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
, I), namely the base monoid
, and several structure morphisms involving
[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
One of the structure maps is the comultiplication
-bimodule morphism and induces an internal
One further requires (rather involved) compatibility requirements between the comultiplication
Some important examples are analogues of associative bialgebroids in the situations involving completed tensor products.