nLab Hopf monoid

Hopf monoids

Hopf monoids


A Hopf monoid is the generalization of a Hopf algebra to an arbitrary symmetric monoidal category.


We work in an arbitrary symmetric monoidal category 𝒞\mathcal{C} (although one can generalize to a braided monoidal category or even a duoidal category). Recall that a bimonoid in 𝒞\mathcal{C} is an object HH 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 HHH \otimes H; e.g., if σ\sigma is the symmetry and μ\mu the monoid multiplication on HH, the multiplication on HHH \otimes H is

HHHH1σ1HHHHμμHHH \otimes H \otimes H \otimes H \stackrel{1 \otimes \sigma \otimes 1}{\to} H \otimes H \otimes H \otimes H \stackrel{\mu \otimes \mu}{\to} H \otimes H

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 HH such that there exists a map s:HHs:H\to H, called the antipode, such that the following diagram commutes

H ϵ I η H Δ m HH 1s HH. \array{ H & \xrightarrow{\epsilon} & I & \xrightarrow{\eta} & H\\ ^{\Delta}\downarrow &&&& \uparrow^{m}\\ H\otimes H & & \xrightarrow{1\otimes s} & & H\otimes H. }

as does the analogous diagram with 1s1\otimes s replaced by s1s\otimes 1.



Closed monoidal structure on modules

For any bimonoid HH in 𝒞\mathcal{C}, the category Mod HMod_H of HH-modules inherits a monoidal structure such that the forgetful functor (“fiber functor”) Mod H𝒞Mod_H \to \mathcal{C} is strong monoidal; see bimonoid. If HH is moreover a Hopf monoid and 𝒞\mathcal{C} is a closed monoidal category, then Mod HMod_H 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 HH-modules MM and NN, we define an HH-module structure on the internal-hom Hom(M,N)Hom(M,N) in 𝒞\mathcal{C} to be the adjunct of the following composite:

HHom(M,N)M ΔHHHom(M,N)M HHom(M,N)HM 11s1HHom(M,N)HM actHHom(M,N)M evalHN actN. \begin{aligned} H \otimes Hom(M,N) \otimes M &\xrightarrow{\Delta} H\otimes H \otimes Hom(M,N) \otimes M\\ &\xrightarrow{\cong} H \otimes Hom(M,N)\otimes H \otimes M\\ &\xrightarrow{1\otimes 1\otimes s\otimes 1} H \otimes Hom(M,N)\otimes H \otimes M\\ &\xrightarrow{act} H \otimes Hom(M,N)\otimes M\\ &\xrightarrow{eval} H \otimes N\\ &\xrightarrow{act} N. \end{aligned}

If 𝒞\mathcal{C} is cartesian and HH is a group object, then this is the “conjugation” action, with gHg\in H sending f:MNf:M\to N to mgf(g 1m)m\mapsto g\cdot f(g^{-1}\cdot m). Diagram chases show that this makes Hom(M,N)Hom(M,N) an HH-module and Mod HMod_H 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:

Last revised on October 6, 2023 at 07:07:14. See the history of this page for a list of all contributions to it.