Dagger categories of orthosets and the complex Hilbert spaces
Jan Paseka, Thomas Vetterlein
TL;DR
The paper develops a categorical framework in which quantum-structural content is captured by orthosets and adjointable maps forming a dagger category. By introducing three baseline hypotheses and a functor ${\mathsf L}$ to uniform orthomodular spaces over a star-field $F$, it shows how orthoset data yields an orthomodular-space description and, under Solér-type conditions, reduces to Hilbert spaces over $\mathbb R$, $\mathbb C$, or $\mathbb H$. With additional hypotheses (extending to (H3')–(H5)) it proves that the resulting dagger category is unitarily dagger equivalent to the dagger category of complex Hilbert spaces, i.e., ${\mathcal Hil}_{\mathbb C}$. The construction yields both faithfulness and fullness (in the Archimedean case) of the representation, thereby providing a principled, categorical path from elementary orthoset structures to the standard Hilbert-space formalism used in quantum theory. This framework strengthens foundations for quantum foundations by deriving Hilbert-space structure from elementary, compositional orthogonality data via a rigorous dagger-categorical approach.
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.