nLab Hopf adjunction

Definition

Let $C$ and $D$ be monoidal categories, and $F \,\colon\, C \rightleftarrows D \,\colon\, G$ a comonoidal adjunction, i.e. an adjunction $F\dashv G$ in the 2-category of colax monoidal functors. (By doctrinal adjunction, this is actually equivalent to requiring that $G$ is a strong monoidal functor.) This adjunction is a Hopf adjunction if the canonical morphisms

$$F(x \otimes G y) \longrightarrow F x \otimes y$$

$$F(G y \otimes x) \longrightarrow y \otimes F x$$

are isomorphisms for any $x\in C$ and $y\in D$.

Of course, if $C$, $D$, $F$, and $G$ are symmetric, then it suffices to ask for one of these. If $C$ and $D$ are moreover cartesian monoidal, then any adjunction is comonoidal, and the condition is also (mis?)named Frobenius reciprocity.

Properties

• If $C$ and $D$ are closed, then by the calculus of mates, saying that $F\dashv G$ is Hopf is equivalent to asking that $G$ be a closed monoidal functor, i.e. preserve internal-homs up to isomorphism.

• If $F\dashv G$ is a Hopf adjunction, then its induced monad $G F$ is a Hopf monad. Conversely, the Eilenberg-Moore adjunction of a Hopf monad is a Hopf adjunction.

Related concepts

• adjunction

• Frobenius reciprocity

• Hopf monad

• Hopf algebra

References

• Alain Bruguières, Steve Lack, Alexis Virelizier, Hopf monads on monoidal categories, Adv. Math. 227 2 (2011) 745-800 [arXiv:1003.1920, doi:10.1016/j.aim.2011.02.008]

• Adriana Balan, On Hopf adjunctions, Hopf monads and Frobenius-type properties, Appl. Categ. Structures 25 5 (2017) 747-774 [arxiv:1411.2236, doi:10.1007/s10485-016-9428-0]

• Harshit Yadav, Frobenius monoidal functors from (co)Hopf adjunctions [arXiv:2209.15606v2]