Categorical aspects of polar decomposition

Polar decomposition unquestionably provides a notion of factorization in the category of Hilbert spaces. But it does not fit existing categorical notions, mainly because its factors are not closed under composition. We observe that the factors are images of functors. This leads us to consider notions of factorization that emphasize reconstruction of the composite category from the factors. We find that the category of Hilbert spaces is a composition of a totally disconnected abelian groupoid and a finitely accessible category. This separates probabilistic and complementarity aspects in quantum mechanics.