symmetric monoidal (∞,1)-category of spectra
A bialgebra (or bigebra) is both an algebra and a coalgebra, where the operations of either one are homomorphisms for the other. A bialgebra structure on an associative algebra is precisely such as to make its category of modules into a monoidal category equipped with a fiber functor.
A bialgebra is one of the ingredients in the concept of Hopf algebra.
A bialgebra is a monoid in the category of coalgebras. Equivalently, it is a comonoid in the category of algebras. Equivalently, it is a monoid in the category of comonoids in Vect — or equivalently, a comonoid in the category of monoids in Vect.
More generally, a bimonoid in a monoidal category is a monoid in the category of comonoids in — or equivalently, a comonoid in the category of monoids in . So, a bialgebra is a bimonoid in .
The structure of a bialgebra on an associative algebra equips its category of modules with the structure of a monoidal category and a monoidal fiber functor. In fact that construction is an equivalence. This is the statement of Tannaka duality for bialgebras. For instance (Bakke)
|monoid/associative algebra||category of modules|
|sesquialgebra||2-ring = monoidal presentable category with colimit-preserving tensor product|
|bialgebra||strict 2-ring: monoidal category with fiber functor|
|Hopf algebra||rigid monoidal category with fiber functor|
|hopfish algebra (correct version)||rigid monoidal category (without fiber functor)|
|weak Hopf algebra||fusion category with generalized fiber functor|
|quasitriangular bialgebra||braided monoidal category with fiber functor|
|triangular bialgebra||symmetric monoidal category with fiber functor|
|quasitriangular Hopf algebra (quantum group)||rigid braided monoidal category with fiber functor|
|triangular Hopf algebra||rigid symmetric monoidal category with fiber functor|
|supercommutative Hopf algebra (supergroup)||rigid symmetric monoidal category with fiber functor and Schur smallness|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
Notions of bialgebra with further structure notably include Hopf algebras and their variants.
Tannaka duality for bialgebras