Category of braided sets, extensions and 2-analogues

A braided set is the same thing as a solution of the set-theoretic Yang-Baxter equation. It is important to rephrase this in a categorical language from the point of view of natural questions of morphisms, extensions and simple objects in this family. I will tell about several results in the problem of constructing extensions of braided sets and how this problem can be generalized to 2-braided categories, how to build extensions of sets with solutions of the Zamolodchikov tetrahedron equation.