Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions

Matthew Di Meglio, Chris Heunen

Abstract: We characterise the category of finite-dimensional Hilbert spaces and linear contractions using simple category-theoretic axioms that do not refer to norms, continuity, dimension, or real numbers. Our proof directly relates limits in category theory to limits in analysis, using a new variant of the classical characterisation of the real numbers instead of Solèr's theorem.