Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu

Archivum Mathematicum (2024)

  • Issue: 2, page 61-99
  • ISSN: 0044-8753

Abstract

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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 ℵ 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- ℵ 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.

How to cite

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Chirvasitu, Alexandru. "Metric enrichment, finite generation, and the path coreflection." Archivum Mathematicum (2024): 61-99. <http://eudml.org/doc/299266>.

@article{Chirvasitu2024,
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.},
author = {Chirvasitu, Alexandru},
journal = {Archivum Mathematicum},
keywords = {complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal hom},
language = {eng},
number = {2},
pages = {61-99},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Metric enrichment, finite generation, and the path coreflection},
url = {http://eudml.org/doc/299266},
year = {2024},
}

TY - JOUR
AU - Chirvasitu, Alexandru
TI - Metric enrichment, finite generation, and the path coreflection
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 2
SP - 61
EP - 99
AB - 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.
LA - eng
KW - complete metric space; path metric; intrinsic metric; gluing; convex; monoidal closed; enriched; tensored; locally presentable; colimit; internal hom
UR - http://eudml.org/doc/299266
ER -

References

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