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