This paper presents a conceptual and efficient geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice \(\mathbb {Z}^n\) . It is shown that, under the typical assumption of Haag duality, the monoidal \(C^*\) -categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of \(\mathbb {Z}^n\) . Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder \(\mathbb {R}^1\times \mathbb {S}^{n-1}\) . While the sphere \(\mathbb {S}^{n-1}\) arises geometrically as the angular coordinates of cones, the origin of the line \(\mathbb {R}^1\) is analytic and rooted in Haag duality. The usual braided (for \(n=2\) ) or symmetric (for \(n\ge 3\) ) monoidal \(C^*\) -categories of superselection sectors are recovered by removing a point of the sphere \(\mathbb {R}^1\times (\mathbb {S}^{n-1}\setminus \text {pt}) \cong \mathbb {R}^n\) and using the equivalence between \(\mathbb {E}_n\) -algebras and locally constant prefactorization algebras defined on open disks in \(\mathbb {R}^n\) . The non-trivial homotopy groups of spheres induce additional algebraic structures on these \(\mathbb {E}_n\) -monoidal \(C^*\) -categories, which in the case of \(\mathbb {Z}^2\) is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle \(\mathbb {S}^1\) . The locally constant prefactorization algebra structures discovered in this work generalize, under some mild geometric conditions, to other discrete spaces and thereby provide a clear link between the geometry of the localization regions and the algebraic structures on the category of superselection sectors.
