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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-4.23195pt}{-16.5041pt}\pgfsys@lineto{-31.3046pt}{2.20627pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.82265}{0.56854}{-0.56854}{-0.82265}{-31.46912pt}{2.31996pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.4815pt}{-16.888pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\langle
c_{2}\rangle[1]}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (51)

where $S^{-1}$ induces the quasi-inverse translation $T^{-1}=H^{0}(S^{-1})$ in
$H^{0}(\mathcal{A})$. With help of the isomorphisms (3.2), we identified
$\langle
c_{3}\rangle\\!\in\\!\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{2},Y_{0})[1]$
$\cong\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{2},SY_{0})$ and $\langle
S^{1}(c_{1})\rangle\in\textup{Hom}_{H^{0}(\mathcal{A})}(SY_{0},SY_{1})$
$\cong\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{0},Y_{1})\cong\textup{Hom}_{H^{0}(\mathcal{A})}(SY_{0},Y_{1})[1]$,
respectively $\langle(S^{-1})^{1}(c_{3})\rangle\in$
$\textup{Hom}_{H^{0}(\mathcal{A})}(S^{-1}Y_{2},S^{-1}Y_{0})[1]$
$\cong\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{2},Y_{0})[1]\cong\textup{Hom}_{H^{0}(\mathcal{A})}(S^{-1}Y_{2},Y_{0})$
and $\langle
c_{2}\rangle\in\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{1},Y_{2})\cong\textup{Hom}_{H^{0}(\mathcal{A})}(Y_{1},S^{-1}Y_{2})[1]$.

As we mentioned right after Definition 3.16, the vanishing of $c=c_{1}$ in
(36) implies that $\mathcal{Y}_{2}\cong\mathcal{C}one(c_{1})\cong
SS\mathcal{Y}_{0}\oplus\mathcal{Y}_{1}$ (in $H(\mathsf{Q})$). Using that $S$
quasi-represents $SS$, we get the following interesting implications (the last
two can be deduced analogously from the diagrams (51) and (37)):

$\langle c_{1}\rangle=0\implies Y_{2}\cong SY_{0}\oplus
Y_{1}\;,\qquad\qquad\langle c_{2}\rangle=0\implies Y_{0}\cong Y_{1}\oplus
S^{-1}Y_{2}\;,$ $\langle c_{3}\rangle=0\implies Y_{1}\cong Y_{2}\oplus
Y_{0}\;.$

Finally, we return to the concepts of triangulated envelope and derived
category of an $A_{\infty}$-category, proving their existence and uniqueness
up to equivalence.

######  Lemma 3.49.

Let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a cohomologically full
and faithful $A_{\infty}$-functor, $\mathcal{B}$ a triangulated c-unital
$A_{\infty}$-category generated by the image objects of $\mathcal{F}$. Then
there exists a quasi-equivalence
$\tilde{\mathcal{F}}:Tw\mathcal{A}\rightarrow\mathcal{B}$ such that
$\tilde{\mathcal{F}}|_{\mathcal{A}}\cong\mathcal{F}$ in
$H^{0}(fun(\mathcal{A},\mathcal{B}))$.

###### Proof.

By Lemma 3.46, $\mathcal{B}\hookrightarrow Tw\mathcal{B}$ is a quasi-
equivalence, for which we can then find a quasi-inverse
$\mathcal{K}:Tw\mathcal{B}\rightarrow\mathcal{B}$ (as in Theorem 2.31). Define
the $A_{\infty}$-functor $\tilde{\mathcal{F}}\coloneqq\mathcal{K}\circ
Tw\mathcal{F}:Tw\mathcal{A}\rightarrow\mathcal{B}$, clearly satisfying
$\tilde{\mathcal{F}}|_{\mathcal{A}}\cong\mathcal{K}\circ\mathcal{F}=\mathcal{F}$.
By Lemma 3.30, $Tw\mathcal{F}$ is cohomologically full and faithful, and hence
so is $\tilde{\mathcal{F}}$.

As to cohomological essential surjectivity: by assumption, any
$Y\in\textup{obj}(\mathcal{B})$ is isomorphic in
$\textup{obj}(H^{0}(\mathcal{B}))$ to an image object
$\mathcal{F}(X)=\tilde{\mathcal{F}}(X)$ for some
$X\in\textup{obj}(\mathcal{A})$, possibly after applying mapping cones and
shifts, which remain in the image of $\tilde{\mathcal{F}}$ because
$H(Tw\mathcal{F})$ preserves exact triangles (see Corollary 3.41). This
amounts to say that $\tilde{\mathcal{F}}$ is cohomologically essentially
surjective. Therefore, it is a quasi-equivalence, restricting to $\mathcal{F}$
as an object of $H^{0}(fun(\mathcal{A},\mathcal{B}))$. ∎

######  Proposition 3.50.

Every (non-empty) c-unital $A_{\infty}$-category $\mathcal{A}$ has a
triangulated envelope $(\tilde{\mathcal{B}},\mathcal{G})$. Given another such,
say $(\mathcal{B},\mathcal{F})$, there is a quasi-equivalence
$\tilde{\mathcal{F}}:\tilde{\mathcal{B}}\rightarrow\mathcal{B}$ such that
$\tilde{\mathcal{F}}\circ\mathcal{G}\cong\mathcal{F}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{A},\mathcal{B}))\big{)}$.

###### Proof.

The pair $(\tilde{\mathcal{B}},\mathcal{G})\coloneqq(Tw\mathcal{A},\imath)$ is
a valid choice of triangulated envelope for $\mathcal{A}$: $Tw\mathcal{A}$ is
triangulated, $\mathcal{A}\equiv\imath(\mathcal{A})$ generates it and the
embedding $\imath:\mathcal{A}\hookrightarrow Tw\mathcal{A}$ is cohomologically
full and faithful (by construction).

Moreover, given another envelope $(\mathcal{B},\mathcal{F})$, by Lemma 3.49
there exists a quasi-equivalence
$\tilde{\mathcal{F}}:Tw\mathcal{A}\rightarrow\mathcal{B}$ such that indeed
$\tilde{\mathcal{F}}|_{\mathcal{A}}=\tilde{\mathcal{F}}\circ\imath:\mathcal{A}\rightarrow\mathcal{B}$
is isomorphic to $\mathcal{F}$ in $H^{0}(fun(\mathcal{A},\mathcal{B}))$. ∎

######  Definition 3.51.

Let $\mathcal{A}$ be a c-unital $A_{\infty}$-category. We call the derived
category $\mathsf{D^{b}}(\mathcal{A})\coloneqq H^{0}(Tw\mathcal{A})$ the
bounded derived category of $\mathcal{A}$.151515This notation and terminology
is purely strategical: as mentioned in Definition 3.7, “bounded” just refers
to the fact that we started with bounded cochain complexes in the localization
construction, here sidestepped!

Putting together Corollary 3.32 and Proposition 3.39, one easily deduces (see
also [Sei13, Lemma 2.4]):

######  Corollary 3.52.

Let $\mathcal{A}$ and $\mathcal{B}$ be c-unital $A_{\infty}$-categories which
are quasi-equivalent. Then $\mathsf{D^{b}}(\mathcal{A})$ and
$\mathsf{D^{b}}(\mathcal{B})$ are equivalent triangulated categories.

####  3.7 Example of triangulated $A_{\infty}$-category: $mod(\mathcal{A})$

One prominent example of triangulated $A_{\infty}$-category is that of
c-unital $A_{\infty}$-modules over a c-unital $A_{\infty}$-category:
$\mathsf{Q}=mod(\mathcal{A})$. Let us briefly justify this, first by
generalizing the definition of abstract mapping cone.

######  Definition 3.53.

Let $\mathcal{M}_{0},\mathcal{M}_{1}\in\textup{obj}(\mathsf{Q})$, and let
$t\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{M}_{0},\mathcal{M}_{1})$ be a
degree 0 cocycle (that is, a module homomorphism). Its abstract mapping cone
$\mathcal{C}=\mathcal{C}one(t)\in\textup{obj}(\mathsf{Q})$ is given by
$\mathcal{C}(X)\coloneqq\mathcal{M}_{0}(X)[1]\oplus\mathcal{M}_{1}(X)\in\textup{obj}(\mathsf{Ch})$
for $X\in\textup{obj}(\mathcal{A})$, with graded maps

$\displaystyle\mu_{\mathcal{C}}^{d}:\;$
$\displaystyle\mathcal{C}(X_{d-1})\\!\otimes\\!\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1})\\!\otimes\\!...\\!\otimes\\!\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\mathcal{C}(X_{0})[2-d]$
$\displaystyle((b_{0},b_{1}),a_{d-1},...)\\!\mapsto\\!(\mu_{\mathcal{M}_{0}}^{d}(b_{0},a_{d-1},...),\mu_{\mathcal{M}_{1}}^{d}(b_{1},a_{d-1},...)\\!+\\!t^{d}(b_{0},a_{d-1},...)),$

for $d\geq 1$ and $X_{0},...,X_{d-1}\in\textup{obj}(\mathcal{A})$.

Upon setting $\mathcal{M}_{k}\coloneqq\Upsilon(Y_{k})=\mathcal{Y}_{k}$ and
$t\coloneqq\Upsilon^{1}(c)\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{Y}_{0},\mathcal{Y}_{1})$,
we recover Definition 3.16. Similarly, an adaptation of what we saw for
$\mathcal{A}$ and $Tw\mathcal{A}$ shows (cf. [Sei08, Lemma 3.35]):

######  Lemma 3.54.

Given a module homomorphism
$t\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{M}_{0},\mathcal{M}_{1})$, the
corresponding abstract mapping cone $\mathcal{C}=\mathcal{C}one(t)$ comes with
the canonical inclusion
$\iota\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{M}_{1},\mathcal{C})$ and
projection $\pi\in\textup{hom}_{\mathsf{Q}}^{1}(\mathcal{C},\mathcal{M}_{0})$
making the triangle

${\mathcal{M}_{0}}$${\mathcal{M}_{1}}$${\mathcal{C}}$$\scriptstyle{\langle
t\rangle}$$\scriptstyle{\langle\iota\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$

exact in $H^{0}(\mathsf{Q})$. In particular, any morphism $\langle t\rangle$
in $H^{0}(\mathsf{Q})$ can be completed to an exact triangle as above.

Existence of a shift functor $SS:\mathsf{Q}\rightarrow\mathsf{Q}$ which is in
particular a quasi-equivalence was already proved in Section 3.2. Together
with last lemma, this shows that $\mathsf{Q}=mod(\mathcal{A})$ is a
triangulated $A_{\infty}$-category.

######  Remark 3.55.

Considering
$\mathcal{F}=\Upsilon_{\mathcal{A}}:\mathcal{A}\rightarrow\tilde{\mathsf{Q}}$
(cohomologically full and faithful), where
$\tilde{\mathsf{Q}}\subset\mathsf{Q}$ is the triangulated full
$A_{\infty}$-subcategory generated by the image objects of
$\Upsilon_{\mathcal{A}}$, we obtain another canonical triangulated envelope
for $\mathcal{A}$. Indeed, by Proposition 3.50, there exists a quasi-
equivalence $\tilde{\Upsilon}:Tw\mathcal{A}\rightarrow\tilde{\mathsf{Q}}$ such
that $\tilde{\Upsilon}\circ\imath\cong\Upsilon_{\mathcal{A}}$ in
$H^{0}(fun(\mathcal{A},\tilde{\mathsf{Q}}))$. One can prove it to be exactly
$\tilde{\Upsilon}=\imath^{*}\circ\Upsilon_{Tw\mathcal{A}}$ (with pullback
$\imath^{*}:mod(Tw\mathcal{A})\rightarrow mod(\mathcal{A})$ as in Definition
2.37). We obtain the commutative (up to isomorphism) diagram

$\leavevmode\hbox to102.39pt{\vbox
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.05554pt\hfil&\hfil\hskip 39.38597pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.386pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
7.08333pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.08333pt\hfil&\hfil\hskip 52.69098pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-40.691pt}{4.22366pt}\pgfsys@lineto{-40.691pt}{-13.2542pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\tilde{\Upsilon}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-8.83548pt}{-22.2361pt}\pgfsys@lineto{-33.0077pt}{-22.2361pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.25015pt}{-28.5333pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\imath^{*}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad,$

which is precisely the one given in Remark 2.38. (See [Kel01, section 7.5],
where the author suggests this can be used to characterize $Tw\mathcal{A}$ by
a universal property.)

###  4 Split-generation and twisting

####  4.1 Category theoretic preliminaries

######  Definition 4.1.

Let $\mathsf{C}$ be a linear category over $\mathbb{K}$. Consider an
idempotent endomorphism $p$ of some $Y\in\textup{obj}(\mathsf{C})$, that is, a
morphism $p\in\textup{Hom}_{\mathsf{C}}(Y,Y)$ such that $p\circ p=p$. A triple
$(Z,r,k)$, where $Z\in\textup{obj}(\mathsf{C})$,
$r\in\textup{Hom}_{\mathsf{C}}(Y,Z)$ (retraction) and
$k\in\textup{Hom}_{\mathsf{C}}(Z,Y)$ (inclusion), is called the image of $p$
if fulfilling

$r\circ k=\textup{id}_{Z}\qquad\text{and}\qquad k\circ r=p\,.$

Accordingly, we say that $p$ splits if such a triple exists. We encode the
data in the following diagram:

${Z}$${Y\\!}$$\scriptstyle{k}$$\scriptstyle{\textup{id}_{Z}}$$\scriptstyle{r}$$\scriptstyle{p}$
(52)

The category $\mathsf{C}$ is split-closed if all its idempotent endomorphisms
split.

Note that the image of an idempotent $p$, if it exists, is unique: given
another suitable triple $(Z^{\prime},r^{\prime},k^{\prime})$, $r^{\prime}\circ
k:Z\rightarrow Z^{\prime}$ is an isomorphism of objects with inverse $r\circ
k^{\prime}$.

Observe also that if $\mathsf{C}$ is additive (for example, triangulated) and
split-closed, given an idempotent $p\in\textup{Hom}_{\mathsf{C}}(Y,Y)$, we may
consider the pair $Z\coloneqq\text{im}(p)$ and
$Z^{\perp}\coloneqq\text{im}(\textup{id}_{Y}-p)$, yielding the decomposition
$Y=Z\oplus Z^{\perp}$.

######  Definition 4.2.

A split-closure of a category $\mathsf{C}$ is a pair $(\mathsf{D},\mathsf{F})$
where $\mathsf{D}$ is a split-closed category and
$\mathsf{F}:\mathsf{C}\rightarrow\mathsf{D}$ a fully faithful embedding such
that any $Z\in\textup{obj}(\mathsf{D})$ is the image of an idempotent
endomorphism $p\in\textup{Hom}_{\mathsf{D}}(\mathsf{F}(X),\mathsf{F}(X))$ for
some $X\in\textup{obj}(\mathsf{C})$.

######  Remark 4.3.

Any category $\mathsf{C}$ has a tautological split-closure $\bar{\mathsf{C}}$,
called Karoubi completion (or idempotent completion). It is defined as
follows: objects are pairs $(Y,p)$ for $Y\in\textup{obj}(\mathsf{C})$ and
$p\in\textup{Hom}_{\mathsf{C}}(Y,Y)$ an idempotent endomorphism (every object
of $\mathsf{C}$ has at least one such, namely $\textup{id}_{Y}$), and
morphisms in $\textup{Hom}_{\bar{\mathsf{C}}}((Y_{0},p_{0}),(Y_{1},p_{1}))$
$\coloneqq p_{1}\circ\textup{Hom}_{\mathsf{C}}(Y_{0},Y_{1})\circ p_{0}$ are
triples $(p_{0},f,p_{1})\coloneqq p_{1}\circ f\circ p_{0}$ for
$f\in\textup{Hom}_{\mathsf{C}}(Y_{0},Y_{1})$ fulfilling $f=p_{1}\circ f\circ
p_{0}$. Compositions is the obvious one induced by $\mathsf{C}$, and identity
morphisms are all $\textup{id}_{(Y,p)}\coloneqq(p,p,p)=p$. The full and
faithful embedding is $\mathsf{E}:\mathsf{C}\rightarrow\bar{\mathsf{C}}$,
$Y\mapsto(Y,\textup{id}_{Y})$ and
$p\mapsto(\textup{id}_{Y},p,\textup{id}_{Y})$.

Indeed, $\bar{\mathsf{C}}$ is split-closed: for any idempotent
$\bar{p}=(p,s,p)\\!\in\\!\textup{Hom}_{\bar{\mathsf{C}}}((Y,p),(Y,p))$ (where
$s\in\textup{Hom}_{\mathsf{C}}(Y,Y)$ necessarily fulfills $s=p\circ s\circ p$
and $s=s\circ p\circ s$), $Z\coloneqq(Y,\textup{id}_{Y})$,
$r\coloneqq(p,s,\textup{id}_{Y}):(Y,p)\rightarrow Z$ and
$k\coloneqq(\textup{id}_{Y},s,p):Z\rightarrow(Y,p)$ fulfill $r\circ
k=\textup{id}_{Y}\circ s\circ\textup{id}_{Y}=\mathsf{E}(s)$ and $k\circ
r=p\circ s\circ p=\bar{p}$.

In fact, one can prove that any split-closure $(\mathsf{D},\mathsf{F})$ of
$\mathsf{C}$ is equivalent to the Karoubi completion
$(\bar{\mathsf{C}},\mathsf{E})$. Moreover, if $\mathsf{C}$ is triangulated, so
is $\bar{\mathsf{C}}$, along with the embedding $\mathsf{E}$. Here is a more
specific result.

######  Lemma 4.4.

Let $\mathsf{B}=(\mathsf{B},T)$ be a split-closed triangulated category,
$\mathsf{A}\subset\mathsf{B}$ a triangulated full subcategory (thus closed
under translations and exact triangles). Let
$\tilde{\mathsf{A}}\subset\mathsf{B}$ be the full subcategory of all objects
isomorphic to a direct summand of an object in $\mathsf{A}$ (particularly,
$\tilde{\mathsf{A}}\cong\bar{\mathsf{A}}$ because the latter is split-closed).
Then $\tilde{\mathsf{A}}$ is triangulated.

###### Proof.

Firstly, note that we are working with additive categories, so that it makes
sense to talk about direct sum decompositions (cf. above). By assumption, any
$\tilde{Y}_{i}\in\textup{obj}(\tilde{\mathsf{A}})$ appears in some
$Y\cong\tilde{Y}_{i}\oplus\tilde{Y}_{i}^{\perp}\in\textup{obj}(\mathsf{A})$.
Hence, we can define $r_{i},k_{i}$ to be the corresponding retraction and
inclusion, so that $r_{i}\circ k_{j}=\textup{id}_{\tilde{Y}_{i}}$.

By axiom (T1) for triangulated categories (see Definition 3.2), we can
complete the morphisms $\tilde{c}:\tilde{Y}_{0}\rightarrow\tilde{Y}_{1}$ and
$c\coloneqq k_{1}\circ\tilde{c}\circ r_{0}:Y_{0}\rightarrow Y_{1}$ to exact
triangles, involving some third objects $\tilde{Y}_{2}$ respectively $Y_{2}$.
These form two diagrams as in Lemma 3.3 (one with inclusions $k_{0},k_{1}$ as
vertical arrows, the other with retractions $r_{0},r_{1}$), whose leftmost
squares commute, by definition of $c$. Therefore, we can find complementing
morphisms $k_{2}:\tilde{Y}_{2}\rightarrow Y_{2}$ and
$r_{2}:Y_{2}\rightarrow\tilde{Y}_{2}$ making the whole diagrams commute.
Composing them, we obtain:

${\tilde{Y}_{0}}$${\tilde{Y}_{1}}$${\tilde{Y}_{2}}$${T(\tilde{Y}_{0})}$${\tilde{Y}_{0}}$${\tilde{Y}_{1}}$${\tilde{Y}_{2}}$${T(\tilde{Y}_{0})}$$\scriptstyle{r_{0}\circ
k_{0}}$$\scriptstyle{\tilde{c}}$$\scriptstyle{r_{1}\circ
k_{1}}$$\scriptstyle{r_{2}\circ k_{2}}$$\scriptstyle{T(r_{0}\circ
k_{0})}$$\scriptstyle{\tilde{c}}$

Since the first two vertical arrows are isomorphisms (being identity
morphisms), a second application of Lemma 3.3 tells us that $r_{2}\circ k_{2}$
is an isomorphim as well, thus invertible, and moreover commutativity of the
diagram forces it to be the identity $\textup{id}_{\tilde{Y}_{2}}$. Hence,
taking as endomorphism $p\coloneqq k_{2}\circ
r_{2}\in\textup{Hom}_{\mathsf{A}}(Y_{2},Y_{2})$, which is clearly idempotent,
we see that it splits via $(\tilde{Y}_{2},r_{2},k_{2})$, making
$\tilde{Y}_{2}$ isomorphic to a direct summand of $Y_{2}$. By definition, this
means that $\tilde{Y}_{2}\in\textup{obj}(\tilde{\mathsf{A}})$, so that
$\tilde{\mathsf{A}}$ wholly contains the exact triangle pictured above.
Therefore, $\tilde{\mathsf{A}}$ is a triangulated category. ∎

####  4.2 Abstract images

Let us translate the theory we have just seen in the language of
$A_{\infty}$-modules. Thereto, identify our ground field $\mathbb{K}$ as the
trivial $A_{\infty}$-category with structure
$\textup{obj}(\mathbb{K})\coloneqq\\{\ast\\}$ and
$\textup{hom}_{\mathbb{K}}(\ast,\ast)$ $\coloneqq\\{\textup{id}_{\ast}\\}$,
thus coinciding with its cohomological category,
$H(\mathbb{K})\cong\mathbb{K}$.

######  Definition 4.5.

Let $\mathcal{A}$ be an $A_{\infty}$-category, $\mathsf{Q}=mod(\mathcal{A})$.
An idempotent up to homotopy for $Y\in\textup{obj}(\mathcal{A})$ is a non-
unital $A_{\infty}$-functor $\wp:\mathbb{K}\rightarrow\mathcal{A}$ such that
$\wp(\ast)=Y$ and
$\wp^{d}\equiv\wp^{d}(\textup{id}_{\ast},...,\textup{id}_{\ast})\in\textup{hom}_{\mathcal{A}}^{1-d}(Y,Y)$
for $d\geq 1$, fulfilling the corresponding equations (10):

$\sum_{\begin{subarray}{c}1\leq r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-18.0mu\mu_{\mathcal{A}}^{r}(\wp^{s_{r}},...,\wp^{s_{1}})=\begin{cases}\wp^{d-1}&\text{if
}d\text{ is even}\\\ 0&\text{if }d\text{ is odd}\end{cases},$

in $\textup{hom}_{\mathcal{A}}^{2-d}(Y,Y)$.

One can show the following relation between idempotent endomorphisms and
idempotents up to homotopy (see [Sei08, Lemma 4.2]).

######  Lemma 4.6.

Let $p\in\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)$ be idempotent,
$Y\in\textup{obj}(\mathcal{A})$. Then there is an idempotent up to homotopy
$\wp:\mathbb{K}\rightarrow\mathcal{A}$ for $Y$ such that
$\langle\wp^{1}\rangle=p$.

We work towards the description of $\wp$ as image of some $A_{\infty}$-module,
similarly to what we did in previous section for idempotents $p$.

######  Definition 4.7.

Let $\wp:\mathbb{K}\rightarrow\mathcal{A}$ be an idempotent up to homotopy for
some $Y\in\textup{obj}(\mathcal{A})$. Then we define the abstract image of
$\wp$ to be the $A_{\infty}$-module $\mathcal{Z}\in\textup{obj}(\mathsf{Q})$
which to any $X\in\textup{obj}(\mathcal{A})$ associates
$\mathcal{Z}(X)\coloneqq\textup{hom}_{\mathcal{A}}(X,Y)[q]$, a graded vector
space of formal polynomials in one variable $q$ of degree $-1$. Writing
$\delta_{q}$ for the operator of formal differentiation, so that
$\delta_{q}(q^{0})=0$ and $\delta_{q}(q^{k})=q^{k-1}$, the composition maps
are

$\mu_{\mathcal{Z}}^{1}(b(q))\coloneqq\mkern-18.0mu\sum_{\begin{subarray}{c}r\geq
2\\\ s_{2},...,s_{r}\geq
1\end{subarray}}\mkern-18.0mu\delta_{q}^{s_{2}+...+s_{r}}\mu_{\mathcal{A}}^{r}(\wp^{s_{r}},...,\wp^{s_{2}},b(q))+(-1)^{|b|}\frac{b(q)\\!-\\!b(-q)}{2q}\in\mathcal{Z}(X_{0})[1],$
$\displaystyle\mu_{\mathcal{Z}}^{d}:\;$
$\displaystyle\mathcal{Z}(X_{d-1})\otimes\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\mathcal{Z}(X_{0})[2-d],$
$\displaystyle(b(q),a_{d-1},...,a_{1})\mapsto\mkern-24.0mu\sum_{\begin{subarray}{c}r\geq
d+1\\\ s_{d+1},...,s_{r}\geq
1\end{subarray}}\mkern-21.0mu\delta_{q}^{s_{d+1}+...+s_{r}}\mu_{\mathcal{A}}^{r}(\wp^{s_{r}},...,\wp^{s_{d+1}},b(q),a_{d-1},...,a_{1})\,,$

for $d>1$.

Recurring to filtrations of $\mathcal{Z}(X)$ and studying the associated
spectral sequences, one eventually obtains the following characterizations on
the level of cohomological modules (recall Definition 2.22).

######  Lemma 4.8.

Consider the abstract image $\mathcal{Z}$ of an idempotent up to homotopy
$\wp$ for some $Y\in\textup{obj}(\mathcal{A})$, and choose
$p\in\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)$ such that
$\langle\wp^{1}\rangle=p$ is idempotent (possible by Lemma 4.6).

  1. i.

The cohomological module
$H(\mathcal{Z})\in\textup{obj}\big{(}\mathsf{Mod}^{0}(H(\mathcal{A}))\big{)}$161616Since
we are implicitly assuming c-unitality of $\mathcal{A}$, indeed
$H(\mathcal{Z}):H(\mathcal{A})^{opp}\rightarrow\mathsf{GradVect}$ is a
contravariant unital functor, that is, an object of
$\mathsf{Mod}^{0}(H(\mathcal{A}))$ (cf. Definition 2.22). is described as
follows: $H(\mathcal{Z})(X)=H(\mathcal{Z}(X))\cong
p\circ\textup{Hom}_{H(\mathcal{A})}(X,Y)\in\textup{obj}(\mathsf{GradVect})$
for any $X\in\textup{obj}(\mathcal{A})$ (in particular, $H(\mathcal{Z})$ is a
direct summand of $H(\Upsilon(Y))$), with module multiplication

$H(\mathcal{Z})(X_{1})\otimes\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{1})\rightarrow
H(\mathcal{Z})(X_{0}),\,(p\circ\langle a_{1}\rangle,\langle
a_{0}\rangle)\mapsto p\circ(\langle a_{1}\rangle\circ\langle a_{0}\rangle).$

  2. ii.

There is an isomorphism
$\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Z},\mathcal{M})\cong
H(\mathcal{M})(Y)\circ p$ as graded groups for any
$\mathcal{M}\in\textup{obj}(\mathsf{Q})$ (which is natural with respect to
left composition with module homomorphisms). Together with the previous point,
it follows that $\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Z},\mathcal{Z})\cong
p\circ\textup{Hom}_{H(\mathcal{A})}(Y,Y)\circ p$.

######  Corollary 4.9.

Consider the abstract image $\mathcal{Z}$ of an idempotent up to homotopy
$\wp$ for some $Y\in\textup{obj}(\mathcal{A})$, and choose
$p\in\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)$ such that
$\langle\wp^{1}\rangle=p$is idempotent. Then $\mathcal{Z}$ is the image of
$\langle\Upsilon^{1}(\wp^{1})\rangle$ in $H^{0}(\mathsf{Q})$, in the sense of
Definition 4.1. In particular, $\mathcal{Z}$ only depends on the class $p$
(and not its representative $\wp^{1}$), up to isomorphism in
$H^{0}(\mathsf{Q})$.

###### Proof.

Writing $\mathcal{Y}=\Upsilon(Y)\in\textup{obj}(\mathsf{Q})$ for the Yoneda
module, observe that $p\in H(\mathcal{Y}(Y))=H(\mathcal{Y})(Y)$. We check that
the class
$\langle\Upsilon^{1}(\wp^{1})\rangle=H(\Upsilon)\langle\wp^{1}\rangle=H(\Upsilon)(p)\in\textup{Hom}_{H(\mathsf{Q})}^{0}(\mathcal{Y},\mathcal{Y})$
is an idempotent endomorphism of $\mathcal{Y}$ in $H^{0}(\mathsf{Q})$. This is
actually immediate from functoriality of $H(\Upsilon)$ and idempotency of $p$:
$H(\Upsilon)(p)\circ H(\Upsilon)(p)=H(\Upsilon)(p\circ p)=H(\Upsilon)(p)$.
Applying $H(\Upsilon)$ to the diagram (52), we obtain:

${\Upsilon(Z)}$${\mathcal{Y}\\!}$$\scriptstyle{\kappa}$$\scriptstyle{\textup{id}_{\Upsilon(Z)}}$$\scriptstyle{\rho}$$\scriptstyle{\langle\Upsilon^{1}(\wp^{1})\rangle}$
(53)

in $H^{0}(\mathsf{Q})$, where we set $\rho\coloneqq H(\Upsilon)(r)$ and
$\kappa\coloneqq H(\Upsilon)(k)$. Recall that $\textup{id}_{\Upsilon(Z)}$ is
induced by the strict unit
$e_{\Upsilon(Z)}\in\textup{hom}_{\mathsf{Q}}(\Upsilon(Z),\Upsilon(Z))$ as
described in Section 2.8. The two characterizing relations for images are
trivially fulfilled, again by functoriality of $H(\Upsilon)$.

It remains to show that $\mathcal{Z}\cong\Upsilon(Z)$ in $H^{0}(\mathsf{Q})$.
By Lemma 4.8, we have
$\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Z},\Upsilon(Z))\cong
H(\Upsilon(Z)(Y))\circ p=\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Z)\circ p$. Thus,
an isomorphism in the right-hand side suffices: one such is given by $r\circ
p$, with inverse $k\in\text{Hom}_{H(\mathcal{A})}^{0}(Z,Y)$. ∎

As a consequence, $H^{0}(\mathsf{Q})$ is a split-closed ordinary category: any
idempotent endomorphism in $H^{0}(\mathsf{Q})$ identifies an idempotent in
$H^{0}(\mathcal{A})$ (again, by bijectivity and functoriality of
$H(\Upsilon)$), whose image in $H^{0}(\mathsf{Q})$ Corollary 4.9 just proved
to split.

####  4.3 The split-closed derived category

######  Definition 4.10.

An $A_{\infty}$-category $\mathcal{A}$ is split-closed if the abstract image
$\mathcal{Z}$ of any idempotent up to homotopy is quasi-represented by some
$Y\in\textup{obj}(\mathcal{A})$, that is, if we can find an isomorphism
$\langle t\rangle:\Upsilon(Y)=\mathcal{Y}\xrightarrow{{}_{\sim}}\mathcal{Z}$
in $H^{0}(\mathsf{Q})$.

A split-closure of $\mathcal{A}$ is a pair $(\mathcal{B},\mathcal{F})$ where
$\mathcal{B}$ is a split-closed $A_{\infty}$-category and
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ a cohomologically full and
faithful $A_{\infty}$-functor such that for all
$Z\in\textup{obj}(\mathcal{B})$ there exists a $Y\in\textup{obj}(\mathcal{A})$
making $Z$ the image of some idempotent
$p\in\textup{Hom}_{H(\mathcal{B})}(\mathcal{F}(Y),\mathcal{F}(Y))$ in
$H^{0}(\mathcal{B})$ (in the classical sense).

######  Lemma 4.11.

An $A_{\infty}$-category $\mathcal{A}$ is split-closed if and only if
$H^{0}(\mathcal{A})$ is (in the classical sense).

###### Proof.

If $H^{0}(\mathcal{A})$ is split-closed and we are given an idempotent up to
homotopy $\wp:\mathbb{K}\rightarrow\mathcal{A}$ with
$\wp(\ast)=Y\in\textup{obj}(\mathcal{A})$, we first apply Lemma 4.6 to find an
idempotent $p=\langle\wp^{1}\rangle\in\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)$,
which splits, by assumption. Applying $H(\Upsilon)$ to the corresponding
diagram (52), we know that $Z$ is a valid quasi-representative for the
abstract image $\mathcal{Z}$ of $\wp$ (the necessary isomorphism $\langle
t\rangle\in\textup{Hom}_{H(\mathsf{Q})}(\Upsilon(Z),\mathcal{Z})$ is the one
constructed in Corollary 4.9). Therefore, $\mathcal{A}$ is split-closed.

Conversely, assume that $\mathcal{A}$ is split-closed and consider an
idempotent $p=\langle\wp^{1}\rangle\in\textup{Hom}_{H(\mathcal{A})}(Y,Y)$
(using again Lemma 4.6). By assumption, there exists some
$X\in\textup{obj}(\mathcal{A})$ whose Yoneda module
$\mathcal{X}\in\textup{obj}(\mathsf{Q})$ quasi-represents $\mathcal{Z}$ of
$\wp$. By Lemma 4.8,
$\textup{Hom}_{H(\mathsf{Q})}^{0}(\mathcal{Z},\mathcal{X})\cong
H^{0}(\mathcal{X})(Y)\circ p=\textup{Hom}_{H(\mathcal{A})}^{0}(Y,X)\circ p$,
hence we can find an isomorphism $r=\tilde{r}\circ p:Y\rightarrow X$ in
$H^{0}(\mathcal{A})$. Take $Z\coloneqq\text{im}(r)$ and $k\coloneqq p\circ
r^{-1}|_{Z}$, then we have $k\circ r=p$ and $r\circ k=\tilde{r}\circ p\circ
r^{-1}|_{Z}=\textup{id}_{Z}$. Therefore, any $p$ splits via such a triple
$(Z,r,k)$. This makes $H^{0}(\mathcal{A})$ split-closed. ∎

######  Corollary 4.12.

Let $\mathcal{A}$ and $\mathcal{B}$ be $A_{\infty}$-categories,
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ an $A_{\infty}$-functor. The
pair $(\mathcal{B},\mathcal{F})$ is a split-closure of $\mathcal{A}$ if and
only if $(H^{0}(\mathcal{B}),H^{0}(\mathcal{F}))$ is a split-closure of
$H^{0}(\mathcal{A})$.

Since $H^{0}(\mathsf{Q})$ is split-closed, we immediately deduce that
$\mathsf{Q}$ is a split-closed $A_{\infty}$-category. Therein lies the unique
split-closure of $\mathcal{A}$:

######  Proposition 4.13.

Every $A_{\infty}$-category $\mathcal{A}$ has a split-closure
$(\tilde{\mathcal{B}},\tilde{\mathcal{F}})$. Given another such, say
$(\mathcal{B},\mathcal{F})$, there is a quasi-equivalence
$\mathcal{G}:\mathcal{B}\rightarrow\tilde{\mathcal{B}}$ such that
$\mathcal{G}\circ\mathcal{F}\cong\tilde{\mathcal{F}}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{A},\tilde{\mathcal{B}}))\big{)}$.

###### Proof.

Let $\Pi\mathcal{A}\subset\mathsf{Q}$ be the full $A_{\infty}$-subcategory
whose objects are all those $A_{\infty}$-modules isomorphic to abstract images
of some idempotent up to homotopy:

$\textup{obj}(\Pi\mathcal{A})\coloneqq\\{\mathcal{M}\in\textup{obj}(\mathsf{Q})\mid\exists\wp\text{
with }\mathcal{Z}\cong\mathcal{M}\text{ in }H^{0}(\mathsf{Q})\\}\,.$

Notice that every Yoneda module
$\mathcal{Y}=\Upsilon(Y)\in\textup{obj}(\mathsf{Q})$ lies in $\Pi\mathcal{A}$:
by Lemma 4.6, there exists some $\wp$ for the idempotent endomorphism
$\textup{id}_{Y}$ such that
$\textup{id}_{Y}=\langle\wp^{1}\rangle\in\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)$,
and whose abstract image $\mathcal{Z}$ fulfills
$\textup{Hom}_{H(\mathcal{A})}^{0}(Y,Y)=H^{0}(\mathcal{Y})(Y)\circ\textup{id}_{Y}\cong\textup{Hom}_{H(\mathsf{Q})}^{0}(\mathcal{Z},\mathcal{Y})$
(by Lemma 4.8); under this isomorphism, $\textup{id}_{Y}$ is necessarily
mapped to an isomorphism $\mathcal{Z}\xrightarrow{\sim}\mathcal{Y}$ in
$H^{0}(\mathsf{Q})$, so that $\mathcal{Y}\in\textup{obj}(\Pi\mathcal{A})$, as
claimed.

By Corollary 4.12,
$(\tilde{\mathcal{B}},\tilde{\mathcal{F}})\coloneqq(\Pi\mathcal{A},\Upsilon)$
is a valid split-closure of $\mathcal{A}$ because
$(H^{0}(\Pi\mathcal{A}),H^{0}(\Upsilon))$ is for $H^{0}(\mathcal{A})$: any
idempotent endomorphism in $H^{0}(\Pi\mathcal{A})$ splits (argue as in Lemma
4.11) and $H^{0}(\Upsilon)$ is a full and faithful embedding such that any
$\mathcal{M}\cong\mathcal{Z}\in\textup{obj}(H^{0}(\Pi\mathcal{A}))$ is the
image of an idempotent endomorphism of the form
$\langle\Upsilon^{1}(\wp^{1})\rangle$ (as in diagram (53)).

Consider now another split-closure $(\mathcal{B},\mathcal{F})$ of
$\mathcal{A}$. By assumption, $\mathcal{F}$ is c-unital, thus admits a
(strictly unital) pullback functor
$\mathcal{F}^{*}:mod(\mathcal{B})\rightarrow\mathsf{Q}$ as in Definition 2.37.
Let
$\mathcal{G}\coloneqq\mathcal{F}^{*}\circ\Upsilon_{\mathcal{B}}:\mathcal{B}\rightarrow\Pi\mathcal{A}\subset\mathsf{Q}$.
Looking at the diagram of Remark 2.38, which commutes up to isomorphism, we
deduce that $\mathcal{G}\circ\mathcal{F}\cong\Upsilon_{\mathcal{A}}$ in
$H^{0}(fun(\mathcal{A},\Pi\mathcal{A}))$. In turn, this implies that
$\mathcal{G}$ is cohomologically full and faithful if restricted to the image
objects of $\mathcal{F}$ (since $\Upsilon_{\mathcal{A}}$ is), and thus on the
whole of $\mathcal{B}$ as well, by definition of split-closure. Just as we
argued above, we can prove that any object of $\Pi\mathcal{A}$ is isomorphic
to some image object of $\mathcal{G}$, providing essential surjectivity. This
makes $\mathcal{G}$ a quasi-equivalence, invertible by Theorem 2.19, and hence
identifying any two split-closures of $\mathcal{A}$ up to quasi-equivalence. ∎

######  Lemma 4.14.

The split-closure of a triangulated $A_{\infty}$-category $\mathcal{A}$ is
triangulated.

###### Proof.

In Section 3.7 we argued that $\mathsf{Q}$ is a triangulated
$A_{\infty}$-category, hence so is $H^{0}(\mathsf{Q})$ as an ordinary category
(by Proposition 3.47), also split-closed by Corollary 4.9. In Remark 3.55 we
identified $\mathcal{A}$ with $\tilde{\mathsf{Q}}\subset\mathsf{Q}$, the
triangulated full $A_{\infty}$-subcategory generated by all Yoneda modules
$\mathcal{Y}=\Upsilon(Y)$, then also objects of the triangulated full
subcategory $H^{0}(\mathcal{A})\subset H^{0}(\mathsf{Q})$. Moreover, by
definition, $H^{0}(\Pi\mathcal{A})\subset H^{0}(\mathsf{Q})$ is the full
subcategory of all objects isomorphic to a direct summand of some
$\mathcal{Y}\in\textup{obj}(H^{0}(\mathcal{A}))$ (by Lemma 4.8, since the
cohomological module $H(\mathcal{Z})$ of any abstract image appears in the
decomposition of some $H(\mathcal{Y})$). Applying Lemma 4.4, it follows that
$H^{0}(\Pi\mathcal{A})$ is a triangulated category as well.

But then we easily deduce that $\Pi\mathcal{A}$ itself is a triangulated
$A_{\infty}$-category: it is of course non-empty, any object is isomorphic to
a shifted one in $H^{0}(\Pi\mathcal{A})$ (because it is in
$H^{0}(\mathsf{Q})$, which contains $H^{0}(\Pi\mathcal{A})$ as full
subcategory) and any morphism in $H^{0}(\Pi\mathcal{A})$ can be completed to
an exact triangle (it can in $H^{0}(\mathsf{Q})$, and all its constituents
must then belong to the triangulated $H^{0}(\Pi\mathcal{A})$; consequently,
the preimage under $\Pi\mathcal{A}\hookrightarrow\mathsf{Q}$ of this exact
triangle is still exact in $H^{0}(\Pi\mathcal{A})$, by Corollary 3.44). ∎

######  Definition 4.15.

Let $\mathcal{B}$ be a split-closed triangulated $A_{\infty}$-category,
$\mathcal{A}\subset\mathcal{B}$ a full non-empty $A_{\infty}$-subcategory. The
smallest split-closed triangulated full $A_{\infty}$-subcategory
$\tilde{\mathcal{B}}\subset\mathcal{B}$ which is closed under isomorphism and
contains $\mathcal{A}$ is called the (triangulated) subcategory of
$\mathcal{B}$ split-generated by $\mathcal{A}$.

Specifically, $\textup{obj}(\tilde{\mathcal{B}})$ can be constructed from
$\textup{obj}(\mathcal{A})$ by forming all possible mapping cones and shifts,
iterating and finally taking images of all possible idempotent endomorphisms.

$\mathcal{A}$ split-generates $\mathcal{B}$ if
$\tilde{\mathcal{B}}=\mathcal{B}$ (then $\mathcal{B}$ is a split-closure of
$\mathcal{A}$; moreover, $\mathcal{A}$ also generates $\mathcal{B}$ in the
sense of Definition 3.22).

######  Definition 4.16.

Let $\mathcal{A}$ be an $A_{\infty}$-category and consider the split-closure
$(\Pi(Tw\mathcal{A}),\Upsilon_{Tw\mathcal{A}})$ of the standard triangulated
envelope $(Tw\mathcal{A},\imath)$ of $\mathcal{A}$ (by construction,
$\Pi(Tw\mathcal{A})$ is split-generated by $\mathcal{A}$). We call
$\mathsf{D^{\pi}}(\mathcal{A})\coloneqq H^{0}(\Pi(Tw\mathcal{A}))$ the split-
closed derived category of $\mathcal{A}$.

######  Corollary 4.17.

Let $\mathcal{B}$ be a split-closed triangulated $A_{\infty}$-category which
is split-generated by a full $A_{\infty}$-subcategory
$\mathcal{A}\subset\mathcal{B}$. Then there is a quasi-equivalence
$\Pi(Tw\mathcal{A})\rightarrow\mathcal{B}$, so that $H^{0}(\mathcal{B})$ is
equivalent to $\mathsf{D^{\pi}}(\mathcal{A})$ as a split-closed triangulated
category.

###### Proof.

Together with the inclusion $\mathcal{A}\hookrightarrow\mathcal{B}$,
$\mathcal{B}$ is a valid triangulated envelope and split-closure of
$\mathcal{A}$. The corresponding uniqueness statements (Propositions 3.50 and
4.13) yield the result. ∎

As a direct consequence of Corollary 3.52, one can show:

######  Corollary 4.18.

Let $\mathcal{A}$ and $\mathcal{B}$ be c-unital $A_{\infty}$-categories which
are quasi-equivalent. Then $\mathsf{D^{\pi}}(\mathcal{A})$ and
$\mathsf{D^{\pi}}(\mathcal{B})$ are equivalent split-closed triangulated
categories.

Combining last two corollaries, we obtain:

######  Corollary 4.19.

Let $\mathcal{A}$ and $\mathcal{B}$ be c-unital $A_{\infty}$-categories split-
generated by full $A_{\infty}$-subcategories
$\tilde{\mathcal{A}}\subset\mathcal{A}$ respectively
$\tilde{\mathcal{B}}\subset\mathcal{B}$ which are quasi-equivalent. Then
$\mathsf{D^{\pi}}(\mathcal{A})$ and $\mathsf{D^{\pi}}(\mathcal{B})$ are
equivalent split-closed triangulated categories.

####  4.4 Twisting

To round-off our discussion of triangulation and give a little more grounding
to the theory of split-generation — which we will eventually exploit in the
context of Fukaya categories — we take a step back to abstract mapping cones
for $A_{\infty}$-modules. However, we opt for a less detailed exposition. More
about twisting can be found in [Sei08, chapter 5].

######  Definition 4.20.

Let $\mathcal{A}$ be an $A_{\infty}$-category and
$Y\in\textup{obj}(\mathcal{A})$. We define the abstract twist functor
$\mathcal{T}_{Y}:\mathsf{Q}\rightarrow\mathsf{Q}$ along $Y$ to be the
$A_{\infty}$-functor mapping:

  * –

$\mathcal{M}\in\textup{obj}(\mathsf{Q})$ to
$\mathcal{C}=\mathcal{T}_{Y}(\mathcal{M})\in\textup{obj}(\mathsf{Q})$, the
$A_{\infty}$-module sending $X\in\textup{obj}(\mathcal{A})$ to
$\mathcal{C}(X)\coloneqq(\mathcal{M}(Y)\otimes\textup{hom}_{\mathcal{A}}(X,Y))[1]\oplus\mathcal{M}(X)\in\textup{obj}(\mathsf{GradVect})$,
with graded maps

$\mu_{\mathcal{C}}^{1}(c_{0}\otimes
b,c_{1})\coloneqq\big{(}(-1)^{|b|-1}\mu_{\mathcal{M}}^{1}(c_{0})\otimes
b+c_{0}\otimes\mu_{\mathcal{A}}^{1}(b),\mu_{\mathcal{M}}^{1}(c_{1})+\mu_{\mathcal{M}}^{2}(c_{0},b)\big{)}\;,$
$\displaystyle\mkern-36.0mu\mu_{\mathcal{C}}^{d}((c_{0}\otimes
b,c_{1}),a_{d-1},...)\coloneqq\big{(}c_{0}\otimes\mu_{\mathcal{A}}^{d}(b,a_{d-1},...),\,$
$\displaystyle\mu_{\mathcal{M}}^{d}(c_{1},a_{d-1},...)\,+$
$\displaystyle\mu_{\mathcal{M}}^{d+1}(c_{0},b,a_{d-1},...)\big{)}\;,$

for $d>1$.

  * –

$t\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$ to
$\tilde{t}=\mathcal{T}_{Y}^{1}(t)\in\textup{hom}_{\mathsf{Q}}(\mathcal{T}_{Y}(\mathcal{M}_{0}),\mathcal{T}_{Y}(\mathcal{M}_{1}))$
given by

$\displaystyle\tilde{t}^{1}(c_{0}\otimes
b,c_{1})\coloneqq\big{(}(-1)^{|b|-1}t^{1}(c_{0})\otimes
b,t^{1}(c_{1})+t^{2}(c_{0},b)\big{)}\;,$
$\displaystyle\tilde{t}^{d}((c_{0}\otimes
b,c_{1}),a_{d-1},...)\coloneqq\big{(}0,t^{d}(c_{1},a_{d-1},...)+t^{d+1}(c_{0},b,a_{d-1},...)\big{)}\;,$

for $d>1$, while $\mathcal{T}_{Y}^{d}\coloneqq 0$ for $d>1$.

Given two $Y_{0},Y_{1}\in\textup{obj}(\mathcal{A})$, we write
$T_{Y_{0}}(Y_{1})\in\textup{obj}(\mathcal{A})$ for any quasi-represent- ative
of $\mathcal{T}_{Y_{0}}(\mathcal{Y}_{1})$ and call it the twist of $Y_{1}$
along $Y_{0}$.

######  Definition 4.21.

Let $\mathcal{M},\,\mathcal{Y}=\Upsilon(Y)\in\textup{obj}(\mathsf{Q})$, for
$Y\in\textup{obj}(\mathsf{Q})$. We define theabstract evaluation morphism
$\epsilon\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{M}(Y)\otimes\mathcal{Y},\mathcal{M})$
as the module homomorphism given by $\epsilon^{d}(c_{0}\otimes
b,a_{d-1},...,a_{1})\coloneqq\mu_{\mathcal{M}}^{d+1}(c_{0},b,a_{d-1},...,a_{1})$.
Then $\mathcal{T}_{Y}(\mathcal{M})=\mathcal{C}one(\epsilon)$, as in Definition
3.53, fitting into the exact triangle

${\mathcal{M}(Y)\otimes\mathcal{Y}}$${\mathcal{M}}$${\mathcal{T}_{Y}\mathcal{M}}$$\scriptstyle{\langle\epsilon\rangle}$$\scriptstyle{\langle\iota\rangle}$$\scriptstyle{\langle\pi\rangle[1]}$

in $H^{0}(\mathsf{Q})$. Assuming $H(\mathcal{A})$ to be closed under finite
direct sums and shifts, and given two
$Y_{0},Y_{1}\in\textup{obj}(\mathcal{A})$ such that
$\textup{Hom}_{H(\mathcal{A})}(Y_{0},Y_{1})$ is finite-dimensional (so that we
may tensor by it as in Section 3.2), there is an obvious evaluation morphism
$\langle\textup{ev}\rangle:\textup{Hom}_{H(\mathcal{A})}(Y_{0},Y_{1})\otimes
Y_{0}\rightarrow Y_{1}$ such that
$H(\Upsilon)\langle\textup{ev}\rangle=\langle\epsilon\rangle$. If the twisting
$T_{Y_{0}}(Y_{1})$ exists, then the triangle

${\textup{Hom}_{H(\mathcal{A})}(Y_{0},Y_{1})\otimes
Y_{0}}$${Y_{1}}$${T_{Y_{0}}(Y_{1})}$$\scriptstyle{\langle\textup{ev}\rangle}$$\scriptstyle{\langle
i\rangle}$$\scriptstyle{\langle p\rangle[1]}$

maps to the preceding one under $H(\Upsilon)$ for the choice
$\mathcal{M}=\mathcal{Y}_{1}$, and is therefore exact (since preimage of an
exact triangle; see Corollary 3.44).

######  Remark 4.22.

  * •

If $\mathcal{A}$ is triangulated and $H(\mathcal{A})$ has finite-dimensional
Hom-spaces, $\langle\textup{ev}\rangle$ has standard mapping cone
$S(H(\mathcal{Y}_{1})(Y_{0})\otimes Y_{0})\oplus Y_{1}$. Using Lemma 3.3, with
vertical arrows equal to $H(\Upsilon)$, this cone maps to
$\mathcal{T}_{Y_{0}}(\mathcal{Y}_{1})$, always existing. Hence,
$T_{Y_{0}}(Y_{1})\coloneqq S(H(\mathcal{Y}_{1}(Y_{0}))\otimes Y_{0})\oplus
Y_{1}$ is well defined for all $Y_{0},Y_{1}\in\textup{obj}(\mathcal{A})$,
allowing us to determine the twist functor
$T_{Y}:\mathcal{A}\rightarrow\mathcal{A}$ as the $A_{\infty}$-functor making
the diagram

${\mathcal{A}}$${\mathcal{A}}$${\mathsf{Q}}$${\mathsf{Q}}$$\scriptstyle{T_{Y}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathcal{T}_{Y}}$

commute (up to isomorphism in $H^{0}(fun(\mathcal{A},\mathsf{Q}))$).

  * •

Recalling the map
$\tilde{\Upsilon}=\imath^{*}\circ\Upsilon_{Tw\mathcal{A}}:Tw\mathcal{A}\rightarrow\mathsf{Q}$
from Remark 3.55, the last diagram tells us — after operating the
substitutions $\mathcal{A}$←$Tw\mathcal{A}$ and $\Upsilon$←$\tilde{\Upsilon}$
— that
$\tilde{\Upsilon}(T_{Y_{0}}(Y_{1}))\cong\mathcal{T}_{Y_{0}}(\tilde{\Upsilon}(Y_{1}))\in\textup{obj}(H^{0}(\mathsf{Q}))$,
for any $Y_{0}\in\textup{obj}(\mathcal{A})$ and
$Y_{1}\in\textup{obj}(Tw\mathcal{A})$.

  * •

For $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ a cohomologically full and
faithful $A_{\infty}$-functor, and $Y_{0},Y_{1}\in\textup{obj}(\mathcal{A})$
such that both $T_{Y_{0}}(Y_{1})$ and
$T_{\mathcal{F}(Y_{0})}(\mathcal{F}(Y_{1}))$ exist, there is a canonical
isomorphism $\mathcal{F}(T_{Y_{0}}(Y_{1}))\cong
T_{\mathcal{F}(Y_{0})}(\mathcal{F}(Y_{1}))$ in $H^{0}(\mathcal{B})$, actually
extendible to $\mathcal{F}\circ T_{Y_{0}}\cong
T_{\mathcal{F}(Y_{0})}\circ\mathcal{F}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{A},\mathcal{B}))\big{)}$.

Here is a criterion which detects objects of $\mathcal{A}$ quasi-representing
a given abstract twisted module of $\mathsf{Q}$.

######  Lemma 4.23.

Let $\mathcal{A}$ be an $A_{\infty}$-category,
$Y_{0},Y_{1},Y_{2}\in\textup{obj}(\mathcal{A})$. Assume there are a cocycle
$c\in\textup{hom}_{\mathcal{A}}(Y_{1},Y_{2})$ and a map
$k:\textup{hom}_{\mathcal{A}}(Y_{0},Y_{1})\rightarrow\textup{hom}_{\mathcal{A}}(Y_{0},Y_{2})[-1]$
which satisfy
$\mu_{\mathcal{A}}^{1}(k(c_{0}))-k(\mu_{\mathcal{A}}^{1}(c_{0}))+\mu_{\mathcal{A}}^{2}(c,c_{0})=0$
for all $c_{0}\in\textup{hom}_{\mathcal{A}}(Y_{0},Y_{1})$. Then the module
homomorphism
$t:\mathcal{T}_{Y_{0}}(\mathcal{Y}_{1})\rightarrow\mathcal{Y}_{2}$ given by

$\displaystyle t^{d}((c_{0}\otimes b,c_{1}),a_{d-1},...,a_{1})\coloneqq\;$
$\displaystyle\mu_{\mathcal{A}}^{d+1}(c,c_{1},a_{d-1},...)+\mu_{\mathcal{A}}^{d+1}(k(c_{0}),b,a_{d-1},...)$
$\displaystyle\\!+\mu_{\mathcal{A}}^{d+2}(c,c_{0},b,a_{d-1},...)$

is well defined, fulfills
$\langle\mu_{\mathsf{Q}}^{2}(t,\iota)\rangle=\langle\Upsilon^{1}(c)\rangle\in\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Y}_{1},\mathcal{Y}_{2})$,
and — given suitable additional data (see [Sei08, Lemma 5.3]) — induces an
isomorphism $\mathcal{T}_{Y_{0}}(\mathcal{Y}_{1})$ $\cong\mathcal{Y}_{2}$ in
$H^{0}(\mathsf{Q})$.

Of course, the twist operation can be applied iteratively. Consider a
triangulated $A_{\infty}$-category $\mathcal{A}$ whose Hom-spaces are all
finite-dimensional. Choose a set
$\\{Y_{1},...,Y_{m}\\}\subset\textup{obj}(\mathcal{A})$ and, starting from
$X\coloneqq X_{m}\in\textup{obj}(\mathcal{A})$, define subsequently
$X_{k-1}\coloneqq T_{Y_{k}}(X_{k})\in\textup{obj}(\mathcal{A})$, which sit
into exact triangles like that of Definition 4.21 each. Using axiom (T3) for
triangulated categories, these can be composed to produce another exact
triangle of the form

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where $Y\in\text{A}$ is suitably constructed from the spaces
$\textup{Hom}_{H(\mathcal{A})}(Y_{k},X_{k})\otimes Y_{k}$, while $\langle
i_{k}\rangle:X_{k}\rightarrow X_{k-1}$ are canonical inclusions. As we
observed in Remark 3.48, the vanishing of $\langle
i_{1}\rangle\circ...\circ\langle i_{m}\rangle$ implies
$X\oplus(T_{Y_{1}}\circ...\circ T_{Y_{m}})(X)[-1]\cong Y$. One builds upon
this to prove the following result.

######  Corollary 4.24.

Let $\mathcal{A}$ be a triangulated $A_{\infty}$-category whose Hom-spaces are
all finite-dimensional.

  1. i.

Suppose there are $Y_{1},...,Y_{m}\in\textup{obj}(\mathcal{A})$ such that
$(T_{Y_{1}}\circ...\circ T_{Y_{m}})(X)\cong
0\in\textup{obj}(H^{0}(\mathcal{A}))$ (the zero object of
$H^{0}(\mathcal{A})$) for all $X\in\textup{obj}(\mathcal{A})$. Then
$\\{Y_{1},...,Y_{m}\\}$ generates $\mathcal{A}$, which is actually split-
closed.

  2. ii.

Suppose $\mathcal{A}$ is readily split-closed and there are
$Y_{1},...,Y_{m}\in\textup{obj}(\mathcal{A})$, $\sigma\neq 0$ such that
$(T_{Y_{1}}\circ...\circ T_{Y_{m}})(X)\cong
S^{\sigma}X\in\textup{obj}(H^{0}(\mathcal{A}))$ for all
$X\in\textup{obj}(\mathcal{A})$. Then $\\{Y_{1},...,Y_{m}\\}$ split-generates
$\mathcal{A}$.

######  Remark 4.25.

We can actually improve the description of twist functors upon considering
twisted complexes in $Tw\mathcal{A}$, where $\mathcal{A}$ is chosen strictly
unital.

Pick $Y_{0},Y_{1}\in\textup{obj}(Tw\mathcal{A})$ and consider
$(\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1}),\partial\\!\coloneqq\\!\mu_{Tw\mathcal{A}}^{1})$,
whose dual complex is
$\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})^{\vee}\coloneqq\textup{hom}_{Tw\mathcal{A}}(\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1}),Y_{1})$
with differential $\partial^{\vee}$ given by
$(\partial^{\vee}(a^{\vee}))(b)\coloneqq(-1)^{|a^{\vee}|+1}a^{\vee}(\partial(b))$,
for $a^{\vee}\in\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})^{\vee}$ and
$b\in\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})$. We can take the tensor
product as in Definition 3.33:

$\displaystyle\textup{hom}_{Tw\mathcal{A}}(\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})\otimes
Y_{0},Y_{1})$
$\displaystyle=\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})^{\vee}\otimes\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1})$
$\displaystyle\cong\text{End}_{\mathbb{K}}(\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1}))\;,$

which has differential $D:a^{\vee}\otimes b\mapsto a^{\vee}(\cdot)b$, and
$\textup{ev}\cong\textup{id}\in\text{End}_{\mathbb{K}}(\textup{hom}_{Tw\mathcal{A}}(Y_{0},Y_{1}))$
is a cocycle with respect to it. Therefore, we can look at the twisted mapping
cone $Cone(\textup{ev})$, turning under $H(\Upsilon)$ into
$\mathcal{C}one(\epsilon)=\mathcal{T}_{Y_{0}}(\mathcal{Y}_{1})$ (cf.
Definition 4.21). This motivates us to set

$T_{Y_{0}}(Y_{1})\coloneqq Cone(\textup{ev})=\bigoplus_{i\in
I}S^{-|b^{i}|+1}Y_{1}\oplus Y_{0}\;,$

where $\\{b^{i}\\}_{i\in I}$ is a basis of
$\textup{hom}_{\mathcal{A}}(Y_{0},Y_{1})$.

Finally, we mention the adjoint notion of the twist operation.

######  Definition 4.26.

Let $\mathcal{A}$ be an $A_{\infty}$-category,
$Y_{0},Y_{1}\in\textup{obj}(\mathcal{A})$. The $A_{\infty}$-module
$\mathcal{C}^{\vee}\in\textup{obj}(\mathsf{Q})$ is given by

$\displaystyle\mathcal{C}^{\vee}(X)\coloneqq\textup{hom}_{\mathcal{A}}(X,Y_{1})\oplus\textup{Hom}_{\mathbb{K}}(\textup{hom}_{\mathcal{A}}(Y_{1},Y_{0}),\textup{hom}_{\mathcal{A}}(X,Y_{0}))[-1]\;,$
$\displaystyle\mu_{\mathcal{C}^{\vee}}^{1}(c,\beta)\coloneqq(\mu_{\mathcal{A}}^{1}(c),\mu_{\mathcal{A}}^{1}\circ\beta+(-1)^{|\beta|-1}\beta\circ\mu_{\mathcal{A}}^{1}+\mu_{\mathcal{A}}^{2}(\cdot,c))\;,$
$\displaystyle\mu_{\mathcal{C}^{\vee}}^{d}((c,\beta),a_{d-1},...)\coloneqq(\mu_{\mathcal{A}}^{d}(c,a_{d-1},...),\mu_{\mathcal{A}}^{d}(\beta(\cdot),a_{d-1},...)+\mu_{\mathcal{A}}^{d+1}(\cdot,c,a_{d-1},...))\;.$

for $d>1,X\in\textup{obj}(\mathcal{A})$.

We write $T_{Y_{0}}^{\vee}(Y_{1})$ for any quasi-representative of
$\mathcal{C}^{\vee}$ and call it the adjoint twist of $Y_{1}$ along $Y_{0}$.
One can show that $T_{Y_{0}}(T_{Y_{0}}^{\vee}(Y))\cong Y$ in
$H^{0}(\mathcal{A})$ for all $Y\in\textup{obj}(\mathcal{A})$ (indeed,
$T_{Y_{0}}^{\vee}$ is actually an inverse of $T_{Y_{0}}$ in
$H^{0}(fun(\mathcal{A},\mathcal{A}))$, up to isomorphism).

As was the case for the normal twist, taking $\mathcal{A}$ to be triangulated
with finite-dimensional Hom-spaces makes $T_{Y_{0}}^{\vee}(Y_{1})$ always
well-defined, thus fitting into the exact triangle which results after
“dualization”:

${\textup{Hom}_{H(\mathcal{A})}(Y_{1},Y_{0})^{\vee}\otimes
Y_{0}}$${Y_{1}}$${T_{Y_{0}}^{\vee}(Y_{1})}$$\scriptstyle{\langle
p^{\vee}\rangle[1]}$$\scriptstyle{\langle\textup{ev}^{\vee}\rangle}$$\scriptstyle{\langle
i^{\vee}\rangle}$

Taking again $\mathcal{A}$ to be strictly unital, $T_{Y_{0}}^{\vee}(Y_{1})$
can be identified in $Tw\mathcal{A}$ as the mapping cone of the “dual”
evaluation morphism $\langle\textup{ev}^{\vee}\rangle$ (in parallel with last
remark).

##  Part II Fukaya Category Theory

Fukaya categories were first studied by Kenji Fukaya in his Morse homotopy,
$A_{\infty}$-category and Floer homologies [Fuk93] from 1993, one year prior
to Kontsevich’s exploit ([Kon94]), building on the original work Morse theory
for Lagrangian intersections [Flo88] by Andreas Floer. Fukaya successfully
packed a lot of geometric information within an algebraic structure, preparing
the ground for a richer interpretation of mirror symmetry.

Indeed, a Fukaya category $\mathscr{F}(M)$ can be associated to any (specific
enough) symplectic manifold $M$, its objects being suitably “decorated”
compact Lagrangian submanifolds, whose modes of intersection are dictated by
the $A_{\infty}$-structure. Specifically, the definition makes use of a
particular version of Floer theory, the Lagrangian (intersection) Floer
homology, which studies solutions to Cauchy–Riemann problems with boundary
conditions given by configurations of intersecting Lagrangian submanifolds.

The underlying language is that of symplectic and complex geometry. Moreover,
as is true of the other Floer-flavoured theories, a lot of functional analysis
is involved. And this is our major compromise as well: we will not insist on
the analytical subtleties which would unavoidably take us on a tangent.

Before moving on, we should also mention that the classical analytical
approach from above originally adopted by Fukaya has been substantially
revised in the meantime, ultimately replaced by the groundbreaking Lagrangian
intersection Floer theory: anomaly and obstruction [FOOO09] by Fukaya, Oh,
Ohta and Ono (last updated in 2009), which explains how to define Fukaya
categories in full generality. Though tempting, this is unfortunately a luxury
we cannot afford, due to the even more advanced analysis and algebra required.

These words of warning set aside, our complete agenda for this part reads as
follows:

  * •

In Chapter 5, we give an overview of Lagrangian Floer homology. After
introducing the basic notions from symplectic and complex geometry, we
anticipate the definition of Fukaya category of a symplectic manifold so to
motivate each decoration required for its objects; we immediately tackle also
a couple aspects of the general framework.
Then we introduce Floer complexes and explain how their boundary operator is
defined through the count of $J$-holomorphic strips solving some specific
Cauchy–Riemann equations with Lagrangian boundary conditions. Next, we discuss
Maslov indices and how to give complexes and Lagrangian submanifolds an
integer grading. Then we address the first of two major obstructions to the
well-definiteness of the boundary operator, namely that of transversality of
the moduli spaces formed by the $J$-holomorphic strips, which is solved by
applying a Hamiltonian perturbation to the Cauchy–Riemann problem. Moreover,
we rectify the issue of compactness of the moduli spaces by imposing a
topological constraint on the Lagrangian submanifolds.
All together, these assumptions enable us to properly define the Floer
differential and its resulting cohomology, and prove the latter to be
independent of the perturbative data. Lastly, we observe the advantage of
dealing with exact Lagrangian submanifolds, which are a priori unobstructed
and thus get us rid of most technical burden.

  * •

In Chapter 6, we construct the $A_{\infty}$-structure of Fukaya categories and
highlight the geometrical peculiarities making them very interesting cases of
$A_{\infty}$-categories. We start by adapting the Cauchy–Riemann problem to
fit an additional Lagrangian submanifold, and see how this defines the Floer
product, which has the desirable properties of a second order
$A_{\infty}$-composition map. Then we generalize the procedure for any given
number of Lagrangian submanifolds, fine-tuning the perturbation data, so to
produce higher order composition maps fulfilling the
$A_{\infty}$-associativity equations. At last, we give the formal definition
of Fukaya category (and a few warning remarks), prove it to be c-unital and
summarize what we already know from Part I.
Next, we describe two geometric constructions, Dehn twists and Lagrangian
connected sums, which algebraically are just suitable mapping cones in the
$A_{\infty}$-category of twisted complexes — a highlight of Fukaya categories
— and later justify the necessity of taking split-closures by looking at the
example of the 2-torus. We conclude with a glance at the wrapped version of
Fukaya category one must resort to when dealing with exact not necessarily
compact Lagrangian submanifolds.

  * •

In Chapter 7, we inject what learned about Fukaya categories into the context
of Homological Mirror Symmetry, paying regard to the historical path which led
to it. Thereto, we start by giving a very superficial description of what
string theory is and how some physical dualities observed there inspired
mathematical research to solve longstanding questions, eventually giving birth
to the new branch of Mirror Symmetry. The latter we discuss next, importing
all the language from complex algebraic topology and differential geometry
(Dolbeault cohomology, Hodge theory, Calabi–Yau manifolds) necessary to state
the main mirror symmetry conjecture.
Finally, we collect all relevant notions from algebraic geometry (coherent
sheaves) and homological algebra (Ext-groups), and suitably extend the
definition of Fukaya category in order to state Kontsevich’s homological
conjecture, backed by some specific cases we list. A quick look at the
geometrical SYZ conjecture, hinting how to “explicitly” fabricate mirror pairs
of Calabi–Yau manifolds, completes our journey.

The main reference for Chapters 5 and 6 is Denis Auroux’s introductory course
in Fukaya categories, summarized in [Aur13]. Basic material entering the frame
via symplectic geometry and Floer theory is distributed among many more
sources, most prominently [Sil12], [McS94], [Mer14] and [AS09] (always with an
eye on [Sei08]). Chapter 7 instead borrows, among others, from [Aur09],
[Fuk02a] and [GHJ03]. Once again, preliminary aspects of the theory are
embedded within the flow of our discussion, but can be safely skipped by the
expert reader.

###  5 Lagrangian Floer (co)homology

####  5.1 Preliminaries from symplectic and complex geometry

In this section, we introduce in a brief if somewhat dense manner all
geometric notions underlying the theory of Fukaya categories. Let us start
from the branch of symplectic geometry (the reader is referred to [Sil12] for
more details).

######  Definition 5.1.

A symplectic manifold $(M,\omega)$ of dimension $2n$ is a smooth manifold
$M^{2n}$ equipped with a symplectic form
$\omega\in\Omega^{2}(M)=\Gamma(\Lambda^{2}(T^{*}M))$, that is, a closed non-
degenerate 2-form (thus canonically identifying, for each $x\in M$,
antisymmetric bilinear maps $\omega_{x}:T_{x}M\times
T_{x}M\rightarrow\mathbb{R}$ such that $\omega_{x}(v,w)=0$ for all $w\in
T_{x}M$ implies $v=0\in T_{x}M$). Closedness means that $d\omega=0$, with
respect to the exterior differential operator
$d:\Omega^{\bullet}(M)\rightarrow\Omega^{\bullet+1}(M)$.

$M$ is exact if $\omega$ is an exact 2-form, that is, if there exists some
$\nu\in\Omega^{1}(M)$ such that $\omega=d\nu$. In case $M$ is compact, one can
prove that $\omega$ is never exact, meaning that $\langle\omega\rangle\in
H_{dR}^{2}(M)$ is a non-trivial class in de Rham cohomology.

We observe that the very definition of $\omega$ forces $M$ to be even-
dimensional and oriented.

######  Definition 5.2.

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold. A Lagrangian
submanifold $L\subset M$ is a maximal isotropic submanifold of $M$, that is,
one of dimension $\frac{1}{2}$dim$(M)=n$ fulfilling $\omega|_{L}=0$ (or
equivalently, $\iota^{*}\omega=0$ for the inclusion $\iota:L\hookrightarrow
M$).

Assuming $M$ is exact with $\omega=d\nu$ for some $\nu\in\Omega^{1}(M)$, $L$
is itself exact if the restriction $\nu|_{L}$ (closed as
$d(\nu|_{L})=(d\nu)|_{L}=0$) is an exact 1-form, that is, if there exists some
smooth function $f\in C^{\infty}(L)$ such that $\nu|_{L}=df$.

######  Remark 5.3.

As a particular case, a vector space $V$ is called symplectic when endowed
with an antisymmetric non-degenerate map $\omega:V\times
V\rightarrow\mathbb{R}$ (called symplectic structure). Then a linear subspace
$W\subset V$ is Lagrangian if $W=W^{\perp}\coloneqq\\{v\in
V\mid\omega(v,w)=0\;\forall w\in W\\}$. This is an element of the Lagrangian
Grassmannian $LGr(V)\coloneqq\\{W\subset V\mid W\text{ linear Lagrangian
subspace}\\}$, a smooth manifold (diffeomorphically) realizable as the
homogeneous space $\text{U}(n)/\text{O}(n)$ (that of unitary matrices modulo
the orthogonal ones).

For example, in $V\coloneqq\mathbb{R}^{2n}$ endowed with
$\omega_{0}\\!\in\\!\Omega^{2}(\mathbb{R}^{2n})$ represented by
$\big{(}\begin{smallmatrix}0&\mathds{1}_{n}\\\
-\mathds{1}_{n}&0\end{smallmatrix}\big{)}$, a subspace
$W\subset\mathbb{R}^{2n}$ is Lagrangian if and only if it is $n$-dimensional
and $\omega_{0}((x_{1},y_{1}),(x_{2},y_{2}))\coloneqq x_{1}\cdot
y_{2}-x_{2}\cdot y_{1}=0$ for all $(x_{1},y_{1}),(x_{2},y_{2})\in W$. In this
case, we simply write $LGr(n)$ instead of $LGr(\mathbb{R}^{2n})$. Recall also
the symplectic group
$\text{Sp}(2n,\mathbb{R})\\!=\\!\\{B\\!\in\\!\text{GL}(2n,\mathbb{R})\mid\omega_{0}(B(v),B(w))=\omega_{0}(v,w)\;\;\forall
v,w\\!\in\\!\mathbb{R}^{2n}\\}\equiv$ $\\{B\in\mathbb{R}^{2n\times 2n}\mid
B^{\text{T}}\omega_{0}B=\omega_{0}^{\text{T}}\\}$.

Clearly, if $(M^{2n},\omega)$ is a symplectic manifold, then
$(T_{x}M,\omega_{x})$ is a symplectic vector space with $LGr(T_{x}M)\cong
LGr(n)$ for every $x\in M$ (we say that the vector bundle $TM\rightarrow M$ is
symplectic). For any point $x\in L$ of a Lagrangian submanifold $L^{n}$ holds
$T_{x}L\in LGr(T_{x}M)$, by definition.

We also recollect the fundamentals of Hamiltonian dynamics.

######  Definition 5.4.

Let $(M,\omega)$ be a symplectic manifold. To each smooth function $H\in
C^{\infty}(M)$ we can assign a vector field $X_{H}\in\mathfrak{X}(M)$
satisfying
$i_{X\\!_{H}}(\omega)=\omega(\,\cdot\,,X_{H})=dH:\mathfrak{X}(M)\rightarrow
C^{\infty}(M)$. $X_{H}$ is called the Hamiltonian vector field associated to
the Hamiltonian function $H$.

If the Hamiltonian is time-dependent, $H\in C^{\infty}(I\times M)$ with
$H(t,\,\cdot\,)\equiv H_{t}(\cdot)\in C^{\infty}(M)$ and $I=[0,1]$ (or
$\mathbb{R}$), then it yields a family of Hamiltonian vector fields
$\\{X_{H_{t}}\\}_{t\in I}\subset\mathfrak{X}(M)$, and we can associate to
$X_{H}$ (assumed complete) its Hamiltonian “flow”
$\phi_{H}:I\\!\times\\!M\rightarrow M$, which satisfies
$\frac{d}{dt}|_{t}(\phi_{H}^{t})=X_{H_{t}}\\!\circ\\!\phi_{H}^{t}$and
$\phi_{H}^{0}=\textup{id}$ (we write
$\phi_{H}(t,\,\cdot\,)=\phi_{H}^{t}(\cdot)$). In particular, we call
$\phi_{H}^{1}$ the Hamiltonian diffeomorphism generated by $H$. This is an
element of the group

$\text{Ham}(M,\omega)\\!\coloneqq\\!\\{\phi\\!\in\\!\text{Diff}(M)\mid\phi=\phi_{H}^{1}\text{
Ham.\,diffeo.\;of\;some }H\\!\in\\!C^{\infty}(I\times M)\\}\,.$

Denoting by
$\text{Symp}(M,\omega)\coloneqq\\{\phi\in\text{Diff}(M)\mid\phi^{*}\omega=\omega\\}$
the group of symplectomorphisms, we have the following inclusions of
subgroups:

$\text{Ham}(M,\omega)\leq\text{Symp}(M,\omega)\leq\text{Diff}(M).$

######  Definition 5.5.

Let $M$ be a smooth manifold, $L_{0},L_{1}\subset M$ compact submanifolds.

  * •

We say that $L_{0}$ and $L_{1}$ intersect transversely (or are transverse) if
$T_{x}L_{0}\pitchfork T_{x}L_{1}$, that is, $T_{x}L_{0}\oplus
T_{x}L_{1}=T_{x}M$ as vector spaces, for all $x\in L_{0}\cap L_{1}\subset M$.
Then we write $L_{0}\pitchfork L_{1}$.

  * •

A (smooth) isotopy between $L_{0}$ and $L_{1}$ is a compact submanifold
$\tilde{L}\subset M\times[0,1]$ such that
$\tilde{L}\cap(M\times\\{i\\})=L_{i}\times\\{i\\}$ for $i=0,1$ and
$\tilde{L}\pitchfork(M\times\\{t\\})$ for all $t\in[0,1]$.
Then $L_{0}$ and $L_{1}$ are also ambient isotopic, that is, there exists a
path $\phi:t\in[0,1]\mapsto\phi_{t}\in\text{Diff}(M)$ such that
$\phi_{0}=\textup{id}_{M}$ and $\phi_{1}(L_{0})=L_{1}$.

  * •

Assume now that $M=(M,\omega)$ is also symplectic and $L_{0},L_{1}$
Lagrangian. The isotopy $\tilde{L}$ from above is a Lagrangian isotopy if
additionally each $\tilde{L}\;\cap\;(M\times\\{t\\})\subset M\times\\{t\\}$ is
a Lagrangian submanifold, whence it follows that
$(\pi_{1}^{*}\omega)|_{\tilde{L}}=\theta\wedge dt\in\Omega^{2}(\tilde{L})$ for
$\pi_{1}:M\times[0,1]\rightarrow M$ and some $\theta\in\Omega^{1}(\tilde{L})$.
Then $L_{0}$ and $L_{1}$ are also symplectic isotopic, that is, there exists a
path $\phi:t\in[0,1]\mapsto\phi_{t}\in\text{Symp}(M,\omega)$ such that
$\phi_{0}=\textup{id}_{M}$ and $\phi_{1}(L_{0})=L_{1}$.

  * •

The Lagrangian isotopy $\tilde{L}$ is in particular a Hamiltonian isotopy if
actually $\theta\wedge dt=d(Hdt)=dH\wedge dt$ for some Hamiltonian function
$H\in C^{\infty}(\tilde{L})$. Choosing above171717We can be slightly more
precise: let
$\text{Symp}_{0}(M,\omega)\coloneqq\\{\varphi\in\text{Symp}(M,\omega)\mid\exists\phi:[0,1]\rightarrow\text{Symp}(M,\omega)\text{
s.t. }\phi_{0}=\textup{id}_{M},\,\phi_{1}=\varphi\\}$. Then
$\phi_{H}:[0,1]\rightarrow\text{Symp}_{0}(M,\omega)$ (see [Mer14, Lemma 2.7]
for a proof). $\phi\coloneqq\phi_{H}:[0,1]\rightarrow\text{Symp}(M,\omega)$,
we obtain that $L_{0}$ and $L_{1}$ are Hamiltonian isotopic.

The study of Lagrangian Floer homology requires some knowledge of complex
geometry as well.

######  Definition 5.6.

Let $M$ be a smooth manifold. An almost complex structure on $M$ is a tensor
field $J\in\mathcal{T}^{1,1}(M)\cong\Gamma(\text{End}(TM))$ such that
$J_{x}\circ J_{x}=-\textup{id}_{T_{x}M}$ for all $x\in M$. Then $M=(M,J)$ is
an almost complex manifold and $\pi:(TM,J)\rightarrow M$ a complex vector
bundle.

If moreover $M\\!=\\!(M,\omega)$ is a symplectic manifold, $J$ is said to be
$\omega$-compatible if
$g_{J}(\,\cdot\,,\,\cdot\,)\coloneqq\omega(J\cdot\,,\,\cdot\,)$
defines181818Warning: literature mostly adopts the alternative definition,
namely $g_{J}(\,\cdot\,,\,\cdot\,)\coloneqq\omega(\,\cdot\,,J\cdot\,)$!
Observe that if $M$ is endowed with a Riemannian metric $g$ (always possible)
such that $g(u,v)=\omega(u,Jv)$, then this relation can be used to define a
$\omega$-compatible $J$ on $M$. In particular, symplectic manifolds can always
be regarded as almost complex manifolds. a Riemannian metric on $M$ (that is,
for each $x\in M$ the map
$g_{J_{x}}(\,\cdot\,,\,\cdot\,)\coloneqq\omega_{x}(J_{x}\cdot\,,\,\cdot\,):T_{x}M\times
T_{x}M\rightarrow\mathbb{R}$ is a positive definite inner product, whence
$\omega_{x}(J_{x}(v),J_{x}(w))=\omega_{x}(v,w)$ for all $v,w\in T_{x}M$).

We denote by $\mathfrak{J}(M,\omega)$ the (nonempty) set of all
$\omega$-compatible almost complex structures on $M$ (it is path-connected,
cf. [Sil12, Corollary 12.9]).

In Chapter 7, when discussing homological mirror symmetry, we will deal with
complex manifolds, which are analytic $n$-dimensional (over $\mathbb{C}$)
manifolds locally homeomorphic to $\mathbb{C}^{n}$ through charts whose
transition functions are holomorphic. We can interpret them as real smooth
$2n$-dimensional manifolds. They are naturally equipped with an almost complex
structure, that is, they are automatically almost complex manifolds (see
[Sil12, Proposition 15.2]), though the converse is not true a priori (cf.
Lemma 7.1).

The maps we are most interested in when discussing Floer theory areJ-
holomorphic (or pseudoholomorphic) curves. A thorough description of them is
provided by [Wen14, sections 2.1–2.2].

######  Definition 5.7.

Let $M=(M,J)$ be a smooth manifold and $\Sigma=(\Sigma,j)$ a Riemann surface
(that is, a complex analytic 1-dimensional manifold), where $J$ and $j$ are
fixed almost complex structures. Then a $J$-holomorphic curve is a smooth map
$u:\Sigma\rightarrow M$ fulfilling $(du)_{x}\circ
j_{x}=J_{x}\circ(du)_{x}:T_{x}\Sigma\rightarrow T_{u(x)}M$ for all
$x\in\Sigma$.

Using that $J$ squares to $-\textup{id}$, this condition is equivalent to the
Cauchy–Riemann equation $\bar{\partial}_{j,J}(u)\coloneqq\frac{1}{2}(du+J\circ
du\circ j)=0$ (which reduces to the homonymous pair of equations from basic
complex analysis after choosing $M=\Sigma\coloneqq\mathbb{C}$ and
$J=j\coloneqq\big{(}\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\big{)}$).
In case $\bar{\partial}_{j,J}(u)=v$ for some perturbative bundle morphism
$0\neq v:T\Sigma\rightarrow TM$, we talk about $(j,J,v)$-holomorphic curve
satisfying a perturbed Cauchy–Riemann equation. In particular, if:

  * •

$\Sigma=\mathbb{R}\times[0,1]\subset\mathbb{R}^{2}\cong\mathbb{C}$, we call
$u$ a $J$-holomorphic strip;

  * •

$\Sigma=\mathbb{D}^{2}\coloneqq\\{z\in\mathbb{C}\mid|z|\leq
1\\}\subset\mathbb{C}$, $u$ is a $J$-holomorphic disk;

  * •

$\Sigma=\mathbb{C}P^{1}\equiv\mathbb{C}\sqcup\\{\infty\\}(\simeq\mathbb{S}^{2}\subset\mathbb{R}^{3})$,
then $u$ is a $J$-holomorphic sphere.

Since $\mathbb{R}\times[0,1]$ is biholomorphic to
$\mathbb{D}^{2}\setminus\\{\pm 1\\}$ by the Riemann Mapping Theorem, strips
solving the (unperturbed!) Cauchy–Riemann equation can be extended to disks.

######  Definition 5.8.

Let $(M,\omega,J)$ be a symplectic almost complex manifold, $(\Sigma,j)$ a
Riemann surface. Then we can assign to any $J$-holomorphic curve
$u:\Sigma\rightarrow M$ its symplectic area (or energy):

$\text{Area}_{g_{J}}(u)\equiv
E(u)\coloneqq\int_{\Sigma}u^{*}\omega=\int_{\Sigma}|du|^{2}\geq 0\;,$ (54)

where $|du|^{2}\equiv|du|_{g_{J}}^{2}=g_{J}(du,du)$. In particular, $E(u)>0$
whenever $u$ is non-constant.

For a $J$-holomorphic strip $u$ parametrized by (orthogonal) coordinates
$(s,t)\equiv s+it\in\mathbb{D}^{2}\setminus\\{\pm 1\\}$, we can clearly see
the geometric interpretation of symplectic area: the Cauchy–Riemann equation
becomes $J\partial_{s}u-\partial_{t}u=0$, and hence the area spanned by the
orthogonal vectors $\partial_{s}u$ and $\partial_{t}u$ is
$|\partial_{s}u|\cdot|\partial_{t}u|=|\partial_{s}u|^{2}=g_{J}(\partial_{s}u,\partial_{s}u)=\omega(J\partial_{s}u,\partial_{s}u)=\omega(\partial_{t}u,\partial_{s}u)$;
integration over all $z=s+it\in\mathbb{D}^{2}\setminus\\{\pm 1\\}$ yields
exactly (54). The same reasoning applies to any Riemann surface, by choosing
holomorphic local coordinates $(s,t)$ as above.

####  5.2 The Fukaya category: a sneak peek

There are several definitions of Fukaya category, which differ by the set of
data and constraints one wishes to attach to its objects, starting with the
underlying manifold itself.

We work in one of the most simplified such settings, namely a symplectic
manifold $M^{2n}=(M,\omega)$ with 2-torsion first Chern class, $2c_{1}(TM)=0$.
Before going into the details, let us give a brief overview of all the
necessary ingredients which will guide our construction; a thorough exposition
is provided in the ensuing sections.

Objects of the (compact) Fukaya category $\mathscr{F}(M)$ of $M$ are compact
Lagrangian submanifolds $L^{n}\subset M$ decorated with the following data:

  * •

a vanishing Maslov class $H^{1}(L;\mathbb{Z})$ — which we can talk about only
by assuming that $2c_{1}(TM)=0$ in the first place — thus enabling us to equip
each $L$ with a “grading” and consequently make Floer complexes graded;

  * •

when char$(\mathbb{K})\neq 2$, a choice of orientation and spin structure for
each $L$, eventually relative to some fixed $\langle b\rangle\in
H^{2}(M;\mathbb{Z}_{2})$;

  * •

$\langle\omega\rangle\cdot\pi_{2}(M,L)=0$ for each $L$, which gets us rid of
“transversality issues” preventing Floer cohomology from being well-defined;

  * •

for every pair of Lagrangian submanifolds $(L,L^{\prime})$, as well as for all
tuples $(L_{0},...,L_{k})$, the choice of a time-dependent almost complex
structure $J\in C^{\infty}([0,1],\mathfrak{J}(M,\omega))$ and Hamiltonian
$H_{L,L^{\prime}}\in C^{\infty}([0,1]\times M,\mathbb{R})$ which is consistent
with all perturbation data, in order to once again take care of transversality
and thus obtain well-defined composition maps.

We also require that Hamiltonian isotopic Lagrangian submanifolds (cf.
Definition 5.5) yield isomorphic objects in the Fukaya category (or rather, in
its associated cohomological category).

Morphism spaces are Floer complexes,
$\textup{hom}_{\mathscr{F}(M)}(L,L^{\prime})\coloneqq
CF^{\bullet}(L,L^{\prime})$, and the choice of suitable perturbation data as
mentioned above allows us to produce composition maps between these spaces.

At this point, the reader may have already guessed what we are aiming for:
Fukaya categories have the structure of $\mathbb{Z}$-graded
$A_{\infty}$-categories! However, unless we are working in the exact
symplectic framework — which we will eventually describe in Section 5.8 — the
choice of ground field is no longer arbitrary: for the morphisms in
$\mathscr{F}(M)$ to be well-defined, we must work over the Novikov field.

######  Definition 5.9.

Let $\mathbb{K}$ be any field. The Novikov ring over $\mathbb{K}$ is

$\Lambda_{\mathbb{K}}^{\prime}\coloneqq\Bigg{\\{}\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\,\bigg{|}\,a_{i}\in\mathbb{K},\,\lambda_{i}\in\mathbb{R}_{\geq
0}\text{ such that }\lim_{i\to\infty}\lambda_{i}=\infty\Bigg{\\}}\,.$

Then the Novikov field is simply its fraction field:

$\Lambda_{\mathbb{K}}\coloneqq\text{Frac}(\Lambda_{\mathbb{K}}^{\prime})=\Bigg{\\{}\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}\,\bigg{|}\,a_{i}\in\mathbb{K},\,\lambda_{i}\in\mathbb{R}\text{
such that }\lim_{i\to\infty}\lambda_{i}=\infty\Bigg{\\}}\,.$ (55)

Indeed, the higher-order composition maps $\mu_{\mathscr{F}(M)}^{d}$ “count”
certain pseudo-holomorphic polygons, encoding them as coefficients of power
series which generally do not converge. To remedy this issue, we are forced to
work over a field of formal power series such as $\Lambda_{\mathbb{K}}$. As
already hinted, assuming $M$ and all Lagrangian submanifolds to be exact, a
suitable rescaling allows us to eliminate the formal variable $T$, so that we
can directly consider coefficients in $\mathbb{K}$.

An aspect worth addressing immediately is our standing assumption that
$2c_{1}(TM)=0$. We just provide a formal definition, encouraging the reader to
look up in the literature for more details (see for example [MS74]).

######  Definition 5.10.

Let $M^{2n}=(M,\omega,J)$ be an almost complex symplectic manifold, and hence
$\pi:TM\rightarrow M$ a complex vector bundle of rank $2n$. The Chern classes
$c_{k}(TM)\in H^{2k}(M;\mathbb{Z})$ of $M$ are implicitly defined to be the
coefficients in the expression

$\text{det}\Bigg{(}\frac{it\Omega}{2\pi}+\mathds{1}_{n}\Bigg{)}=\sum_{k\geq
0}c_{k}(TM)t^{k}\;.$

Here $t$ is a formal variable and
$\Omega=d\varpi+\frac{1}{2}[\varpi,\varpi]\in\Omega^{2}(P,\mathbb{K}^{2n\times
2n})$ is the curvature form of the principal $\text{GL}(2n)$-bundle $P$
associated191919The principal bundle $P$ is the frame bundle $\text{Fr}(TM)$
of $TM$, whose fibers are given by
$\text{Fr}(TM)_{x}\coloneqq\\{f:\mathbb{K}^{2n}\rightarrow T_{x}M\mid f\text{
linear isomorphism}\\}$. to $TM$, identifying a $C^{\infty}(P)$-bilinear
alternating map $\mathfrak{X}(P)\times\mathfrak{X}(P)\rightarrow
C^{\infty}(P,\mathbb{K}^{2n\times 2n})$ of which we can indeed consider the
standard determinant.

Note that, given another almost complex structure
$\tilde{J}\in\mathfrak{J}(M,\omega)$, one can show that
$c_{1}(TM,J)=c_{1}(TM,\tilde{J})$ (similarly for the other $c_{k}$’s), and
therefore unambiguously write $c_{1}(TM)\in H^{2}(M;\mathbb{Z})$ for the first
Chern class of $M$.

[frame]Henceforth, we assume that the symplectic manifold $M$ fulfills
$2c_{1}(TM)=0$,
and consider only compact Lagrangian submanifolds $L\subset M$.

####  5.3 The Floer complex

As apparent from the previous section, to understand Fukaya categories we must
first understand Lagrangian Floer theory. The rest of Chapter 5 is devoted to
gain an essential grasp on this subject. However, the reader should be aware
that the following exposition deliberately takes the many underlying
analytical results and constructions for granted; whenever sensitive, we
provide technical comments so to orientate the more expert eyes. Detail-filled
accounts on (general) Floer theory can easily be found throughout literature
(see for example [Sal97] or [Mer14]); here we limit ourselves to the concise
version of [Aur13, section 1].

Firstly, we should mention that Lagrangian Floer homology can be seen as an
infinite-dimensional analogue of Morse homology202020For an overview on Morse
theory consult for example [Mer14, chapter 1]. for the action functional on
the universal cover of
$\mathcal{P}(L_{0},L_{1})\coloneqq\\{\gamma:[0,1]\rightarrow M\mid\gamma(0)\in
L_{0},\,\gamma(1)\in L_{1}\\}$, given any two Lagrangian submanifolds
$L_{0},L_{1}\subset M$ (see [Aur13, section 1.1]). We will return to this in
Section 5.5.

However, the infinite-dimensional framework makes the analytical problems too
hard to tackle, so that one actually resorts to an alternative point of view:
moduli spaces of pseudo-holomorphic strips.

######  Definition 5.11.

Let $M=(M,\omega)$ be a symplectic manifold equipped with some
$J\in\mathfrak{J}(M,\omega)$, and let $L_{0},L_{1}\subset M$ be two compact
Lagrangian submanifolds such that $L_{0}\pitchfork L_{1}$ (whence $|L_{0}\cap
L_{1}|<\infty$, because $L_{0}\cap L_{1}$ is a 0-dimensional compact set, by
definition). Then the Floer complex of $(L_{0},L_{1})$ is the free
$\Lambda_{\mathbb{K}}$-module generated by $\mathcal{X}(L_{0},L_{1})\coloneqq
L_{0}\cap L_{1}$,

$CF(L_{0},L_{1})\coloneqq\mkern-18.0mu\bigoplus_{p\in\mathcal{X}(L_{0},L_{1})}\mkern-18.0mu\Lambda_{\mathbb{K}}\cdot
p\;.$ (56)

Of course, Definition 5.11 lacks information about the corresponding boundary
operator $\partial:CF(L_{0},L_{1})\rightarrow CF(L_{0},L_{1})$. This is given
by “counting” $J$-holomorphic strips in $M$ with boundary on $L_{0}$ and
$L_{1}$. More precisely, the coefficient of $q$ in $\partial(p)$ depends on
the space of solutions
$u:\mathbb{R}\times[0,1]\simeq\mathbb{D}^{2}\setminus\\{\pm
1\\}\subset\mathbb{C}\rightarrow M,(s,t)\equiv(s+it)\mapsto u(s,t)$ of the
Cauchy–Riemann equation

$\bar{\partial}_{J}(u)\coloneqq\partial_{s}u+J(u)\partial_{t}u=0$ (57)

(where $J(u)\equiv J_{u(\cdot,\cdot)}$) with boundary conditions

$\left\\{\begin{array}[]{llll}u(s,0)\in L_{0}&\text{ and }&u(s,1)\in
L_{1}&\forall s\in\mathbb{R}\\\
\\!\\!\\!\displaystyle\lim_{s\to-\infty}\\!\\!u(s,t)=q&\text{ and
}&\\!\\!\\!\displaystyle\lim_{s\to+\infty}\\!\\!u(s,t)=p&\forall
t\in[0,1]\end{array}\right.\,,$ (58)

and the finite energy constraint212121When treating the perturbed
Cauchy–Riemann equation (66), one can in fact prove that $E(u)<\infty$ is
equivalent to the asymptotic conditions $\lim_{s\to+\infty}u(s,t)=p$ and
$\lim_{s\to-\infty}u(s,t)=q$ from (58); see [Sal97, Proposition 1.21].

$E(u)=\int_{\mathbb{R}\times[0,1]}\mkern-18.0muu^{*}\omega=\frac{1}{2}\int_{-\infty}^{+\infty}\int_{0}^{1}|\partial_{s}u|^{2}dsdt\overset{!}{<}\infty\;.$
(59)

(equivalently a bound on the symplectic area). Any solution $u$ is indeed a
$J$-holomorphic map (compare to Definition 5.7), as depicted in Figure 2.

Figure 2: A $J$-holomorphic strip $u$ belonging to
$\widehat{\mathcal{M}}(p,q;[u],J)$. [Source:[Aur13]]

######  Definition 5.12.

Let $M=(M,\omega,J)$ be an almost complex symplectic manifold and
$L_{0},L_{1}\subset M$ compact Lagrangian submanifolds such that
$L_{0}\pitchfork L_{1}$. Consider a solution
$u:\mathbb{R}\times[0,1]\rightarrow M$ of the Cauchy–Riemann problem (57) with
constraints (58) and (59), thus yielding a homotopy class
$[u]\\!\in\\!\pi_{2}(M,L_{0}\cup L_{1})$
($=\textup{Hom}_{\mathsf{hTop}}((B^{2},\mathbb{S}^{1}),(M,L_{0}\cup L_{1}))$,
for pointed pairs of topological spaces
$(B^{2},\mathbb{S}^{1})\simeq(\mathbb{D}^{2}\setminus\\{\pm
1\\},\mathbb{R}\times\\{0\\}\cup\mathbb{R}\times\\{1\\})$ and $(M,L_{0}\cup
L_{1})$).

For fixed $p,q\in\mathcal{X}(L_{0},L_{1})$, we denote by
$\widehat{\mathcal{M}}(p,q;[u],J)$ the moduli space of solutions to (57)–(59)
representing $[u]$ and set
$\mathcal{M}(p,q;[u],J)\\!\coloneqq\\!\widehat{\mathcal{M}}(p,q;[u],J)/\mathbb{R}$.

We observe that $\mathbb{R}$ acts freely on $\widehat{\mathcal{M}}(p,q;[u],J)$
by translation in the $s$-coordinate: if
$u(s,t)\in\widehat{\mathcal{M}}(p,q;[u],J)$, then $u_{\tilde{s}}(s,t)\coloneqq
u(s-\tilde{s},t)\in\widehat{\mathcal{M}}(p,q;[u],J)$ for all
$\tilde{s}\in\mathbb{R}$ as well. This is taken into account in the definition
of $\mathcal{M}(p,q;[u],J)$.

There are now three issues hindering our attempt to define a boundary operator
for $CF(L_{0},L_{1})$. Namely, for $\widehat{\mathcal{M}}(p,q;[u],J)$ to be a
“manageable” space, we need it to have the following properties:

  * •

Transversality, which ensures that $\widehat{\mathcal{M}}(p,q;[u],J)$ can be
given the structure of a finite-dimensional smooth manifold, whose dimension
is given by the Maslov index of $[u]$.

  * •

Compactness, which is achieved only in absence of “energy build-ups” making
$\widehat{\mathcal{M}}(p,q;[u],J)$ ill-defined.

  * •

Orientability, which should be consistent with the choice of orientation of
the underlying Lagrangian submanifolds.

However, none of these requisites are a priori met, and hence we need
additional assumptions.

Let us start by briefly addressing orientability. The following definition is
kept intentionally vague, since part of the aforementioned technicalities we
would rather avoid. The pragmatic reader may skip it.

######  Definition 5.13.

Let $L^{n}\subset M^{2n}$ be a Lagrangian submanifold. Assume $L$ is oriented,
that is, endowed with (the class of) a nowhere-vanishing $n$-form
$\theta_{L}\in\Omega^{n}(L)$, a volume form. Observe that, since smooth
manifolds are in particular cell complexes, we can consider skeleton
filtrations thereof. Then $L$ is:

  * •

a spin manifold if we can trivialize the tangent bundle $TL$ over the
1-skeleton $L^{(1)}$ in a way extendable to the 2-skeleton $L^{(2)}$. The
homotopy class of such trivialization is called a spin structure for $L$, in
fact existing if and only if the second Stiefel–Whitney class $w_{2}(TL)\in
H^{2}(L;\mathbb{Z}_{2})$ — an invariant of $TL$ — vanishes.

  * •

a relatively spin manifold relative to some fixed $\langle b\rangle\in
H^{2}(M;\mathbb{Z}_{2})$ if $w_{2}(TL)=\langle b\rangle|_{L}$.

Assuming $L_{0}$ and $L_{1}$ are oriented and spin, one can show that their
spin structures yield a canonical orientation for
$\widehat{\mathcal{M}}(p,q;[u],J)$ (this is proved in [Sei08, chapter 12]).
However, these assumptions are unnecessary if $\text{char}(\mathbb{K})=2$.

[frame]Henceforth, we will always assume that all Lagrangian submanifolds come
with an (implicit) choice of orientation and spin structure.

####  5.4 The Maslov index

The Maslov index of the homotopy class $[u]\in\pi_{2}(M,L_{0}\cup L_{1})$,
where $u$ solves (57)–(59), determines the dimension of
$\widehat{\mathcal{M}}(p,q;[u],J)$. Actually, the situation is more subtle
(the unfamiliar reader should skip the following remark):

######  Remark 5.14.

The Cauchy–Riemann problem (57)–(59) is actually a Fredholm problem, meaning
that the linearization $D_{\bar{\partial}_{J},u}$ of $\bar{\partial}_{J}$ at
any zero $u$ (that is, solution) is a Fredholm operator. We can assign to
$[u]$ its Fredholm index
$\text{ind}([u])\coloneqq\text{dim}(\text{ker}(D_{\bar{\partial}_{J},u}))-\text{dim}(\text{coker}(D_{\bar{\partial}_{J},u}))$,
which in fact coincides with the Maslov index we will compute below. We remark
that Hamiltonian Floer theory is actually discussed in terms of another
quantity, the Conley–Zehnder index.

Recall from Remark 5.3 the definition of Lagrangian Grassmannian $LGr(n)$,
diffeomorphic to $\text{U}(n)/\text{O}(n)$. It can be shown that the square of
the determinant map
$\text{det}:\text{U}(n)/\text{O}(n)\rightarrow\mathbb{S}^{1}$ induces an
isomorphism $\pi_{1}(LGr(n))\cong\pi_{1}(\mathbb{S}^{1})=\mathbb{Z}$.

######  Definition 5.15.

The Maslov index of a loop $\gamma:[0,1]\rightarrow LGr(n)$ based at some
$W\in LGr(n)$ is given by the winding number $\text{ind}(\gamma)\coloneqq
w(\gamma,W)\in\mathbb{Z}$ of
$\text{im}(\text{det}^{2}\circ\gamma)\subset\mathbb{S}^{1}$ (identifying
$LGr(n)\equiv\text{U}(n)/\text{O}(n)$).

On the other hand, given two paths $l_{0},l_{1}:[0,1]\rightarrow LGr(n)$ with
$l_{0}(0)\pitchfork l_{1}(0)$ and $l_{0}(1)\pitchfork l_{1}(1)$, define
$l\coloneqq(l_{0},l_{1}):t\\!\in\\![0,1]\mapsto(l_{0}(t),l_{1}(t))\\!\in\\!LGr(n)\times
LGr(n)$. The Maslov index of the paths $l_{0},l_{1}$ is

$\text{ind}(l_{0},l_{1})\coloneqq\big{|}\text{im}(l)\cap\overline{LGr(n)}\big{|}\,,$
(60)

where $\overline{LGr(n)}\coloneqq\\{(W,W^{\prime})\in LGr(n)\times LGr(n)\mid
W\,\cancel{\pitchfork}\,W^{\prime}\\}$. Put differently,
$\text{ind}(l_{0},l_{1})$ is the (signed) number of times $l_{0}(t)$ is not
transverse to $l_{1}(t)$, for increasing $0\leq t\leq 1$.

Let us parse the geometric meaning of Maslov index for paths with help of an
easy example.

######  Example 5.16.

For simplicity, we take
$l_{0}(t)\coloneqq\mathbb{R}^{n}\subset\mathbb{R}^{2n}\cong\mathbb{C}^{n}$ to
be constant, and define
$l_{1}(t)\coloneqq(e^{i\varphi_{1}(t)}\mathbb{R})\times...\times(e^{i\varphi_{n}(t)}\mathbb{R})$
so that all angles $\varphi_{k}$ vary uniformly from $-\pi/2$ to $\pi/2$. The
Maslov index of $l_{0}$ and $l_{1}$ is precisely $\text{ind}(l_{0},l_{1})=n$.
This is best seen for the case $n=1$, where $l_{0}=x\text{-axis}$: the
straight line $l_{1}(t)$ rotates counterclockwise from
$l_{1}(0)=y\text{-axis}$ (where $\varphi_{1}(0)=-\pi/2$) to
$l_{1}(1)=y\text{-axis}$ (where $\varphi_{1}(1)=\pi/2$), intersecting $l_{0}$
transversely at all times, except at $\varphi(1/2)=0$, where the two lines
actually coincide. $\blacklozenge$

Consider now a $J$-holomorphic strip $u:\mathbb{R}\times[0,1]\rightarrow M$
solving the Cauchy–Riemann problem (57)–(59). Write $q=u(-\infty,t)$ and
$p=u(\infty,t)$ for any $t\in[0,1]$. Then the smooth map $u$ yields the
symplectic pullback
bundle$u^{*}TM=\\{((s,t),(x,v))\in(\mathbb{R}\times[0,1])\times TM\mid
u(s,t)=x\\}$, and we can define the Maslov index of $u$ as follows.

######  Definition 5.17.

After choosing a trivialization for $u^{*}TM\rightarrow\mathbb{R}\times[0,1]$
(see for example [Mer14, chapter 5]), we define two paths
$l_{i}(s)\coloneqq((u|_{\mathbb{R}\times\\{i\\}})^{*}TL_{i})|_{(-s,i)}$ $\cong
T_{u(-s,i)}L_{i}$ for $s\in\overline{\mathbb{R}}=[-\infty,+\infty]$ and
$i=0,1$, whose image actually lies in $LGr(n)$ (by Remark 5.3) and such that
each $l_{i}$ goes from $T_{p}L_{i}$ to $T_{q}L_{i}$, satisfying
$l_{0}(-\infty)\pitchfork l_{1}(-\infty)$ and $l_{0}(\infty)\pitchfork
l_{1}(\infty)$ (because $L_{0}\pitchfork L_{1}$).

The Maslov index of the $J$-holomorphic strip $u$ is just the Maslov index of
the two paths $l_{0}$ and $l_{1}$,
$\text{ind}([u])\coloneqq\text{ind}(l_{0},l_{1})$, as in Definition 5.15
(which is still valid for the enlarged domain $\overline{\mathbb{R}}$).

We remark that $\text{ind}(l_{0},l_{1})$ is actually independent from the
chosen trivialization for $u^{*}TM$, and invariant under conjugation by an
element in $\text{Sp}(2n,\mathbb{R})$. This makes $\text{ind}([u])$ well-
defined. Moreover, it is additive under concatenation (further properties,
along with a slightly modified construction, can be found in [Sei99, section
4]). Let us see the equivalent formulation as index of a suitable loop.

######  Definition 5.18.

Consider $\lambda_{0},\lambda_{1}\in LGr(n)$ such that
$\lambda_{0}\pitchfork\lambda_{1}$. Identify the unique transformation
$A\in\text{Sp}(2n,\mathbb{R})$ such that
$A(\lambda_{0})=\mathbb{R}^{n}\subset\mathbb{C}^{n}$ and
$A(\lambda_{1})=(i\mathbb{R})^{n}\subset\mathbb{C}^{n}$. Then we call
$\lambda:t\in[0,1]\mapsto A^{-1}((e^{-i\pi t/2}\mathbb{R})^{n})\in LGr(n)$ the
canonical short path from $\lambda_{0}=\lambda(0)$ to
$\lambda_{1}=\lambda(1)$.

######  Definition 5.19.

Let $u:\mathbb{R}\times[0,1]\rightarrow M$ be a $J$-holomorphic strip solving
the Cauchy–Riemann problem (57)–(59). Define the paths
$l_{0},l_{1}:\overline{\mathbb{R}}\rightarrow LGr(n)$ as in the paragraph
above, and consider canonical short paths
$\lambda_{p},\lambda_{q}:[0,1]\rightarrow LGr(n)$ from $T_{p}L_{0}$ to
$T_{p}L_{1}$ respectively from $T_{q}L_{0}$ to $T_{q}L_{1}$, as in Definition
5.18.

We thus have a loop $\gamma_{u}\coloneqq\overline{l_{0}}\ast\lambda_{p}\ast
l_{1}\ast\overline{\lambda_{q}}$ in $LGr(n)$ based at $T_{q}L_{0}$ (bars
denote inverse paths). The Maslov index of $u$ coincides with the
corresponding winding number,

$\text{ind}([u])\coloneqq\text{ind}(\gamma_{u})=w(\gamma_{u},T_{q}L_{0})\,.$
(61)

######  Example 5.20.

Take $M=\mathbb{R}^{2}$. Then the Maslov index of the $J$-holomorphic strip
$u$ depicted in Figure 2 can be computed using the following trick: we choose
a trivialization $u^{*}TM$ making the path $l_{0}(s)=T_{u(-s,0)}L_{0}$ equal
the constant tangent line $\mathbb{R}\times\\{0\\}$, and without loss of
generality parametrizing $L_{1}$ as the downwards open parabola segment
$\\{(x,-x^{2})\in\mathbb{R}^{2}\mid x\in[q,p]\\}$ (refer to Figure 2 for
clarity).

It is then obvious that the path $l_{1}(s)=T_{u(-s,1)}L_{1}$ is non-transverse
to $l_{0}$ only once, namely at $s=0$ (where the tangent line $l_{1}(0)$ is
parallel to $\mathbb{R}\times\\{0\\}$).Therefore, completing the loop at
$T_{q}L_{0}$ with canonical short paths as in Definition 5.19, it holds that
ind($[u]$)=1. $\blacklozenge$

Let us discuss how Floer complexes can be equipped with a
$\mathbb{Z}$-grading. Here is where our assumption of a 2-torsion first Chern
class comes into play.

Consider the fibre bundle $LGr(TM)\rightarrow M$ of fibres
$LGr(TM)_{x}\coloneqq LGr(T_{x}M)$ $\cong LGr(n)$ (this is the bundle
$P\times_{\text{Sp}(2n,\mathbb{R})}LGr(n)$, where $P$ is the associated
symplectic frame bundle, a principal $\text{Sp}(2n,\mathbb{R})$-bundle over
$M$), each having universal cover
$\upsilon_{x}:\widetilde{LGr}(T_{x}M)\rightarrow LGr(T_{x}M)$. Then
$2c_{1}(TM)=0$ allows222222Unfortunately, this is yet another technical bit
which falls out of scope. It is addressed in [Fuk02b, Lemma 2.6] or [Sei08,
section (11j)], in the context of Lagrangian branes. us to glue all these
covers to form a covering space $\widetilde{LGr}(TM)\rightarrow LGr(TM)$ such
that $\widetilde{LGr}(TM)_{x}=\widetilde{LGr}(T_{x}M)$ for all $x\in M$.
Elements of $\widetilde{LGr}(TM)$ are called graded Lagrangian planes in $TM$.

Now, given the section $s\in\Gamma(L,LGr(TM)),\;x\in L\mapsto T_{x}L\in
LGr(T_{x}M)$ on some Lagrangian submanifold $L\subset M$, we may wonder what
is the obstruction to lifting $s$ to some section
$\tilde{s}\in\Gamma(L,\widetilde{LGr}(TM))$, so to produce the commutative
diagram

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It turns out that we need the Maslov class of $L$ to vanish:

######  Definition 5.21.

Let $(M,\omega)$ be a symplectic manifold with $2c_{1}(TM)=0$, so that to each
$W\in LGr(T_{x}M)$ we can assign a “phase” $\phi(W)\in\mathbb{S}^{1}$ and
choose a lift $\tilde{\phi}(W)\in\mathbb{R}$ fulfilling $\phi(W)=e^{2\pi
i\tilde{\phi}(W)}$. Then, given a Lagrangian submanifold $L\subset M$, we can
define its “phase function” $\phi_{L}:L\rightarrow\mathbb{S}^{1}$ by
$\phi_{L}(x)\coloneqq\phi(T_{x}L)$.

The Maslov class of $L$ is232323Observe that, $\mathbb{Z}$ being divisible,
$\text{Ext}(H_{0}(L),\mathbb{Z})$ vanishes, and hence
$H^{1}(L;\mathbb{Z})\cong\textup{Hom}(H_{1}(L),\mathbb{Z})$. Now, working on
connected components of $L$, the Hurewicz Theorem tells us that
$H_{1}(L)\cong\pi_{1}(L)^{ab}=\pi_{1}(L)/[\pi_{1}(L),\pi(L)]$, and abelianity
of $\mathbb{Z}$ guarantees that
$\textup{Hom}(\pi_{1}(L),\mathbb{Z})\cong\textup{Hom}(\pi_{1}(L)^{ab},\mathbb{Z})$.
precisely

$\mu_{L}\equiv[\phi_{L}]\coloneqq\pi_{1}(\phi_{L})\in\textup{Hom}(\pi_{1}(L),\pi_{1}(\mathbb{S}^{1}))\cong
H^{1}(L;\mathbb{Z})\,.$ (63)

Moreover, the choice of a map $\tilde{\phi}_{L}:L\rightarrow\mathbb{R}$
lifting $\phi_{L}$, that is, fitting into the commutative diagram

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