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 .
Bialgebras are precisely those sesquialgebras whose product --bimodule is induced from an algebra homomorphism and whose unit - bimodule is induced from an algebra homomorphism .
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)
Tannaka duality for categories of modules over monoids/associative algebras
| monoid/associative algebra | category of modules |
|---|---|
| -algebra | -2-module |
| 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 |
2-Tannaka duality for module categories over monoidal categories
| monoidal category | 2-category of module categories |
|---|---|
| -2-algebra | -3-module |
| Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
| monoidal 2-category | 3-category of module 2-categories |
|---|---|
| -3-algebra | -4-module |
Notions of bialgebra with further structure notably include Hopf algebras and their variants.
Tannaka duality for bialgebras
On bialgebras in locally presentable categories:
Last revised on October 28, 2023 at 05:22:32. See the history of this page for a list of all contributions to it.