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 (C, $$\otimes$$, I,s) admitting coequalizers commuting with the monoidal product $$\otimes$$. It consists of two monoids in the monoidal category (C, $$\otimes$$, 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. The coequalizers are needed to introduce the tensor product $$\otimes_A$$ 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 $$\Delta:H\to H\otimes_A H$$ 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 $$H\otimes H$$.

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