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 $\epsilon(l,j)=(-1)^{j\sum_{0\leq s\leq l-1}deg(a_{s})}$.

A functor $F:{\cal C}\to{\cal D}$ between non-unital $A_{\infty}$-pre-categories is given by the following data:

1) A map of classes of objects $\phi:{\cal C}\to{\cal D}$, such that $\phi^{n}({\cal C}_{n}^{tr})\subset{\cal D}_{n}^{tr}$.

2) For any transversal sequence of objects $(X_{0},...,X_{n}),\,n\geq 1$ in ${\cal C}$, a morphism of graded $k$-modules

These data satisfy the following property: the sequence $f_{n},n\geq 1$ defines an $A_{\infty}$-morphism $\oplus_{i<j}Hom_{{\cal C}}(X_{i},X_{j})\to\oplus_{i<j}Hom_{{\cal D}}(\phi(X_{i}),\phi(X_{j}))$.

Let ${\cal C}$ be a non-unital $A_{\infty}$-pre-category, $(X_{1},X_{2})\in{\cal C}_{2}^{tr}$. We say that $f\in Hom^{0}(X_{1},X_{2})$ (zero stands for degree) is a quasi-isomorphism if $m_{1}(f)=0$, and for any objects $X_{0}$ and $X_{3}$ such that $(X_{0},X_{1},X_{2})\in{\cal C}_{3}^{tr}$ and $(X_{1},X_{2},X_{3})\in{\cal C}_{3}^{tr}$ one has: $m_{2}(f,\cdot):Hom(X_{0},X_{1})\to Hom(X_{0},X_{2})$ and $m_{2}(\cdot,f):Hom(X_{2},X_{3})\to Hom(X_{1},X_{3})$ are quasi-isomorphisms of complexes.

An $A_{\infty}$-pre-category is a non-unital $A_{\infty}$-pre-category ${\cal C}$, satisfying the following extension property:

For any finite collection of transversal sequences $S_{1},...,S_{m}$ in ${\cal C}$ and an object $X$ there exist objects $X_{+}$ and $X_{-}$ and quasi-isomorphisms $f_{-}:X_{-}\to X$, $f_{+}:X\to X_{+}$ such that extended sequences $(X_{-},S_{1},...,S_{m},X_{+}),1\leq i\leq m$ are transversal.

Let ${\cal C}$ and ${\cal D}$ be $A_{\infty}$-pre-categories over $k$. An $A_{\infty}$-functor $F:{\cal C}\to{\cal D}$ is a functor between the corresponding non-unital $A_{\infty}$-pre-categories such that $F$ takes quasi-isomorphisms in ${\cal C}$ to quasi-isomorphisms in ${\cal D}$.

An $A_{\infty}$-functor $F:{\cal C}\to{\cal D}$ between $A_{\infty}$-pre-categories is called an $A_{\infty}$-equivalence functor if:

a) Every object $Y\in{\cal D}$ is quasi-isomorphic to an object $\phi(X),X\in{\cal C}$.

b) The functor induces quasi-isomorphisms of non-unital $A_{\infty}$-algebras of morphisms, corresponding to all transversal sequences of objects.

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.

An $A_{\infty}$-category with strict identity morphisms is a non-unital $A_{\infty}$-category ${\cal C}$, such that for any object $X$ there exists an element $1=1_{X}\in Hom^{0}(X,X)$ (identity morphism) such that $m_{2}(1,f)=m_{2}(f,1)=f$ and $m_{n}(f_{1},...,1,...,f_{n})=0,n\neq 2$ for any morphisms $f,f_{1},...,f_{n}$.