A unified framework for higher Hilbert spaces

Categorifications of vector spaces such as linear categories have been used extensively in modern representation theory. Categorifications of Hilbert spaces are less well-studied, even though they are important inputs for unitary higher representation theory and extended reflection-positive quantum field theory. Several distinct definitions have been proposed in the literature. For example there are dagger categories (quantum information theory and categorical quantum mechanics), W*-categories (operator theory and von Neumann algebras), bi-involutive categories, Baez 2-Hilbert spaces, etcetera. Some of these have positivity built in (e.g. by being enriched over the category of Hilbert spaces or by having vector space-valued inner products), others don’t even require linearity and are purely abstract categorical notions.

I propose a general framework parameterized by subgroups G of O(n). More specifically, for every subgroup G of O(n) I will define what is a “finite-dimensional n-vector space with G-Hermitian form”. I will illustrate how to tell for some specific G when such a form is “positive definite”. I will explain how this recovers the usual notion of Hilbert space for n=1. For n=2 I will recover previously established notions of 2-Hilbert space. This suggests a notion of n-Hilbert space for all n.