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 and the total monoid , and several structure morphisms involving and as first axiomatized by G. Böhm.[1] The coequalizers are needed to introduce the tensor product of (internal) bimodules over the base monoid; this tensor product is consequently (a part of) a monoidal structure on the category of -bimodules. In the axiomatics, appears to be an -bimodule in a specific way. One of the structure maps is the comultiplication which is an -bimodule morphism and induces an internal -coring structure on . One further requires (rather involved) compatibility requirements between the comultiplication and the monoid structures on and .

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

See also

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References

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  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