Abstract
In this chapter an analysis is carried out which is analogous to, but more general than that in Chap. 4. The category of vector spaces in Chap. 4 is replaced by the category of bimodules over some algebra B; or, isomorphically, the category of left modules over B ⊗ B op. Those endofunctors on it are considered which are induced, as in Example 2.5 4, by the B ⊗ B op-module tensor product with a fixed B ⊗ B op-bimodule A. The monad structures on this functor \(A \otimes _{B \otimes B^{\mathsf {op}}} -\) are related to the algebra homomorphisms B ⊗ B op → A.
The monoidal structure of the category of B-bimodules is explained in Example 3.2 5. The opmonoidal structures with respect to it on the endofunctor \(A \otimes _{B \otimes B^{\mathsf {op}}} -\), are related to the so-called B|B-coring structures on A. This results in a bijection between the bimonads with underlying functor \(A \otimes _{B \otimes B^{\mathsf {op}}} -\) on the category of B-bimodules; and the bialgebroids over the base algebra B. The bijection is shown to restrict to Hopf monads on one hand; and Hopf algebroids on the other hand.
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Böhm, G. (2018). (Hopf) Bialgebroids. 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_5
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