Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces

Abstract

We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.