Axioms for the category of finite-dimensional Hilbert spaces and linear contractions

I will explain the motivation and main ideas behind recent joint work with Chris Heunen (arXiv:2401.06584) that characterises the category of finite-dimensional Hilbert spaces and linear contractions. The axioms are about simple category-theoretic structures and properties. In particular, they do not refer to norms, continuity, dimension, or real numbers. The proof is noteworthy for the new way that the scalars are identified as the real or complex numbers. Instead of resorting to Solèr’s theorem, which is an opaque result underpinning similar characterisations of other categories of Hilbert spaces, suprema of bounded increasing sequences of scalars are explicitly constructed using directed colimits of contractions. To keep the talk accessible, I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed.