On strict higher \(C^*\)-categories. (English. French summary) Zbl 1440.18041

Bertozzini, Paolo; Conti, Roberto; Lewkeeratiyutkul, Wicham; Suthichitranont, Noppakhun

Cah. Topol. Géom. Différ. Catég. 61, No. 3, 239-348 (2020).

Summary: 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 proposed for a relaxed form of the exchange property for higher \(C^*\)-categories that avoids the Eckmann-Hilton collapse and hence allows the construction of expolicit 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.

MSC:

18N10	2-categories, bicategories, double categories
46M15	Categories, functors in functional analysis
46M99	Methods of category theory in functional analysis
16D90	Module categories in associative algebras

Keywords:

\(C^*\)-category; Fell bundle; involutive category; higher category; hypermatrix