The other $A_{\infty}$-pre-category we are interested in will be a differential-graded category (dg-category for short). In other words, it is an $A_{\infty}$-category with strict identity morphisms and vanishing compositions $m_{n},n\geq 3$. We will call it de Rham category of $Y$ and denote by $DR(Y)$. Objects of $DR(Y)$ are same as for $M(Y)$. They are pairs $(f,\rho)$, where $f:Y\to{\bf R}$ is a smooth function and $\rho$ is a local system on $Y$. Morphisms are complexes defined by the formula

$$Hom_{DR(Y)}((f_{0},\rho_{0}),(f_{1},\rho_{1}))=\Gamma(Y,\wedge^{\ast}T^{\ast}_{Y}\otimes Hom(\rho_{0},\rho_{1})).$$

Notice that the space of morphisms does not depend on $f_{0}$ and $f_{1}$. The composition of morphisms is defined in the obvious way: in a local trivialization of $\rho_{0}$ and $\rho_{1}$ it is given by the product of matrices with the coefficients in $\Omega^{\ast}(Y)$.

Now we can formulate the main result of this section.

The proof of the theorem will occupy the rest of the section. First, we will discuss a version of formulas from “homological perturbation theory” (see [GS], [Me]). They will give an $A_{\infty}$-structure on a subcomplex of a dg-algebra. Then we will discuss an approach to the proof based on the ideas of [HL]. It seems plausible that an alternative proof (but, presumably, much more difficult) can be obtained within the framework of Witten complex, using methods of [BZ].