# Categories of Orthosets and Adjointable Maps

## TL;DR

This paper introduces orthosets with 0 and studies maps between them that preserve orthogonality, focusing on their categorical properties and applications to Hilbert spaces.

## Contribution

The paper defines and analyzes adjointable maps between orthosets with 0 and explores the dagger category structure of irredundant orthosets.

## Key findings

- Adjointable maps between orthosets with 0 preserve orthogonality and have unique adjoints.
- The category of irredundant orthosets with 0 can be structured as a dagger category.
- This framework provides a new perspective for studying Hilbert spaces through orthosets.

## Abstract

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation \documentclass[12pt]{minimal}
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\begin{document}$$\perp $$\end{document}⊥, 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 \documentclass[12pt]{minimal}
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\begin{document}$$f :X \rightarrow Y$$\end{document}f:X→Y between orthosets with 0 possesses the adjoint \documentclass[12pt]{minimal}
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\begin{document}$$g :Y \rightarrow X$$\end{document}g:Y→X if, for any \documentclass[12pt]{minimal}
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\begin{document}$$x \in X$$\end{document}x∈X and \documentclass[12pt]{minimal}
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\begin{document}$$y \in Y$$\end{document}y∈Y, \documentclass[12pt]{minimal}
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\begin{document}$$f(x) \perp y$$\end{document}f(x)⊥y if and only if \documentclass[12pt]{minimal}
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\begin{document}$$x \perp g(y)$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{O}\mathcal{S}$$\end{document}OS of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory \documentclass[12pt]{minimal}
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\begin{document}$$\mathcalligra {i}\mathcal{O}\mathcal{S}$$\end{document}iOS of irredundant orthosets with 0. \documentclass[12pt]{minimal}
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\begin{document}$$\mathcalligra {i}\mathcal{O}\mathcal{S}$$\end{document}iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. \documentclass[12pt]{minimal}
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\begin{document}$$\mathcalligra {i}\mathcal{O}\mathcal{S}$$\end{document}iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of Hilbert spaces.