Hopf monads on monoidal categories
By Alain Bruguières, Stephen Lack, Alexis Virelizier. 2010
134 citations recorded by the source corpus.
We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguieres and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedlerʼs Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8f7697bd1cf15efd73b2e8f80b91b594935a5cbb
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In Algebra and Number Theory
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Citations in the corpus
- Skew monoidales, skew warpings and quantum categories (2012, unknown): “ M are left skew monoidales in Comod(V ). An outline of the two constructions is as follows. Start with a quantum category (C,A) in V . Then we have a monoidal comonad A: Ce −→Ce. The left fusion map [4, 6, 26] for Ais vℓ= (p·(A⊗A) p·(1=A⇒⊗δ) p·(A⊗A)·(1⊗A) φ2=·(1⇒⊗A) A·p·(1⊗A)) . (6.32) From the cal”
- Skew-monoidal categories and bialgebroids (2012, background): “The so-called fusion operator [7] associated to a bimonad 〈O,ω, ι〉 is the natural transformation”