### Axioms for the Category of Hilbert Spaces

#### Posted by Tom Leinster

Chris Heunen and Andre Kornell have found an axiomatic characterization of the category of Hilbert spaces!

Chris Heunen and Andre Kornell, Axioms for the category of Hilbert spaces. arXiv:2109.07418, 2021.

A bit more precisely, they axiomatize the monoidal dagger category of Hilbert spaces: $\mathbf{Hilb}$ equipped with the operations of tensor product and taking the adjoint.

Their paper appeared on the arXiv this week, and uses Solèr’s theorem, which John wrote about here in 2010.

I don’t have time to write more, but I wanted to make sure that Café readers don’t miss this treat.

*Edit:* Chris has now written a more substantial post on his and Kornell’s paper.

## Re: Axioms for the Category of Hilbert Spaces

There is a typo in the article. When defining the dagger on page 1, the condition should be $id_H^\dagger = id_H$, not $id_H^\dagger = id_A$.