Categories of Orthosets and Adjointable Maps
| Authors | |
|---|---|
| Year of publication | 2025 |
| Type | Article in Periodical |
| Magazine / Source | International Journal of Theoretical Physics |
| MU Faculty or unit | |
| Citation | |
| web | https://link.springer.com/article/10.1007/s10773-025-06031-4 |
| Doi | https://doi.org/10.1007/s10773-025-06031-4 |
| Keywords | Orthoset; Orthogonality space; Hermitian space; Hilbert space; Dagger category |
| Attached files | |
| Description | 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. |
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