Two $A$-pre-categories ${\cal C}$ and ${\cal D}$ are called equivalent if there exists a finite sequence of $A_{\infty}$-pre-categories $({\cal C}_{0},\dots,{\cal C}_{n}),\,{\cal C}_{0}={\cal C},\,{\cal C}_{0}={\cal D}$ such that for every $i,\,0\leq i\leq k-1$ there exists an $A_{\infty}$-equivalence functor from ${\cal C}_{i}$ to ${\cal C}_{i+1}$ or vice versa.