Abstract
This chapter continues the survey of the theoretical background, turning to more specific topics. Monoidal categories are introduced and monads on them are studied. The general theory of lifting (explained in Chap. 2) is applied to the functors and natural transformations constituting the monoidal structure of the base category. A bijection is proven between the liftings of the monoidal structure of the base category to the Eilenberg-Moore category of a monad; and opmonoidal structures on the monad. An opmonoidal monad is termed a bimonad. If the base category is closed monoidal, then a sufficient and necessary condition is obtained for the lifting of the closed structure as well, in the form of the invertibility of a canonical natural transformation. A Hopf monad is defined as a bimonad for which this natural transformation is invertible.
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Böhm, G. (2018). (Hopf) Bimonads. In: Hopf Algebras and Their Generalizations from a Category Theoretical Point of View. Lecture Notes in Mathematics, vol 2226. Springer, Cham. https://doi.org/10.1007/978-3-319-98137-6_3
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DOI: https://doi.org/10.1007/978-3-319-98137-6_3
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