arXiv 1709.093392017-09math.OAmath.CT

论严格高阶 C*-范畴

On Strict Higher C*-categories

Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Noppakhun Suthichitranont

备注：: 57 pages, AMS-LaTeX2e, very expanded introduction, several typos corrected, results unchanged; this is a reformatted version (only for arXiv) of the published paper, with two additional clarifying notes (in footnote 48 and 57)
发表信息：: Cahiers de Topologie et Géométrie Différentielle Catégoriques (2020) LXI-3:239-348

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AI总结：

本文定义了严格对合高阶范畴、严格高阶 C*-范畴及高阶 Fell 丛，提出松弛交换性质以避免 Eckmann-Hilton 坍缩，从而构造出源于超矩阵和超 C*-代数的非平凡非交换示例，并探讨其在非交换几何与关系量子理论中的应用。

摘要：

我们给出了严格对合高阶范畴（dagger 范畴的纵向范畴化）、严格高阶 C*-范畴以及高阶 Fell 丛（在任意对合高阶拓扑范畴上）的定义。我们提出了一种针对高阶（C*）范畴的交换性质的松弛形式，该形式避免了 Eckmann-Hilton 坍缩，从而允许构造出源于超矩阵和本文定义的超 C*-代数研究的显式非平凡“非交换”示例。此外，还探讨了严格高阶范畴的通常 globular 和 cubical 设定的替代方案。这些非交换高阶 C*-范畴的应用预计将出现在非交换几何中的态射研究以及关系量子理论的代数表述中。

英文摘要：

We provide definitions for strict involutive higher categories (a vertical categorification of dagger categories), strict higher C*-categories and higher Fell bundles (over arbitrary involutive higher topological categories). We put forward a proposal for a relaxed form of the exchange property for higher (C*)-categories that avoids the Eckmann-Hilton collapse and hence allows the construction of explicit non-trivial "non-commutative" examples arising from the study of hypermatrices and hyper-C*-algebras, here defined. Alternatives to the usual globular and cubical settings for strict higher categories are also explored. Applications of these non-commutative higher C*-categories are envisaged in the study of morphisms in non-commutative geometry and in the algebraic formulation of relational quantum theory.

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