A directed $A_{\infty}$-pre-category is an $A_{\infty}$-pre-category such that

a) There is bijection of the class of objects and the set integer numbers: ${\cal C}\simeq{{\bf Z}}$. We denote by $X_{i}$ the object corresponding to $i\in{\bf Z}$.

b) Transversal sequences are $(X_{i_{1}},...,X_{i_{n}}),i_{1}<i_{2}<...<i_{n}$.