  We extend the formalism of Hopf cyclic cohomology to the context of braided
categories. For a Hopf algebra in a braided monoidal abelian category we
introduce the notion of stable anti-Yetter-Drinfeld module. We associate a
para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a
braided modular pair in involution in the sense of Connes and Moscovici. When
the braiding is symmetric the full formalism of Hopf cyclic cohomology with
coefficients can be extended to our categorical setting.
