Manuals in orthogonal categories
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 over \({\mathcal M}\). The resulting category of \({\mathcal M}\)-empirical sets has many topos- like features except for the existence of exponentials. The final paper applies the above ideas to algebraic geometry by considering instead pairs \((X, {\mathcal A})\) where \(X\) is a topological space and \({\mathcal A}\) is a sheaf of abelian groups on \(X\). Again, morphisms are pairs \((f, f^\vee)\) where \(f : X \to Y\) is a continuous map and \(f^\vee : {\mathcal B} \to f_* ({\mathcal A})\) is a homomorphism of sheaves of sheaves of abelian groups. This leads to the consideration of a category of sheaves of sheaves of abelian groups.
category of sheaves of sheaves of abelian groups
scientific article; zbMATH DE number 763176
scientific article