Unitary TQFTs, Unitary Disk-Like $n$-Categories, and Higher Hilbert Spaces

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 $(n + 1)\text{D}$ 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 $(1 + 1)\text{D}$ TQFTs in terms of $H^$-Morita equivalence classes of $H^$-algebras, and for $n = 2$ we categorify this to classify oriented $(2 + 1)\text{D}$ unitary TQFTs by $H^$-Morita equivalence classes of $H^$-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.

Recommended citation: G. Wesley. "Unitary TQFTs, Unitary Disk-Like \(n\)-Categories, and Higher Hilbert Spaces" (2026). arXiv:2609.19713.
Download Paper