Braided monoidal categories have applications in various situations. In particular, their universal R-matrices give solutions of the quantum Yang-Baxter equation and representations of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras and Drinfeld centres of monoidal categories. On the other hand, universal K-matrices, leading to solutions of the reflection equation and representations of braid groups of type B are much less well understood and there are no general constructions for them. We will describe a construction of reflective centers of module categories. It gives rise to a quantum double construction for universal K-matrices, which can be viewed as a type B analog of Drinfeld’s double construction. This is a joint work with Robert Laugwitz and Chelsea Walton.