one is the $A_\infty$-category of twisted complexes $A^{\mathit{tw}}$, introduced in [@bondal-kapranov91; @kontsevich94]. One can carry out this enlargement in two steps. First, consider the additive envelope $A^{\oplus}$, whose objects are formal expressions $$\label{eq:formal-sum}
X = \bigoplus_{i \in I} F^i \otimes X^i[-\sigma^i]$$ where $I$ is a finite set, the $X^i$ are objects of $A$, formally shifted by degrees $\sigma^i \in {\mathbb{Z}}$, and the $F^i$ are finite-dimensional vector spaces. Morphisms in $A^{\oplus}$ can be thought of as matrices, whose entries are morphisms in $A$ tensored with maps of vector spaces. Correspondingly, the $A_\infty$-products combine those of $A$ with composition of linear maps and matrix multiplication (with auxiliary signs due to the shift). In the second step, one defines a twisted complex to be an object $X \in \mathit{Ob}\,A^{\oplus}$ equipped with an additional differential. This differential $\delta_X \in \mathit{hom}_{A^{\oplus}}^1(X,X)$ is an endomorphism which is strictly decreasing with respect to some filtration of $X$ by sub-objects, and which satisfies the equation $$\label{eq:generalized-maurer-cartan}
\mu^1_{A^{\oplus}}(\delta_X) + \mu^2_{A^{\oplus}}(\delta_X,\delta_X) + \cdots = 0.$$ $A^{\mathit{tw}}$ is an $A_\infty$-category with the same general properties as $A$, and which contains $A$ as a full subcategory. It is closed under shifts and mapping cones, and is characterized up to quasi-equivalence as the 