[Paper Review] Generalised bialgebras and entwined monads and comonads

Muriel Livernet, Bachuki Mesablishvili+1 more|arXiv (Cornell University)|Mar 17, 2014

Advanced Topics in Algebra8 references4 citations

TL;DR

This paper extends Loday's Rigidity Theorem for generalised bialgebras within a categorical framework of entwined monads and comonads. By establishing a correspondence between mixed bimodules and entwined structures, the authors prove that a comonad morphism induced by an entwining is an isomorphism if and only if the associated functor is an equivalence, thereby generalising the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems to this setting.

ABSTRACT

Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework.

Motivation & Objective

• To generalise Loday's structure theorems for generalised bialgebras using categorical tools of entwined monads and comonads.
• To clarify the relationship between generalised bialgebras and mixed bimodules over bimonads in a category-theoretic setting.
• To extend the Rigidity Theorem to a broader categorical framework, unifying results from operad theory and distributive laws.
• To provide a categorical foundation for the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the context of entwined structures.
• To demonstrate that the equivalence of categories induced by entwinings implies the Hopf monad property, generalising known results on infinitesimal bialgebras.

Proposed method

• Utilises the Eilenberg-Moore category of modules over a monad and comodules over a comonad to model algebraic and coalgebraic structures.
• Applies the theory of distributive laws between monads and comonads to define entwined structures between algebras and coalgebras.
• Introduces a functor $ K: ext{Alg}({rak T}_{rak A}) o ext{Coalg}({rak G}_{rak C}) $ that maps algebras to comodules via an entwining, establishing a correspondence.
• Employs a comonad morphism $ t: ext{Free} o ext{Cofree} $ induced by the entwining, and proves its isomorphism via the condition $ ext{Id} = ext{Id} \circ \varepsilon $.
• Applies the equivalence criterion from [14, 3.1] to show that if the composite $ ext{Id} \circ \varepsilon $ is an isomorphism, then the comonad morphism is an isomorphism.
• Verifies that the entwining satisfies the required commutative diagrams (e.g., for $ \lambda $), ensuring compatibility between multiplication, comultiplication, and the entwining.

Experimental results

Research questions

• RQ1 How can Loday’s Rigidity Theorem for generalised bialgebras be extended to a categorical framework involving entwined monads and comonads?
• RQ2 What conditions ensure that a functor between the category of algebras and the category of comodules is an equivalence of categories?
• RQ3 In what way do distributive laws between monads and comonads generalise the structure of bialgebras and Hopf algebras?
• RQ4 How does the entwining structure between a monad and a comonad relate to the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems?
• RQ5 Under what categorical conditions does the existence of a comonad morphism imply that the associated monad is a Hopf monad?

Key findings

• The functor $ K $, mapping algebras over a monad $ {rak T}_{rak A} $ to comodules over a comonad $ {rak G}_{rak C} $, is an equivalence of categories if and only if the composite $ ext{Id} \circ \varepsilon $ is an isomorphism.
• The comonad morphism $ t $ induced by the entwining is an isomorphism if and only if the natural transformation $ \varphi $, defined as $ \mathscr{M}(\varepsilon_V) \circ \delta_V $, is an isomorphism.
• The condition $ \varphi = \text{Id} $ holds for all $ V $, implying $ \varphi $ is an isomorphism, which in turn implies $ t $ is an isomorphism.
• The equivalence of categories implies that the monad $ {rak H} $ is a Hopf monad, generalising the Hopf algebra structure in the categorical setting.
• The theory recovers Loday and Ronco’s Rigidity Theorem for infinitesimal bialgebras, showing that such bialgebras are freely and cofreely generated by their primitive parts.
• The entwining $ \lambda $ defined on the tensor algebra satisfies the required commutative diagrams, confirming compatibility between multiplication, comultiplication, and the entwining.