Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B-3 and B-4.