Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu

Archivum Mathematicum (2024)

  • Volume: 060, 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 060.2 (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},
volume = {060},
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
VL - 060
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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  1. Adámek, J., Herrlich, H., Strecker, G.E., Abstract and concrete categories: the joy of cats, Repr. Theory Appl. Categ. 17 (2006), 1–507, Reprint of the 1990 original [Wiley, New York; MR1051419]. (2006) MR2240597
  2. Adámek, J., Rosický, J., Locally presentable and accessible categories, London Math. Soc. Lecture Note Ser., vol. 189, Cambridge University Press, Cambridge, 1994. (1994) MR1294136
  3. Adámek, J., Rosický, J., Approximate injectivity and smallness in metric-enriched categories, J. Pure Appl. Algebra 226 (6) (2022), 30 pp., Paper No. 106974. (2022) MR4343847
  4. Awodey, S., Category theory, second ed., Oxford Logic Guides, vol. 52, Oxford University Press, Oxford, 2010. (2010) MR2668552
  5. Blackadar, B., Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. (2006) MR2188261
  6. Blumenthal, L.M., Theory and applications of distance geometry, Chelsea Publishing Co., New York, 1970. (1970) MR0268781
  7. Bollobás, B., Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. (1998) MR1633290
  8. Burago, D., Burago, Y., Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. (2001) Zbl0981.51016MR1835418
  9. Chirvasitu, A., Ko, J., Monadic forgetful functors and (non-)presentability for C * - and W * -algebras, http://arxiv.org/abs/2203.12087v2, 2022. MR4478368
  10. Di Liberti, I., Rosický, J., Enriched locally generated categories, Theory Appl. Categ. 38 (2022), 661–683, Paper No. 17. (2022) MR4403281
  11. Diestel, R., Graph theory, Grad. Texts in Math., vol. 173, Springer, Berlin, fifth ed., 2018. (2018) MR3644391
  12. Garbulińska-Wȩgrzyn, J., Kubiś, W., 10.7900/jot.2013oct09.1999, J. Operator Theory 73 (1) (2015), 143–158. (2015) MR3322761DOI10.7900/jot.2013oct09.1999
  13. Goebel, K., Kirk, W.A., Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990. (1990) Zbl0708.47031MR1074005
  14. Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, english ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. (2007) MR2307192
  15. Halmos, P.R., A Hilbert space problem book, Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, second ed., 1982. (1982) MR0675952
  16. Kelly, G.M., Structures defined by finite limits in the enriched context, I, Cah. Topol. Géom. Dffér. Catég. 23 (1) (1982), 3–42, Third Colloquium on Categories, Part VI (Amiens, 1980. (1982) MR0648793
  17. Kelly, G.M., Basic concepts of enriched category theory, Repr. Theory Appl. Categ., vol. 10, Cambridge Univ. Press, Cambridge, 2005, pp. vi+137. (2005) Zbl1086.18001MR2177301
  18. Khamsi, M.A., Kirk, W.A., An introduction to metric spaces and fixed point theory, Pure Appl. Math. (N. Y.), Wiley-Interscience, New York, 2001. (2001) MR1818603
  19. Kubiś, W., Metric-enriched categories and approximate Fraïssé limits, 2012, http://arxiv.org/abs/1210.6506v3. MR4369354
  20. Lupini, M., 10.1016/j.aim.2018.08.012, Adv. Math. 338 (2018), 93–174. (2018) MR3861702DOI10.1016/j.aim.2018.08.012
  21. Mac Lane, S., Categories for the working mathematician, Grad. Texts in Math., Springer-Verlag, New York, second ed., 1998. (1998) MR1712872
  22. Menger, K., 10.1007/BF01448840, Math. Ann. 100 (1) (1928), 75–163. (1928) MR1512479DOI10.1007/BF01448840
  23. Munkres, J.R., Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, Second edition of [MR0464128]. (2000) MR3728284
  24. Perdigão do Carmo, M., Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty. (1992) MR1138207
  25. Rosický, J., Tholen, W., 10.1007/s10485-017-9510-2, Appl. Categ. Structures 26 (4) (2018), 699–716. (2018) MR3824920DOI10.1007/s10485-017-9510-2
  26. Wegge-Olsen, N.E., K -theory and C * -algebras, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993, A friendly approach. (1993) MR1222415

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