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A cobraided bialgebra is a bialgebra $H$ equipped with a convolution-invertible bilinear form $r:H\otimes H\to k$ satisfying the following identities, in Sweedler notation:Convolution invertibility means there is $\bar{r}$ with $\sum r(a_{(1)},b_{(1)})\bar{r}(a_{(2)},b_{(2)})=\epsilon (a)\epsilon (b)$, and likewise with $r,\bar{r}$ interchanged. This is a coquasitriangular structure. On right comodules it induces the braidingThe third identity makes this a comodule map, the first two give its two hexagon identities, and the convolution inverse gives the inverse braiding.

For unital algebras over a field $A,B$, a measuring coalgebra is a coalgebra $C$ together with a linear map $p:C\to {\mathrm{Hom}}_k(A,B)$ such that, in Sweedler notation,Thus a group-like element acts by a unital algebra homomorphism over a field, while skew-primitive elements give twisted product rules.

The universal measuring coalgebra $P(A,B)$ has a universal measuring map $\pi :P(A,B)\to {\mathrm{Hom}}_k(A,B)$ with this property: for every measuring map $p$ from a coalgebra $C$, there is a unique coalgebra homomorphism $f:C\to P(A,B)$ satisfying $\pi f=p$. The uniqueness is among coalgebra homomorphisms; $\pi$ itself need not be an injective map of vector spaces.

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.