$$\label{vaidiv} (:ab:)\circ_n
c=\sum_{k\geq0}{1\over k!}:(\partial^ka)(b\circ_{n+k}c):
+(-1)^{|a||b|}\sum_{k\geq0}b\circ_{n-k-1}(a\circ_k c) .$$

Category $\cR$
==============

In [@LL] we considered a certain category $\cR$ of vertex algebras, together with a functor from $\cR$ to the category of supercommutative rings.

Let $\cR$ be the category of vertex algebras $\cA$ equipped with a $\mathbb{Z}_{\geq 0}$-filtration $$\label{goodi} \cA_{(0)}\subset\cA_{(1)}\subset\cA_{(2)}\subset \cdots,\ \ \ \cA = \bigcup_{k\geq 0}
\cA_{(k)}$$ such that $\cA_{(0)} = \mathbb{C}$, and for all $a\in \cA_{(k)}$, $b\in\cA_{(l)}$, we have $$\label{goodii} a\circ_n b\in\cA_{(k+l)},\ \ \ \text{for}\
n<0,$$ $$\label{goodiii} a\circ_n b\in\cA_{(k+l-1)},\ \ \ \text{for}\
n\geq 0.$$ Elements $a(z)\in\cA_{(d)}\setminus \cA_{(d-1)}$ are said to have degree $d$, and morphisms in $\cR$ are vertex algebra homomorphisms that preserve the filtration.

Filtrations on vertex algebras satisfying (\[goodii\])-(\[goodiii\]) were introduced in [@LiII] and are known as [*good increasing filtrations*]{}. Setting $\cA_{(-1)} = \{0\}$, the associated graded object $gr(\cA) = \bigoplus_{k\geq 0}\cA_{(k)}/\cA_{(k-1)}$ is a $\mathbb{Z}_{\geq 0}$-graded associative, supercommutative algebra with a unit $1$ under a product induced by the Wick product on $\cA$. In general, there is no natural linear map $\cA\ra gr (\cA)$, but for each $r\geq 1$ we have the projection $$\label{proj} \phi_r: \cA_{(r)} \ra \cA_{(r)}/\cA_{(r-1)}\subset gr(\cA).$$ Moreover, $gr(\cA)$ has a derivation $\partial$ of degree zero (induced by the operator $\partial = \frac{d}{dz}$ on $\cA$), 