minimal enlargement with this property. In our formulation, it also admits a canonical operation of tensoring a given object with a finite-dimensional vector space.

Suppose that we allow only trivial one-dimensional spaces $F^i = R$. The resulting objects, which can be written more concisely as $X = \bigoplus_{i \in I} X^i[-\sigma^i]$, form a quasi-equivalent full subcategory of $A^{\mathit{tw}}$. We prefer the more general form since it is better suited to later generalizations.

There is a different approach to formal enlargements, through $A_\infty$-modules [@keller99]. Write $C$ for the differential graded category of complexes of $R$-vector spaces whose cohomology is of finite total dimension. A (right) $A$-module with finite cohomology is an $A_\infty$-functor $A^{\mathit{opp}} \rightarrow C$. Concretely, such a module $M$ assigns to each $X \in \mathit{Ob}\,A$ a graded vector space $M(X)$, together with structure maps $$\label{eq:module}
\begin{aligned}
& \mu^1_M: M(X_0) \longrightarrow M(X_0)[1], \\
& \mu^2_M: M(X_1) \otimes \mathit{hom}_A(X_0,X_1) \longrightarrow M(X_0), \\
& \mu^3_M: M(X_2) \otimes \mathit{hom}_A(X_1,X_2) \otimes \mathit{hom}_A(X_0,X_1) \longrightarrow M(X_0)[-1], \\
& \dots
\end{aligned}$$ satisfying the $A_\infty$-module equations ([@keller99] or [@seidel04 Section 1j]), and such that $(M(X),\mu^1_M) \in \mathit{Ob}\,C$ for all $X$. We require $M$ to be strictly unital, which means that $\mu^2_M(m,e_{X_0}) = m$, and $\mu^d_M(m,a_{d-1},\dots,a_1) = 0$ whenever $d 