Metric enrichment, finite generation, and the path coreflection. (English) Zbl 07830506
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 I. Di Liberti and J. Rosický [Theory Appl. Categ. 38, 661–683 (2022; Zbl 1492.18006)]. 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.
MSC:
| 54E50 | Complete metric spaces |
| 54E40 | Special maps on metric spaces |
| 51F30 | Lipschitz and coarse geometry of metric spaces |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
| 18C35 | Accessible and locally presentable categories |
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L09 | Free products of \(C^*\)-algebras |
Keywords:
complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal homCitations:
Zbl 1492.18006References:
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