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Keywords:
complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal hom
complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal hom
Summary:
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.
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