Categories of orthosets and adjointable maps
Jan Paseka, Thomas Vetterlein

TL;DR
This paper explores the structure of orthosets with 0, focusing on adjointable maps, their properties, and the categorical framework, with applications to projective Hilbert spaces and linear maps.
Contribution
It introduces a categorical perspective on orthosets with 0, characterizes adjointable maps, and develops a framework for analyzing projective Hilbert spaces.
Findings
Orthosets with 0 generalize subspace structures in Hilbert spaces.
Adjointable maps correspond to bounded linear operators in Hilbert spaces.
The category iOS forms a dagger category with applications to quantum structures.
Abstract
An orthoset is a non-empty set together with a symmetric and irreflexive binary relation , called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is orthogonal to every element. The collection of subspaces of a Hilbert space that are spanned by a single vector provides a motivating example. We say that a map between orthosets with 0 possesses the adjoint if, for any and , if and only if . We call in this case adjointable. For instance, any bounded linear map between Hilbert spaces induces a map with this property. We discuss in this paper adjointability from several perspectives and we put a particular focus on maps preserving the orthogonality relation. We moreover investigate the category OS of all orthosets…
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Taxonomy
TopicsFuzzy and Soft Set Theory
