Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
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 is actually equivalent to requiring that is a strong monoidal functor.) This adjunction is a Hopf adjunction if the canonical morphisms
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 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]
Last revised on April 26, 2023 at 10:01:47. See the history of this page for a list of all contributions to it.