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Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers

Johannes Flake, Robert Laugwitz, Sebastian Posur

TL;DR

The paper develops a general framework in which an ambiadjunction F ⊣ G ⊣ F, with G strong monoidal, yields a Frobenius monoidal functor F when projection formulas are invertible. It then lifts these structures to Drinfeld centers, giving braided Frobenius monoidal functors under stronger invertibility, via endomorphism and center formalisms. The authors apply the theory to Hopf-algebra morphisms, showing Ind_φ and related functors become Frobenius monoidal and, on centers, braided Frobenius monoidal when a Frobenius tr: H → K is a bicomodule morphism; Yetter–Drinfeld centers offer explicit braided Frobenius lifts. The results produce numerous new examples of braided Frobenius monoidal functors and unify several known center-lifting phenomena under a projection-formula criterion, with concrete Hopf-algebra realizations and counterexamples illustrating the necessity of the bicomodule condition for the center lift.

Abstract

We identify general conditions, formulated using the projection formula morphisms, for a functor that is simultaneously left and right adjoint to a strong monoidal functor to be a Frobenius monoidal functor. Moreover, we identify stronger conditions for the adjoint functor to extend to a braided Frobenius monoidal functor on Drinfeld centers building on our previous work in [arXiv:2402.10094]. As an application, we construct concrete examples of (braided) Frobenius monoidal functors obtained from morphisms of Hopf algebras via induction.

Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers

TL;DR

The paper develops a general framework in which an ambiadjunction F ⊣ G ⊣ F, with G strong monoidal, yields a Frobenius monoidal functor F when projection formulas are invertible. It then lifts these structures to Drinfeld centers, giving braided Frobenius monoidal functors under stronger invertibility, via endomorphism and center formalisms. The authors apply the theory to Hopf-algebra morphisms, showing Ind_φ and related functors become Frobenius monoidal and, on centers, braided Frobenius monoidal when a Frobenius tr: H → K is a bicomodule morphism; Yetter–Drinfeld centers offer explicit braided Frobenius lifts. The results produce numerous new examples of braided Frobenius monoidal functors and unify several known center-lifting phenomena under a projection-formula criterion, with concrete Hopf-algebra realizations and counterexamples illustrating the necessity of the bicomodule condition for the center lift.

Abstract

We identify general conditions, formulated using the projection formula morphisms, for a functor that is simultaneously left and right adjoint to a strong monoidal functor to be a Frobenius monoidal functor. Moreover, we identify stronger conditions for the adjoint functor to extend to a braided Frobenius monoidal functor on Drinfeld centers building on our previous work in [arXiv:2402.10094]. As an application, we construct concrete examples of (braided) Frobenius monoidal functors obtained from morphisms of Hopf algebras via induction.

Paper Structure

This paper contains 21 sections, 47 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Background and related work
  4. Description of results
  5. Summary of content
  6. Preliminaries
  7. Frobenius monoidal functors
  8. Ambiadjunctions and Frobenius functors
  9. From Frobenius functors to Frobenius monoidal functors
  10. Frobenius monoidal functors between endomorphism categories
  11. A context for the projection formulas in an ambiadjunction
  12. When do monoidal functors induce ambiadjunctions of (bi)module categories?
  13. Conditions for Frobenius functors to be Frobenius monoidal functors
  14. Frobenius monoidal functors on Drinfeld centers
  15. Centers and endomorphism categories
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that $\mathrm{rproj}^{F, G \dashv F}$ and $\mathrm{rproj}^{F, F \dashv G}$ are mutual inverses. Then, $F\colon \mathcal{D}\to \mathcal{C}$ with $\operatorname{lax}^F$ and $\operatorname{oplax}^F$ is a Frobenius monoidal functor.

Theorems & Definitions (120)

  • Theorem 1: See \ref{['theorem:frobenius_functors_frobenius_monoidal']}
  • Theorem 2: See \ref{['thm:ZF']}
  • Theorem 3: See \ref{['theorem:relation_frobenius_monoidal_and_ambi']}
  • Theorem 4: See \ref{['theorem:strong_monoidal_to_ambiadjunction']}
  • Corollary 5: See \ref{['prop:Hopf-right']}, \ref{['cor:Ind-Frob']}, and \ref{['cor:ZInd-Frob']}
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 110 more