AI 中文总结
本文通过范畴化Baez的高阶线性代数，为3-希尔伯特空间提供正交基等工具，赋予其C*-3-范畴自富集结构，证明酉Yoneda引理与Deligne积的酉版本性质，推进高阶希尔伯特空间理论研究。
AI 中文摘要
在我们之前的论文[arXiv:2410.05120]中，我们引入了有限维3-希尔伯特空间的概念，它是对Baez的2-希尔伯特空间的范畴化。在本文中，通过进一步对Baez的高阶线性代数进行范畴化，我们为处理3-希尔伯特空间提供了有用工具，包括广义标量乘法、正交基以及算子的酉伴随。我们利用这些工具为3-希尔伯特空间的C*-3-范畴赋予了自富集结构。我们证明了3-希尔伯特空间的酉Yoneda引理/Riesz表示定理：Yoneda嵌入是等距等价。最后，我们定义了3-希尔伯特空间上的Deligne积的酉版本，并证明它满足等距版本的折叠技巧。
英文摘要
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.
Comments39 pages, many TikZ figures. Comments welcome!
