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

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