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On the metrical and quantalic versions of the *-autonomous category of sup-lattices

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Walter Tholen

In 1984, Joyal and Tierney presented the category Sup of complete lattices and their suprema-preserving maps as a *-autonomous category in the sense of Barr.
Work on this paper was motivated by the question whether the Joyal-Tierney proof may be extended to a metrical context, so that the order of the lattice gets replaced by a generalized metric in the sense of Lawvere. The affirmative answer we give relies crucially on working with not necessarily symmetric metrics. It applies not only to small separated and cocomplete categories enriched in the Boolean quantale 2 (reproducing Sup), or in the Lawvere quantale [0,∞] (producing the category we were looking for), but in any commutative and unital quantale V. Benefitting from previous work by Stubbe, Hofmann, and others, with rather explicit constructions of its tensor product and the internal hom we give an alternative proof that the resulting category V-Sup is *-autonomous, a result first established by Eklund, Gutiérrez García, Höhle, and Kortelainen in 2018 from a predominantly order-theoretic perspective.

Keywords:
sup-lattice, quantale, enriched category, cocomplete, *-autonomous category, dualizing object

2020 MSC:
16T05, 16E10, 18G20, 18A40

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 7, pp 227-248.

Published 2024-04-09.

http://www.tac.mta.ca/tac/volumes/40/7/40-07.pdf

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