One can check that the only sequences $\{ p = p_0, p_{1}, \ldots, p_{k} = q \}$ which contribute to ${\mathscr H}(p,q)$ are those for which there is $i < k$ such that $\deg(p_j) = \deg(p)$ for all $j \leq i$ and all other intersection points have degree equal to $\deg(q)$.  In this case, we have a collection of exceptional solutions together with one (regular) $J_{t,1-s_i}$-holomorphic strip connecting $p_{i}$ and $p_{i+1}$, and it is clear that we can allow $s_i$ to vary, giving us precisely a $1$-dimensional moduli space.


There is a natural bordification $\overline{{\mathscr H}}(p,q)$ of ${\mathscr H}(p,q)$ which allows broken curves, as well as the possibility that $s_0 = 0$ or $s_1 =1$.    The proof that ${\mathscr F}_{{\mathbf J}}$ is a chain map has now been reduced to an analysis of the boundary of the bordification $\overline{{\mathscr H}}(p,q)$.  Two of the  boundary strata correspond to having $s_0 = 0$, and $\deg(p_1) = \deg(p) +1$ or, alternatively, $s_1 = 1$ and $\deg(p_{k-1}) + 1 = \deg(q)$.  It is rather clear that these terms correspond to
\[ \mu_{1} \circ {\mathscr F}_{{\mathbf J}} + {\mathscr F}_{{\mathbf J}} \circ \mu_1.\]
We must therefore argue that $\overline{{\mathscr H}}(p,q)$  is compact, and that the contributions of all other boundary strata vanish.  Using part (2) of Proposition \ref{morse_floer_compactness}, we see that $J_{t,s}$ holomorphic curves do not approach the boundary, so we can apply the usual compactness arguments to $\overline{{\mathscr H}}(p,q)$.  As to the other boundary strata, we observe that they correspond to a broken curve consisting of two components, one exceptional and the other regular, occurring at some fixed time $s_i$.  Note that if $u_{i}$ is a regular curve at time $s_i$, and $u_{i+1}$ is an exceptional curve at time $s_{i+1}$, we obtain exactly such a broken curve if we consider a family where $s_i$ converges to $s_{i+1}$.  However, the usual gluing procedure yields another a family of curves converging to this broken curve.  Taken together, these contributions cancel, proving that ${\mathscr F}_{{\mathbf J}}$ is a chain map.



The homotopy method was extended by Fukaya, Oh, Ohta, and Ono in their joint work \cite{FOOO} to produce $A_{\infty}$-functors; an account in the exact setting appears in \cite{seidel-book}*{Section 10e}.  The end result is that, using the analogue of Proposition \ref{morse_floer_compactness} in the presence of more than one Lagrangian, we can conclude that the pre-categories 
\[  {\mathscr T} \operatorname{Fuk}(\ctorus{n},M_{t,1}; J_{t,1}), \, \,   {\mathscr T} \operatorname{Fuk}(\ctorus{n},M_{t,1}; J_{t,0}). \]
are $A_{\infty}$ equivalent.



Continuing along the same train of thought, we consider the family of almost complex structures $\epsilon J$.  Part (3) of Proposition \ref{morse_floer_compactness} implies a compactness theorem for families of $\epsilon J$-holomorphic curves which for any $\epsilon_0$ project to the complement of the $\delta^{t}_3$ neighbourhood of $\partial {\mathscr Q}_{t,1}$.  The assumption that $2\delta^t_{3} < \delta^{t}_2$ implies an equivalence between the chain-level multiplicative structures defined using (i) $J$-holomorphic curves, and (ii) those $\epsilon J$ holomorphic cures whose projecting to ${\mathscr Q}_{t,1}$ is contained in the complement of the $\delta^{t}_3$ neighbourhood of $\partial {\mathscr Q}_{t,1}$.


Further, by Fukaya and Oh's theorem, the moduli spaces of $\epsilon J$ holomorphic discs correspond, for $\epsilon$ small enough, to gradient trajectories of $f^t$.  Since $\delta^{t}_{3} < \delta^t_{1}$, it is sufficient to count holomorphic curves which project to the complement of the  $\delta^{t}_3$  neighbourhood of $\partial {\mathscr Q}_{t,1}$.   Running time backwards (and applying the homotopy method one last time) we can relate the count of gradient trajectories for $f^t$ to the one for the original function $f$.


To implement these ideas in the language of pre-categories, one has to introduce an intermediate pre-category whose objects are families of function $f^{t}$ satisfying the conditions of Proposition \ref{morse_floer_compactness}, and whose morphisms are the Floer complexes at time $t = t_0$.  By forgetting the family, we obtain an $A_{\infty}$ quasi-equivalent to ${\mathscr T} \operatorname{Fuk}(\ctorus{n},M_{t_0,1})$, while the use of homotopy maps defines a $A_\infty$ functor to $\operatorname{Morse}^{{\mathbb Z}^{n}}({\mathscr Q}_{t_0,1})$ which one can easily see is an $A_{\infty}$ equivalence by an energy filtration argument.   Leaving the details to the reader,  we conclude
\begin{thm}
The categories ${\mathscr T} \operatorname{Fuk}(\ctorus{n},M_{t_0,1})$ and $\operatorname{Morse}^{{\mathbb Z}^{n}}({\mathscr Q}_{t0,1})$ are equivalent as $A_{\infty}$-pre-categories. \hfill \qedsymbol
\end{thm}
We now return to the practice of dropping the subscripts from $M_{t,1}$ and all associated objects.










\section{HMS for toric varieties: Morse = \v{C}ech} \label{morse=cech-section}

In this section, we shall prove the equivalence of the category $\operatorname{Morse}^{{\mathbb Z}^{n}}({\mathscr Q})$ with a DG category whose objects are holomorphic line bundles on $X$, and whose morphisms are \v{C}ech complexes.   We will use the following version of \v{C}ech theory:

Given a sheaf of vector spaces (or free abelian groups) ${\mathscr S}$, and an ordered cover $\{ O_{k} \}_{k=0}^{N}$ of a (nice) space, such that ${\mathscr S}$ is acyclic on all iterated intersections of the elements of our cover, we consider the \v{C}ech complex 
\[  \check{C}^{i}({\mathscr S})  \equiv \bigoplus_{k_0 < \ldots < k_i} \Gamma( O_{k_0} \cap \cdots \cap O_{k_i},{\mathscr S} ), \]
with the differential on $\Gamma( O_{k_0} \cap \cdots \cap O_{k_i},{\mathscr S} ) $ defined by
\begin{equation} d \phi = \sum_{\stackrel{k}{k_0 < \ldots < k_j < k < k_{j+1} < \ldots < k_i}} (-1)^{j+1} \phi|_{ O_{k_0} \cap \cdots O_{ k_j} \cap O_{k} \cap  O_{k_{j+1}} \cap \cdots \cap O_{k_i} }. \end{equation}
Further, we define a cup product
\[ \check{C}^{i}({\mathscr S}_1) \otimes  \check{C}^{j}({\mathscr S}_2) \to  \check{C}^{i+j}({\mathscr S}_1 \otimes  {\mathscr S}_2) \] 
which, on $\Gamma( O_{k_0} \cap \cdots \cap O_{k_i},{\mathscr S}_1 )  \otimes \Gamma( O_{l_0} \cap \cdots \cap O_{l_j},{\mathscr S}_2 ) $ is defined by
\begin{equation} \label{cech-product} \phi \otimes \psi \mapsto 
\begin{cases} \phi \otimes  \psi|_{ O_{k_0} \cap \cdots \cap O_{k_i} \cap  O_{l_1} \cap \cdots \cap O_{l_j}}  & \textrm{if } k_i = l_0 \\
0 & \textrm{otherwise.}  \end{cases} \end{equation}
It is easy to check that the above cup product is associative.  Since our covers will be labeled by the cells of $Q$, we will assume that we have chosen an ordering of these cells.

\subsection{Another cover of the moment polytope} \label{sec:covers