Axiomatising the dagger category of complex Hilbert spaces
Jan Paseka, Thomas Vetterlein
TL;DR
This work addresses foundational issues in quantum mechanics by axiomatizing the dagger category of complex Hilbert spaces in purely categorical terms. It introduces five axioms (H1–H5) that shape biproducts, directed colimits of dagger monomorphisms, object decomposition, a distinguished dagger simple object, and a unitary square-root symmetry, enabling reconstruction without a monoidal structure. The main result shows any dagger category satisfying these axioms is unitarily dagger equivalent to $\mathcalHil_{\mathbb{C}}$, leveraging Solér's theorem to force the underlying field to $\mathbb{C}$ and Bikchentaev to achieve fullness. Compared to prior characterisations, this approach offers a simpler, more interpretable pathway to a categorical foundation for complex Hilbert spaces and quantum theory.
Abstract
We axiomatise the dagger category of complex Hilbert spaces and bounded linear maps, using exclusively purely categorical conditions. Our axioms are chosen with the aim of an easy interpretability: two of them describe the composition of objecs, two further ones deal with the decomposition of objects, and a final axiom expresses a symmetry property. The categorical reconstruction of complex Hilbert spaces addresses foundational issues in quantum physics. We present a simplified alternative to recent characterisations.