Manuals in orthogonal categories
type: Item
instance of: scholarly article
title: Manuals in orthogonal categories
zbMATH Open document ID: 0830.18004
DOI: 10.1007/BF00672803
author: Hirokazu Nishimura
published in: International Journal of Theoretical Physics
publication date: 1996-01-28T00:00:00Z
review text: These three papers [see also the following two articles] center around the author's two notions of manual of Boolean locales and orthogonal category. This work represents a categorical generalization of \textit{D. J. Foulis} and \textit{C. H. Randall}'s theory of manuals of operations [cf. J. Math. Phys. 13, 1667-1675 (1972; Zbl 0287.60002); ibid. 14, 1472-1480 (1973; Zbl 0287.60003)] in studying the operational foundations of the epistemological sciences, including quantum mechanics. The author believes that the logical aspects of this are best represented by manuals of Boolean locales. The definition is too lengthy to reproduce here, but to give the reader some idea, a manual is a small subcategory \({\mathcal M}\) on the category of Boolean locales with various conditions imposed on it centered around the notion of \({\mathcal M}\)-orthogonality. Two Boolean locales \(X\) and \(Y\) are \({\mathcal M}\)-orthogonal iff \(X\) and \(Y\) are part of a coproduct diagram lying in \({\mathcal M}\). The author then proceeds to generalize to the definition of an orthogonal category \({\mathcal O}\) and an \({\mathcal O}\)-manual. These ideas are then applied to looking at sheaves of Hilbert spaces. The second paper, after reviewing the key parts of the first, proceeds to consider a category whose objects are pairs \((X, {\mathcal A})\) where \(X\) is a Boolean locale and \({\mathcal A}\) is a sheaf over \(X\). A morphism from \((X, {\mathcal A})\) to \((Y, {\mathcal B})\) is a pair \((f, f^\vee)\) where \(f : X \to Y\) is a morphism of Boolean locales and \(f^\vee : {\mathcal B} \to f_* ({\mathcal A})\) is a sheaf map. This resulting category is an orthogonal category \({\mathcal S}\) and if \({\mathcal M}\) is a manual of Boolean locales, an \({\mathcal M}\)-empirical set is a functor \(F : {\mathcal M} \to {\mathcal S}\) such that the first component of \(F(X)\) is \(X\) itself (for all \(X\) in \({\mathcal M})\). The paper then investigates this empirical set theory ove…
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 03G12
zbMATH DE Number: 763176
zbMATH Keywords: Boolean locales
zbMATH Keywords: orthogonal category
zbMATH Keywords: manuals of operations
zbMATH Keywords: quantum mechanics
zbMATH Keywords: sheaves of Hilbert spaces
zbMATH Keywords: empirical set theory
zbMATH Keywords: category of sheaves of sheaves of abelian groups
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763176