Here we will define the Morse category of smooth functions $M(Y)$ following [FuO]. It will be an $A_{\infty}$-pre-category over ${{\bf C}}$. Objects of $M(Y)$ are pairs $(f,\rho)$, where $f:Y\to{\bf R}$ is a smooth function, and $\rho$ is a local system of finite-dimensional complex vector spaces on $Y$. Before defining the transversality of objects, we will define the transversality of functions.

Suppose we are given a sequence of smooth functions $(f_{0},...,f_{k}),k\geq 2$ such that all $f_{i}-f_{j},i\neq j$ are Morse functions, and a sequence of critical points $x_{i}\in Cr(f_{i}-f_{i+1}),0\leq i\leq k-1,x_{k}\in Cr(f_{0}-f_{k})$. We will use oriented binary planar trees in order to describe certain moduli spaces associated with such sequences. Let us fix a planar trivalent tree $T$ with $k+1$ tails vertices. Among the tail vertices we choose one and call it the root vertex. Let us orient edges of $T$ along the shortest paths towards the root. Thus, $T$ becomes a binary tree considered as an oriented tree . We depict $T$ inside of the standard unit disc $D\subset{{\bf R}}^{2}$ in such a way that tail vertices of $T$ belong to $\partial D$, and connected components of $D\setminus T$ are cyclically numbered from $0$ to $k$ in the clockwise order. We assume that numbers attached to two regions near the root vertex are $0$ and $k$.

We define a gradient immersion of $T$ into $Y$ as a continuous map $j:T\to Y$ such that:

1) The restriction $j|_{e}$ is an orientation preserving homeomorphism of the edge $e$ onto an interval in the gradient line of $f_{l(e)}-f_{r(e)}$, where the label $l(e)$ (resp. $r(e)$) corresponds to the region of $D\setminus T$ which is left (resp. right) to $e$.

2) Each tail vertex $v$ is mapped to the point $x_{v}\in Cr(f_{l_{v}}-f_{r_{v}})$, where $l_{v}$ (resp. $r_{v}$) is the label of the region which is left (resp. right) to the only tail edge containing $v$.

We will need immersed binary trees (let us call them gradient trees) in order to define compositions and transversal sequences in the Morse $A_{\infty}$-pre-category. These structures can be defined in terms of certain varieties, which we are going to describe now.

Suppose that we are given a sequence of functions $(f_{0},...,f_{k}),k\geq 2$ and critical points $(x_{0},\dots,x_{k})$ as above, and a binary planar tree $T$. Let us consider the manifold $Y(T)=Y^{V_{i}(T)}$, where $V_{i}(T)$ is the set of internal vertices of $T$. We are going to define several submanifolds in $Y(T)$. For each tail vertex $v_{m},0\leq m\leq k-1$ we define $Z_{v_{m}}=\pi_{\hat{v}_{m}}^{-1}(U_{x_{m}})$ and for $m=k$ we define $Z_{v_{k}}=\pi_{\hat{v}_{k}}^{-1}(S_{x_{k}})$. Here $\hat{v}_{l}$ denotes the second endpoint of the edge of $T$ containing $v_{l}$, and $\pi_{v}:Y(T)\to Y$ is the canonical projection on the factor corresponding to $v\in V_{i}(T)$.

For pair $(f_{i},f_{j})$ we define a subset $Z_{i,j}\subset Y\times Y$, consisting of pairs $(y_{1},y_{2})$ such that $y_{1}\neq y_{2}$ and $y_{2}=e^{t\,grad(f_{i}-f_{j})}y_{1}$ for some $t>0$. Then $Z_{i,j}$ is a non-compact submanifold of $Y\times Y$.

An edge $e$ of $T$ we call internal if both endpoints of it are internal vertices. The set of internal edges we denote by $E_{i}(T)$. For each internal edge $e\in E_{i}(T)$, which separates two regions labeled by $l(e)$ (left) and $r(e)$ (right), we define a submanifold $Z_{e}=\pi_{e}^{-1}(Z_{l(e),r(e)})$, where $\pi_{e}:Y(T)\to Y\times Y$ is the natural projection.

It follows from the definitions that the space of gradient immersions of a given $T$ as above, up to homeomorphisms preserving tails, can be identified with ${\cal M}(T;f_{0},...,f_{k};x_{0},...,x_{k}):=(\cap_{0\leq m\leq k}Z_{v_{m}})\cap(\cap_{e\in E_{i}(T)}Z_{e})\subset Y^{V_{i}(T)}$.

It can be proven (see [Fu1]) that there exists a subset of second Baire category in $(C^{\infty}(Y))^{{\bf Z}}$ such that for any element $(f_{i})_{i\in{\bf Z}}$ of this set and for any strictly increasing sequence of integers $i_{0}<\dots<i_{k})$ and for any planar tree $T$ with $k+1$ tails, the sequence $(f_{i_{0}},\dots,f_{i_{k}})$ is $T$-transversal.

For any two transversal objects $W_{0}=(f_{0},\rho_{0})$ and $W_{1}=(f_{1},\rho_{1})$ we define the space of morphisms $Hom_{M(Y)}(W_{0},W_{1})$ as the Morse complex $M^{\ast}(Y,f_{0}-f_{1},\rho_{0}^{\ast}\otimes\rho_{1})$. Now we define the $A_{\infty}$-structure on $M(Y)$.

The map $m_{1}:Hom((f_{0},\rho_{0}),(f_{1},\rho_{1}))\to Hom((f_{0},\rho_{0}),(f_{1},\rho_{1}))[1]$ is the standard differential in the Morse-Smale complex. Higher compositions $m_{k}$ where $k\geq 2$ for transversal sequences of objects are linear maps

Each $m_{k}$ is defined as a sum $m_{k}=\sum\pm m_{k,T}$ where $T$ runs through the set of isomorphism classes of oriented binary planar trees with $(k+1)$ tails. Let us describe the summands $m_{k,T}$. For simplicity we will give the formulas in the case when all local systems are trivial of rank one.

Let us fix critical points $x_{i}\in Cr(f_{i}-f_{i+1}),0\leq i\leq k-1,y_{k}\in Cr(f_{0}-f_{k})$, such that $\sum_{0\leq i\leq k-1}ind(x_{i})=ind(y_{k})+2-k$, and orientations of manifolds $S_{x_{i}},0\leq i\leq k$. It follows from the definition of a transversal sequence that the moduli space of gradient trees ${\cal M}(T;f_{0},...,f_{k};y_{0},...,y_{k})$ is an oriented compact zero-dimensional manifold.

For local systems of higher ranks one proceeds as in the case of Fukaya categories, using flat connections in order to define an analog of the holonomy of local systems.

One can obtain slightly different formulas for $m_{k}$ in the following way. For any point $\gamma\in{\cal M}(T;f_{0},...,f_{k};y_{0},...,y_{k})$ we define the weight

Here $var_{\gamma}(f_{l(e)}-f_{r(e)})>0$ is a variation of $f_{l(e)}-f_{r(e)}$ along the gradient line $\gamma(e)$, which is defined such as follows: $var_{\gamma}(f_{l(e)}-f_{r(e)})=(f_{l(e)}-f_{r(e)})(y_{max})-(f_{l(e)}-f_{r(e)})(y_{min})$, where $y_{max}$ and $y_{min}$ are the endpoints of $\gamma(e)$, such that $(f_{l(e)}-f_{r(e)})(y_{max})-(f_{l(e)}-f_{r(e)})(y_{min})>0$. After extension of scalars to ${{\bf C}}_{\varepsilon}$ one can choose another basis in $Hom_{M(Y)}(W_{0},W_{1})$, namely $[y]_{new}=[y]exp({(f_{0}(y)-f_{1}(y))\over{\varepsilon}})$ for $y\in Cr(f_{0}-f_{1})$. Then the formulas for $m_{k}$ will be modified. The contribution of each $\gamma$ will be multiplied by $w_{\gamma}$. The formulas will be similar to those for the Fukaya-Oh category (see Section 5.2).