Weak Factorization and Product Systems Over Groupoids
Jon Bannon, Alina Vdovina
math.OA
Aug 26, 2026 · v1
TL;DR
An accompanying Lean 4/Mathlib development verifies the paper's categorical and combinatorial results, including weak factorization and strict-splitting characterizations; the operator-algebraic parts are not formalized.
Abstract
We give a product-system description of generalized higher-rank graphs with the weak factorization property. For such a graph with degree-zero groupoid G, we prove that weak factorization is equivalent to multiplication inducing coherent bijections between balanced products of its homogeneous G-G-bisets. Consequently, generalized higher-rank graphs with fixed degree-zero groupoid are equivalent to normalized product systems of groupoid bisets over N^k. For countable left-cancellative graphs, these bisets linearize canonically to product systems of C*(G)-correspondences. Finite alignment implies compact alignment, and the resulting Nica-Toeplitz algebra agrees canonically with Spielberg's full category algebra. Under row-finiteness modulo G and the no-sources condition, the corresponding Cuntz-Pimsner quotient is the boundary groupoid algebra; with injective left actions, the same conclusion holds for the Cuntz-Nica-Pimsner algebra. We also characterize the R-condition by the existence of a strict multiplicative splitting and relate such splittings to higher-rank graph/groupoid Zappa-Szep products. A cancellative rank-two example shows that strict splittings need not exist.
Problem
The paper seeks a product-system description of Lawson–Vdovina generalized higher-rank graphs with the weak factorization property. It also relates their operator algebras to Spielberg's category and boundary groupoid algebras.
Approach
Weak factorization is shown to be equivalent to multiplication inducing coherent bijections between balanced products of homogeneous groupoid bisets. These bisets are linearized to product systems of C*(G)-correspondences, which are then compared with Nica–Toeplitz and Cuntz–Pimsner algebras. A Lean 4 development (Lean 4.28.0, Mathlib v4.28.0) checks the categorical and combinatorial part, including normal forms and the equivalence between the R-condition and strict splitting.
Results
Generalized higher-rank graphs with fixed degree-zero groupoid are equivalent to normalized product systems of bisets over N^k. Under suitable conditions, the Nica–Toeplitz algebra matches Spielberg's category algebra, and the Cuntz–Pimsner quotient is the boundary groupoid algebra. A cancellative rank-two example shows that strict splittings need not exist.
