A bialgebroid may be viewed as a multiobject generalization of a concept of a bialgebra, or a possibly noncommutative generalization of a space-algebra dual version of the concept of an internal category in spaces.
This entry is about “associative” bialgebroid, see also the different concept of a Lie bialgebroid.
When a monoidal category has a fiber functor to a category of vector spaces over a field, one tries to “reconstruct” the category as the category of representations of the endomorphism object of a fiber functor. One often does not have a fiber functor to vector spaces but only to bimodules over some base algebra . Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over and the category is equivalent to the category of representations of that bialgebroid.
Given a unital (possibly noncommutative) ring an -bialgebroid is an --bimodule (object of ) equipped with a structure of a comonoid in (i.e. an -coring) and of a monoid in (i.e. an -ring), where is the enveloping ring of ; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad is opmonoidal. The category of -comodules is by definition the category of comodules over the underlying -coring.
All modules and morphisms will be over a fixed ground commutative ring .
A left -bialgebroid is an -ring , together with the -bimodule map “comultiplication” , which is coassociative and counital with a counit , such that
(i) the -bimodule structure used on is , where and are the algebra maps induced by the unit of the -ring
(ii) the coproduct corestricts to the Takeuchi product and the corestriction is a -algebra map, where the Takeuchi product has a multiplication induced factorwise
(iii) is a left character on the -ring
Notice that is in general not an algebra, just an -bimodule.
The definition of a right -bialgebroid differs by the -bimodule structure on given instead by and the counit is a right character on the -coring ( and can be interchanged in the last requirement).
Related notions: Hopf algebroid
The commutative case is rather classical. See for example the appendix to
The first version of a bialgebroid over a noncommutative base was more narrow:
A modern generality, but in different early formalism, is due Takeuchi (who was motivated to generalize the results from the Sweedler’s paper), under the name of -bialgebra (as it involves the -product, nowdays called Takeuchi product):
Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi’s -bialgebra (though differently axiomatized there):
Modern treatments are in