Nuclearity in the category of complete semilattices
type: Item
instance of: scholarly article
title: Nuclearity in the category of complete semilattices
zbMATH Open document ID: 0666.18007
DOI: 10.1016/0022-4049(89)90028-5
author: K. A. Rowe
author: Denis Higgs
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 06D10
zbMATH DE Number: 4089775
zbMATH Keywords: symmetric monoidal closed category
zbMATH Keywords: nuclear objects
zbMATH Keywords: autonomous category
zbMATH Keywords: complete join semi-lattices
zbMATH Keywords: nuclear morphisms
zbMATH Keywords: tight maps
zbMATH Keywords: completely distributive lattices
reviewed by: J. Koslowski
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90028-5
OpenAlex ID: W2051110641
cites work: Tensor Products and Bimorphisms
cites work: Q3943005
cites work: Q5620723
cites work: Reflexivity for Subspace Maps and Linear Spaces of Operators
cites work: Q3673517
cites work: Produits tensoriels topologiques et espaces nucléaires
cites work: Q5654206
cites work: Completely distributive latices
cites work: Autonomous categories and duality of functors
cites work: Q5523691
cites work: Q4693151
cites work: A Galois problem for mappings
cites work: Tight Galois Connections and Complete Distributivity
cites work: Nuclearity
cites work: Q4049119
cites work: The structure of Galois connections
cites work: The tensor product of distributive lattices
description: scientific article; zbMATH DE number 4089775