mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an internal bialgebroid is a structure which generalizes the notion of an

associative bialgebroid In mathematics, if L is an associative algebra over some ground field ''k'', then a left associative L-bialgebroid is another associative ''k''-algebra H together with the following additional maps: an algebra map \alpha:L\to H called the source ma ...

to the setup where the ambient symmetric monoidal category of vector spaces is replaced by any abstract

symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...

(''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.Gabriella Böhm, Internal bialgebroids, entwining structures and corings, in: Algebraic structures and their representations, 207–226, Contemp. Math. 376, Amer. Math. Soc. 2005
Cornell University Library, retrieved 11 September, 2017
/ref> 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.

References

{{context, date=September 2017 Bialgebras

See also

References