Hopf algebras and their generalizations from a category theoretical point of view
These lecture notes provide a unified category theoretical introduction to the notion of Hopf algebra and many of its generalizations. The approach has as its point of departure the fact that the category of modules over such algebraic object turns out to be a closed monoidal category and this structure is strictly preserved by some forgetful functor, the target of which is responsible for each of the specific generalizations. One of the main achievements of the book is to derive the axioms defining a given Hopf algebraic structure from the mentioned feature of its category of modules. \par The strategy consists in regarding the category of modules over an algebra, as well as its generalizations, as the Eilenberg-Moore category of some induced monad such that the monoidal structure of the base category is lifted to the Eilenberg-Moore category, the last condition being a reformulation of the condition that the structure is preserved by a forgetful functor. \par Along its seven chapters the book discusses (Hopf) bimonads on monoidal categories, (Hopf) bialgebras over a field, (Hopf) bialgebroids over an arbitrary associative and unital base algebra $B$, weak (Hopf) bialgebras and (Hopf) bimonoids in so-called \textit{duoidal categories}, which are categories endowed with two compatible monoidal structures introduced by \textit{M. Aguiar} and \textit{S. Mahajan} [Monoidal functors, species and Hopf algebras. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1209.18002)] under the name \textit{2-monoidal category}: such (Hopf) bimonoids include as particular examples Hopf monoids in braided monoidal categories, small groupoids, Hopf algebroids over a commutative base algebra, weak Hopf algebras, Hopf monads and certain coalgebra-enriched categories called \textit{Hopf categories}. \par The book offers a self-contained presentation, starting from the basic notions of categories and functors. The introduction of the different algebraic structures is illustrated with several examples and there is an extensive list of bibliographical references. The solutions to the proposed exercises are provided at the end of the book.
scientific article; zbMATH DE number 6927920