nLab Hopf adjunction



2-Category theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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

F(xGy)Fxy F(x \otimes G y) \to F x \otimes y
F(Gyx)yFx F(G y \otimes x) \to y \otimes F x

are isomorphisms for any xCx\in C and yDy\in D.

Of course, if CC, DD, FF, and GG are symmetric, then it suffices to ask for one of these. If CC and DD are moreover cartesian monoidal, then any adjunction is comonoidal, and the condition is also (mis?)named Frobenius reciprocity.


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

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


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

  • Adriana Balan, On Hopf adjunctions, Hopf monads and Frobenius-type properties, Appl. Categ. Structures 25 (2017), no. 5, 747–774. arxiv

Last revised on December 6, 2017 at 18:00:54. See the history of this page for a list of all contributions to it.