Research

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.

In our previous article [arXiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.