On bicrossed product of fusion categories and exact factorizations

We introduce the notion of a matched pair of fusion rings and fusion categories, generalizing the one for groups. Using this concept, we define the bicrossed product of fusion rings and fusion categories and we construct exact factorizations for them. This concept generalizes the bicrossed product, also known as external Zappa-Sz\'ep product, of groups. We also show that every exact factorization of fusion rings can be presented as a bicrossed product. With this characterization, we describe the adjoint subcategory and universal grading group of an exact factorization of fusion categories. We give explicit fusion rules and associativity constraints for examples of fusion categories arising as a bicrossed product of combinations of Tambara-Yamagami categories and pointed fusion categories. These examples are new to the best of the knowledge of the authors.