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An algebra over a commutative ring that is also a coalgebra, whose comultiplication and counit preserve the multiplication and unit. Equivalently it is a bimonoid in $k$-modules.

With the convention $m\ast \gamma =\gamma \ast n$ and invertible bialgebra scalar cocycle $\gamma$ for $n$, the same coalgebra has $n=\bar{\gamma }\ast m\ast \gamma$, or equivalently $m=\gamma \ast n\ast \bar{\gamma }$. Thus $n(a,b)=\sum \bar{\gamma }(a_{(1)},b_{(1)})m(a_{(2)},b_{(2)})\gamma (a_{(3)},b_{(3)})$. Specifying the equation prevents opposite twist conventions from being conflated.

For a comonad, a coalgebra is a map $a:A\to GA$ satisfying ${ϵ}_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. A morphism $f:(A,a)\to (B,b)$ obeys $bf=Gfa$. This is distinct from the tensor-based notion of a coalgebra in algebra.

A right comodule is an object $M$ with a coaction $\rho :M\to M\otimes C$ satisfying the coassociativity and counit laws. This definition extends from coalgebras to comonoids in a monoidal category.

For a coalgebra $C$ and a unital associative algebra over a commutative ring $A$, maps $f,g:C\to A$ have convolution $f\ast g=m_A(f\otimes g){\Delta }_C$, with unit $j_A{\epsilon }_C$. The two-sided convolution inverse of $1_H$ is the antipode.

On the monoidal category of modules over the commutative ring $k$, consider the monad $T=H{\otimes }_k-$ coming from the unit and multiplication of the bialgebra $H$. Its opmonoidal functor structure has comparison mapsHere and below Sweedler notation abbreviates the comultiplication $\Delta h=\sum h_{(1)}\otimes h_{(2)}$. The opmonoidal associativity and unit axioms are the coassociativity and counit laws of the coalgebra. The unit and multiplication of a monad are opmonoidal natural transformations because the bialgebra axioms sayThe Eilenberg-Moore category of this opmonoidal monad is the category of left $H$-modules: a monad-algebra map $H{\otimes }_{k}M\to M$ is exactly a unital associative action.

Applying the preceding construction gives the diagonal bialgebra action and the unit actionThe usual associators and unitors for the tensor product of modules are $H$-linear, and the underlying tensor product is exactly ${\otimes }_k$. The forgetful functor into $k$-modules is strict monoidal. The base need not be a field; the modules need be neither flat modules nor finitely generated modules.