A non-unital $A_{\infty}$-category ${\cal C}$ over $k$ is given by the following data:

1) A class of objects $Ob({\cal C})$.

2) For any two objects $X_{1}$ and $X_{2}$ a ${{\bf Z}}$-graded $k$-module of morphisms $Hom(X_{1},X_{2})$.

3) For any sequence of objects $X_{0},...,X_{n}$, $n\geq 1$, a morphism of $k$-modules (called a 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 for any sequence of objects $X_{0},...,X_{N}$, $N\geq 0$ the graded $k$-module $A=A(X_{0},...,X_{N}):=\oplus_{i,j}Hom(X_{i},X_{j})$, equipped with the direct sum of the compositions $m_{n},n\geq 1$, is a non-unital $A_{\infty}$-algebra.