2007/10/05 by Bachuki Mesablishvili, Mesablishvili, Bachuki, Robert Wisbauer +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.CT #math.QA
paper · pdf · doi:10.48550/arxiv.0710.1163
openalex publication_date 2007/10/05 · arxiv created 2008/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to \em monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category \A and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on \A: we define a \em bimonad on \A as an endofunctor B which is a monad and a comonad with an entwining λ:BB→ BB satisfying certain conditions. This λ is also employed to define the category \ABB of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ:BB→ BB satisfying the Yang-Baxter equation (\em local prebraidings) which induce an entwining λ and lead to an extension of the theory of \em braided Hopf algebras. An antipode is defined as a natural transformation S:B→ B with special properties and for categories \A with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between \A and the category \ABB of B-bimodules. This is a general form of the \em Fundamental Theorem of Hopf algebras.