Categories of Orthosets and Adjointable Maps

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⊥</mml:mo></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi></mml:mrow> </mml:math> between orthosets with 0 possesses the adjoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>→</mml:mo> <mml:mi>X</mml:mi></mml:mrow> </mml:math> if, for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Y</mml:mi></mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⊥</mml:mo> <mml:mi>y</mml:mi></mml:mrow> </mml:math> if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi> <mml:mo>⊥</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . We call <i>f</i> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mi>S</mml:mi></mml:mrow> </mml:math> of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi> <mml:mi>O</mml:mi> <mml:mi>S</mml:mi></mml:mrow> </mml:math> of irredundant orthosets with 0. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi> <mml:mi>O</mml:mi> <mml:mi>S</mml:mi></mml:mrow> </mml:math> can be made into a dagger category, the dagger of a morphism being its unique adjoint. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi> <mml:mi>O</mml:mi> <mml:mi>S</mml:mi></mml:mrow> </mml:math> contains dagger subcategories of various sorts and provides in particular a framework for the investigation of Hilbert spaces.