluing, convex, monoidal closed, enriched, tensored, locally presentable, colimit, internal hom.
DOI: 10.5817/AM2024-2-61