Internal bialgebroid explained

In mathematics, an internal bialgebroid is a structure which generalizes the notion of an associative bialgebroid to the setup where the ambient symmetric monoidal category of vector spaces is replaced by any abstract symmetric monoidal category (C,

⊗ , I,s) admitting coequalizers commuting with the monoidal product ⊗ . It consists of two monoids in the monoidal category (C, ⊗ , I), namely the base monoid A and the total monoid H , and several structure morphisms involving A and H as first axiomatized by G. Böhm.^[1] The coequalizers are needed to introduce the tensor product ⊗ _{A} of (internal) bimodules over the base monoid; this tensor product is consequently (a part of) a monoidal structure on the category of A -bimodules. In the axiomatics, H appears to be an A -bimodule in a specific way. One of the structure maps is the comultiplication \Delta:H\to H ⊗ _{A} H which is an A -bimodule morphism and induces an internal A -coring structure on H . One further requires (rather involved) compatibility requirements between the comultiplication \Delta and the monoid structures on H and H ⊗ H .

Some important examples are analogues of associative bialgebroids in the situations involving completed tensor products.

See also

• Bialgebra

Notes and References

1. Gabriella Böhm, Internal bialgebroids, entwining structures and corings, in: Algebraic structures and their representations, 207–226, Contemp. Math. 376, Amer. Math. Soc. 2005. Cornell University Library, retrieved 11 September, 2017