Article
MSC:
18A30,
18C35,
18D15,
18D20,
46L05,
46L09,
51F30,
54E40,
54E50 | MR 4729650 | Zbl 07830506 | DOI: 10.5817/AM2024-2-61
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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.
References:
[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]. MR 2240597
[2] Adámek, J., Rosický, J.: Locally presentable and accessible categories. London Math. Soc. Lecture Note Ser., vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136
[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. MR 4343847
[4] Awodey, S.: Category theory. second ed., Oxford Logic Guides, vol. 52, Oxford University Press, Oxford, 2010. MR 2668552
[5] Blackadar, B.: Operator algebras. Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. MR 2188261
[6] Blumenthal, L.M.: Theory and applications of distance geometry. Chelsea Publishing Co., New York, 1970. MR 0268781
[7] Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. MR 1633290
[8] Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. MR 1835418 | Zbl 0981.51016
[9] Chirvasitu, A., Ko, J.: Monadic forgetful functors and (non-)presentability for C$^*$- and W$^*$-algebras. http://arxiv.org/abs/2203.12087v2, 2022. MR 4478368
[10] Di Liberti, I., Rosický, J.: Enriched locally generated categories. Theory Appl. Categ. 38 (2022), 661–683, Paper No. 17. MR 4403281
[11] Diestel, R.: Graph theory. Grad. Texts in Math., vol. 173, Springer, Berlin, fifth ed., 2018. MR 3644391
[12] Garbulińska-Wȩgrzyn, J., Kubiś, W.: A universal operator on the Gurariĭ space. J. Operator Theory 73 (1) (2015), 143–158. DOI 10.7900/jot.2013oct09.1999 | MR 3322761
[13] Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990. MR 1074005 | Zbl 0708.47031
[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. MR 2307192
[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. MR 0675952
[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. MR 0648793
[17] Kelly, G.M.: Basic concepts of enriched category theory. Repr. Theory Appl. Categ., vol. 10, Cambridge Univ. Press, Cambridge, 2005, pp. vi+137. MR 2177301 | Zbl 1086.18001
[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. MR 1818603
[19] Kubiś, W.: Metric-enriched categories and approximate Fraïssé limits. 2012, http://arxiv.org/abs/1210.6506v3 MR 4369354
[20] Lupini, M.: Fraïssé limits in functional analysis. Adv. Math. 338 (2018), 93–174. DOI 10.1016/j.aim.2018.08.012 | MR 3861702
[21] Mac Lane, S.: Categories for the working mathematician. Grad. Texts in Math., Springer-Verlag, New York, second ed., 1998. MR 1712872
[22] Menger, K.: Untersuchungen über allgemeine Metrik. Math. Ann. 100 (1) (1928), 75–163. DOI 10.1007/BF01448840 | MR 1512479
[23] Munkres, J.R.: Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000, Second edition of [MR0464128]. MR 3728284
[24] Perdigão do Carmo, M.: Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty. MR 1138207
[25] Rosický, J., Tholen, W.: Approximate injectivity. Appl. Categ. Structures 26 (4) (2018), 699–716. DOI 10.1007/s10485-017-9510-2 | MR 3824920
[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. MR 1222415
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