Manifestly unitary higher Hilbert spaces

Source abstract

Abstract Higher idempotent completion gives a formal inductive construction of the ‐category of finite‐dimensional ‐vector spaces starting with the complex numbers. We propose a manifestly unitary construction of low‐dimensional higher Hilbert spaces, formally constructing the ‐3‐category of 3‐Hilbert spaces from Baez's 2‐Hilbert spaces, which itself forms a 3‐Hilbert space. We prove that the forgetful functor from 3‐Hilbert spaces to 3‐vector spaces is fully faithful but not essentially surjective.