These lecture notes provide a self-contained introduction to a wide range of generalizations of
Hopf algebras. Multiplication of their modules is described by replacing the category of vector
spaces with more general monoidal categories thereby extending the range of applications.Since
Sweedler's work in the 1960s Hopf algebras have earned a noble place in the garden of
mathematical structures. Their use is well accepted in fundamental areas such as algebraic
geometry representation theory algebraic topology and combinatorics. Now similar to having
moved from groups to groupoids it is becoming clear that generalizations of Hopf algebras must
also be considered. This book offers a unified description of Hopf algebras and their
generalizations from a category theoretical point of view. The author applies the theory of
liftings to Eilenberg-Moore categories to translate the axioms of each considered variant of a
bialgebra (or Hopf algebra) to a bimonad (or Hopf monad) structure on a suitable functor.
Covered structures include bialgebroids over arbitrary algebras in particular weak bialgebras
and bimonoids in duoidal categories such as bialgebras over commutative rings semi-Hopf group
algebras small categories and categories enriched in coalgebras. Graduate students and
researchers in algebra and category theory will find this book particularly useful. Including a
wide range of illustrative examples numerous exercises and completely worked solutions it is
suitable for self-study.
