nLab
Hopf adjunction

Context

2-Category theory

Monoidal categories

Contents

Definition

Let CC and DD be monoidal categories, and F:CD:GF\colon 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 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.

Properties

  • 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.

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
Revised on June 14, 2011 14:23:39 by Urs Schreiber (131.211.233.220)