Let $k$ be a ${\bf Z}$-graded commutative associative ring as before. A non-unital $A_{\infty}$-pre-category over $k$ is defined by the following data:

a) A class of objects ${\cal C}$.

b) For any $n\geq 1$ a subclass ${\cal C}_{n}^{tr}$ of ${\cal C}^{n}$, ${\cal C}_{1}^{tr}={\cal C}$, called the class of transversal sequences.

c) For $(X_{1},X_{2})\in{\cal C}_{2}^{tr}$ a ${{\bf Z}}$-graded $k$-module of morphisms $Hom(X_{1},X_{2})$.

d) For a transversal sequence of objects $(X_{0},...,X_{n})$, $n\geq 0$, a morphism of $k$-modules (composition map) $m_{n}:\otimes_{0\leq i\leq n-1}Hom(X_{i},X_{i+1})\to Hom(X_{0},X_{n})[2-n]$.

It is required that a subsequence $(X_{i_{1}},...,X_{i_{l}}),i_{1}<i_{2}<...<i_{l}$ of a transversal sequence $(X_{1},...,X_{n})$ is transversal, and that the composition maps satisfy the same system of equations as for non-unital $A_{\infty}$-categories. Explicitly:

$\sum_{i+j=n+1}\sum_{0\leq l\leq i}\epsilon(l,j)m_{i}(a_{0},...,a_{l-1},m_{j}(a_{l},...,a_{l+j}),a_{l+j+1},...,a_{n})=0$,
where $a_{m}\in Hom(X_{m},X_{m+1})$, and $\epsilon(l,j)=(-1)^{j\sum_{0\leq s\leq l-1}deg(a_{s})}$.