Noncommutative Geometry Seminar

Although (monoidal) C*-categories have been a basic instrument in
algebraic quantum field theory for more than thirty years, very little
direct interaction exists between research in non-commutative operator
algebras and the thriving area of higher category theory (that meanwhile
developed along lines mainly inspired by higher homotopy theory that might
not be suitable for the non-commutative case).

We introduce definitions for strict n-C*-categories (possibly equipped
with involutions of different arbitrary depth) and, in order to recover
many natural non-trivial examples (whose enveloping convolution algebras
are hypermatrices and hyper-C*-algebras), we propose a non-commutative
version of exchange property avoiding the Eckmann-Hilton collapse. This
might open the way for a vertical categorification of functional analysis
and spectral theory.

Among the several possible applications of non-commutative higher
C*-categories, we like to speculate on the notion of morphism in Connes'
non-commutative geometry; the algebraic formulation of Rovelli's
relational quantum mechanics and our proposed modular approach to quantum
gravity (arXiv: 1007.4094).
Part of this work is a joint collaboration with Roberto Conti, Wicharn
Lewkeeratiyutkul and Noppakhun Suthichitranont.