Internal bialgebroid
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] admitting coequalizers commuting with the monoidal product. It consists of two monoids in the monoidal category, namely the base monoid and the total monoid, and several structure morphisms involving and as first axiomatized by G. Böhm. The coequalizers are needed to introduce the tensor product of bimodules over the base monoid; this tensor product is consequently 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 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.