Weak Factorization and Product Systems Over Groupoids

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.