Characterisations for the category of Hilbert spaces

Category theory is an algebraic framework based on the composition of functions. Categories consist of objects and morphisms between objects. A dagger category is a type of category which has a notion of reversibility for each morphism. A monoidal category is one which allows the joining of objects and of morphisms in parallel, rather than in series as with composition. This joining is done in such a way as to satisfy certain coherence conditions.

The categories of real and of complex Hilbert spaces with bounded linear maps are dagger monoidal categories and have received purely categorical characterisations by Chris Heunen and Andre Kornell. This characterisation is achieved through Soler's theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions.

The Heunen-Kornell characterisation makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only characterises the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.