The notation $A^{\text{cop}}$.

Question

Let $A$ be an algebra. Then $A^{\text{op}}$ is the algebra with multiplication defined by $a \cdot b = b \circ a$, where $b \circ a$ is the multiplication in $A$.

Let $A$ be a coalgebra. How to define the comultiplication in $A^{\text{cop}}$? Thank you very much.

Answer 1

I assume you work over a commutative ring $R$.

Then, multiplication is a map from the tensor product $\mu:A\otimes A\to A$, satisfying associativity. Its opposite is defined as the composition $\mu\circ\sigma$ where $\sigma:A\otimes A\to A\otimes A$ is the exchange $a\otimes b\mapsto b\otimes a$.

Now, comultiplication is a map to the tensor product $\Delta:A\to A\otimes A$ (satisfying coassociativity), and thus its opposite should be the composition $\sigma\circ\Delta$.