On Strict Higher C*-categories
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Noppakhun Suthichitranont
TL;DR
The paper develops a framework for strict involutive higher categories with a relaxed non-commutative exchange to avoid Eckmann–Hilton collapse, enabling non-trivial higher C*-categorical structures. It introduces strict globular $n$-categories, non-commutative exchange, and strict higher involutions, and extends these to higher algebroids and Fell bundles, culminating in hyper-C*-algebras and hypermatrices as concrete examples. The work provides a pathway toward higher functional analysis, higher Gel′fand–Naĭmark theory, and potential applications to non-commutative geometry and relational quantum theory, with detailed constructions of higher C*-categories of Longo–Roberts type and their convolution/bimodule realizations. It also outlines substantial future directions, including spectral reconstructions of non-commutative spaces, higher Morita theory, and quantum-relational formulations, enriching the mathematical apparatus for non-commutative topology and quantum field theory.
Abstract
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.