Axioms for the Category of Hilbert Spaces

We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure such as probabilities, convexity, complex numbers, continuity, or dimension. We'll discuss the axioms, sketch the proof of the theorem, and survey open questions, further directions, and context. (Based on joint work with Andre Kornell arxiv:2109.07418.)