On functors between module categories for associative algebras and for $\mathbb{N}$-graded vertex algebras

We prove that the weak associativity for modules for vertex algebras are equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requiring that these two formulas instead of the commutator formula hold, we construct a functor $S$ from the category of modules for Zhu's algebra of a vertex operator algebra $V$ to the category of $\mathbb{N}$-gradable weak $V$-modules. We prove that $S$ has a universal property and the functor $T$ of taking top levels of $\mathbb{N}$-gradable weak $V$-modules is a left inverse of $S$. In particular, $S$ is equal to a functor implicitly given by Zhu and explicitly constructed by Dong, Li and Mason and we obtain a new construction without using relations corresponding to the commutator formula. The hard part of this new construction is a technical theorem stating roughly that in a module for Zhu's algebra, the relation corresponding to the residue formula mentioned above can in fact be obtained from the relations corresponding to the action of Zhu's algebra.