  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.
