Categories of Orthosets and Adjointable Maps.

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 f : X → Y between orthosets with 0 possesses the adjoint g : Y → X if, for any x ∈ X and y ∈ Y , f ( x ) ⊥ y if and only if x ⊥ 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 O S of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory i O S of irredundant orthosets with 0. i O S can be made into a dagger category, the dagger of a morphism being its unique adjoint. i O S contains dagger subcategories of various sorts and provides in particular a framework for the investigation of Hilbert spaces.