Relation between bicrossed products and crossed extensions of fusion categories

Abstract: We show that all crossed extensions defined by Natale can be recovered as duals of bicrossed products of fusion categories. As an application, we prove that any exact factorization between a pointed fusion category $\operatorname{vec}_G$ and a fusion category $\mathcal{C}$ can be realized as a bicrossed product $\operatorname{vec}_G\bowtie \mathcal{C}$.