symmetric monoidal (∞,1)-category of spectra
A Hopf monoid is the generalization of a Hopf algebra to an arbitrary symmetric monoidal category.
We work in an arbitrary symmetric monoidal category (although one can generalize to a braided monoidal category or even a duoidal category). Recall that a bimonoid in is an object which is both a monoid object and a comonoid object in a compatible way, i.e. the comultiplication and the counit are morphisms of monoids, or equivalently the multiplication and the unit are morphisms of comonoids. (To make sense of this description, one requires the symmetry in order to define the appropriate monoid/comonoid structure on ; e.g., if is the symmetry and the monoid multiplication on , the multiplication on is
where we are writing as if the monoidal product is strict. One can get away with an ambient context of a monoidal category provided that the monoid/comonoid structures in question can be viewed as living in the center of the monoidal category.)
A Hopf monoid is a bimonoid such that there exists a map , called the antipode, such that the following diagram commutes
as does the analogous diagram with replaced by .
A Hopf monoid in is precisely a Hopf algebra.
In a cartesian monoidal category, every monoid object is a bimonoid in a unique way. Such a bimonoid is a Hopf monoid exactly when the monoid object is a group object. More generally, see the corresponding remark there.
For any bimonoid in , the category of -modules inherits a monoidal structure such that the forgetful functor (“fiber functor”) is strong monoidal; see bimonoid. If is moreover a Hopf monoid and is a closed monoidal category, then is also closed, and the forgetful functor is strong closed (preserves internal homs).
This is a special case of Tannaka duality for monoids/algebras, see at structure on algebras and their module categories - table.
Specifically, given -modules and , we define an -module structure on the internal-hom in to be the adjunct of the following composite:
If is cartesian and is a group object, then this is the “conjugation” action, with sending to . Diagram chases show that this makes an -module and a closed monoidal category.
This result can be generalized to algebras over any Hopf monad (in the sense of Bruguières-Lack-Virelizier).
A few references emphasizing applications to combinatorics:
Marcelo Aguiar and Swapneel Mahajan. Monoidal Functors, Species and Hopf Algebras, 2010. (author pdf)
Marcelo Aguiar and Swapneel Mahajan. Hopf monoids in the category of species. Contemporary Mathematics 585 (2013), 17–124. (author pdf)
Marcelo Aguiar and Federico Ardila. Hopf monoids and generalized permutahedra. arXiv:1709.07504
Last revised on October 6, 2023 at 07:07:14. See the history of this page for a list of all contributions to it.