Turaev bicategories and generalized Yetter-Drinfel’d modules in 2-categories

Abstract

We introduce Turaev bicategories and group braided Turaev bicategories. On one hand, the latter generalize the notions of braided Turaev categories, introduced at the turn of the millennium and originally called “braided crossed group categories” by Turaev himself, and on the other hand, the former generalize the notion of bicategories. 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 a Turaev 2-category for bimonads in 2-categories as a Turaev extension of the 2-category of bimonads, generalizing the corresponding Turaev category 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. Moreover, two examples of proper Turaev bicategories are constructed, one of which is group braided.