Geodesic
Metric enrichment, finite generation, and the path coreflection
Archivum mathematicum, Tome 60 (2024) no. 2, pp. 61-99.

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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.
DOI : 10.5817/AM2024-2-61
Classification : 18A30, 18C35, 18D15, 18D20, 46L05, 46L09, 51F30, 54E40, 54E50
Keywords: complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal hom 
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Chirvasitu, Alexandru. Metric enrichment, finite generation, and the path coreflection. Archivum mathematicum, Tome 60 (2024) no. 2, pp. 61-99. doi : 10.5817/AM2024-2-61. http://geodesic.mathdoc.fr/articles/10.5817/AM2024-2-61/

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