Injective symmetric quantaloid-enriched categories. (English) Zbl 1533.18007
A quantaloid [K. I. Rosenthal, The theory of quantaloids. Harlow: Addison Wesley Longman (1996; Zbl 0845.18003)] \(\mathcal{Q}\)is a category enriched in the monoidal-closed category \(\boldsymbol{Sup}\)[A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck. Providence, RI: American Mathematical Society (AMS) (1984; Zbl 0541.18002)] of complete lattices and sup-preserving maps. Considering \(\mathcal{Q}\)as a base for enrichment, a theory of \(\mathcal{Q}\)-categories, \(\mathcal{Q}\)-functors and \(\mathcal{Q}\)-distributors were developed [K. I. Rosenthal, The theory of quantaloids. Harlow: Addison Wesley Longman (1996; Zbl 0845.18003); I. Stubbe, Theory Appl. Categ. 14, 1–45 (2005; Zbl 1079.18005); Theory Appl. Categ. 16, 283–306 (2006; Zbl 1119.18005)]. Furthermore, if \(\mathcal{Q}\)is involutive, it makes sense to consider symmetric \(\mathcal{Q}\)-categories [Zbl 0498.18007, Zbl 1230.06007].
This paper is concerned with injectivity [J. Adámek et al., Abstract and concrete categories. The joy of cats. New York etc.: John Wiley & Sons, Inc. (1990; Zbl 0695.18001); J. M. Maranda, Trans. Am. Math. Soc. 110, 98–135 (1964; Zbl 0121.26601)] in the category of symmetric \(\mathcal{Q}\)-categories, wishing to find the categorical interpretation of injective metric spaces in the framework of quantaloid-enriched categories. It is well known [N. Aronszajn and P. Panitchpakdi, Pac. J. Math. 6, 405–439 (1956; Zbl 0074.17802); J. R. Isbell, Comment. Math. Helv. 39, 65–76 (1964; Zbl 0151.30205); R. Espínola and M. A. Khamsi, in: Handbook of metric fixed point theory. Dordrecht: Kluwer Academic Publishers. 391–435 (2001; Zbl 1029.47002)] that injective metric spaces in the category of (classical) metric spaces and non-expansive maps are precisely hyperconvex metric spaces. Furthermore, the injective hull of a metric space was firstly constructed by J. R. Isbell [Comment. Math. Helv. 39, 65–76 (1964; Zbl 0151.30205)], which was later characterized by A. W. M. Dress [Adv. Math. 53, 321–402 (1984; Zbl 0562.54041)] as the tight span. This paper looks deeply into the categorical meaning of the concepts of hyperconvexity and tight span, providing a far reaching extension of the above results in the context of quantaloid-enriched categories.
This paper is concerned with injectivity [J. Adámek et al., Abstract and concrete categories. The joy of cats. New York etc.: John Wiley & Sons, Inc. (1990; Zbl 0695.18001); J. M. Maranda, Trans. Am. Math. Soc. 110, 98–135 (1964; Zbl 0121.26601)] in the category of symmetric \(\mathcal{Q}\)-categories, wishing to find the categorical interpretation of injective metric spaces in the framework of quantaloid-enriched categories. It is well known [N. Aronszajn and P. Panitchpakdi, Pac. J. Math. 6, 405–439 (1956; Zbl 0074.17802); J. R. Isbell, Comment. Math. Helv. 39, 65–76 (1964; Zbl 0151.30205); R. Espínola and M. A. Khamsi, in: Handbook of metric fixed point theory. Dordrecht: Kluwer Academic Publishers. 391–435 (2001; Zbl 1029.47002)] that injective metric spaces in the category of (classical) metric spaces and non-expansive maps are precisely hyperconvex metric spaces. Furthermore, the injective hull of a metric space was firstly constructed by J. R. Isbell [Comment. Math. Helv. 39, 65–76 (1964; Zbl 0151.30205)], which was later characterized by A. W. M. Dress [Adv. Math. 53, 321–402 (1984; Zbl 0562.54041)] as the tight span. This paper looks deeply into the categorical meaning of the concepts of hyperconvexity and tight span, providing a far reaching extension of the above results in the context of quantaloid-enriched categories.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
| 18F75 | Quantales |
Keywords:
quantaloid; enriched category; symmetry; injective object; injective hull; essential embedding; \(\Omega\)-set; partial metric spaceCitations:
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