Journal of Noncommutative Geometry

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Volume 9, Issue 1, 2015, pp. 121–159
DOI: 10.4171/JNCG/189

Published online: 2015-04-13

Presheaves of symmetric tensor categories and nets of $C^*$-algebras

Ezio Vasselli[1]

(1) Università di Roma La Sapienza, Italy

Motivated by algebraic quantum field theory, we study presheaves of symmetric tensor categories defined over the base of a space, intended as a spacetime. Any section of a presheaf (that is, any “superselection sector”, in the applications that we have in mind) defines a holonomy representation whose triviality is measured by Cheeger–Chern–Simons characteristic classes, and a non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an embedding in a presheaf of full subcategories of the one of Hilbert spaces, the section category of a presheaf is a Tannaka-type dual of a locally constant group bundle (the “gauge group”), which may not exist and in general is not unique. This leads to the notion of gerbe of $C*$-algebras, defined on the given base.

Keywords: Net of $C*$-algebras, presheaf, duality, gerbe

Vasselli Ezio: Presheaves of symmetric tensor categories and nets of $C^*$-algebras. J. Noncommut. Geom. 9 (2015), 121-159. doi: 10.4171/JNCG/189