Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
Let and be monoidal categories, and a comonoidal adjunction , i.e. an adjunction in the 2-category of colax monoidal functors. (By doctrinal adjunction, this implies that is a strong monoidal functor.) This adjunction is a Hopf adjunction if the canonical morphisms
F(x \otimes G y) \to F x \otimes y
F(G y \otimes x) \to y \otimes F x
are isomorphisms for any and .
Of course, if , , , and are symmetric, then it suffices to ask for one of these. If and are moreover cartesian monoidal, then any adjunction is comonoidal, and the condition is also (mis?)named Frobenius reciprocity.
If and are closed, then by the calculus of mates, saying that is Hopf is equivalent to asking that be a closed monoidal functor, i.e. preserve internal-homs up to isomorphism.
If is a Hopf adjunction, then its induced monad 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
Revised on June 14, 2011 14:23:39
by Urs Schreiber