AI 中文总结
本文提出用完备有限酉圆盘状$n$-范畴定义有限$(n+1)$-希尔伯特空间，验证了$n=1,2$情形，并由此分类了低维酉TQFT。
AI 中文摘要
我们引入了酉圆盘状$n$-范畴的概念，这是一种配备了反射结构和球面迹的圆盘状$n$-范畴，该球面迹诱导正定配对。由于有限酉圆盘状$n$-范畴被定义为完全扩展的$(n+1)$维酉TQFT的局部场数据，我们提出将一个完备的有限酉圆盘状$n$-范畴作为对所有$n$的有限$(n+1)$-希尔伯特空间的定义。对于$n=1$和$n=2$，我们验证了这一提议，证明了完备的有限酉圆盘状1-范畴和2-范畴分别与有限2-和3-希尔伯特空间等距等价，并且这些等价是函子性的。对于$n=1$，我们恢复了Schommer-Pries关于定向$(1+1)$维TQFT分类的酉细化，该分类基于$\nathrm{H}^*$-代数的$\nathrm{H}^*$-Morita等价类；对于$n=2$，我们将其范畴化，以通过$\nathrm{H}^*$-多融合范畴的$\nathrm{H}^*$-Morita等价类来分类定向$(2+1)$维酉TQFT。在此过程中，我们研究了完全不完全的3-希尔伯特空间，证明了枢轴dagger 2-范畴的一个严格化结果，并定义了圆盘状$n$-范畴之间的函子与高阶变换。
英文摘要
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.
Comments115 pages, many TikZ figures. Comments welcome!
