A bialgebra (or bigebra) is both an algebra and a coalgebra, where the operations of either one are homomorphisms for the other. 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 .