Nuclearity in the category of complete semilattices
In Can. Math. Bull. 31, No.2, 227-235 (1988; Zbl 0664.18007), the second author had introduced nuclear objects in a symmetric monoidal closed, or autonomous category, guided by a characterization of finite-dimensional vector spaces. The present paper extends this notion to morphisms in autonomous categories, a concept that for Banach spaces had been considered by \textit{A. Grothendieck} [Mem. Am. Math. Soc. 16 (1955; Zbl 0064.355 resp. Zbl 0055.097)]. f: \(A\to B\) is called nuclear, if its exponential transpose n(f): \(I\to [A,B]\) factors through the natural map \(\phi\) : \(B\otimes [A,I]\to [A,B]\). (Here I denotes the base object of the monoidal structure.) Objects are called nuclear, if their identity morphisms have this property. After recalling that the category CJSL of complete join semi-lattices is autonomous, the authors proceeds to identify the nuclear morphisms in CLSL as the tight maps introduced by \textit{G. N. Raney} [Trans. Am. Math. Soc. 97, 418-426 (1961; Zbl 0098.027)], and hence the nuclear objects as the completely distributive lattices.
scientific article; zbMATH DE number 4089775
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