vector field. Indeed, one might think of the original Hochschild cohomology class as determining a vector field on the “moduli spaces of objects in $A$” (making this rigorous requires machinery far beyond that deployed here; interested readers are referred to [@toen-vaquie07]).

We work over an algebraically closed field $R$ of characteristic $0$. While the condition of algebraic closedness is perhaps mostly for the sake of familiarity, the restriction on the characteristic is crucial, since we will be using differentiation (and in particular Lemma \[th:locally-free\]). Sign conventions for $A_\infty$-algebras and associated structures usually follow [@seidel04]. Those for twisted complexes are specifically as in [@seidel04 Remark 3.26]. All categories are assumed to be small.

$A_\infty$-categories
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Fix an $A_\infty$-category $A$ over $R$. This is assumed to be ${\mathbb{Z}}$-graded, strictly unital, and proper. The units (identity endomorphisms) are denoted by $e_X \in \mathit{hom}_A^0(X,X)$. Write $H(A)$ for the associated cohomology level category, and by $H^0(A)$ the version where only morphisms of degree $0$ are allowed. Properness means that the morphisms in $H(A)$ are graded vector spaces of finite total dimension. Two objects of $A$ are called quasi-isomorphic if they become isomorphic in $H^0(A)$. There are various canonical formal enlargements of $A$. Possibly the simplest 