A dual theory of Zhu's theory

We establish a dual theory of Zhu’s $A(\mathcal{V})$-theory, i.e., for a graded vertex operator coalgebra $V$, the coassociative coalgebra $C(V)$ are investigated, any admissible $V$-comodule gives a $C(V)$-comodule, and vice versa. We also prove that $V$ is corational, which means every admissible $V$-comodule is completely reducible, if and only if its dual vertex operator algebra $V^{\mathrm{\prime }}$ is rational.