Dagger categories of orthosets and the complex Hilbert spaces

TL;DR

This paper explores the structure of orthosets, a generalization of Hilbert spaces, and investigates dagger categories of orthosets, establishing conditions for their equivalence to categories of complex Hilbert spaces and bounded linear maps.

Contribution

It introduces a categorical framework for orthosets and characterizes when these categories are equivalent to categories of Hilbert spaces, extending the understanding of adjoint pairs in this context.

Findings

Orthosets generalize Hilbert space orthogonality.

Dagger categories of orthosets are studied with adjoint pairs.

Conditions for equivalence to Hilbert space categories are provided.

Abstract

An orthoset is a non-empty set $X$ together with a symmetric binary relation $\perp$ and a constant $0$ such that $x \not\perp x$ for any $x \neq 0$, and $0 \perp x$ for any $x$. Maps $f \colon X \to Y$ and $g \colon Y \to X$ between orthosets are said to form an adjoint pair if, for any $x \in X$ and $y \in Y$, $f(x) \perp g$ if and only if $x \perp g(x)$. Hilbert spaces, equipped with the usual orthogonality relation and the zero vector, provide the motivating examples of orthosets. The usual adjoints of bounded linear maps between Hilbert spaces are adjoints also in our sense. We investigate dagger categories of orthosets and maps between them, requiring that any morphism and its dagger form an adjoint pair. We indicate conditions under which such a category is unitarily dagger equivalent to the dagger category of complex Hilbert spaces and bounded linear maps.