theHigherGeometer on Nostr: This is cool. Another categorical characterisation of the category of Hilbert spaces, ...

This is cool. Another categorical characterisation of the category of Hilbert spaces, this time in a way that doesn't assume the existence of a monoidal structure on the category, which means that it also is able to capture quaternionic Hilbert spaces. The real and complex Hilbert spaces are then the cases when there is a(n appropriate) monoidal structure.

Stephen Lack, Shay Tobin, "A characterisation for the category of Hilbert spaces"

Abstract: The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr'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 quaternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quaternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.
https://arxiv.org/abs/2412.03776