  We study completely positive maps and injectivity for Yetter-Drinfeld
algebras over compact quantum groups, and module categories over rigid
C*-tensor categories. This gives a generalization of Hamana's theory of
injective envelopes to the framework of dynamical systems over quantum groups.
As a byproduct we also establish a duality between the Yetter-Drinfeld algebras
and certain bimodule categories with central generators.
