Enriched Topology and Asymmetry

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be variable-basis . This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces enriched by many-valued preorders, spaces which are non-commutative when the underlying lattice-theoretic base is equipped with a non-commutative (semi-)tensor product. Of special interest are crisp and many-valued specialization preorders generated by many-valued topological spaces, orders having these consequences for many-valued spaces: they characterize the well-established L-T0 separation axiom, define the L-T1(1) separation axiom-logically equivalent under appropriate lattice-theoretic conditions to the L-T1 axiom of T. Kubiak, and define an apparently new L-T1(2) separation axiom. Along with the consequences of such ideas for many-valued spectra, these orders show that asymmetry has a home in many-valued topology comparable in at least some respects to its home in traditional topology

Injective symmetric quantaloid-enriched categories

We characterize injective objects, injective hulls and essential embeddings in the category of symmetric categories enriched in a small, integral and involutive quantaloid. In particular, injective partial metric spaces are precisely formulated.Comment: 27 pages, final versio

Quantale-valued Cauchy tower spaces and completeness

[EN] Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.Jäger, G.; Ahsanullah, TMG. (2021). Quantale-valued Cauchy tower spaces and completeness. Applied General Topology. 22(2):461-481. https://doi.org/10.4995/agt.2021.15610OJS461481222J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. 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