  The Yang-Baxter equation plays a fundamental role in various areas of
mathematics. Its solutions, called braidings, are built, among others, from
Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive
structures, and from crossed modules of groups. In the present paper these
three sources of solutions are unified inside the framework of Yetter-Drinfe' d
modules over a braided system. A systematic construction of braiding structures
on such modules is provided. Some general categorical methods of obtaining such
generalized Yetter-Drinfel'd (=GYD) modules are described. Among the braidings
recovered using these constructions are the Woronowicz and the Hennings
braidings on a Hopf algebra. We also introduce the notions of crossed modules
of shelves / Leibniz algebras, and interpret them as GYD modules. This yields
new sources of braidings. We discuss whether these braidings stem from a
braided monoidal category, and discover several non-strict pre-tensor
categories with interesting associators.
