  We introduce Turaev bicategories and Turaev pseudofunctors. On the one hand,
they generalize the notions of Turaev categories (and Turaev functors),
introduced at the turn of the millennium and originally called "crossed group
categories" by Turaev himself, and the notions of bicategories and
pseudofunctors, on the other. For bimonads in 2-categories, which we defined in
one of our previous papers, we introduce generalized Yetter-Drinfel`d modules
in 2-categories. These generalize to the 2-categorical setting the generalized
Yetter-Drinfel`d modules (over a field) of Panaite and Staic, and thus also in
particular the anti Yetter-Drinfel`d modules, introduced by
Hajac-Khalkhali-Rangipour-Sommerhauser as coefficients for the cyclic
cohomology of Hopf algebras, defined by Connes and Moscovici. We construct
Turaev 2-category for bimonads in 2-categories as a Turaev extension of the
2-category of bimonads. This Turaev 2-category generalizes the Turaev category
of generalized Yetter-Drinfel`d modules of Panaite and Staic. We also prove in
the 2-categorical setting their results on pairs in involution, which in turn
go back to modular pairs in involution of Connes and Moscovici.
