\section{Weakly filtered $A_{\infty}$-theory}
\label{s:wf-ai-theory}

In this section we develop a general framework for weakly filtered
$A_{\infty}$-categories, with an emphasis on weakly filtered modules
over such categories. In our context ``weakly filtered'' generally
means that the morphisms in the category are filtered chain complexes
but the higher $A_{\infty}$-operations do not necessarily preserve
these filtrations. Rather they preserve them up to prescribed errors
which we call {\em discrepancies}. In the same vein one can consider
also weakly filtered $A_{\infty}$-functors and modules. Related
notions of filtered $A_{\infty}$-structures have been considered in
the literature (e.g.~\cite{FO3:book-vol1, FO3:book-vol2}), but the
existing theory seems to differ from ours in its scope
and applications.

Below we will cover only the most basic concepts of
$A_{\infty}$-theory in the weakly filtered setting. In particular we
will not go into the topics of derived categories, split closure or
generation in the weakly filtered framework. Our main goal is in fact
much more modest: to provide an effective description of iterated
cones of modules in the weakly filtered setting in terms of weakly
filtered twisted complexes.

Some readers may find the details of the weakly filtered setting
somewhat overwhelming, especially in what concerns keeping track of
the discrepancies. If one assumes all the discrepancies to vanish, the
theory becomes ``genuinely filtered'' and is easier to
follow. However, the additional difficulty due to the weakly filtered
setting is in large superficial. Indeed, significant parts of the
theory developed in this section do not become easier if one
works in the genuinely filtered setting, except in terms of notational convenience. 
We also remark that, as far
as we know, a good part of the theory developed in this section, particularly the study 
of iterated cones, is new even
in the genuinely filtered case. The reason for developing the theory
in the weakly filtered setting (rather than filtered) has to do with
the geometric applications we aim at which have to do with Fukaya
categories of symplectic manifolds. For technical reasons, the weakly
filtered framework fits better with the standard implementations of
these categories.

\subsection{Weakly filtered $A_{\infty}$-categories}
\label{sb:wf-ai-categ}

In the following we will often deal with sequences
\label{pg:conv-seq}
$\bme = (\epsilon_1, \ldots, \epsilon_d, \ldots)$ of real numbers that
we will refer to as discrepancies. We will use the following
abbreviations and conventions. For two sequences $\bme$, $\bme'$ we
write $\bme \leq \bme'$ in order to say that
$\epsilon_d \leq \epsilon'_d$ for all $d$. For $c \in \mathbb{R}$ we
write $\bme + c$ for the sequence
$(\epsilon_1+c, \ldots, \epsilon_d+c, \ldots)$.  For a finite number
of sequences $\bme^{(1)}, \ldots, \bme^{(r)}$ we define
$\max\{\bme^{(1)}, \ldots, \bme^{(r)} \}$ to be the sequence
$\bme = (\epsilon_1, \ldots, \epsilon_d, \ldots)$ with
$\bme_d := \max\{\epsilon^{(1)}_d, \ldots, \epsilon^{(r)}_d\}$.


Fix a commutative ring $R$, which for simplicity we will henceforth
assume to be of characteristic $2$ (i.e. $2r=0$ for all $r \in R$).
Unless otherwise stated all tensor products will be taken over $R$.

The $A_{\infty}$-theory developed below will be carried out in the
ungraded framework. Also, in contrast to standard texts on the subject
such as~\cite{Se:book-fukaya-categ}, we will work in a homological
(rather than cohomological) setting, following the conventions
from~\cite{Bi-Co:lcob-fuk}.

Let $\mathcal{A}$ be an $A_{\infty}$-category over $R$. To simplify
notation, in what follows we will denote the morphisms between two
objects $X,Y \in \textnormal{Ob}(\mathcal{A})$ by
$C(X,Y) := \hom_{\mathcal{A}}(X,Y)$. We denote the composition maps of
$\mathcal{A}$ by $\mu^{\mathcal{A}}_d$, $d \geq 1$.

Let
$\bm{\epsilon}^{\mathcal{A}} = (\epsilon_1^{\mathcal{A}},
\epsilon_2^{\mathcal{A}}, \ldots, \epsilon_d^{\mathcal{A}}, \ldots)$
be an infinite sequence of non-negative real numbers, with
$\epsilon_1^{\mathcal{A}}=0$. We call $\mathcal{A}$ a {\em weakly
  filtered} $A_{\infty}$-category with discrepancy
$\leq \bm{\epsilon}^{\mathcal{A}}$ if the following holds:
\begin{enumerate}
\item For every $X, Y \in \textnormal{Ob} (\mathcal{A})$, $C(X,Y)$ is
  endowed with an increasing filtration of $R$-modules indexed by the
  real numbers. We denote by $C^{\leq \alpha}(X,Y) \subset C(X,Y)$ the
  part of the filtration corresponding to $\alpha \in \mathbb{R}$. By
  {\em increasing filtration} we mean that
  $C^{\leq \alpha'}(X,Y) \subset C^{\leq \alpha''}(X,Y)$ for every
  $\alpha'\leq \alpha''$.
\item The $\mu_d$-operation preserves the filtration up to an
  ``error'' of $\epsilon_d^{\mathcal{A}}$. More precisely, for every
  $X_0, \ldots, X_d \in \textnormal{Ob}(\mathcal{A})$ and
  $\alpha_1, \ldots, \alpha_d \in \mathbb{R}$ we have:
  $$\mu_d \bigl( C^{\leq \alpha_1}(X_0, X_1) \otimes \cdots \otimes C^{\leq
    \alpha_d}(X_{d-1},X_d) \bigr) \subset C^{\leq \alpha_1 + \cdots +
    \alpha_d + \epsilon_d^{\mathcal{A}}} (X_0, X_d).$$
\end{enumerate}
Note that since $\epsilon_1^{\mathcal{A}}=0$, $\mu_1^{\mathcal{A}}$
preserves the filtration, hence each $C^{\leq \alpha}(X,Y)$,
$\alpha \in \mathbb{R}$, is a sub-complex of $C(X,Y)$. Note also that
the discrepancy is not uniquely defined -- in fact we can always
increase it if needed. Namely, if
$\bm{\epsilon}' = (\epsilon'_1=0, \epsilon'_2, \ldots, \epsilon'_d,
\ldots)$ is another sequence like $\bm{\epsilon}^{\mathcal{A}}$ but
with $\bmea \leq \bme'$ then $\mathcal{A}$ is also weakly filtered
with discrepancy $\leq \bm{\epsilon}'$.

By analogy with symplectic topology we will often refer to the index of
the filtration as an {\em action} and say that elements of
$C^{\leq \alpha}(X,Y)$ have action $\leq \alpha$.

\subsubsection{Unitality} \label{sbsb:unitality-A} Let $\mathcal{A}$
be a weakly filtered $A_{\infty}$-category and assume that
$\mathcal{A}$ is homologically unital (h-unital for
short). \label{pg:h-unital-categ} We say that $\mathcal{A}$ is
h-unital in the weakly filtered sense if the following holds. There
exists $u^{\mathcal{A}} \in \mathbb{R}_{\geq 0}$ such that for every
$X \in \textnormal{Ob}(\mathcal{A})$ we have a cycle
$e_X \in C^{\leq u^{\mathcal{A}}}(X,X)$ representing the homology unit $[e_X] \in H(C(X,X), \mu^{\mathcal{A}}_1)$. We view the choices of
$e_X$, $X \in \textnormal{Ob}(\mathcal{A})$ and $u^{\mathcal{A}}$ as
part of the data of a weakly filtered h-unital
$A_{\infty}$-category. We call $u^{\mathcal{A}}$ the discrepancy of
the units.

Occasionally we will have to impose the following additional
assumption on $\mathcal{A}$.

\newcommand{\lbue}{U^e} \hypertarget{h:asmp-ue}{\noindent
  \textbf{Assumption~$\lbue$.}}  \label{l:assm-ue} Let $\mathcal{A}$
be a weakly filtered $A_{\infty}$-category which is h-unital in the
weakly filtered sense. Let
$2u^{\mathcal{A}} + \epsilon_2^{\mathcal{A}} \leq \zeta \in
\mathbb{R}$. We say that $\mathcal{A}$ satisfies
Assumption~$\lbue(\zeta)$ if for every
$X \in \textnormal{Ob}(\mathcal{A})$ we have:
$$\mu_2^{\mathcal{A}}(e_X, e_X) = e_X + \mu_1^{\mathcal{A}}(c)$$ for some 
$c \in C^{\leq \zeta}(X,X)$. Put in different words, the assumption
$\lbue$ says that $[e_X] \cdot [e_X] = [e_X]$ in
$H_*(C^{\leq \zeta}(X,X))$, where $\cdot$ stands for the product
induced by $\mu_2^{\mathcal{A}}$ in homology. (The superscript $e$ in
$\lbue$ indicates that the assumption deals with the cycles $e_X$
representing the units.) Below we will sometimes write
$\mathcal{A} \in \lbue(\zeta)$ to say that $\mathcal{A}$ satisfies
Assumption~$\lbue(\zeta)$.

\subsection{Typical classes of examples} \label{sb:typ-exp} Before we
go on with the general algebraic theory of weakly filtered
$A_{\infty}$-structures, it might be useful to make a short digression
in order to exemplify what types of filtrations will actually occur in
our applications. We resume with the general algebraic theory
in~\S\ref{sb:functors} below.

The weakly filtered $A_{\infty}$-categories that will appear in our
applications are Fukaya categories associated to symplectic
manifolds. They will mostly be of the following types, described
in~\S\ref{sbsb:filtration-action} - \ref{sbsb:families-class} below.

\subsubsection{Filtrations induced by an ``action'' functional on the
  generators} \label{sbsb:filtration-action} In this class of weakly
filtered $A_{\infty}$-categories the morphisms $C(X,Y)$ between two
objects are assumed to be free $R$-modules with a distinguished basis
$B(X,Y)$, i.e.  $C(X,Y) = \bigoplus_{b \in B(X,Y)} Rb$. We also have a
function $\mathbf{A}: B(X,Y) \longrightarrow \mathbb{R}$, which (by
analogy to symplectic topology) we call the {\em action function},
defined for every $X, Y \in \textnormal{Ob}(\mathcal{A})$, and this
function induces the filtration, namely:
$$C^{\leq \alpha}(X,Y) = \bigoplus_{b \in B(X,Y), \mathbf{A}(b)\leq \alpha} Rb.$$
We will mostly assume that $C(X,Y)$ has finite rank and that $R$ is a
field.

\subsubsection{Filtration coming from the Novikov
  ring} \label{sbsb:filtration-nov} Here we fix a commutative ring $A$ and
consider the (full) Novikov ring over $A$:
\begin{equation} \label{eq:Nov-ring}
  \Lambda = \Bigl\{ \sum_{k=0}^{\infty} a_k T^{\lambda_k} \mid a_k \in A, 
  \lim_{k \to \infty} \lambda_k = \infty \Bigr\},
\end{equation}
as well as the positive Novikov ring:
\begin{equation} \label{eq:Nov-ring-0}
  \Lambda_0 = \Bigl\{ \sum_{k=0}^{\infty} a_k T^{\lambda_k} \mid a_k \in A, 
  \lambda_k\geq 0, \lim_{k \to \infty} \lambda_k = \infty \Bigr\}.
\end{equation}

The weakly filtered $A_{\infty}$-categories $\mathcal{A}$ of the type
discussed here are defined over $\Lambda$, but the weakly filtered
structure is only over the ring $R = \Lambda_0$. 

As in~\S\ref{sbsb:filtration-action} above, we have
$C(X,Y) = \bigoplus_{b \in B(X,Y)} \Lambda b$. The filtration on
$C(X,Y)$ is then defined by
$$C^{\leq \alpha}(X,Y) = 
\bigoplus_{b \in B(X,Y)} T^{-\alpha} \Lambda_0 b.$$ Note that
$C^{\leq a}(X,Y)$ is not a $\Lambda$-module but rather a
$\Lambda_0$-module.

We will mostly assume that $B(X,Y)$ are finite (hence $C(X,Y)$ have
finite rank) and that $A$ is a field (in which case $\Lambda$ is a
field too).

\subsubsection{Mixed filtration} \label{sbsb:filtration-mixed} In some
situations the filtrations on our $A_{\infty}$-categories occur as
combination of~\S\ref{sbsb:filtration-action}
and~\S\ref{sbsb:filtration-nov} above.  More specifically, we have
$C(X,Y) = \Lambda B(X,Y)$ as in~\S\ref{sbsb:filtration-nov} and an
action functional $\mathbf{A}: B(X,Y) \longrightarrow \mathbb{R}$ as
in~\S\ref{sbsb:filtration-action}. We then extend $\mathbf{A}$ to a
functional
$\mathbf{A}: C(X,Y) = \Lambda \cdot B(X,Y) \longrightarrow \mathbb{R}
\cup \{-\infty\}$ by first setting $\mathbf{A}(0) = -\infty$. Then for
$P(T) \in \Lambda$ and $b \in B(X,Y)$ we define:
$$\mathbf{A}(P(T)b) := -\lambda_0 + \mathbf{A}(b),$$ where
$\lambda_0 \in \mathbb{R}$ is the minimal exponent that appears in the
formal power series of $P(T) \in \Lambda$, i.e.
$P(T) = a_0 T^{\lambda_0} + \sum_{i=1}^{\infty} a_i T^{\lambda_i}$
with $a_0 \neq 0$ and $\lambda_i > \lambda_0$ for every $i\geq 1$.
Finally, for a general non-trivial element
$c = P_1(T)b_1 + \cdots + P_l(T)b_l \in C(X,Y)$, define
$$\mathbf{A}(c) = \max \{ \mathbf{A}(P_k(T) b_k) \mid 1\leq k \leq
l\}.$$ The filtration on $C(X,Y)$ is then induced by $\mathbf{A}$:
$$C^{\leq \alpha}(X,Y) = 
\{c \in C(X,Y) \mid \mathbf{A}(c) \leq \alpha \}.$$ It is easy to see
that $C^{\leq \alpha}(X,Y)$ is a $\Lambda_0$-module.

\subsubsection{Families of weakly filtered
  $A_{\infty}$-categories} \label{sbsb:families-class}

The weakly filtered $A_{\infty}$-categories in our applications will
naturally occur in families $\{ \mathcal{A}_s \}_{s \in \mathcal{P}}$
parametrized by choices of auxiliary structures $s$ needed to define
the $A_{\infty}$-structure. The parameter $s$ will typically vary over
a subset $\mathcal{P} \subset E \setminus \{0\}$ where $E$ is a
neighborhood of $0$ in a Banach (or Fr\'{e}chet) space.  The subset
$\mathcal{P}$ will usually be residual (in the sense of Baire) so that
$0$ is in the closure of $\mathcal{P}$.

Typically all the members of the family
$\{ \mathcal{A}_s\}_{s \in \mathcal{P}}$ will be mutually
quasi-equivalent (see~\cite[Section~10]{Se:book-fukaya-categ} for
several approaches to families of $A_{\infty}$-categories). Of course,
in the weakly filtered setting the quasi-equivalences between
different $\mathcal{A}_s$'s are supposed to bear some compatibility
with respect to the weakly filtered structures on the
$\mathcal{A}_s$'s.

Apart from the above, in our applications the families
$\{\mathcal{A}_s \}_{s \in \mathcal{P}}$ will enjoy the following
additional property which will be crucial. The bounds
$\bme^{\mathcal{A}_s}$ for the discrepancies of the $\mathcal{A}_s$'s
can be chosen such that:
$$\lim_{s \to 0} \epsilon_d^{\mathcal{A}_s} = 0, \; \forall d.$$
Moreover, the categories $\mathcal{A}_s$ will mostly be h-unital with
discrepancy of units $u^{\mathcal{A}_s}$ and satisfy
Assumption~$\hyperlink{h:asmp-ue}{\lbue(\zeta_s)}$. The latter two
quantities will satisfy
$$\lim_{s \to 0} u^{\mathcal{A}_s} = \lim_{s \to 0} \zeta_s = 0.$$

Below we will encounter further notions in the framework of weakly
filtered $A_{\infty}$-categories such as weakly filtered functors and
modules. Each of these comes with its own discrepancy sequence $\bme$. In our
applications everything will occur in families and we will usually
have $\lim_{s \to 0} \epsilon_d(s) = 0$ for each $d$.

While the algebraic theory below is developed without a priori
assumptions on the size of discrepancies, it might be useful to view
the discrepancies as quantities that can be made arbitrarily small.

\subsubsection{The case of Fukaya
  categories} \label{sbsb:fukaya-class} The general description
in~\S\ref{sbsb:families-class} applies to the case of Fukaya
categories which will be central in our applications. More
specifically, in order to define the $A_{\infty}$-structure of Fukaya
categories one has to make choices of perturbation data (e.g. choices
of almost complex structures as well as Hamiltonian perturbation --
see e.g.~\cite[Sections~8, 9]{Se:book-fukaya-categ}). The space
$\mathcal{P}$ will consist of those perturbation data that are regular
(or admissible). This is normally a 2'nd category subset of the space
of all perturbations $E$.  The discrepancies occur as ``error''
curvature terms (associated to the perturbations) when defining the
$\mu_d$-operations. These discrepancies can be made arbitrarily small
(for a fixed $d$) by choosing smaller and smaller perturbations. The
same holds for discrepancy of the units and the $\zeta_s$'s.


\subsection{Weakly filtered $A_{\infty}$-functors and modules} \label{sb:functors}
Let $\mathcal{A}$, $\mathcal{B}$ two weakly filtered
$A_{\infty}$-categories and
$\mathcal{F}: \mathcal{A} \longrightarrow \mathcal{B}$ an
$A_{\infty}$-functor. Let
$\bme^{\mathcal{F}} = (\epsilon_1^{\mathcal{F}},
\epsilon_2^{\mathcal{F}}, \ldots, \epsilon_d^{\mathcal{F}}, \ldots)$
be a sequence of non-negative real numbers. In contrast to $\bmea$ and
$\bme^{\mathcal{B}}$ we do allow here that
$\epsilon_1^{\mathcal{F}} \neq 0$. We say that $\mathcal{F}$ is a
weakly filtered $A_{\infty}$-functor with discrepancy
$\leq \bme^{\mathcal{F}}$ if for all
$X_0, \ldots, X_d \in \textnormal{Ob}(\mathcal{A})$ and
$\alpha_1, \ldots, \alpha_d \in \mathbb{R}$ we have:
\begin{equation} \label{eq:wf-func}
  \mathcal{F}_d \bigl( C_{\mathcal{A}}^{\leq \alpha_1}(X_0, X_1) 
  \otimes \cdots \otimes C_{\mathcal{A}}^{\leq \alpha_d}(X_{d-1},X_d)
  \bigr) \subset C_{\mathcal{B}}^{\leq \alpha_1 + \cdots + \alpha_d +
    \epsilon_d^{\mathcal{F}}} (\mathcal{F}X_0, \mathcal{F}X_d).
\end{equation}
Here we have denoted by $C_{\mathcal{A}}$ and $C_{\mathcal{B}}$ the
$\hom$'s in $\mathcal{A}$ and $\mathcal{B}$ respectively and by
$\mathcal{F}_d$ the higher order terms of the functor $\mathcal{F}$.

There is also a notion of weakly filtered natural transformations
between weakly filtered functors but we will not go into this now as
our main focus will be on a special case -- weakly filtered modules
and weakly filtered morphisms between them.

\begin{rem} \label{r:wf-func} One could attempt to generalize the
  notion of weakly filtered functors to allow also an additional shift
  $\rho \in \mathbb{R}$ in the action filtration, namely replace the
  right-hand side of~\eqref{eq:wf-func} by
  $C_{\mathcal{B}}^{\leq \alpha_1 + \cdots + \alpha_d + \rho +
    \epsilon_d^{\mathcal{F}}} (\mathcal{F}X_0, \mathcal{F}X_d)$. While
  this might make sense theoretically, it does not seem to be in line
  with our approach (viewing the discrepancies as small errors) since
  the $A_{\infty}$-identities defining $A_{\infty}$-functors do not
  have the correct ``homogeneity'' properties with respect to $\rho$.
\end{rem}

\subsubsection{Weakly filtered modules} \label{sb:mod} Let $\mathcal{A}$
be a weakly filtered $A_{\infty}$-category with discrepancy
$\bm{\epsilon}^{\mathcal{A}}$. Let $\mathcal{M}$ be an
$\mathcal{A}$-module with composition maps $\mu^{\mathcal{M}}_d$,
$d \geq 1$. Let
$\bm{\epsilon}^{\mathcal{M}} = (\epsilon^{\mathcal{M}}_1,
\epsilon^{\mathcal{M}}_2, \ldots, \epsilon^{\mathcal{M}}_d, \ldots)$
be an infinite sequence of non-negative real numbers with
$\epsilon^{\mathcal{M}}_1=0$. We say that $\mathcal{M}$ is weakly
filtered with discrepancy $\leq \bm{\epsilon}^{\mathcal{M}}$ if the
following holds:
\begin{enumerate}
\item For every $X \in \textnormal{Ob}(\mathcal{A})$, $\mathcal{M}(X)$
  is endowed with an increasing filtration
  $\mathcal{M}^{\leq \alpha}(X)$ indexed by $\alpha \in \mathbb{R}$.
\item The $\mu^{\mathcal{M}}_d$-operation respects the filtration up
  to an ``error'' of $\epsilon^{\mathcal{M}}_d$. Namely, for all
  $X_0, \ldots, X_{d-1} \in \textnormal{Ob}(\mathcal{A})$ and
  $a_1, \ldots, a_d \in \mathbb{R}$ we have:
  $$\mu^{\mathcal{M}}_d \bigl( C^{\leq \alpha_1}(X_0, X_1) \otimes 
  \cdots \otimes C^{\leq \alpha_{d-1}}(X_{d-2}, X_{d-1}) \otimes
  \mathcal{M}^{\leq \alpha_d}(X_{d-1}) \bigr) \subset
  \mathcal{M}^{\leq \alpha_1 + \cdots + \alpha_d +
    \epsilon^{\mathcal{M}}_d} (X_0).$$
\end{enumerate}
Again, since $\epsilon^{\mathcal{M}}_1=0$, every
$(\mathcal{M}^{\leq \alpha}(X), \mu^{\mathcal{M}}_1)$ is a sub-complex
of $(\mathcal{M}(X), \mu^{\mathcal{M}}_1)$.

\begin{rems} \label{r:modules} 
  \begin{enumerate}
  \item It is easy to see that weakly filtered $\mathcal{A}$-modules
    are the same as weakly filtered functors
    $\mathcal{F}: \mathcal{A} \longrightarrow
    Ch_{\textnormal{f}}^{\textnormal{opp}}$ (having some
    discrepancy). Here $Ch_{\textnormal{f}}$ is the dg-category of
    filtered chain complexes (of $R$-modules) and
    $Ch_{\textnormal{f}}^{\textnormal{opp}}$ stands for its opposite
    category. (Note that $Ch_{\textnormal{f}}$ and
    $Ch_{\textnormal{f}}^{\textnormal{opp}}$ are in fact {\em
      filtered} dg-categories, i.e. they have discrepancies $0$.)  The
    correspondence between weakly filtered functors and weakly
    filtered modules is the same as in the ``unfiltered''
    case~\cite[Section~(1j)]{Se:book-fukaya-categ}. Note that if
    $\mathcal{F}: \mathcal{A} \longrightarrow
    Ch_{\textnormal{f}}^{\textnormal{opp}}$ has discrepancy
    $\leq \bme^{\mathcal{F}}$ then the weakly filtered module
    $\mathcal{M}$ corresponding to it has discrepancy
    $\leq \bme^{\mathcal{M}}$ with
    $\epsilon^{\mathcal{M}}_d = \epsilon^{\mathcal{F}}_{d-1}$ for
    every $d \geq 2$.
  \item In what follows it would be sometimes convenient to fix one
    discrepancy which applies to several weakly filtered modules
    together. In that case we will denote it by $\bmemm$ rather than
    by $\bm{\epsilon}^{\mathcal{M}}$ (the superscript $m$ stands for
    ``modules''). Note that we can always increase the bound on
    discrepancy in the sense that if $\mathcal{M}$ has discrepancy
    $\leq \bm{\epsilon}^{\mathcal{M}}$ than it also has discrepancy
    $\leq \bm{\epsilon}$ for every sequence $\bm{\epsilon}$ with
    $\bme \geq \bme^{\mathcal{M}}$. Therefore, whenever dealing with a
    finite collection of weakly filtered modules we can bound from
    above all their discrepancies by a single sequence.
  \end{enumerate}
\end{rems}


Next we define morphisms between weakly filtered
$\mathcal{A}$-modules. Let $\mathcal{M}_0, \mathcal{M}_1$ be two
weakly filtered $\mathcal{A}$-modules, both with discrepancy
$\leq \bm{\epsilon}^m$. Let
$f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ be a pre-module
homomorphism. We write $f=(f_1, \ldots, f_d, \ldots)$ where the
$f_d$-component is a an $R$-linear map
$$f_d: C(X_0, X_1) \otimes \cdots C(X_{d-2}, X_{d-1}) \otimes 
\mathcal{M}_0(X_{d-1}) \longrightarrow \mathcal{M}_1(X_0).$$ Let
$\alpha \in \mathbb{R}$ and
$\bm{\epsilon}^f = (\epsilon^f_1, \ldots, \epsilon^f_d, \ldots)$ be a
vector of non-negative real numbers. In contrast to
$\bm{\epsilon}^{\mathcal{A}}$ and $\bm{\epsilon}^m$ we {\em do allow}
that $\epsilon^f_1 \neq 0$. We say that $f$ shifts action by
$\leq \rho$ and has discrepancy $\leq \bm{\epsilon}^f$ if for every
$d$, the map $f_d$ shifts action by not more than $\rho+\epsilon_d^f$,
namely:
$$f_d \bigl( C^{\leq \alpha_1} (X_0, X_1) \otimes \cdots 
C^{\leq \alpha_{d-1}}(X_{d-2}, X_{d-1}) \otimes \mathcal{M}_0^{\leq
  \alpha_d}(X_{d-1}) \bigr) \subset \mathcal{M}_1^{a_1+\cdots+a_d +
  \rho + \epsilon_d^f}(X_0).$$ We will generally refer to such $f$'s
as weakly filtered pre-module homomorphisms. 

As before, if $\rho \leq \rho'$ and $\bme^f \leq \bme$ then $f$ also
shifts filtration by $\leq \rho'$ and has discrepancy
$\leq \bm{\epsilon}$.

We will now attempt to define a filtration on the totality of
pre-module homomorphisms. Denote by
$\hom(\mathcal{M}_0, \mathcal{M}_1)$ the pre-module homomorphisms
$\mathcal{M}_0 \longrightarrow \mathcal{M}_1$. The filtration will
depend on an additional ``discrepancy'' parameter $\bmeh$ which is a
sequence
$\bm{\epsilon}^h = (\epsilon^h_1, \epsilon^h_2, \ldots, \epsilon^h_d,
\ldots)$ of non-negative real numbers (the superscript $h$ stands for
``homomorphisms''). Again, we {\em do not} assume here that
$\epsilon^h_1$ is $0$. Our filtration is indexed by $\mathbb{R}$ and
is defined as follows. The part of the filtration corresponding to
$\rho \in \mathbb{R}$ is denoted by
$\hom^{\leq \rho; \bm{\epsilon}^h}(\mathcal{M}_0, \mathcal{M}_1)$ and
consists of all pre-module homomorphisms
$f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ which shift action by
not more than $\rho$ and have discrepancy $\leq \bmeh$. Clearly this
yields an increasing filtration on
$\hom(\mathcal{M}_0, \mathcal{M}_1)$. Note however, that in general
this filtration might not be exhaustive since not every pre-module
homomorphism must be weakly filtered.

A few words are in order about the structure of the totality of weakly
filtered $\mathcal{A}$-modules. Recall that $\mathcal{A}$-modules form
a dg-category $\textnormal{mod}_{\mathcal{A}}$ with composition maps
$\mu_1^{\textnormal{mod}}$, $\mu_2^{\textnormal{mod}}$
(See~\cite[Section~(1j)]{Se:book-fukaya-categ} for the
definitions). We would like to view the weakly filtered modules (say
of given discrepancy) as a sub-category and define a weakly filtered
structure on it. However, when trying do that one encounters the
following problem. For general choices of
$\bm{\epsilon}^{\mathcal{A}}$, $\bm{\epsilon}^{m}$ and
$\bm{\epsilon}^h$ and two weakly filtered modules $\mathcal{M}_0$,
$\mathcal{M}_1$ with discrepancy $\leq \bm{\epsilon}^m$ the
differential $\mu_1^{\textnormal{mod}}$ does not preserve
$\hom^{\leq \rho; \bm{\epsilon}^h}(\mathcal{M}_0,
\mathcal{M}_1)$. Nevertheless it is possible to correct this problem
by restricting the choice of $\bm{\epsilon}^h$ as follows:

\medskip \newcommand{\lbe}{\mathcal{E}}
\hypertarget{h:asmp-e}{\noindent \textbf{Assumption~$\lbe$.}}  
\label{pp:assump-E}
A sequence
$\bm{\varepsilon} = (\varepsilon_1, \ldots, \varepsilon_d, \ldots)$ is
said to satisfy Assumption~$\lbe$ if for every $d \geq 1$ we have
$\varepsilon_d \geq \max \bigl\{ \epsilon^m_i + \varepsilon_j, \,
\epsilon_i^{\mathcal{A}} + \varepsilon_j \mid i+j=d+1 \bigr\}$.

Sometimes we will need to emphasize the dependence of
Assumption~$\lbe$ on the choices of $\bmea$ and $\bmemm$ in which case
we will refer to it as 


Assumption~\hyperlink{h:asmp-e}{$\lbe(\bmemm, \bmea)$}. Alternatively
we will sometimes write
$\bm{\varepsilon} \in~\hyperlink{h:asmp-e}{\lbe(\bmemm, \bmea)}$.

\medskip

An inspection of definition of $\mu_1^{\textnormal{mod}}$ (see
e.g.~\cite[Section~(1j)]{Se:book-fukaya-categ}) shows that if
$\bm{\epsilon}^h$ satisfies Assumption~\hyperlink{h:asmp-e}{$\lbe$}
then $\hom^{\leq \rho; \bm{\epsilon}^h}(\mathcal{M}_0, \mathcal{M}_1)$
is preserved by $\mu_1^{\textnormal{mod}}$ hence is a chain complex.

The following will be useful later on:
\begin{lem} \label{l:choice-eps-1} Given $\bm{\epsilon}^{\mathcal{A}}$
  and $\bm{\epsilon}^m$ it is always possible to find
  $\bm{\varepsilon}$ that satisfies
  Assumption~\hyperlink{h:asmp-e}{$\lbe$}. Moreover, there exists a
  sequence of real numbers $\{A_d\}_{d \in \mathbb{N}}$ which is
  universal in the sense that it does not depend on $\bmea$ or
  $\bmemm$ and has the following property: for any given sequence
  $\bm{\delta} = (\delta_1, \ldots, \delta_d, \ldots)$ of non-negative
  real numbers there exists an $\bm{\varepsilon}$ that satisfies
  {Assumption~\hyperlink{h:asmp-e}{$\lbe$}} and such that for all $d$:
  \begin{equation} \label{eq:ineq-Ad} \delta_d \leq \varepsilon_d \leq
    A_d \sum_{j=1}^d (\ea_j + \emm_j + \delta_j).
  \end{equation}
\end{lem}

\begin{rem} \label{r:universal-1} The bound from above on
  $\varepsilon_d$ in~\eqref{eq:ineq-Ad} becomes useful if for every
  $d$ all the quantities
  $\epsilon^{\mathcal{A}}_j, \epsilon^m_j, \delta_j$,
  $j=1, \ldots, d$, can simultaneously be made arbitrarily small. This
  will indeed be the case in our applications where the weakly
  filtered $A_{\infty}$-categories and modules appear in families -
  see~\S\ref{sbsb:families-class} for further details.
\end{rem}

\begin{proof}[Proof of Lemma~\ref{l:choice-eps-1}]
  One can easily construct $\varepsilon_d$ and $A_d$ inductively:
  start with $\varepsilon_1 := \delta_1$ then set
  $\varepsilon_2 := \max\{\epsilon^m_2+\varepsilon_1,
  \epsilon^{\mathcal{A}}_2 + \varepsilon_1, \delta_2\}$ and so
  on. (Note that $\epsilon^{\mathcal{A}}_1=\epsilon^m_1=0$ so that the
  inequality in Assumption~\hyperlink{h:asmp-e}{$\lbe$} is obviously
  satisfied for $i=1$, $j=d$.)
\end{proof}

\begin{rems} \label{r:eh}
  \begin{enumerate}
  \item If $\bm{\varepsilon}$ satisfies
    {Assumption~\hyperlink{h:asmp-e}{$\lbe$}} then so does
    $\widetilde{\bm{\varepsilon}}: = \bm{\varepsilon} +
    \textnormal{const}$.
  \item When dealing with $\hom^{\leq \rho; \bmeh}$ we can always
    arrange that $\epsilon^h_1=0$ by applying the following
    procedure. Suppose that $\bm{\epsilon}^h$ satisfies
    Assumption~\hyperlink{h:asmp-e}{$\lbe$}. Put
    $$\widetilde{\rho} := \rho + \epsilon^h_1, \quad 
    \widetilde{\epsilon}^h_d := \epsilon^h_d - \epsilon^h_1.$$ Note
    that $\widetilde{\epsilon}^h_1 = 0$,
    $\widetilde{\epsilon}^h_d \geq 0$ and that
    $\widetilde{\bm{\epsilon}}^h$ still satisfies
    Assumption~\hyperlink{h:asmp-e}{$\lbe$}. It is easy to see that
    $$\hom^{\leq \widetilde{\rho};
      \widetilde{\bm{\epsilon}}^h}(\mathcal{M}_0, \mathcal{M}_1) =
    \hom^{\leq \rho; \bm{\epsilon}^h}(\mathcal{M}_0,
    \mathcal{M}_1).$$ \label{i:rems-eh}
  \end{enumerate}
\end{rems}

We now turn to the $\mu_2^{\tmod}$ operation. Let $\mathcal{M}_0$,
$\mathcal{M}_1$, $\mathcal{M}_2$ be weakly filtered
$\mathcal{A}$-modules with discrepancy $\leq \bmemm$. Let
$f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$,
$g: \mathcal{M}_1 \longrightarrow \mathcal{M}_2$ be two weakly
filtered pre-module homomorphism with
$f \in \hom^{\leq \rho^f; \bme^f}(\mathcal{M}_0, \mathcal{M}_1)$,
$g \in \hom^{\leq \rho^g; \bme^g}(\mathcal{M}_1, \mathcal{M}_2)$. Set
$\varphi := \mu_2^{\tmod}(f,g): \mathcal{M}_0 \longrightarrow
\mathcal{M}_2$. A simple calculation shows that $\varphi$ is weakly
filtered and that
$\varphi \in \hom^{\leq \rho^f + \rho^g; \bme^f *
  \bme^g}(\mathcal{M}_0, \mathcal{M}_2)$, where the sequence of
discrepancies $\bme^f * \bme^g$ is defined as:
\begin{equation} \label{eq:disc-f*g}
  (\bme^f * \bme^g)_d = \max \{\epsilon_i^f + \epsilon_j^g \mid i+j=d+1\}.
\end{equation}
Moreover, a simple calculation shows that if
$\bme^f, \bme^g \in~\assmpe$ then the same holds for
$\bme^f * \bme^g$.

\subsubsection{Do weakly filtered $\mathcal{A}$-modules form a
  dg-category?} \label{sbsb:wfm-categ} Fix $\bmemm$ and
$\bmeh \in \assmpe$. As explained in the preceding section, for weakly
filtered $\mathcal{A}$-modules
$\mathcal{M}_0, \mathcal{M}_1, \mathcal{M}_2$ with discrepancy
$\leq \bmemm$, the operation $\mu_2^{\tmod}$ satisfies:
$$\mu_2^{\tmod}: \hom^{\leq \rho'; \bmeh}(\mathcal{M}_0, \mathcal{M}_1) \otimes 
\hom^{\leq \rho''; \bmeh}(\mathcal{M}_1, \mathcal{M}_2)
\longrightarrow \hom^{\leq \rho'+\rho''; \bmeh * \bmeh}(\mathcal{M}_0,
\mathcal{M}_2),$$ where $\bmeh*\bmeh$ is defined
by~\eqref{eq:disc-f*g}.  Thus in order for the $\mu_2^{\tmod}$
operation to have values in
$\hom^{\rho'+\rho''; \bmeh}(\mathcal{M}_0, \mathcal{M}_2)$ we need
$\bmeh*\bmeh \leq \bmeh$. It is easy to see that such an inequality
cannot hold unless $\eh_1=0$ (the latter condition being only a
necessary one).

One way to solve this problem is based on the fact that
$$\hom^{\leq \rho'+\rho''; \bmeh * \bmeh}(\mathcal{M}_0, \mathcal{M}_2)
= \hom^{\leq \rho' + \rho'' + \eh_1; \bmeh*\bmeh-\eh_1}(\mathcal{M}_0,
\mathcal{M}_2).$$ Therefore, we only need to arrange that
$\bmeh*\bmeh-\eh_1 \leq \bmeh$. Unwrapping this gives the following
system of inequalities. For all $d\geq 1$:
\begin{equation} \label{eq:eheh-eh1} \eh_d +\eh_1 \geq \eh_i + \eh_j,
  \; \; \forall \; i+j=d+1.
\end{equation}

It is easy to see (by induction on $d$) that sequences $\bmeh$
satisfying both Assumption~$\assmpen$ and~\eqref{eq:eheh-eh1} do
exist. In fact, Lemma~\ref{l:choice-eps-1} continues to hold (for
$\bm{\varepsilon} = \bmeh$) with the stronger assertion that $\bmeh$
satisfies both $\lbe$ as well as~\eqref{eq:eheh-eh1}.

Now let $\bmeh$ be a sequences that satisfies both $\lbe$
and~\eqref{eq:eheh-eh1}. We define a dg-category of weakly filtered
modules $\tmod^{\textnormal{wf}}_{\mathcal{A}}(\bmea, \bmemm, \bmeh)$
as follows. (The superscript wf stands for ``weakly filtered''). The
objects of this category are the weakly filtered $\mathcal{A}$-modules
with discrepancy $\leq \bmemm$. The morphisms
$\hom^{\bmeh}(\mathcal{M}_0, \mathcal{M}_1)$ between two such modules
$\mathcal{M}_0, \mathcal{M}_1$ are defined by:
$$\hom^{\bmeh}(\mathcal{M}_0, \mathcal{M}_1) := 
\bigcup_{\rho \in \mathbb{R}} \hom^{\leq \rho; \bmeh}(\mathcal{M}_0,
\mathcal{M}_1).$$ We endow
$\tmod^{\textnormal{wf}}_{\mathcal{A}}(\bmea, \bmemm, \bmeh)$ with the
operations $\mu_1^{\tmod}$ and $\mu_2^{\tmod}$ coming from the
dg-category $\rmod_{\mathcal{A}}$. Finally, we consider the obvious
filtration on the $\hom^{\bmeh}$'s given by $\hom^{\leq \rho; \bmeh}$,
$\rho \in \mathbb{R}$.

The following is easy to check.
\begin{prop} \label{p:wf-mod-categ} With the structures defined above
  $\tmod^{\textnormal{wf}}_{\mathcal{A}}(\bmea, \bmemm, \bmeh)$ is a
  weakly filtered dg-category with discrepancy
  $\leq (0, \eh_1, 0, \ldots, 0, \ldots)$. In particular, if $\eh_1=0$
  this category is a filtered dg-category.
\end{prop}

\begin{rems}
  \begin{enumerate}
  \item One can attempt to enlarge this category by taking unions over
    different choices of $\bmeh$ (say with fixed $\eh_1$) and possibly
    get a category that does not depend on $\bmeh$, but we will not do
    that here.
  \item The category
    $\tmod^{\textnormal{wf}}_{\mathcal{A}}(\bmea, \bmemm, \bmeh)$ is
    in general not triangulated. The reason is that while cones of
    morphisms in that category are weakly filtered they might have
    higher discrepancy (see~\S\ref{sb:wf-mc} below). One can attempt
    to rectify this issue by enlarging this category to contain weakly
    filtered modules of arbitrary discrepancy and labeling each
    pre-module morphisms by a pair of quantities indicating the bounds
    on their action shift and their discrepancy. We will not pursue
    this direction in this paper since it will not be needed for our
    applications.
  \item With the above definitions one can see that the Yoneda
    embedding restricts to a weakly filtered functor from
    $\mathcal{A}$ to the category of weakly filtered
    $\mathcal{A}$-modules with discrepancy $\leq \bmea$ and where
    $\bmeh$ is taken to be any sequences
    in~$\hyperlink{h:asmp-e}{\lbe(\bmea,\bmea)}$, with the additional
    condition that
    $\eh_d \geq \epsilon^{\mathcal{A}}_{d+1} \; \forall d$.
  \end{enumerate}
\end{rems}

At the same time, for the applications intended in this paper, we will
not really need the weakly filtered modules to form a
dg-category. Therefore we will not insist that $\bmeh$ satisfies
condition~\eqref{eq:eheh-eh1}. In contrast, Assumption~$\assmpen$ will
continue to play an important role since it assures that the
$\hom^{\leq \rho; \bmeh}$'s are preserved by $\mu_1^{\tmod}$. Thus we
will mostly continue to assume~$\assmpen$.

\subsubsection{Action-shifts} \label{sbsb:action-shifts} Let
$\mathcal{M}$ be a weakly filtered module over a weakly filtered
$A_{\infty}$-category $\mathcal{A}$. Let $\nu_0 \in \mathbb{R}$.
Define a new weakly filtered $\mathcal{A}$-module
$S^{\nu_0}\mathcal{M}$ to be the same module as $\mathcal{M}$ only
that its filtration is shifted by $\nu_0$, namely:
$$(S^{\nu_0}\mathcal{M})^{\leq \alpha}(N) := 
\mathcal{M}^{\leq \alpha + \nu_0}(N) \; \; \forall \; N \in
\textnormal{Ob}(\mathcal{A}), \; \alpha \in \mathbb{R}.$$ Clearly
$S^{\nu_0}\mathcal{M}$ has the same discrepancy as $\mathcal{M}$. We
call $S^{\nu_0}\mathcal{M}$ the ``action-shift of $\mathcal{M}$ by
$\nu_0$''.

In what follows we will use the same notation $S^{\nu_0}$ also for
action shifts of other filtered objects such as filtered chain
complexes or more generally filtered $R$-modules.

\subsubsection{Homologically unital
  $\mathcal{A}$-modules} \label{sbsb:hu-mod} We have already discussed
h-unital $A_{\infty}$ categories in the weakly filtered sense on
page~\pageref{pg:h-unital-categ}. In what follows we will sometimes
need an analogous, yet somewhat stronger, notion for modules.

\medskip \newcommand{\lbh}{U} \hypertarget{h:asmp-H}{\noindent
  \textbf{Assumptions~$\lbh_w$ and~$\lbh_s$.}}  Let $\mathcal{A}$ be a
weakly filtered $A_{\infty}$-category with discrepancy $\leq
\bmea$. Assume that $\mathcal{A}$ is h-unital in the weakly filtered
sense as defined in~\S\ref{sb:wf-ai-categ}, i.e. we have
$u^{\mathcal{A}} \geq 0$ and choices of cycles
$e_X \in C^{\leq u^{\mathcal{A}}}(X,X)$ for every
$X \in \textnormal{Ob}(\mathcal{A})$ representing the units in
homology.

Let $\mathcal{M}$ be a weakly filtered $\mathcal{A}$-module with
discrepancy $\leq \bmemm$, and let
$u^{\mathcal{A}} + \epsilon_2^{\mathcal{M}} \leq \kappa \in
\mathbb{R}$. We say that $\mathcal{M}$ satisfies
{Assumption~$\lbh_w(\kappa)$} if for every
$X \in \textnormal{Ob}(\mathcal{A})$ and every $\alpha \in \mathbb{R}$
the map
\begin{equation} \label{eq:mu-2-e}
  \mathcal{M}^{\leq \alpha}(X) \longrightarrow 
  \mathcal{M}^{\leq \alpha + \kappa}(X), \quad b \longmapsto
  \mu_2^{\mathcal{M}}(e_X, b)
\end{equation}
induces in homology the same map as the one induced by the inclusion
$\mathcal{M}^{\leq \alpha}(X) \longrightarrow \mathcal{M}^{\leq \alpha
  + \kappa}(X)$. Note that in particular, $\mathcal{M}$ is an h-unital
module.

Sometimes the module $\mathcal{M}$ will be a Yoneda module
$\mathscr{Y}$ associated to an object
$Y \in \textnormal{Ob}(\mathcal{A})$. In that case we will sometimes
write $Y \in \lbh_w(\kappa)$ instead of
$\mathscr{Y} \in \lbh_w(\kappa)$. Note that in this case the map
in~\eqref{eq:mu-2-e} becomes
\begin{equation*} \label{eq:mu-2-e-2} C^{\leq \alpha}(Y, X)
  \longrightarrow C^{\leq \alpha + \kappa}(Y, X), \quad b \longmapsto
  \mu_2^{\mathcal{A}}(e_X, b).
\end{equation*}

There is also a homotopical version of $\lbh_w$ which is
stronger. Namely, if we replace Assumption~$\lbh_w$ by the stronger
one requiring that~\eqref{eq:mu-2-e} is chain homotopic to the
inclusion, then we say that $\mathcal{M}$ satisfies
Assumption~$\lbh_s(\kappa)$. (The subscripts $s$ and $w$ stand for
``strong'' and ``weak'' -- $\lbh_w$ is weaker than $\lbh_s$.) For
modules $\mathcal{M}$ satisfying~$\lbh_w(\kappa)$ we will sometimes
write $\mathcal{M} \in \lbh_w(\mathcal{M})$ and similarly for $\lbh_s$.

\begin{rem} \label{r:us-uw} If the ground ring $R$ is a field then
  $U_s(\kappa) = U_w(\kappa)$. This is so because for chain complexes
  $C_{\bullet}, D_{\bullet}$ over a field, two chain maps
  $C_{\bullet} \longrightarrow D_{\bullet}$ induce the same map in
  homology iff they are chain homotopic. Or in other words, the
  canonical map
  $$H_*(\hom(C_{\bullet}, D_{\bullet})) \longrightarrow \hom(H_*(C),
  H_*(D))$$ is an isomorphism.
\end{rem}

\subsubsection{Pulling back weakly filtered
  modules} \label{sbsb:pull-back-mod} Let $\mathcal{A}, \mathcal{B}$
be two weakly filtered $A_{\infty}$-categories and
$\mathcal{F}: \mathcal{A} \longrightarrow \mathcal{B}$ a weakly
filtered $A_{\infty}$-functor with discrepancy
$\leq \bme^{\mathcal{F}}$. Let $\mathcal{M}$ be a weakly filtered
$\mathcal{B}$-module with discrepancy $\leq \bme^{\mathcal{M}}$.
Consider the $\mathcal{A}$-module $\mathcal{F}^*\mathcal{M}$ which is
obtained by pulling back $\mathcal{M}$ via $\mathcal{F}$.
We filter $\mathcal{F}^*\mathcal{M}$ by setting
$$(\mathcal{F}^*\mathcal{M})^{\leq \alpha}(N) = 
\mathcal{M}^{\leq \alpha}(\mathcal{F} N).$$ The following can be
easily proved.
\begin{lem} \label{l:pull-back-M} The module
  $\mathcal{F}^*\mathcal{M}$ is weakly filtered with discrepancy
  $\leq \bme^{\mathcal{F}^*\mathcal{M}}$, where
  $$\epsilon_d^{\mathcal{F}^*\mathcal{M}} = 
  \max \bigl\{ \epsilon_{s_1}^{\mathcal{F}} + \cdots +
  \epsilon_{s_k}^{\mathcal{F}} + \epsilon_{k+1}^{\mathcal{M}} \bigm| 1
  \leq k \leq d-1, \;\; s_1 + \cdots + s_k = d-1 \bigr\}, \; \;
  \forall \; d \geq 2.$$ In particular, if the higher order terms of
  $\mathcal{F}$ vanish, i.e. $\mathcal{F}_s = 0$ for all $s \geq 2$,
  then
  $$\epsilon_d^{\mathcal{F}^*\mathcal{M}} = (d-1)
  \epsilon_1^{\mathcal{F}} + \epsilon_d^{\mathcal{M}}, \; \; \forall
  \; d.$$
\end{lem}

Let $\mathcal{M}_0, \mathcal{M}_1$ be two weakly filtered
$\mathcal{B}$-modules and
$f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ a weakly filtered
module homomorphism that shifts action by $\leq \rho$ and has
discrepancy $\leq \bme^f$. Pulling back $\mathcal{M}_0, \mathcal{M}_1$
and $f$ by $\mathcal{F}$ we obtain a homomorphism of
$\mathcal{A}$-modules
$\mathcal{F}^*f: \mathcal{F}^*\mathcal{M}_0 \longrightarrow
\mathcal{F}^* \mathcal{M}_1$. The following can be easily verified.
\begin{lem} \label{l:pull-back-f} The module homomorphism
  $\mathcal{F}^*f$ is weakly filtered with action shift $\leq \rho$
  and discrepancy $\leq \bme^{\mathcal{F}^*f}$, where
  $\epsilon_1^{\mathcal{F}^*f} = \epsilon_1^f$ and
  $$\epsilon_d^{\mathcal{F}^*f} = 
  \max \bigl\{ \epsilon_{s_1}^{\mathcal{F}} + \cdots +
  \epsilon_{s_k}^{\mathcal{F}} + \epsilon_{k+1}^{f} \bigm| 1 \leq k
  \leq d-1, \;\; s_1 + \cdots + s_k = d-1 \bigr\}, \; \; \forall \; d
  \geq 2.$$ In particular, if the higher order terms of $\mathcal{F}$
  vanish, i.e. $\mathcal{F}_s = 0$ for all $s \geq 2$, then
  $$\epsilon_d^{\mathcal{F}^*f} = (d-1)
  \epsilon_1^{\mathcal{F}} + \epsilon_d^{f}, \; \; \forall \; d.$$
\end{lem}
  
\subsection{Weakly filtered mapping cones} \label{sb:wf-mc} Let
$\mathcal{M}_0$, $\mathcal{M}_1$ be two weakly filtered
$\mathcal{A}$-modules with discrepancies $\leq \bme^{\mathcal{M}_0}$
and $\leq \bme^{\mathcal{M}_1}$ respectively.  Let
$f : \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ be a {\em module
  homomorphism}, i.e. $f$ is a pre-module homomorphisms which is a
cycle $\mu_1^{\textnormal{mod}}(f)=0$. Assume that $f$ shifts action
by $\leq \rho$ and has discrepancy $\leq \bme^{f}$, or in other words
$f \in \hom^{\leq \rho; \bme^f}(\mathcal{M}_0, \mathcal{M}_1)$. We
{\em generally do not} assume that $\bme^f$ satisfies
Assumption~$\hyperlink{h:asmp-e}{\lbe(\bmemm, \bmea)}$ unless
explicitly specified.

Consider the mapping cone $\mathcal{C} := \mathcal{C}one(f)$ viewed as
an $\mathcal{A}$-module and endowed with its standard
$A_{\infty}$-composition maps $\mu_d^{\mathcal{C}}$. We endow
$\mathcal{C}$ with a weakly filtered structure as follows.  For
$X \in \textnormal{Ob}(\mathcal{A})$ and $a \in \mathbb{R}$, put
\begin{equation} \label{eq:fcone} \mathcal{C}^{\leq \alpha}(X) :=
  \mathcal{M}_0^{\leq \alpha - \rho - \epsilon^f_1}(X) \oplus
  \mathcal{M}_1^{\leq \alpha}(X).
\end{equation}
Define \label{pg:disc-cone}
$\bme^{\mathcal{C}} := \max \{\bme^{\mathcal{M}_0},
\bme^{\mathcal{M}_1}, \bme^f-\epsilon^f_1\}$. (See
page~\pageref{pg:conv-seq} for the precise meaning of this notation.)
Then $\mathcal{C}$ is weakly filtered with discrepancy
$\leq \bme^{\mathcal{C}}$. This follows from~\eqref{eq:fcone} and the
fact that
$$\mu^{\mathcal{C}}_d \bigl( a_1, \ldots, a_{d-1}, (b_0, b_1) \bigr) = 
\bigl( \mu_d^{\mathcal{M}_0}(a_1, \ldots, a_{d-1}, b_0), f_d(a_1,
\ldots, a_{d-1}, b_0) + \mu^{\mathcal{M}_1}_d(a_1, \ldots, a_{d-1},
b_1) \bigr).$$

\begin{rem} \label{r:epsN}
  If we assume in addition that
  $\bme^{\mathcal{M}_0}, \bme^{\mathcal{M}_1} \leq \bmemm$ and that
  $\bme^f \in \assmpe$ then we have
  $\bme^f - \epsilon^f_1 \geq \bmemm$, hence
  $\bme^{\mathcal{C}} = \bme^f - \epsilon^f_1$.
\end{rem}

It is important to note that the filtration we have defined on
$\mathcal{C}one(f)$ in~\eqref{eq:fcone} strictly depends on the
choices of $\rho$ and $\epsilon^f_1$. Therefore, whenever these
dependencies are relevant we will denote the weakly filtered cone of
$f$ by
\begin{equation} \label{eq:notation-cone} \mathcal{C}one(f; \rho,
  \bme^f) \; \; \textnormal{or by } \; \tcn ( \mathcal{M}_0
  \xrightarrow{ \; (f; \rho, \bme^f) \;} \mathcal{M}_1).
\end{equation}

\begin{rem} \label{r:cone-filtrations} 
  \begin{enumerate}
  \item The filtration~\eqref{eq:fcone} might look ad hoc since it can
    be shifted in an arbitrary way. However, we opted for the one
    in~\eqref{eq:fcone} so that the inclusion
    $\mathcal{M}_1 \longrightarrow \mathcal{C}$ would be a filtered
    map (rather than just weakly filtered).
  \item If $\rho' \geq \rho$ and $\bme' \geq \bme^f$ then clearly
    $f \in \hom^{\leq \rho'; \bme'}(\mathcal{M}_0, \mathcal{M}_1)$ and
    we can also consider
    $\tcn (\mathcal{M}_0 \xrightarrow{\; (f; \rho', \bme') \;}
    \mathcal{M}_1)$ which is the same $\mathcal{A}$-module but with a
    different weak filtration structure. It is easy to see that the
    identity homomorphism $\tcn(f) \longrightarrow \tcn(f)$ becomes a
    weakly filtered module homomorphism
    $\tcn(f; \rho, \bme^f) \longrightarrow \tcn(f; \rho', \bme')$ that
    shifts action by $\leq \rho'-\rho$ and has discrepancy
    $\leq (\epsilon'_1 - \epsilon^f_1, 0, \ldots, 0, \ldots)$. In the
    other direction the identity map becomes a weakly filtered module
    homomorphism
    $\tcn(f; \rho', \bme') \longrightarrow \tcn(f; \rho, \bme^f)$ that
    shifts action by $\leq \rho-\rho'$ and has discrepancy $0$ (so it
    is in fact a filtered homomorphism).
  \end{enumerate}
\end{rem}

We will now derive several elementary properties of weakly filtered
mapping cones that will be useful later on. We begin with the effect
of action-shifts (see~\S\ref{sbsb:action-shifts}) on mapping cones.
The following follows immediately from the definitions.
\begin{lem} \label{l:action-shift-cone} Let
  $f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ be a weakly
  filtered module homomorphism between two weakly filtered
  $\mathcal{A}$-modules. Assume that $f$ shifts action by $\leq \rho$
  and has discrepancy $\leq \bme^f$. Let $\nu_0 \in \mathbb{R}$. Then
  we have the following equality of weakly filtered
  $\mathcal{A}$-modules:
  \begin{equation*}
    \begin{aligned}
      & S^{\nu_0} \bigl(\tcn(\mathcal{M}_0 \xrightarrow{\; (f; \rho,
        \bme^f) \;} \mathcal{M}_1) \bigr) =
      \tcn(S^{\nu_0}\mathcal{M}_0 \xrightarrow{\; (f; \rho, \bme^f)
        \;} S^{\nu_0}\mathcal{M}_1) =
      \\
      & \tcn(\mathcal{M}_0 \xrightarrow{\; (f; \rho, \bme^f-\nu_0) \;}
      S^{\nu_0}\mathcal{M}_1) = \tcn(\mathcal{M}_0 \xrightarrow{\; (f;
        \rho-\nu_0, \bme^f) \;} S^{\nu_0}\mathcal{M}_1).
    \end{aligned}
  \end{equation*}
\end{lem}

Next, we analyze (a special case of) cones over composition of module
homomorphisms, from the weakly filtered perspective. Let
$f:\mathcal{M}_0 \longrightarrow \mathcal{M}_1$ be as at the beginning
of the present section. Let $\mathcal{M}'_1$ be another weakly
filtered $\mathcal{A}$-module with discrepancy
$\leq \bme^{\mathcal{M}'}$ and let
$\xi: \mathcal{M}_1 \longrightarrow \mathcal{M}'_1$ be a weakly
filtered module homomorphism with
$\xi \in \hom^{\leq s; \bme^{\xi}}(\mathcal{M}_1,
\mathcal{M}'_1)$. Denote by
$$f' = \xi \circ f := \mu_2^{\tmod}(f,\xi): \mathcal{M}_0 \longrightarrow
\mathcal{M}'_1$$ the composition of $f$ and $\xi$. We have
$f' \in \hom^{\rho+s; \bme^{f'}}(\mathcal{M}_0, \mathcal{M}_1)$, where
$$\epsilon^{f'}_d = (\bme^f * \bme^{\xi})_d =: 
\max\{ \epsilon^f_i + \epsilon^{\xi}_j \mid i+j=d+1\}.$$ 


\begin{lem} \label{l:con-f-g} There exists a weakly filtered module
  homomorphism
  $$\psi: \tcn(f;\rho, \bme^f) \longrightarrow \tcn(f'; \rho+s, \bme^{f'})$$
  that shifts action by $\leq s$ and has discrepancy
  $\leq \bme^{\xi}$. The homomorphism $\psi$ fits into the following
  (chain level) commutative diagram of $\mathcal{A}$-modules:
  \begin{equation} \label{eq:triangles-psi} 
    \begin{gathered}
      \xymatrix{ \mathcal{M}_0
        \ar[r]^{f} \ar[d]^{\id} & \mathcal{M}_1 \ar[r]^{\,} \ar[d]^{\xi}
        & \tcn(f)
        \ar[r]^{\,} \ar[d]^{\psi} & T \mathcal{M}_0 \ar[d]^{\id} \\
        \mathcal{M}_0 \ar[r]^{f'} & \mathcal{M}'_1 \ar[r]^{\,} & \tcn(f')
        \ar[r]^{\,} & T \mathcal{M}_0 }
    \end{gathered}
  \end{equation}
  where the horizontal unlabeled maps are the standard inclusion and
  projection maps (with $0$ higher order terms). Moreover, if $\xi$ is
  a quasi-isomorphism then so is $\psi$.
\end{lem}
\begin{proof}
  Simply define $\psi_1(b_0, b_1) = \bigl( b_0, \xi_1(b_1) \bigr)$ and
  for $d\geq 2$:
  $$\psi_d\bigl (a_1, \ldots, a_{d-1}, (b_0, b_1) \bigr) := 
  \bigl(0, \xi_d(a_1, \ldots, a_{d-1}, b_1)\bigr).$$ All the
  statements asserted by the lemma can be verified by direct
  calculation.
\end{proof}


Next we discuss how the weakly filtered mapping cone changes if we
alter the cycle $f$ by a boundary. Assume now that $\mathcal{M}_0$ and
$\mathcal{M}_1$ have both discrepancies $\leq \bmemm$. Fix a sequence
$\bmeh$ that satisfies Assumption~$\assmpe$. Let
$f \in \hom^{\leq \rho; \bmeh}(\mathcal{M}_0, \mathcal{M}_1)$ be a
module homomorphism and $f' = f + \mu_1^{\tmod}(\theta)$ for some
$\theta \in \hom^{\leq \rho; \bmeh}(\mathcal{M}_0,
\mathcal{M}_1)$. Consider the two weakly filtered mapping cones
$\mathcal{C}one(f; \rho, \bmeh)$ and
$\mathcal{C}one(f'; \rho, \bmeh)$.


\begin{lem} \label{l:cone-f-f'} There exists a module homomorphism
  $\vartheta: \mathcal{C}one(f; \rho, \bmeh) \longrightarrow
  \mathcal{C}one(f'; \rho, \bmeh)$ with the following properties:
  \begin{enumerate}
  \item $\vartheta$ is a quasi-isomorphism.
  \item $\vartheta$ does not shift action (i.e. it shifts the action
    filtration by $\leq 0$) and has discrepancy
    $\leq \bme^{\vartheta} := \bmeh - \eh_1$. In particular (since
    $\epsilon^{\vartheta}_1=0$) the chain map
    $$\vartheta_1: \mathcal{C}one(f; \rho, \bmeh)(X) \longrightarrow
    \mathcal{C}one(f'; \rho, \bmeh)(X)$$ preserves the action
    filtration for every $X \in \textnormal{Ob}(\mathcal{A})$.
  \end{enumerate}
\end{lem}

\begin{proof}
  Define
  $\vartheta_1(b_0, b_1) := \bigl( (-1)^{|b_0|-1} b_0, (-1)^{|b_1|}
  b_1 + \theta_1(b_0) \bigr)$ and for $d \geq 2$ define:
  $$\vartheta_d \bigl( a_1, \ldots, a_{d-1}, (b_0, b_1) \bigr) = 
  \bigl(0, \theta_d(a_1, \ldots, a_{d-1}, b_0) \bigr).$$
  Cf.~\cite[Formula~{3.7}, page~35]{Se:book-fukaya-categ}.
  
  Note that in this paper we work with a base ring $R$ of
  characteristic $2$, hence the signs in the preceding formula for
  $\vartheta_1$ can actually be ignored. Nevertheless we included
  them, just as an indication for a possible extension to more general
  rings.
\end{proof}

The next lemma shows that weakly filtered cones are preserved under
pulling back by weakly filtered functors.
\begin{lem} \label{l:pull-back-cone} Let $\mathcal{A}, \mathcal{B}$ be
  two weakly filtered $A_{\infty}$-categories,
  $\mathcal{M}_0, \mathcal{M}_1$ weakly filtered $\mathcal{B}$-modules
  and $f: \mathcal{M}_0 \longrightarrow \mathcal{M}_1$ a weakly
  filtered module homomorphism which shifts action by $\leq \rho$ and
  has discrepancy $\leq \bme^f$. Then we have the following equality
  of weakly filtered $\mathcal{A}$-modules:
  $$\mathcal{F}^* \bigl( 
  \tcn(\mathcal{M}_0 \xrightarrow{\; (f; \rho, \bme^f) \;}
  \mathcal{M}_0) \bigr) = \tcn(\mathcal{F}^* \mathcal{M}_0
  \xrightarrow{\; (\mathcal{F}^*f; \rho, \bme^{\mathcal{F}^*f}) \;}
  \mathcal{F}^*\mathcal{M}_1),$$ where $\bme^{\mathcal{F}^*f}$ is
  given in Lemma~\ref{l:pull-back-f}.
\end{lem}
The proof is straightforward, hence omitted.

We now return briefly to unitality of modules, more specifically to
Assumptions~$\hyperlink{h:asmp-H}{\lbh_s, \lbh_w}$.  The following
lemma shows that these two assumptions are preserved under certain
quasi-isomorphisms of weakly filtered modules.
\begin{lem} \label{l:ass-usw-iso} Let $\mathcal{M}'$, $\mathcal{M}''$
  be weakly filtered $\mathcal{A}$-modules with discrepancies
  $\leq \bme^{\mathcal{M}'}$ and $\leq \bme^{\mathcal{M}''}$
  respectively. Let
  $\phi': \mathcal{M}' \longrightarrow \mathcal{M}''$ be a weakly
  filtered module homomorphism with
  $\phi' \in \hom^{\rho'; \bme^{\phi'}}(\mathcal{M}',
  \mathcal{M}'')$. Let $\phi''$ be a collection of chain maps
  $\phi''_X: \mathcal{M}''(X) \longrightarrow \mathcal{M}'(X)$,
  defined for all $X \in \textnormal{Ob}(\mathcal{A})$, and assume
  $$\phi''_X \bigl( \mathcal{M}''^{\leq \alpha}(X) \bigr) \subset 
  \mathcal{M}'^{\leq \alpha + \rho'' + \epsilon''}(X), \; \; \forall
  \, X \in \textnormal{Ob}(\mathcal{A}), \, \alpha \in \mathbb{R},$$
  for some fixed $\rho'', \epsilon'' \in \mathbb{R}$. (For example, if
  $\phi'': \mathcal{M}'' \longrightarrow \mathcal{M}'$ is a weakly
  filtered module homomorphism with
  $\phi'' \in \hom^{\rho''; \bme^{\phi''}}(\mathcal{M}'',
  \mathcal{M}')$, where $\epsilon_1^{\phi''} \leq \epsilon''$, then
  the assumptions on $\phi''$ are clearly satisfied.) Let
  $\nu, \kappa'' \in \mathbb{R}$ and assume further that:
  \begin{enumerate}
  \item For every $X \in \textnormal{Ob}(\mathcal{A})$ and every
    $\alpha \in \mathbb{R}$ the composition of chain maps
    $$\mathcal{M}'^{\leq \alpha}(X) \xrightarrow{\; \phi''_X \circ \phi'_1 \;}
    \mathcal{M}'^{\leq \alpha + \rho'+\rho''+\epsilon_1^{\phi'} +
      \epsilon''}(X) \xrightarrow{\; \text{inc} \;} \mathcal{M}'^{\leq
      \alpha + \rho'+\rho''+\epsilon_1^{\phi'} + \epsilon''+\nu}(X)$$
    induces in homology the same map as the one induced by the
    inclusion
    $$\mathcal{M}'^{\leq \alpha}(X) \longrightarrow
    \mathcal{M}'^{\leq \alpha + \rho'+\rho''+\epsilon_1^{\phi'} +
      \epsilon''+\nu}(X).$$
    \label{i:phi-phi}
  \item $\mathcal{M}'' \in \hyperlink{h:asmp-H}{\lbh_w(\kappa'')}$.
    \label{i:us-kappa''}
  \end{enumerate}
  Then $\mathcal{M}' \in \hyperlink{h:asmp-H}{\lbh_w(\kappa')}$, where
  \begin{equation} \label{eq:kappa'-1} \kappa' = \rho' + \rho'' +
    \epsilon'' + \max \{ \epsilon_1^{\phi'} +
    u^{\mathcal{A}}+\epsilon_2^{\mathcal{M}'} + \nu,
    \epsilon_1^{\phi'} + \kappa'', \epsilon_2^{\phi'} +
    u^{\mathcal{A}} \}.
  \end{equation}
  
  If we replace~(\ref{i:phi-phi}) above by the assumption that for
  every $X \in \textnormal{Ob}(\mathcal{A})$, $\alpha \in \mathbb{R}$,
  the map
  $\phi''_X \circ \phi'_1: \mathcal{M'}(X) \longrightarrow
  \mathcal{M'}(X)$ is chain homotopic to $\textnormal{id}$ via a chain
  homotopy that shifts action by $\leq \nu$ and instead of
  assuming~(\ref{i:us-kappa''}) we assume
  $\mathcal{M}'' \in \hyperlink{h:asmp-H}{\lbh_s(\kappa'')}$, then
  $\mathcal{M}' \in \hyperlink{h:asmp-H}{\lbh_s(\kappa')}$ with
  \begin{equation} \label{eq:kappa'-2}
    \kappa' = \max\{\rho'+\rho''+\epsilon_1^{\phi'} + 
    \epsilon'' + \kappa'', \rho'+\rho''+\ua +
    \epsilon_2^{\phi'}+\epsilon'', \epsilon_2^{\mathcal{M}'} + \ua +
    \nu\}.
  \end{equation}
\end{lem}

\begin{proof}
  Fix $X \in \textnormal{Ob}(\mathcal{A})$, $\alpha \in \mathbb{R}$
  and let $b \in \mathcal{M}'^{\leq \alpha}(X)$ be a {\em
    cycle}. Since $\phi'$ is a module homomorphism
  (i.e. $\mu_1^{\textnormal{mod}}(\phi')=0$) we have:
  \begin{equation*}  \phi_1'
    \mu_2^{\mathcal{M}'}(e_X, b) = \mu_2^{\mathcal{M}''}(e_X,
    \phi'_1(b)) \pm \mu_1^{\mathcal{M}''} \phi'_2(e_X, b).
  \end{equation*}
  Applying $\phi''_X$ to both sides of this identity we obtain:
  \begin{equation} \label{eq:phi''-phi'-1} \phi''_X \phi_1'
    \mu_2^{\mathcal{M}'}(e_X, b) = \phi''_X \mu_2^{\mathcal{M}''}(e_X,
    \phi'_1(b)) \pm \mu_1^{\mathcal{M}'} \phi''_X \phi'_2(e_X, b).
  \end{equation}
  Since
  $\mu_2^{\mathcal{M}'}(e_X, b) \in \mathcal{M}'^{\leq \alpha + \ua +
    \epsilon_2^{\mathcal{M}'}}(X)$ our assumption on
  $\phi''_X \circ \phi'_1$ implies that there exists
  $x \in \mathcal{M}'^{\leq \alpha + \ua +
    \epsilon_2^{\mathcal{M}'}+\rho'+\rho''+\epsilon_1^{\phi'} +
    \epsilon''+\nu}(X)$ such that
  \begin{equation} \label{eq:phi''-phi'-mu_2}
    \phi''_X \phi_1' \mu_2^{\mathcal{M}'}(e_X, b) =
    \mu_2^{\mathcal{M}'}(e_X, b) - \mu_1^{\mathcal{M}'}(x).
  \end{equation}
  Since $\mathcal{M}'' \in \hyperlink{h:asmp-H}{\lbh_w(\kappa'')}$
  there exists an element
  $y \in \mathcal{M}''^{\leq \alpha +
    \rho'+\epsilon_1^{\phi'}+\kappa''}(X)$ such that
  $$\mu_2^{\mathcal{M}''}(e_X, \phi_1'(b)) = \phi_1'(b) + 
  \mu_1^{\mathcal{M}''}(y).$$ 
  Substituting the last identity together
  with~\eqref{eq:phi''-phi'-mu_2} into~\eqref{eq:phi''-phi'-1} yields:
  \begin{equation} \label{eq:mu-2-e-b} \mu_2^{\mathcal{M}'}(e_X, b) =
    \mu_1^{\mathcal{M}'}(x) + \phi''_X \phi'_1(b) +
    \mu_1^{\mathcal{M}'}(\phi''_X(y)) + \mu_1^{\mathcal{M}'}( \phi''_X
    \phi_2' (e_X, b)).
  \end{equation}
  Using our assumption on $\phi''_X \circ \phi_1'$, we can write the
  2'nd term of~\eqref{eq:mu-2-e-b} as
  $\phi''_X \phi'_1(b) = b + \mu_1^{\mathcal{M}'}(z)$ for some
  $z \in \mathcal{M}'^{\leq \alpha + \rho'+\rho''+\epsilon_1^{\phi'} +
    \epsilon'' + \nu}(X)$. Substituting this in~\eqref{eq:mu-2-e-b} we
  obtain:
  \begin{equation} \label{eq:mu-2-e-b-II} \mu_2^{\mathcal{M}'}(e_X, b)
    = b + \mu_1^{\mathcal{M}'}(x) + \mu_1^{\mathcal{M}'}(z) +
    \mu_1^{\mathcal{M}'}(\phi''_X(y)) + \mu_1^{\mathcal{M}'}( \phi''_X
    \phi_2' (e_X, b)),
  \end{equation}
  where
  \begin{equation*}
    \begin{aligned}
      & x \in \mathcal{M}'^{\leq \alpha + \ua +
        \epsilon_2^{\mathcal{M}'}+\rho'+\rho''+\epsilon_1^{\phi'} +
        \epsilon''+\nu}(X), \quad \phi''_X (y) \in \mathcal{M}''^{\leq
        \alpha +
        \rho'+\rho''+\epsilon_1^{\phi'}+\epsilon''+\kappa''}(X), \\
      & z \in \mathcal{M}'^{\leq \alpha +
        \rho'+\rho''+\epsilon_1^{\phi'} + \epsilon'' + \nu}(X), \quad
      \phi''_X \phi_2' (e_X, b) \in \mathcal{M}'^{\leq \alpha + \ua +
        \rho' + \rho'' + \epsilon'' + \epsilon_2^{\phi'}}.
    \end{aligned}
  \end{equation*}
  The estimate~\eqref{eq:kappa'-1} for $\kappa'$ readily follows.


  We now turn to the proving the second statement, concerning
  $\mathcal{M}' \in \hyperlink{h:asmp-H}{\lbh_s(\kappa')}$.  Let
  $X \in \textnormal{Ob}(\mathcal{A})$ and $b \in \mathcal{M}'(X)$ a
  chain. Since $\phi'$ is a module homomorphism we have the following
  identity:
  \begin{equation} \label{eq:ident-phi'} \phi'_1
    \mu_2^{\mathcal{M}'}(e_X, b) = \mu_2^{\mathcal{M}''}(e_X,
    \phi'_1(b)) - \mu_1^{\mathcal{M}''} \phi'_2(e_X, b) + \phi_2'(e_X,
    \mu^{\mathcal{M}'}_1(b)).
  \end{equation}

  Since $\mathcal{M}'' \in \hyperlink{h:asmp-H}{\lbh_s(\kappa'')}$, we
  can write
  $$\mu_2^{\mathcal{M}''}(e_X, \phi'_1(b)) = b + g \mu_1^{\mathcal{M}''}
  \phi'_1(b) + \mu_1^{\mathcal{M}''} g \phi'_1(b)$$ for some map
  $g:\mathcal{M}''(X) \longrightarrow \mathcal{M}''(X)$ that shifts
  action by $\leq \kappa''$. Substituting this
  into~\eqref{eq:ident-phi'} and then applying $\phi_X''$ to both
  sides of the equation we get:
  \begin{equation} \label{eq:phi''-phi'-s1}
    \begin{aligned}
      \phi''_X \phi'_1 \mu_2^{\mathcal{M}'}(e_X, b) = & \; \phi''_X
      \phi'_1(b) + \phi''_X g \mu_1^{\mathcal{M}''}
      \phi'_1(b) + \phi''_X \mu_1^{\mathcal{M}''} g \phi'_1(b) \\
      & - \mu_1^{\mathcal{M}''} \phi''_X \phi'_2(e_X, b) + \phi''_X
      \phi'_2 (e_X, \mu_1^{\mathcal{M}'}(b)).
    \end{aligned}
  \end{equation}
  By assumption
  $\phi''_X \circ \phi'_1 = id + h \circ \mu_1^{\mathcal{M}'} +
  \mu_1^{\mathcal{M}'} \circ h$ for some map
  $h: \mathcal{M}'(X) \longrightarrow \mathcal{M}'(X)$ that shifts
  action by $\leq \nu$. Thus we can write:
  $$\phi''_X \phi'_1 \mu_2^{\mathcal{M}'}(e_X, b) = \mu_2^{\mathcal{M}'}(e_X, b) + 
  h \mu_1^{\mathcal{M}'} \mu_2^{\mathcal{M}'}(e_X, b) +
  \mu_1^{\mathcal{M}'} h \mu_2^{\mathcal{M}'}(e_X, b).$$ Substituting
  $\phi''_X \circ \phi'_1(b) = b + h \mu_1^{\mathcal{M}'}(b) +
  \mu_1^{\mathcal{M}'} h(b)$ together with the latter equality
  into~\eqref{eq:phi''-phi'-s1} we obtain:
  \begin{equation} \label{eq:mu_2-e-b-s} 
    \begin{aligned}
      \mu_2^{\mathcal{M}'}(e_X, b) = b & + h \mu_1^{\mathcal{M}'}(b) +
      \mu_1^{\mathcal{M}'} h(b) + \phi''_X g \phi'_1
      \mu_1^{\mathcal{M}'}(b) + \mu_1^{\mathcal{M}'}
      \phi''_X g \phi'_1(b) \\
      & - \mu_1^{\mathcal{M}''} \phi''_X \phi'_2(e_X, b) + \phi''_X
      \phi'_2(e_X, \mu_1^{\mathcal{M}'}(b)) \\
      & - h \mu_2^{\mathcal{M}'}(e_X, \mu_1^{\mathcal{M}'}(b)) -
      \mu_1^{\mathcal{M}'} h \mu_2^{\mathcal{M}'}(e_X, b).
    \end{aligned}
  \end{equation}
  Here we have also used the fact that $\phi''_X$ and $\phi'_1$ are
  chain maps and that
  $\mu_1^{\mathcal{M}'} \mu_2^{\mathcal{M}'}(e_X, b) =
  \mu_2^{\mathcal{M}'}(e_X, \mu_1^{\mathcal{M}'}(b))$.  The
  estimate~\eqref{eq:kappa'-2} for $\kappa'$ now easily follows.
\end{proof}


It is known that h-unitality is preserved under mapping
cones~\cite[Section~3e]{Se:book-fukaya-categ}. The following Lemma is
a weakly filtered analog concerning
Assumptions~$\hyperlink{h:asmp-H}{\lbh_s, \lbh_w}$.

\begin{lem} \label{l:assh-cones} Assume that $\mathcal{A}$ satisfies
  Assumption~\hyperlink{h:asmp-ue}{$\lbue(\zeta)$} (see
  Page~\pageref{l:assm-ue}). Let $\mathcal{M}_0$, $\mathcal{M}_1$ be
  weakly filtered $\mathcal{A}$-modules with discrepancies
  $\leq \bme^{\mathcal{M}_0}$ and $\leq \bme^{\mathcal{M}_1}$
  respectively and assume that
  $\mathcal{M}_0 \in
  \hyperlink{h:asmp-H}{\lbh_w(\kappa^{\mathcal{M}_0})}$ and
  $\mathcal{M}_1 \in
  \hyperlink{h:asmp-H}{\lbh_w(\kappa^{\mathcal{M}_1})}$. Let
  $f \in \hom^{\leq \rho; \bme^f}(\mathcal{M}_0, \mathcal{M}_1)$ be a
  module homomorphism. Then the weakly filtered module
  $\tcn(f; \rho, \bme^f)$ satisfies
  Assumption~$\hyperlink{h:asmp-H}{\lbh_w(\kappa)}$, where
  \begin{equation} \label{eq:kappa-cone} \kappa = \max
    \{2\kappa^{\mathcal{M}_0}, 2\kappa^{\mathcal{M}_1},
    2u^{\mathcal{A}} + \epsilon_3^{\mathcal{C}},
    2u^{\mathcal{A}}+2\epsilon_2^{\mathcal{C}},
    \zeta+\epsilon_2^{\mathcal{C}}\},
  \end{equation}
  and
  $\bme^{\mathcal{C}} := \max\{\bme^{\mathcal{M}_0},
  \bme^{\mathcal{M}_1}, \bme^f-\epsilon_1^f\}$. (Recall that
  $\bme^{\mathcal{C}}$ is the standard bound on the discrepancy of
  $\mathcal{C} = \tcn(f; \rho, \bme^f)$ -- see
  Page~\pageref{pg:disc-cone}.) If the ground ring $R$ is a field then
  we actually have
  $\tcn(f; \rho, \bme^f) \in~\hyperlink{h:asmp-H}{\lbh_s(\kappa)}$.
\end{lem}

\begin{proof}[Proof of Lemma~\ref{l:assh-cones}]
  Denote $\mathcal{C} = \tcn (f; \rho, \bme^f)$. Recall that this
  module has discrepancy
  $\leq \bme^{\mathcal{C}} := \max\{\bme^{\mathcal{M}_0},
  \bme^{\mathcal{M}_1}, \bme^f-\epsilon_1^f\}$. Put
  $$\delta := 
  \max \{\ua + \epsilon_2^{\mathcal{C}}, \kappa^{\mathcal{M}_0},
  \kappa^{\mathcal{M}_1} \}, \quad \kappa := \max\{2\delta,
  2\ua+\epsilon_3^{\mathcal{C}}, \zeta + \epsilon_2^{\mathcal{C}}
  \}.$$ It is easy to see that the latter expression for $\kappa$
  coincides with~\eqref{eq:kappa-cone}.

  For an $A_{\infty}$-module $\mathcal{M}$ and
  $X \in \textnormal{Ob}(\mathcal{A})$ we will typically denote by
  $$V_{\mathcal{M}}: H_*(\mathcal{M}^{\leq \alpha}(X)) 
  \longrightarrow H_*(\mathcal{M}^{\leq \alpha + r}(X))$$ the map
  induced in homology by the chain map
  $$v_{\mathcal{M}}: \mathcal{M}^{\leq \alpha}(X) \longrightarrow
  \mathcal{M}^{\leq \alpha + r}(X), \quad b \longmapsto
  \mu_2^{\mathcal{M}}(e_X, b).$$ Here $r \in \mathbb{R}$ is chosen
  such that $\ua + \epsilon_2^{\mathcal{M}} \leq r$ so that
  $v_{\mathcal{M}}$ is well defined with the above given target. We
  will need to consider such maps for different values of $r$, and
  whenever a need to distinguish between them arises we will use
  additional ``decorations'' such as
  $V'_{\mathcal{M}}, V''_{\mathcal{M}}$ etc.

  Fix $\alpha \in \mathbb{R}$, $X \in
  \textnormal{Ob}(\mathcal{A})$. Since
  $$\mathcal{C}^{\leq \alpha}(X) = Cone \bigl( 
  \mathcal{M}_0^{\leq \alpha - \rho -\epsilon_1^f}(X) \xrightarrow{\;
    f_1 \;} \mathcal{M}_1^{\leq \alpha}(X) \bigr)$$ we have a long
  exact sequence in homology:
  $$\cdots \longrightarrow H_k(\mathcal{M}_1^{\leq \alpha}(X)) 
  \xrightarrow{\;\; \iota \;\;} H_k(\mathcal{C}^{\leq \alpha}(X))
  \xrightarrow{\;\; \pi \;\;} H_k(\mathcal{M}_0^{\leq \alpha - \rho
    -\epsilon_1^f}(X)) \longrightarrow \cdots$$ where $\iota$ and
  $\pi$ are the maps in homology induced by the inclusion
  $\mathcal{M}_1(X) \longrightarrow \mathcal{C}(X)$ and the projection
  $\mathcal{C}(X) \longrightarrow \mathcal{M}_0(X)$ respectively.

  Replacing $\alpha$ by $\alpha+\delta$ and by $\alpha+\kappa$ we
  obtain two similar long exact sequences. These three sequences are
  mapped on to the other via maps induced from the inclusions of the
  coming from raising the action level from $\alpha$ to
  $\alpha+\delta$ and then to $\alpha+\kappa$. In particular, the
  degree-$k$ chunks of these exact sequences gives the following
  commutative diagram with exact rows:

  \begin{equation} \label{eq:3-LES}
    \begin{gathered}
      \xymatrix{ H_k \bigl( \mathcal{M}_1^{\leq \alpha}(X) \bigr)
        \ar[r] \ar[d]_{V'_{\mathcal{M}_1} = i'_{\mathcal{M}_1}} &
        H_k(\mathcal{C}^{\leq \alpha}(X)) \ar[r]
        \ar[d]_{V'_{\mathcal{C}}}^{i'_{\mathcal{C}}} &
        H_k(\mathcal{M}_0^{\leq \alpha-\rho-\epsilon_1^f}(X))
        \ar[d]^{V'_{\mathcal{M}_0} = i'_{\mathcal{M}_0} } \\
        H_k(\mathcal{M}_1^{\leq \alpha + \delta}(X)) \ar[r]^{\iota}
        \ar[d]_{V''_{\mathcal{M}_1} = i''_{\mathcal{M}_1}} &
        H_k(\mathcal{C}^{\leq \alpha+\delta}(X)) \ar[r]^{\pi \quad \; }
        \ar[d]_{V''_{\mathcal{C}}}^{i''_{\mathcal{C}}} &
        H_k(\mathcal{M}_0^{\leq \alpha - \rho - \epsilon_1^f +
          \delta}(X)) \ar[d]^{V''_{\mathcal{M}_0} = i''_{\mathcal{M}_0}} \\
        H_k(\mathcal{M}_1^{\leq \alpha + \kappa}(X)) \ar[r] &
        H_k(\mathcal{C}^{\leq \alpha + \kappa}(X)) \ar[r] &
        H_k(\mathcal{M}_0^{\leq \alpha-\rho-\epsilon_1^f+\kappa}(X)) }
    \end{gathered}
  \end{equation}
  The maps $i'_{\mathcal{C}}$, $i''_{\mathcal{C}}$ are induced by the
  corresponding inclusions and similarly for
  $i'_{\mathcal{M}_0}, i''_{\mathcal{M}_0}, i'_{\mathcal{M}_1},
  i''_{\mathcal{M}_1}$. By assumption (and by the choices of $\delta$
  and $\kappa$) we have
  $$V'_{\mathcal{M}_j} = i'_{\mathcal{M}_j}, 
  \quad V''_{\mathcal{M}_j} = i''_{\mathcal{M}_j}, \; \text{for} \;
  j=0,1.$$ Note also that each of the maps $V'_{\mathcal{C}}$ and
  $i'_{\mathcal{C}}$ makes the above diagram commutative, and similarly
  for $V''_{\mathcal{C}}$ and $i''_{\mathcal{C}}$.

  Denote by
  $V'''_{\mathcal{C}}: H_k(\mathcal{C}^{\leq \alpha}(X))
  \longrightarrow H_k(\mathcal{C}^{\leq \alpha + \kappa})$ the map
  induced in homology by
  $v_{\mathcal{C}}: \mathcal{C}^{\leq \alpha}(X) \longrightarrow
  \mathcal{C}^{\leq \alpha+\kappa}(X)$, and by 
  $$i'''_{\mathcal{C}}: H_k(\mathcal{C}^{\leq \alpha}(X))
  \longrightarrow H_k(\mathcal{C}^{\leq \alpha + \kappa})$$ the map
  induced by the inclusion. Clearly we have:
  $$V'''_{\mathcal{C}} = i''_{\mathcal{C}} \circ V'_{\mathcal{C}} = 
  V''_{\mathcal{C}} \circ i'_{\mathcal{C}}.$$

  We will need soon the following proposition.
  \begin{prop} \label{p:v_M} Assume that
    $\mathcal{A} \in \hyperlink{h:asmp-ue}{\lbue(\zeta)}$. Let
    $\mathcal{M}$ be a weakly filtered $\mathcal{A}$-module with
    discrepancy $\leq \bme^{\mathcal{M}}$ and let
    $X \in \textnormal{Ob}(\mathcal{A})$. Then the chain maps
    $$v: \mathcal{M}(X) \longrightarrow \mathcal{M}(X),
    \quad v(b) := \mu_2^{\mathcal{M}}(e_X, b)$$ and
    $v \circ v : \mathcal{M}(X) \longrightarrow \mathcal{M}(X)$ are
    chain homotopic via a chain homotopy that shifts action by
    $\leq \max\{2\ua + \epsilon_3^{\mathcal{M}}, \zeta +
    \epsilon_2^{\mathcal{M}} \}$.
  \end{prop}
  We postpone the proof of this Proposition to later in this section
  and proceed with the proof of the current lemma.

  To prove the lemma, we need to show that
  $V'''_{\mathcal{C}}(x) = i'''_{\mathcal{C}}(x)$ for all
  $x \in H_k(\mathcal{C}^{\leq \alpha}(X))$.  To prove the latter
  equality, we first note that since both $V'_{\mathcal{C}}$ and
  $i'_{\mathcal{C}}$ make diagram~\eqref{eq:3-LES} commutative, we
  have:
  $$V'_{\mathcal{C}}(x) - i'_{\mathcal{C}}(x) \in \ker \pi = 
  \textnormal{image\,} \iota.$$ Now write
  $V'_{\mathcal{C}}(x) - i'_{\mathcal{C}}(x) = \iota(y)$ for some
  $y \in H_k(\mathcal{M}_1^{\leq \alpha + \delta}(X))$. As both
  $V''_{\mathcal{C}}$ and $i''_{\mathcal{C}}$ make
  diagram~\eqref{eq:3-LES} commutative we also have:
  $V''_{\mathcal{C}} \circ \iota (y) = i''_{\mathcal{C}} \circ
  \iota(y)$.  It follows that
  $$V''_{\mathcal{C}} 
  \bigl( V'_{\mathcal{C}}(x) - i'_{\mathcal{C}}(x) \bigr) =
  i''_{\mathcal{C}}(V'_{\mathcal{C}}(x) - i'_{\mathcal{C}}(x)).$$
  Applying Proposition~\ref{p:v_M} with $\mathcal{M} = \mathcal{C}$ we
  obtain
  $$V'''_{\mathcal{C}}(x) - V''_{\mathcal{C}} \circ i'_{\mathcal{C}}(x)
  = i''_{\mathcal{C}} \circ V'_{\mathcal{C}}(x) -
  i'''_{\mathcal{C}}(x).$$ Since
  $V'''_{\mathcal{C}} = V''_{\mathcal{C}} \circ i'_{\mathcal{C}} =
  i''_{\mathcal{C}} \circ V'_{\mathcal{C}}$ the lemma follows.
\end{proof}

It still remains to prove the preceding Proposition.
\begin{proof}[Proof of Proposition~\ref{p:v_M}]
  The $A_{\infty}$-identities for $\mathcal{M}$ (+ the fact that $e_X$
  is a cycle) imply that for every $b \in \mathcal{M}(X)$ we have:
  \begin{equation} \label{eq:ai-M} \mu_1^{\mathcal{M}}
    \mu_3^{\mathcal{M}}(e_X, e_X, b) - \mu_2^{\mathcal{M}}(e_X,
    \mu_2^{\mathcal{M}}(e_X,b)) + \mu_3^{\mathcal{M}}(e_X, e_X,
    \mu_1^{\mathcal{M}}(b)) +
    \mu_2^{\mathcal{M}}(\mu_2^{\mathcal{A}}(e_X, e_X), b) = 0.
  \end{equation}
  Since $\mathcal{A} \in \hyperlink{h:asmp-ue}{\lbue(\zeta)}$ we have
  $\mu_2^{\mathcal{A}}(e_X, e_X) = e_X + \mu_1^{\mathcal{A}}(c)$, for
  some $c \in C^{\leq \zeta}(X,X)$. Substituting this
  in~\eqref{eq:ai-M} together with the fact that
  $\mu_2^{\mathcal{M}}(\mu_1^{\mathcal{A}}(c), b) +
  \mu_2^{\mathcal{M}}(c, \mu_1^{\mathcal{M}}(b)) +
  \mu_1^{\mathcal{M}}\mu_2^{\mathcal{M}}(c,b) = 0$ yields:
  \begin{equation*}
    \mu_2^{\mathcal{M}}(e_X, \mu_2^{\mathcal{M}}(e_X,b)) -
    \mu_2^{\mathcal{M}}(e_X, b) = \mu_1^{\mathcal{M}} h(b) + h
    \mu_1^{\mathcal{M}}(b),
  \end{equation*}
  where
  $h(b) = \mu_3^{\mathcal{M}}(e_X,e_X,b) - \mu_2^{\mathcal{M}}(c,b)$.
  Clearly the chain homotopy $h$ shifts action by not more than
  $\max \{2\ua + \epsilon_3^{\mathcal{M}},
  \zeta+\epsilon_2^{\mathcal{M}} \}$.
\end{proof}

\subsection{The $\lambda$-map} \label{sb:l-map} Let $\mathcal{A}$ be
an $A_{\infty}$-category and $\mathcal{M}$ and
$\mathcal{A}$-module. Let $Y \in \textnormal{Ob}(\mathcal{A})$ and
denote by $\mdly$ the Yoneda module corresponding to $Y$.  Consider
the following map:
\begin{equation} \label{eq:lambda} 
  \begin{aligned}
    & \lambda: \mathcal{M}(Y) \longrightarrow \hom(\mdly,
    \mathcal{M}), \quad c \longmapsto \lambda(c) = \bigl(\lambda(c)_1,
    \lambda(c)_2, \ldots, \lambda(c)_d,
    \ldots\bigr), \\
    & \textnormal{where} \quad \lambda(c)_d \bigl(a_1, \ldots,
    a_{d-1}, b \bigr) = \mu_{d+1}^{\mathcal{M}}(a_1, \ldots, a_{d-1},
    b, c).
  \end{aligned}
\end{equation}
This map was defined by
Seidel~\cite[Section~(1l)]{Se:book-fukaya-categ} in the context of the
Yoneda embedding of $A_{\infty}$-categories. A straightforward
calculation shows that it is a chain map. We will refer to it from now
on as the {\em $\lambda$-map}.

Seidel proves in~\cite[Lemma~2.12]{Se:book-fukaya-categ} that, under
the additional assumptions that $\mathcal{A}$ and $\mathcal{M}$ are
h-unital, the $\lambda$-map is a quasi-isomorphism. Our goal is to
establish a weakly filtered analog of this result.

We begin with a technical assumption on a given object
$Y \in \textnormal{Ob}(\mathcal{A})$.

\newcommand{\lbur}{U^{R,e}} \newcommand{\lbul}{U^{e,L}}
\hypertarget{h:asmp-ur}{\noindent \textbf{Assumption~$\lbur$.}}
\label{l:assm-ur} Let $\kappa \geq \epsilon_2^{\mathcal{A}} + \ua$ be
a real number. We say that $Y$ satisfies Assumption~$\lbur(\kappa)$
(or $Y \in \lbur(\kappa)$ for short) if for every
$X \in \textnormal{Ob}(\mathcal{A})$ the map
$$C(X,Y) \longrightarrow C(X,Y), \quad b \longmapsto \mu_2(b, e_Y)$$
is chain homotopic to the identity via a chain homotopy $h_X$ that
shifts action by $\leq \kappa$. The superscript $R,e$ stand for
``Right''-multiplication with $e_Y$.

\medskip

We now define the right setting for the $\lambda$-map in the weakly
filtered case. Assume that $\mathcal{A}$ and $\mathcal{M}$ are both
weakly filtered with discrepancies $\leq \bmea$ and
$\leq \bme^{\mathcal{M}}$ respectively. Clearly $\mdly$ is also a
weakly filtered module with discrepancy $\leq \bmea$.

Without loss of generality we assume from now on that
$\bme^{\mathcal{M}} \geq \bmea$ so that $\mdly$ can be regarded also
as a weakly filtered module with discrepancy
$\leq \bme^{\mathcal{M}}$. (If needed, we can always increase
$\bme^{\mathcal{M}}$ and $\mathcal{M}$ will continue being weakly
filtered with discrepancy $\leq$ than the increased
$\bme^{\mathcal{M}}$.)

Let $\bmeh$ be any sequence that satisfies
Assumption~\hyperlink{h:asmp-e}{$\lbe$} and assume in addition that:
\begin{equation} \label{eq:eh-emm} \eh_d \geq
  \epsilon^{\mathcal{M}}_{d+1} \; \forall \, d.
\end{equation}

Under these assumptions, the $\lambda$-map restricts to maps:
\begin{equation} \label{eq:lambda-alpha} \lambda: \mathcal{M}^{\leq
    \alpha}(Y) \longrightarrow \hom^{\leq \alpha; \bmeh}(\mdly,
  \mathcal{M}),
\end{equation}
defined for all $\alpha \in \mathbb{R}$.


Since $\bmeh$ satisfies Assumption~\hyperlink{h:asmp-e}{$\lbe$}, the
right-hand side of~\eqref{eq:lambda-alpha} is a chain complex with
respect to $\mu_1^{\tmod}$ and the $\lambda$-map
from~\eqref{eq:lambda-alpha} is a chain map.

Let $\mathcal{A}$, $\mathcal{M}$, $Y$ and $\mdly$ be as at the
beginning of~\S\ref{sb:l-map}. Fix also $\bme^{\mathcal{M}}$, $\bmeh$
as above.  For every $\alpha \in \mathbb{R}$ set
$\mathcal{H}^{\leq \alpha} := \hom^{\leq \alpha; \bmeh}(\mdly,
\mathcal{M})$ and for every $k \geq 1$:
\begin{equation*}
  Q^{\leq \alpha}_{(k)} := \bigl\{ t \in \mathcal{H}^{\leq \alpha} 
  \mid t_1 = \cdots = t_k = 0 \bigr \}, \quad 
  \mathcal{H}^{\leq \alpha}_{(k)} := 
  \mathcal{H}^{\leq \alpha} \big/ Q^{\leq \alpha}_{(k)}.
\end{equation*}
As explained above, the $\lambda$-map restricts to maps
$\lambda^{\alpha} : \mathcal{M}^{\leq \alpha}(Y) \longrightarrow
\mathcal{H}^{\leq \alpha}$ for every $\alpha \in \mathbb{R}$ and we
also have the following induced maps:
$$\lambda^{\alpha}_{(k)} : \mathcal{M}^{\leq \alpha}(Y) 
\longrightarrow \mathcal{H}^{\leq \alpha}_{(k)}$$ defined by composing
$\lambda^{\alpha}$ with the quotient map
$\pi_{(k)}: \mathcal{H}^{\leq \alpha} \longrightarrow
\mathcal{H}^{\leq \alpha}_{(k)}$.

\begin{prop} \label{p:lmap} Suppose that $\mathcal{A}$ is h-unital in
  the weakly filtered sense with discrepancy of units
  $\leq u^{\mathcal{A}}$. Let $\kappa \in \mathbb{R}$ such that
  $\kappa \geq u^{\mathcal{A}} + \epsilon_2^{\mathcal{M}}, \ua +
  \epsilon_2^{\mathcal{A}}$ and assume that
  $\mathcal{M} \in \hyperlink{h:asmp-H}{\lbh_w(\kappa)}$ and
  $Y \in \hyperlink{h:asmp-ur}{\lbur(\kappa)}$.  Let
  $\alpha \in \mathbb{R}$. Fix $1 \leq \ell \in \mathbb{Z}$ and put
  $\alpha' := \alpha + \ell \kappa$. Consider the following
  commutative diagram in cohomology:
  \begin{equation} \label{eq:H-M} 
    \begin{gathered}
      \xymatrix{ H^*
        \bigl(\mathcal{M}^{\leq \alpha}(Y) \bigr) \ar[r]^{\quad
          \lambda^{\alpha}_*} \ar[d]_{i^H} &
        H^* \bigl(\mathcal{H}^{\leq \alpha} \bigr) \ar[d]^{i^H} \\
        H^* \bigl(\mathcal{M}^{\leq \alpha'}(Y) \bigr) \ar[r]^{\quad
          \lambda^{\alpha'}_*} \ar[d]_{\id}&
        H^* \bigl(\mathcal{H}^{\leq \alpha'} \bigr) \ar[d]^{\pi^H_{(\ell)}}\\
        H^* \bigl(\mathcal{M}^{\leq \alpha'}(Y) \bigr) \ar[r]^{\quad
          \lambda^{\alpha'}_{(\ell)*}} & H^*
        \bigl(\mathcal{H}_{(\ell)}^{\leq \alpha'}\bigr) }
      \end{gathered}
  \end{equation}
  where the $i^H$ maps are induced by the inclusions
  $\mathcal{M}^{\leq \alpha}(Y) \longrightarrow \mathcal{M}^{\leq
    \alpha'}(Y)$ and
  $\mathcal{H}^{\leq \alpha} \longrightarrow \mathcal{H}^{\leq
    \alpha'}$ and $\pi^H_{(\ell)}$ is induced by the projection
  $\pi_{(\ell)}: \mathcal{H}^{\leq \alpha'} \longrightarrow
  \mathcal{H}^{\leq \alpha'}_{(\ell)}$. Then for every
  $b \in H^*(\mathcal{H}^{\leq \alpha})$ there exists
  $c \in H^*(\mathcal{M}^{\leq \alpha'}(Y))$ such that
  $\pi^H_{(\ell)} \circ i^H(b) = \lambda^{\alpha'}_{(\ell) *}(c)$. In
  other words, for every cycle $\beta \in \mathcal{H}^{\leq \alpha}$
  there exists a cycle $\gamma \in \mathcal{M}^{\leq \alpha'}(Y)$ such
  that
  \begin{equation} \label{eq:beta-gamma} \beta = \lambda(\gamma) +
    \mu_1^{\tmod}(\theta) + \tau,
  \end{equation}
  for some $\theta \in \mathcal{H}^{\leq \alpha'}$ and some cycle
  $\tau = (\tau_1, \tau_2, \ldots) \in \mathcal{H}^{\leq \alpha'}$
  with $\tau_1 = \cdots = \tau_{\ell} = 0$.
\end{prop}

\begin{proof}
  The proof below follows the general scheme of the proof of
  Lemma~2.12 from of~\cite{Se:book-fukaya-categ}, however the weakly
  filtered setting entails significant adjustments with respect
  to~\cite{Se:book-fukaya-categ}.

  We begin with some preparations regarding the weakly filtered
  version of the $\lambda$-map. Let $\rho \in \mathbb{R}$,
  $d \in \mathbb{N}$. Recall that we have the chain map
  $\lambda^{\rho}: \mathcal{M}^{\leq \rho}(Y) \longrightarrow
  \mathcal{H}_{(d)}^{\leq \rho}$ and consider its mapping cone:
  $$\mathcal{K}_{(d)}^{\rho} = Cone \bigl( \mathcal{M}^{\leq \rho}(Y)
  \xrightarrow{\; \lambda^{\rho} \;} \mathcal{H}_{(d)}^{\leq
    \rho}\bigr).$$ Define a decreasing filtration
  $F^r \mathcal{K}_{(d)}^{\rho}$, $r \in \mathbb{Z}_{\geq 0}$ on this
  chain complex by setting
  \begin{equation} \label{eq:FK}
    F^r \mathcal{K}_{(d)}^{\rho} =
    \begin{cases}
      \mathcal{K}_{(d)}^{\rho}, & \text{if $r = 0$;} \\
      \mathcal{H}_{(d)}^{\leq \rho}, & \text{if $r=1$;} \\
      \bigl\{f \in \mathcal{H}_{(d)}^{\leq \rho} \mid f_1 = \cdots =
      f_{r-1} = 0\bigr\}, & \text{if $2\leq r$.}
    \end{cases}
  \end{equation}
  Note that this is a bounded filtration and we actually have
  $F^r\mathcal{K}_{(d)}^{\rho} = 0$ for every $r \geq d+1$.

  Consider now the cohomological spectral sequence
  $\{E_r^{p,q}(\rho), \partial_r\}_{r \in \mathbb{Z}_{\geq 0}}$
  associated to the filtration $F^{\bullet}$. Since the filtration is
  bounded the spectral sequence converges to
  $H^*(\mathcal{K}_{(d)}^{\rho})$.

  Note also that for $\rho \leq \rho'$ we have an obvious inclusion of
  chain complexes
  $i: \mathcal{K}_{(d)}^{\rho} \longrightarrow
  \mathcal{K}_{(d)}^{\rho'}$. Moreover, this inclusion preserves the
  filtrations $F^{\bullet}$ on the corresponding chain
  complexes. Therefore, $i$ induces a map of spectral
  sequences
  $$i_r^E: E_r^{p,q}(\rho) \longrightarrow E_{r+1}^{p,q}(\rho'), \quad
  \forall \; r \geq 0, \; \forall \; p,q.$$

  We now describe more explicitly the first two pages of
  $E_r^{p,q}(\rho)$. A simple calculation gives the following
  description of the $E_0$-page of this spectral sequence. We have
  $E_0^{p,\bullet}(\rho) = 0$ for $p > d$ and for $p<0$. Next we have:
  $E_0^{0,\bullet}(\rho) = \mathcal{M}^{\leq \rho}(Y)^{\bullet}$,
  where the superscript $^{\bullet}$ stands here for the
  (cohomological) grading of the chain complex
  $\mathcal{M}^{\leq \rho}(Y)$. The differential
  $\partial_0: E_0^{0,q}(\rho) \longrightarrow E_0^{0, q+1}(\rho)$ is
  simply $\mu_1^{\mathcal{M}}$.

  The rest of the columns,
  $1 \leq p \leq d$, are:
  \begin{align} \label{eq:E_0^p}
    & E_0^{p,\bullet}(\rho) = \\
    & \prod_{X_0, \ldots, X_{p-1} \in \textnormal{Ob}(\mathcal{A})}
      \hom_{R}^{\leq \rho+\epsilon_p^h; \bullet} \bigl( C(X_0, X_1)
      \otimes \cdots \otimes C(X_{p-2}, X_{p-1}) \otimes C(X_{p-1},
      Y), \mathcal{M}(X_0) \bigr), \notag
  \end{align}
  where the superscript $^{\bullet}$ stand again for (cohomological)
  grading and $\hom_R^{\leq \rho+\epsilon_p^h}$ stands for $R$-linear
  homomorphism that shift action by not more than
  $\rho+\epsilon_p^h$. (Recall that $\bmeh$ has been fixed at the
  beginning of~\S\ref{sb:l-map} and is used in the definitions of
  $\mathcal{H}^{\leq \rho}$ and $\mathcal{H}^{\leq \rho}_{(d)}$.)  The
  differentials
  $\partial_0: E_0^{p,q}(\rho) \longrightarrow E_0^{p,q+1}(\rho)$ for
  $1\leq p\leq d$ are induced in a standard way from
  $\mu_1^{\mathcal{A}}$ and $\mu_1^{\mathcal{M}}$.

  The $E_1$-page is consequently the following:
  $E_1^{p,\bullet}(\rho) = 0$ for all $p > d$ and for $p<0$. For $p=0$
  we have $E_1^{0,q}(\rho) = H^q(\mathcal{M}^{\leq \rho}(Y))$ for all
  $q$. And for $1\leq p\leq d$ we have:
  \begin{align} \label{eq:E_1^pq}
    & E_1^{p,q}(\rho) = \\ 
    & \prod_{X_0, \ldots, X_{p-1} \in
      \textnormal{Ob}(\mathcal{A})} H^q \Bigl( \hom_{R}^{\leq
      \rho+\epsilon_p^h} \bigl( C(X_0, X_1) \otimes \cdots \otimes
    C(X_{p-2}, X_{p-1}) \otimes C(X_{p-1}, Y), \mathcal{M}(X_0) \bigr)
    \Bigr). \notag
  \end{align}

  We now describe the differentials
  $\partial_1: E_1^{p,q} \longrightarrow E_1^{p+1,q}$ on the
  $E_1$-page.  We start with $p=0$. Let
  $[c] \in E_1^{0,q} = H^q(\mathcal{M}^{\leq \rho}(Y))$, where $c$ is
  a cycle.  Then
  $$\partial_1 [c] \in E_1^{1,q} = 
  \prod_{X \in \textnormal{Ob}(\mathcal{A})} H^q \Bigl(\hom_R^{\leq
    \rho+\epsilon_1^h} \bigl(C(X,Y), \mathcal{M}(X) \bigr) \Bigr)$$ is
  the cycle represented by the following homomorphism:
  $$C(X,Y) \ni b \longmapsto \mu_2^{\mathcal{M}}(b,c) \in \mathcal{M}(X).$$
  It is easy to check that this homomorphism is a cycle and that it
  shifts action by not more than $\rho+\epsilon_1^h$. (The latter hold
  because $\epsilon_1^h \geq \epsilon_2^{\mathcal{M}}$
  by~\eqref{eq:eh-emm}.)  The formula for $\partial_1$ for
  $1\leq p \leq d-1$ is the following. Let $f$ be an element in the
  RHS of~\eqref{eq:E_0^p} which is a cycle. Then
  $\partial_1 [f] =[g] \in E_1^{p+1,q}$, where $g$ is a collection of
  $R$-linear homomorphism
  $$g: C(X_0, X_1) \otimes \cdots \otimes C(X_{p-1}, X_{p}) \otimes
  C(X_{p}, Y) \longrightarrow \mathcal{M}(X_0),$$ defined for all
  objects $X_0, \ldots, X_p \in \textnormal{Ob}(\mathcal{A})$ and is
  given by the following formula:
  \begin{equation} \label{eq:del_1-2}
    \begin{aligned}
      g(a_1, \ldots, a_p, b) \longmapsto & \pm
      \mu_2^{\mathcal{M}}(a_1, f(a_2, \ldots, a_p, b)) \pm f(a_1,
      \ldots, a_{p-1},
      \mu_2^{\mathcal{A}}(a_p, b)) \\
      & + \sum_{n=0}^{p-2} \pm f(a_1, \ldots,
      \mu_2^{\mathcal{A}}(a_{n+1}, a_{n+2}), \ldots, b).
    \end{aligned}
  \end{equation}
  This follows from a direct calculation. See the proof of Lemma~2.12
  in~\cite{Se:book-fukaya-categ} for the precise signs in
  formula~\eqref{eq:del_1-2}. Note also that $g$ shifts action by
  $\leq \max \{\rho + \epsilon^h_{p-1} + \epsilon^{\mathcal{M}}_2,
  \rho + \epsilon^{\mathcal{A}}_{p-1}\} \leq \rho + \epsilon^h_p$,
  where the latter inequality follows from
  Assumption~\hyperlink{h:asmp-e}{$\lbe(\bmemm, \bmea)$}.
  
  Consider now the inclusion
  $i: \mathcal{K}_{(d)}^{\rho} \longrightarrow \mathcal{K}_{(d)}^{\rho
    + \kappa}$. As indicated earlier this induces a map of spectral
  sequences $i^E: E(\rho) \longrightarrow E(\rho+\kappa)$, namely
  $$i^E_r: E_r^{p,q}(\rho) \longrightarrow 
  E_r^{p,q}(\rho+ \kappa), \; \forall r\geq 0.$$

  \begin{claim} \label{cl:homotopic} For every $q$ the chain map
    $i_1^E : E_1^{\bullet, q}(\rho) \longrightarrow E_1^{\bullet,
      q}(\rho+\kappa)$ is chain homotopic to $0$ in the degree range
    $0 \leq \bullet \leq d-1$. In other words, for every $q$ there
    exist homomorphisms
    $S^{p,q}: E_1^{p, q}(\rho) \longrightarrow E_1^{p-1,
      q}(\rho+\kappa)$, defined for all $p$, such that
    \begin{equation} \label{eq:i_1-S_pq}
      i_1^E|_{E_1^{p,q}(\rho)} = \partial_1 \circ S^{p,q} + S^{p+1,q}
      \circ \partial_1, \; \; \forall \, 0\leq p \leq d-1.
    \end{equation}
  \end{claim}
  We postpone the proof of this claim till later in this section and
  continue now with the proof of Proposition~\ref{p:lmap}.

  Claim~\ref{cl:homotopic} implies that
  $i^E_2: E_2^{p,q}(\rho) \longrightarrow E_2^{p,q}(\rho + \kappa)$ is
  the $0$ map for every $0\leq p \leq d-1$ and every $q$. It follows
  that the same holds for the maps
  $i^E_r: E_r^{p,q}(\rho) \longrightarrow E_r^{p,q}(\rho+\kappa)$ for
  every $r \geq 2$.

  Since both the spectral sequences converge after a finite number of
  pages (in fact they collapse at page $r=d+1$) we conclude that
  $i_{\infty}^E : E_{\infty}^{p,q}(\rho) \longrightarrow
  E_{\infty}^{p,q}(\rho+\kappa)$ is $0$ for all $0\leq p \leq d-1$ and
  all $q$. Denote by $F^{\bullet} H^*(\mathcal{K}_{(d)}^{\rho})$ the
  filtration on $H^*(\mathcal{K}_{(d)}^{\rho})$ induced by
  $F^{\bullet} \mathcal{K}_{(d)}^{\rho}$. Since
  $$E_{\infty}^{p,q}(\rho) = F^p H^{p+q}(\mathcal{K}_{(d)}^\rho) /
  F^{p+1}H^{p+q}(\mathcal{K}_{(d)}^{\rho}),$$ and similarly for
  $E_{\infty}^{p,q}(\rho+\kappa)$, we have proved the following:
  \begin{cor} \label{c:i^H-map} The map
    $i^H: H^n(\mathcal{K}_{(d)}^{\rho}) \longrightarrow
    H^n(\mathcal{K}_{(d)}^{\rho+\kappa})$ induced in homology by the
    inclusion
    $i: \mathcal{K}_{(d)}^{\rho} \longrightarrow
    \mathcal{K}_{(d)}^{\rho+\kappa}$ sends
    $F^pH^n(\mathcal{K}_{(d)}^{\rho})$ to
    $F^{p+1}H^n(\mathcal{K}_{(d)}^{\rho+\kappa})$ for every $n$ and
    $0 \leq p \leq d-1$.
  \end{cor}

  We are now in position to conclude the proof of
  Proposition~\ref{p:lmap}. Fix the parameters $\alpha$, $\ell$ and
  $\alpha'$ as in the statement of the proposition. Choose
  $d \gg \ell$ and apply what we have proved above to
  $\mathcal{K}_{(d)}^{\alpha}$ (i.e. take $\rho =
  \alpha$). Corollary~\ref{c:i^H-map}, applied with $p=0$, implies
  that $i^H$ maps $H^n(\mathcal{K}_{(d)}^{\alpha})$ to
  $F^1 H^n(\mathcal{K}_{(d)}^{\alpha+\mathcal{K}^{\mathcal{M}}})$ for
  all $n$. We now apply the Corollary~\ref{c:i^H-map} again, this time
  with $p=1$, $\rho = \alpha + \mathcal{K}^{\mathcal{M}}$ and the
  inclusion
  $\mathcal{K}_{(d)}^{\alpha+\mathcal{K}^{\mathcal{M}}}
  \longrightarrow
  \mathcal{K}_{(d)}^{\alpha+2\mathcal{K}^{\mathcal{M}}}$ and together
  with the previous conclusion infer that $i^H$ maps
  $H^n(\mathcal{K}_{(d)}^{\alpha})$ to
  $F^2 H^n(\mathcal{K}_{(d)}^{\alpha+2\mathcal{K}^{\mathcal{M}}})$ for
  all $n$. (By a slight abuse of notation we have denoted here by the
  same symbol, $i^H$, the maps induced in homology by the following
  different inclusions:
  $\mathcal{K}_{(d)}^{\rho} \longrightarrow
  \mathcal{K}_{(d)}^{\rho+\kappa}$,
  $\mathcal{K}_{(d)}^{\rho+\mathcal{K}^{\mathcal{M}}} \longrightarrow
  \mathcal{K}_{(d)}^{\rho+2\kappa}$ and
  $\mathcal{K}_{(d)}^{\rho} \longrightarrow
  \mathcal{K}_{(d)}^{\rho+2\kappa}$. Below we will continue with this
  notation.)

  Applying the same argument over and over again, $\ell$ times, we
  conclude that the map
  $i^H: H^n(\mathcal{K}_{(d)}^{\alpha}) \longrightarrow
  H^n(\mathcal{K}_{(d)}^{\alpha+ \ell \kappa})$ induced by the
  inclusion
  $\mathcal{K}_{(d)}^{\alpha} \longrightarrow
  \mathcal{K}_{(d)}^{\alpha+ \ell \kappa}$ maps
  $H^n(\mathcal{K}_{(d)}^{\alpha})$ to
  $F^{\ell}H^n(\mathcal{K}_{(d)}^{\alpha+\ell \kappa})$.

  Let now $\beta \in \mathcal{H}^{\leq \alpha}$ be a cycle and denote
  by $\overline{\beta}$ its image in
  $\mathcal{H}_{(d)}^{\leq \alpha'}$, where
  $\alpha' = \alpha + \ell \kappa$. Consider the cycle
  $(0, \overline{\beta}) \in \mathcal{K}_{(d)}^{\alpha'}$. By what we
  have proved before we have
  $[(0, \overline{\beta})] \in F^{\ell}
  H^*(\mathcal{K}_{(d)}^{\alpha'})$.

  It follows that there exists
  $\tau' \in \hom^{\leq \alpha'; \bmeh}(\mathscr{Y}, \mathcal{M})$
  with $\tau'_1 = \cdots = \tau'_{\ell} = 0$ such that
  $[(0, \overline{\beta})] = [(0, \tau')]$ in
  $H^*(\mathcal{K}_{(d)}^{\alpha'})$. Therefore, there exist
  $\gamma \in \mathcal{M}^{\leq \alpha'}(Y)$ and
  $\theta \in \hom^{\leq \alpha'; \bmeh}(\mathscr{Y}, \mathcal{M})$
  such that
  $$(0, \overline{\beta}) = (0, \tau') + 
  \bigl( \mu_1^{\mathcal{M}}(\gamma),
  \lambda_{(d)}^{\alpha'}(\gamma)+\mu_1^{\text{hom}}(\theta) \bigr)$$
  in $\mathcal{K}_{(d)}^{\alpha'}$.  In order to lift the last
  equation from $\mathcal{K}_{(d)}^{\alpha'}$ to
  $Cone \bigl( \mathcal{M}^{\leq \alpha'}(Y) \xrightarrow{\;
    \lambda^{\alpha'} \;} \mathcal{H}^{\leq \alpha'}\bigr)$ we can
  correct the terms beyond order $d$ if necessary by replacing $\tau'$
  with a suitable $\tau$ that coincides with $\tau'$ up to order $d$
  (recall that $d \gg \ell$).

  Summing up, we have proved that there exists a cycle
  $\gamma \in \mathcal{M}^{\leq \alpha'}(Y)$, and a pre-module
  homomorphism
  $\theta \in \hom^{\leq \alpha'; \bmeh}(\mathscr{Y}, \mathcal{M})$
  such that
  $$\beta = \lambda(\gamma) + \mu_1^{\text{mod}}(\theta) + \tau,$$
  where
  $\tau \in \hom^{\leq \alpha'; \bmeh}(\mathscr{Y}, \mathcal{M})$ is a
  cycle with $\tau_1 = \cdots = \tau_{\ell} = 0$, as claimed by the
  Proposition. (The statement in the Proposition concerning
  diagram~\eqref{eq:H-M} is a rephrasing of what we have just proved.)

  This concludes the proof of Proposition~\ref{p:lmap}, modulo the
  proof of Claim~\ref{cl:homotopic} which we will carry out next.
\end{proof}

\begin{proof}[Proof of Claim~\ref{cl:homotopic}]
  Fix $q$. We define the chain homotopy $S^{\bullet, q}$ as
  follows. Define $S^{0,q}=0$. (Note that $E_1^{-1,q}(\rho) = 0$.)
  Next, to define $S^{1,q}$, let
  $$f \in E_0^{1,q}(\rho) = \prod_{X \in \textnormal{Ob}(\mathcal{A})} 
  \hom_R^{\leq \rho+\epsilon_1^h} \bigl( C(X,Y), \mathcal{M}(X)
  \bigr)$$ be a $\partial_0$-cycle. We define
  $$S^{1,q}[f] := [f(e_Y)] \in E_1^{0,q}(\rho + \kappa) = 
  H^q(\mathcal{M}^{\leq \rho + \epsilon_1^h + \kappa}(X)).$$ Since
  $\kappa \geq \ua$, $f(e_Y)$ indeed belongs to
  $E_0^{0,q}(\rho + \kappa)$. Moreover, a straightforward calculation
  shows that $f(e_Y)$ is a $\partial_0$-cycle and that its homology
  class $[f(e_Y)]$ depends only on the homology class
  $[f] \in E_1^{1,q}(\rho)$.

  for the range of degrees $2 \leq p \leq d$ we define $S^{p,q}$ by a
  similar formula: let $f \in E_0^{p,q}(\rho)$, i.e. a collection of
  $R$-linear homomorphism as in~\eqref{eq:E_0^p}. Assume that $f$ is a
  $\partial_0$-cycle. Define $S^{p,q}[f]$ to be the homology class
  $[g] \in E_1^{p-1,q}(\rho + \kappa)$ of the element
  $g \in E_0^{p-1,q}(\rho+\kappa)$ given by:
  \begin{equation*}
    \begin{aligned}
      & g(a_1, \ldots, a_{p-2}, b) = f(a_1, \ldots, a_{p-2}, b, e_Y), \\
      & \forall \; a_i \in C(X_{i-1}, X_i), \; i=1, \ldots, p-2,
      \quad b \in C(X_{p-2}, Y).
    \end{aligned}
  \end{equation*}
  Since $\kappa \geq \ua$, we have $g \in E_0^{p-1,q}(\rho+\kappa)$. A
  straightforward calculation shows that $g$ is a $\partial_0$-cycle and
  moreover its homology class, $[g] \in E_1^{p-1,q}(\rho + \kappa)$
  depends only on the homology class $[f]$ of $f$. This concludes the
  definition of the maps $S^{p,q}$. (For $p$'s outside of the range
  $0,\ldots, d$ we can define $S^{p,q}$ in an arbitrary way.)

  We verify now the identity~\eqref{eq:i_1-S_pq}. We begin with
  $2 \leq p \leq d-1$. Let $f \in E_0^{p,q}(\rho)$ be a cycle. A
  straightforward calculation shows that
  $(\partial_1 S^{p,q} + S^{p+1,q} \partial_1 )[f] = [\widetilde{f}],$
  where
  $$\widetilde{f}(a_1, \ldots, a_{p-1}, b) = f(a_1, \ldots, a_{p-1},
  \mu_2^{\mathcal{A}}(b, e_Y)).$$

  We claim that $[\widetilde{f}] = [f]$ in
  $E_1^{p,q}(\rho+\kappa)$. Indeed, since
  $Y \in \hyperlink{h:asmp-ur}{\lbur(\kappa)}$ there exists a chain
  homotopy $h_{X_{p-1}}: C(X_{p-1}, Y) \longrightarrow C(X_{p-1}, Y)$
  that shifts action by $\leq \kappa$ such that
  $$\mu_2(b, e_Y) = b + h_{X_{p-1}} \mu_1^{\mathcal{A}}(b) + 
  \mu_1^{\mathcal{A}} h_{X_{p-1}}(b), \; \; \forall \; b \in
  C(X_{p-1},Y).$$ Define $\psi \in E_0^{p,q-1}(\rho+\kappa)$ by
  $$\psi(a_1, \ldots, a_{p-1}, b) := f(a_1, \ldots, a_{p-1}, 
  h_{X_{p-1}}(b)).$$ A straightforward calculation shows that
  $\widetilde{f} - f = \partial_0 \psi$, hence $[\widetilde{f}] = [f]$
  in $E_1^{p,q}(\rho+\kappa)$. This proves~\eqref{eq:i_1-S_pq} for
  $2 \leq p \leq d$. A similar argument shows that~\eqref{eq:i_1-S_pq}
  holds also for $p=1$.

  It remains to verify~\eqref{eq:i_1-S_pq} the case $p=0$.  Let
  $m \in \mathcal{M}^{\leq \rho}(Y)$ be a cycle. We have:
  $$(\partial_1 S^{0,q} + S^{1,q} \partial_1)[m] = 
  S^{1,q} \partial_1[m) = (\partial_1[m])(e_Y) =
  [\mu_2^{\mathcal{M}}(e_Y, m)].$$ By assumption
  $\mathcal{M} \in \hyperlink{h:asmp-H}{\lbh_w(\kappa)}$, hence
  $[\mu_2^{\mathcal{M}}(e_Y, m)] = [m]$ in
  $H^q(\mathcal{M}^{\leq \rho + \kappa}(Y))$. This
  proves~\eqref{eq:i_1-S_pq} for $p=0$ and concludes the proof of
  Claim~\ref{cl:homotopic}. 
\end{proof}


\subsection{Structure theorem for weakly filtered iterated cones} \label{sb:wf-ic}

Let $\mathcal{A}$ be a an h-unital weakly filtered
$A_{\infty}$-category with discrepancy $\leq \bmea$ and discrepancy of
units $\ua$. Let $L_0, \ldots, L_r \in \textnormal{Ob}(\mathcal{A})$
and denote by $\mathcal{L}_i$ the Yoneda module associated to $L_i$,
viewed as weakly filtered modules. In this section we analyze iterated
cones in the weakly filtered framework. By iterated cones we mean
modules of the type:
\begin{equation} \label{eq:itcone}
  \tcn (\mathcal{L}_r \xrightarrow{\; \phi_r \;} \tcn(\mathcal{L}_{r-1} 
  \xrightarrow{\; \phi_{r-1} \;} \tcn( \cdots \tcn(\mathcal{L}_2
  \xrightarrow{\; \phi_2 \;} \tcn(\mathcal{L}_1 \xrightarrow{\; \phi_1
    \;} \mathcal{L}_0 )) {\cdot}{\cdot}{\cdot}))).
\end{equation}
The weakly filtered structure is defined by iterating the construction
from~\S\ref{sb:wf-mc}. More precisely, we define a sequence of weakly
filtered $\mathcal{A}$-modules $\mathcal{K}_0, \ldots, \mathcal{K}_r$
as follows. We start by setting $\mathcal{K}_0 := \mathcal{L}_0$ which
is a weakly filtered module with discrepancy
$\leq \bme^{\mathcal{K}_0} := \bmea$. Note that all the modules
$\mathcal{L}_i$ have discrepancy $\leq \bmea$ too. Suppose that
$\phi_1 \in \hom^{\leq \rho_1; \bm{\delta}^{\phi_1}}(\mathcal{L}_1,
\mathcal{K}_0)$ is a module homomorphism, where
$\rho_1 \in \mathbb{R}$ and $\bm{\delta}^{\phi_1}$ is some
sequence. We {\em do not} assume that $\bm{\delta}^{\phi_1}$ satisfies
anything like Assumption~$\assmpen$. We define
$\mathcal{K}_1 = \tcn(\mathcal{L}_1 \xrightarrow{\; (\phi_1; \rho_1,
  \bmd^{\phi_1}) \;} \mathcal{K}_0)$. Since
$\bme^{\mathcal{K}_0} = \bmea$, the discrepancy of $\mathcal{K}_1$ is
$\leq \bme^{\mathcal{K}_1} := \max \{\bmea,
\bmd^{\phi_1}-\delta^{\phi_1}_1\}$.

Let $i\geq 1$ and suppose that we have already defined the weakly
filtered modules $\mathcal{K}_0, \ldots, \mathcal{K}_i$. Let
$\phi_{i+1} : \mathcal{L}_{i+1} \longrightarrow \mathcal{K}_i$ be a
module homomorphism that shifts action by $\leq \rho_{i+1}$ and has
discrepancy $\leq \bmd^{\phi_{i+1}}$. Again, we {\em do not} assume
that $\bmd^{\phi_{i+1}}$ satisfies any assumption of the
type~$\assmpen$. We define
$\mathcal{K}_{i+1} = \tcn( \mathcal{L}_{i+1} \xrightarrow{\;
  (\phi_{i+1}; \rho_{i+1}, \bmd^{\phi_{i+1}})\;} \mathcal{K}_i)$. The
$\mathcal{A}$-module $\mathcal{K}_{i+1}$ has discrepancy
$\leq \bme^{\mathcal{K}_{i+1}} := \max\{ \bme^{\mathcal{K}_i},
\bmd^{\phi_{i+1}}-\delta^{\phi_{i+1}}_1 \}$ because (by induction)
$\bme^{\mathcal{K}_i} \geq \bmea$.  The final $\mathcal{A}$-module
$\mathcal{K}_r$ is precisely the one described by~\eqref{eq:itcone}
and moreover now it also has the structure of a weakly filtered
module.

The following expressions will be used frequently in what follows:
\begin{equation} \label{eq:chi-xi}
  \begin{aligned}
    \chi_{m,d} & := \sum_{j=1}^m \sum_{i=1}^{d+m} \delta_i^{\phi_j} +
    \sum_{i=1}^{d+m} \epsilon_i^{\mathcal{A}},\\
    \xi_q & := \kappa + \sum_{i=1}^{q+3} \epsilon_i^{\mathcal{A}} +
    \sum_{j=1}^q \sum_{i=1}^{q+2} \delta_i^{\phi_j}.
  \end{aligned}
\end{equation}

\begin{thm} \label{t:itcones} Let $\mathcal{K}_i$, $0 \leq i \leq r$
  be as above. Assume that $\mathcal{A}$ is h-unital in the weakly
  filtered sense with discrepancy of units $\leq u^{\mathcal{A}}$. Let
  $\kappa \geq 2\ua + \epsilon_2^{\mathcal{A}}$ be a real number and
  assume that $\mathcal{A}$ and the objects $L_i$ satisfy the
  following two conditions:
  \begin{enumerate}  
  \item $\mathcal{A} \in \hyperlink{h:asmp-ue}{\lbue(\kappa)}$.
  \item For every $0 \leq i \leq r$,
    $L_i \in \hyperlink{h:asmp-ur}{\lbur(\kappa)}$, and
    $L_i \in \hyperlink{h:asmp-H}{\lbh_w(\kappa)}$.
  \end{enumerate}

  Then there exists a weakly filtered $\mathcal{A}$-module
  $\mathcal{M}$ with the following properties:
  \begin{enumerate}
  \item For every $X \in \textnormal{Ob}(\mathcal{A})$, we have
    $\mathcal{M}(X) = \mathcal{K}_r(X)$ as $R$-modules, namely the
    $R$-module $\mathcal{M}(X)$ is a direct sum:
    \begin{equation}
      \label{eq:M-direct-sum}
      \mathcal{M}(X) = C(X, L_0) \oplus C(X, L_1) \oplus \cdots 
      \oplus C(X, L_r).
    \end{equation}
  \item Denote by $\mu_1^{\mathcal{M}}$ the differential of the chain
    complex $\mathcal{M}(X)$. Then the matrix of $\mu_1^{\mathcal{M}}$
    with respect to the splitting~\eqref{eq:M-direct-sum} has the
    following shape:
    $\mu_1^{\mathcal{M}} = (a_{ij})_{0\leq i, j \leq r}$ with
    $a_{i,j}: C(X,L_j) \longrightarrow C(X,L_i)$, where:
    \begin{enumerate}
    \item $a_{i,j}=0$ for every $i>j$. In other words, the matrix of
      $\mu_1^{\mathcal{M}}$ is upper triangular.
    \item
      $a_{i,i} = \mu^{\mathcal{A}}_1: C(X,L_i) \longrightarrow
      C(X,L_i)$.
    \item There exist elements $c_{q,p} \in C(L_q,L_p)$ for all
      $0\leq p < q \leq r$, such that for for every $i<j$ the
      $(i,j)$'th entry of the matrix of $\mu_1^{\mathcal{M}}$ is given
      by:
      \begin{equation} \label{eq:aij} a_{i,j}(-) = \sum_{2 \leq d,\,
          \underline{k}} \mu^{\mathcal{A}}_{d}(-, c_{k_{d}, k_{d-1}},
        \ldots, c_{k_2,k_1}),
      \end{equation}
      where $\underline{k}=(k_1, \ldots, k_d)$ runs over all
      partitions $i=k_1 < k_2 < \cdots < k_{d-1} < k_d=j$. (Note that
      the sum in~\eqref{eq:aij} is finite. In fact, the index $d$ in
      this sum cannot exceed $j-i$ which in turn is $\leq r$.)
    \item $c_{q,p} \in C^{\leq \alpha_{q,p}}(L_q,L_p)$, where
      \begin{equation} \label{eq:alpha_qp}
        \alpha_{q,p} = \rho_q - \rho_p + B_q \xi_q,
      \end{equation}
      where $B_q$ is a universal constant in the sense that it depends
      only on $q$, but not on $\mathcal{A}$, the modules
      $\mathcal{K}_i$ or their discrepancy
      data. (In~\eqref{eq:alpha_qp} and in what follows we use the
      convention that $\rho_0 = 0$.)
      \label{i:univ-const-E}
    \end{enumerate}
  \item There exists a quasi-isomorphism of $\mathcal{A}$-modules
    $\sigma: \mathcal{K}_r \longrightarrow \mathcal{M}$ which shifts
    action by $\leq \rho^{\sigma}$ and has discrepancy
    $\leq \bme^{\sigma}$. The latter quantities admit the following
    estimates:
    \begin{equation} \label{eq:rho-sigma} \rho^{\sigma} \leq C_r
      \xi_r, \quad \epsilon_d^{\sigma} \leq D_{r,d} \chi_{r,d},
    \end{equation}
    where the constants $C_r$ and $\{D_{r,d}\}_{d \in \mathbb{N}}$ are
    universal in the sense mentioned at point~\eqref{i:univ-const-E}
    above.
  \item The 1'st order part
    $\sigma_1: \mathcal{K}_r(X) \longrightarrow \mathcal{M}(X)$ of the
    quasi-isomorphism $\sigma$ is an isomorphism of chain complexes
    for all $X \in \textnormal{Ob}(\mathcal{A})$, and the matrix
    corresponding to $\sigma_1$ with respect to the
    splitting~\eqref{eq:M-direct-sum} (taken both for
    $\mathcal{K}_r(X)$ and $\mathcal{M}(X)$) is upper triangular with
    $\id$-maps along its diagonal. \label{pp:sigma-1-iso}
  \item The inverse
    $\sigma_1^{-1}:\mathcal{M}(X) \longrightarrow \mathcal{K}_r(X)$ of
    $\sigma_1$ is action preserving (i.e. it is filtered and shifts
    action by $\leq 0$).
    \label{i:sigma_1-inv}
  \item For every $0 \leq j \leq r$ the diagonal element
    $$\Delta_j = \textnormal{pr}_{C(X,L_j)} 
    \circ \sigma_1|_{C(X,L_j)}: C(X,L_j) \longrightarrow C(X,L_j)$$ is
    the identity map (as follows from point~\eqref{pp:sigma-1-iso}
    above). However, when the domain inherits filtration from
    $\mathcal{K}_r(X)$ and the target from $\mathcal{M}(X)$ this map
    shifts action by $\leq \rho^{\sigma}$. (Note that for $j\geq 1$,
    $C(X,L_j)$ is in general not a subcomplex of either
    $\mathcal{K}_r(X)$ or of $\mathcal{M}(X)$). For $j=0$, $C(X,L_0)$
    is a subcomplex of both $\mathcal{K}_r(X)$ and of $\mathcal{M}(X)$
    and the two inherited filtrations on $C(X,L_0)$ coincide, hence
    $\Delta_0 = \id$ preserves filtration (i.e. shifts action by
    $\leq 0$).
    \label{pp:Delta_j-Delta_0}
  \end{enumerate}
\end{thm}

\input{proof-main-alg.tex}
{"config": "arxiv", "file": "1806.06630/wf-ainfty.tex"}