Categories of orthosets and adjointable maps

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 $\perp$, 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 $f \colon X \to Y$ between orthosets with 0 possesses the adjoint $g \colon Y \to X$ if, for any $x \in X$ and $y \in Y$, $f(x) \perp y$ if and only if $x \perp g(y)$. We call $f$ 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 with 0 and adjointable maps between them. We especially focus on the full subcategory iOS of irredundant orthosets with 0. iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of projective Hilbert spaces.