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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 \n \n $$f :X \\rightarrow Y$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n :<\/mml:mo>\n X<\/mml:mi>\n \u2192<\/mml:mo>\n Y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> between orthosets with 0 possesses the adjoint \n \n $$g :Y \\rightarrow X$$<\/jats:tex-math>\n \n \n g<\/mml:mi>\n :<\/mml:mo>\n Y<\/mml:mi>\n \u2192<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> if, for any \n \n $$x \\in X$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$y \\in Y$$<\/jats:tex-math>\n \n \n y<\/mml:mi>\n \u2208<\/mml:mo>\n Y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$f(x) \\perp y$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \u22a5<\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> if and only if \n \n $$x \\perp g(y)$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n \u22a5<\/mml:mo>\n g<\/mml:mi>\n (<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We call f<\/jats:italic> 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 \n \n $$\\mathcal{O}\\mathcal{S}$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory \n \n $$\\mathcalligra {i}\\mathcal{O}\\mathcal{S}$$<\/jats:tex-math>\n \n \n i<\/mml:mi>\n O<\/mml:mi>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of irredundant orthosets with\u00a00. \n \n $$\\mathcalligra {i}\\mathcal{O}\\mathcal{S}$$<\/jats:tex-math>\n \n \n i<\/mml:mi>\n O<\/mml:mi>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can be made into a dagger category, the dagger of a morphism being its unique adjoint. \n \n $$\\mathcalligra {i}\\mathcal{O}\\mathcal{S}$$<\/jats:tex-math>\n \n \n i<\/mml:mi>\n O<\/mml:mi>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> contains dagger subcategories of various sorts and provides in particular a framework for the investigation of Hilbert spaces.<\/jats:p>","DOI":"10.1007\/s10773-025-06031-4","type":"journal-article","created":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T09:36:35Z","timestamp":1748338595000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Categories of Orthosets and Adjointable Maps"],"prefix":"10.1007","volume":"64","author":[{"given":"Jan","family":"Paseka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Vetterlein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,5,27]]},"reference":[{"key":"6031_CR1","doi-asserted-by":"crossref","unstructured":"Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. 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