nLab
Hopf adjunction

Definition

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

$$F(x\otimes Gy)\to Fx\otimes y$$F(x \otimes G y) \to F x \otimes y

$$F(Gy\otimes x)\to y\otimes Fx$$F(G y \otimes x) \to 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 $F⊣G$ is Hopf is equivalent to asking that $G$ be a closed monoidal functor, i.e. preserve internal-homs up to isomorphism.

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

References

• Alain Bruguières, Steve Lack, Alexis Virelizier, Hopf monads on monoidal categories, Adv. Math. 227 No. 2, June 2011, pp 745–800, arxiv/0812.2443