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.

Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

“…While any monoidal functor is, in particular, Frobenius monoidal, for general Frobenius monoidal functors, like those considered in this paper, F (V ) ⊗ F (W ) and F (V ⊗ W ) are not isomorphic. However, any Frobenius monoidal functor sends Frobenius algebras in C to Frobenius algebras in D. Frobenius monoidal functors have recently appeared in different contexts in the quantum algebra literature, see, e.g., [16,34,47]. Here, we construct Frobenius monoidal functors to categories of the form Z Vect ω G .…”

Hopf algebraic methods in finite tensor categories: Characters and integrals

“…Recently, non-special symmetric Frobenius algebras are also of interest from the viewpoint of CFTs and topological field theories. With regard to this, some constructions of Frobenius algebras in finite tensor categories are established in [FS23,WY23,Shi23,Yad24].…”