Categories of Orthosets and Adjointable Maps

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 \(f :X \rightarrow Y\) between orthosets with 0 possesses the adjoint \(g :Y \rightarrow X\) 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 \(\mathcal{O}\mathcal{S}\) of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory \(\mathcalligra {i}\mathcal{O}\mathcal{S}\) of irredundant orthosets with 0. \(\mathcalligra {i}\mathcal{O}\mathcal{S}\) can be made into a dagger category, the dagger of a morphism being its unique adjoint. \(\mathcalligra {i}\mathcal{O}\mathcal{S}\) contains dagger subcategories of various sorts and provides in particular a framework for the investigation of Hilbert spaces.