Frobenius monoidal functors from (co)Hopf adjunctions

Abstract

Let U : C → D U:\mathcal {C}\rightarrow \mathcal {D} be a strong monoidal functor between abelian monoidal categories admitting a right adjoint R R , such that R R is exact, faithful and the adjunction U ⊣ R U\dashv R is coHopf. Building on the work of Balan [Appl. Categ. Structures 25 (2017), pp. 747–774], we show that R R is separable (resp., special) Frobenius monoidal if and only if R ( 1 D ) R(\mathbb {1}_{\mathcal {D}}) is a separable (resp., special) Frobenius algebra in C \mathcal {C} . If further, C , D \mathcal {C},\mathcal {D} are pivotal (resp., ribbon) categories and U U is a pivotal (resp., braided pivotal) functor, then R R is a pivotal (resp., ribbon) functor if and only if R ( 1 D ) R(\mathbb {1}_{\mathcal {D}}) is a symmetric Frobenius algebra in C \mathcal {C} . As an application, we construct Frobenius monoidal functors going into the Drinfeld center Z ( C ) \mathcal {Z}(\mathcal {C}) , thereby producing Frobenius algebras in it.