Relation between bicrossed products and crossed extensions of fusion categories

TL;DR

This paper demonstrates that crossed extensions of fusion categories can be understood as duals of bicrossed products, providing a new perspective on their structure and applications in exact factorizations.

Contribution

It establishes a connection between crossed extensions and bicrossed products of fusion categories, extending the understanding of their duality and factorization properties.

Findings

Crossed extensions by Natale are duals of bicrossed products.

Any exact factorization between a pointed fusion category and another fusion category can be realized as a bicrossed product.

Provides a new framework for analyzing fusion category extensions and factorizations.

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 $\operatorname{vec}_G\bowtie \mathcal{C}$.