In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication are presented.
Keywords: Enriched category, continuous t-norm, forward Cauchy weight, distributive law, completely distributive quantale-enriched category, continuous quantale-enriched category
2010 MSC: 18B35, 18D20, 06D10, 06F07
Theory and Applications of Categories, Vol. 35, 2020, No. 3, pp 64-88.