Abstract 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 Zⁿ Z n. It is shown that, under the typical assumption of Haag duality, the monoidal C^* C ∗ -categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Zⁿ 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 R¹ S^n-1 R 1 × S n - 1. While the sphere S^n-1 S n - 1 arises geometrically as the angular coordinates of cones, the origin of the line R¹ R 1 is analytic and rooted in Haag duality. The usual braided (for n=2 n = 2) or symmetric (for n 3 n ≥ 3) monoidal C^* C ∗ -categories of superselection sectors are recovered by removing a point of the sphere R¹ (S^n-1 pt) Rⁿ R 1 × (S n - 1 \ pt) ≅ R n and using the equivalence between Eₙ E n -algebras and locally constant prefactorization algebras defined on open disks in Rⁿ <mml: math xmlns:
Benini et al. (Mon,) studied this question.