[columns=2, title=Index, intoc]

Towards Homological Mirror Symmetry

Alessandro Imparato111Department of Mathematics, ETH Zürich, Autumn
2020–Winter 2021. This work was last updated on December 22, 2022.

This is an expository article on the A-side of Kontsevich’s Homological Mirror
Symmetry Conjecture. We give first a self-contained study of
$A_{\infty}$-categories and their homological algebra, and later restrict to
Fukaya categories, with particular emphasis on the basics of the underlying
Floer theory, and the geometric features therein. Finally, we place the theory
in the context of mirror symmetry, building towards its main predictions.

This article is the author’s Master Thesis, supervised by Prof. Dr. William
John Merry and submitted for the degree Master of Science ETH in Mathematics.
It should be accessible to graduate students.

Key words: $A_{\infty}$-category, split-closed derived category, Lagrangian
submanifold, Lagrangian Floer cohomology, Fukaya category, Mirror Symmetry,
Homological Mirror Symmetry.

###### Contents

  1. Introduction
  2. I Homological Algebra
    1. 1 $A_{\infty}$-algebras
      1. 1.1 Example of non-associativity: $A_{\infty}$-spaces
      2. 1.2 $A_{\infty}$-algebras and morphisms
      3. 1.3 $A_{\infty}$-modules and morphisms
    2. 2 $A_{\infty}$-categories
      1. 2.1 Category theoretic preliminaries
      2. 2.2 $A_{\infty}$-categories and $A_{\infty}$-functors
      3. 2.3 Pre-natural transformations and compositions
      4. 2.4 Interlude: Hochschild cohomology
      5. 2.5 Homological perturbation theory
      6. 2.6 $A_{\infty}$-modules over $A_{\infty}$-categories
      7. 2.7 Strict unitality and cohomological unitality
      8. 2.8 Quasi-equivalences
      9. 2.9 The Yoneda embedding
    3. 3 Triangulation
      1. 3.1 Category theoretic preliminaries
      2. 3.2 Operations on $A_{\infty}$-modules
      3. 3.3 Mapping cones and exact triangles
      4. 3.4 Triangulated $A_{\infty}$-categories and derived categories
      5. 3.5 The $A_{\infty}$-category of twisted complexes
      6. 3.6 The bounded derived category
      7. 3.7 Example of triangulated $A_{\infty}$-category: $mod(\mathcal{A})$
    4. 4 Split-generation and twisting
      1. 4.1 Category theoretic preliminaries
      2. 4.2 Abstract images
      3. 4.3 The split-closed derived category
      4. 4.4 Twisting
  3. II Fukaya Category Theory
    1. 5 Lagrangian Floer (co)homology
      1. 5.1 Preliminaries from symplectic and complex geometry
      2. 5.2 The Fukaya category: a sneak peek
      3. 5.3 The Floer complex
      4. 5.4 The Maslov index
      5. 5.5 Transversality
      6. 5.6 Compactness
      7. 5.7 The Floer differential and Floer cohomology
      8. 5.8 The exact manifold setting
    2. 6 The Fukaya category
      1. 6.1 The Floer product
      2. 6.2 The higher order composition maps
      3. 6.3 The $A_{\infty}$-structure of the Fukaya category
      4. 6.4 Geometric features of Fukaya categories
      5. 6.5 The wrapped Fukaya category
    3. 7 Towards Homological Mirror Symmetry
      1. 7.1 Physical origins of mirror symmetry
      2. 7.2 Mirror symmetry
      3. 7.3 Homological mirror symmetry
    4. Outlook
    5. ‣ II Fukaya Category Theory

### Introduction

At its core, Homological Mirror Symmetry expresses a relation between
symplectic and algebraic geometry through category theoretic structures. This
sentence alone tells us a great deal about the intrinsic beauty of this
subject, and the difficulties its ambitious goal presents.

The conjecture guiding the overall research was first stated by the Russian-
born mathematician Maxim Kontsevich in his famous talk (transcripted in
[Kon94]) at the International Congress of Mathematicians of 1994, held in
Zürich, and builds upon the numerical predictions of Mirror Symmetry, a
mathematical branch evolved from string theory. The latter claims the
existence of a duality between pairs of “mirror” Calabi–Yau manifolds, which
are some specific kind of complex symplectic manifolds. Physically speaking,
these geometrical objects possess the necessary extra dimensions along which
strings vibrations are theorised to occur, and their duality translates into a
congruence of the physical laws therein described.

Kontsevich gave the Mirror Symmetry Conjecture a categorical character,
reformulating it through the language of derived categories, a topic of great
interest to homological algebra. Roughly speaking, the claimed correspondence
now selects only part of the underlying Calabi–Yau structures: namely, given a
pair $(X,X^{\dagger})$ of mirror Calabi–Yau manifolds, there is a duality
between the symplectic structure of $X$ (the “A-side”) — encoded in its Fukaya
category $\mathscr{F}(X)$ — and the algebraic geometric structure of
$X^{\dagger}$ (the “B-side”) — encoded in its category of coherent sheaves
$\mathsf{Coh}(X^{\dagger})$ — and vice versa, the B-side of $X$ reflects the
A-side of $X^{\dagger}$. Explicitly, it is conjectured that there are two
equivalences of triangulated categories,

$\mathsf{D}^{\pi}(\mathscr{F}(X))\cong\mathsf{D^{b}}(\mathsf{Coh}(X^{\dagger}))\qquad\text{and}\qquad\mathsf{D}^{\pi}(\mathscr{F}(X^{\dagger}))\cong\mathsf{D^{b}}(\mathsf{Coh}(X))\,.$

Our goal is to plausibly explain the exact meaning of such expressions.
Actually, the landscape is so rich we will only be able to discuss the A-side,
merely providing some hints of the “bigger picture” described above. Moreover,
that offered is just a polished version of a much more profound and hence
unavoidably insidious theory; we will often work in significantly simplified
settings, especially when dealing with Floer theory in the second part,
encouraging, whenever sensible, a comprehensive look rather than a single in-
depth analysis. Hopefully, such an approach will convey the essentiality of
each pure mathematical branch homological mirror symmetry borrows from. After
all, this synergy between distinct areas of mathematics is perhaps the most
appealing virtue offered.

The work consists of two parts: a general though self-contained treatment of
the abstract algebraic notion of $A_{\infty}$-categories, and a subsequent
specialization to a particular instance thereof, the Fukaya category.
Specifically, we start our journey in Chapter 1 with $A_{\infty}$-algebras,
the prototypes of $A_{\infty}$-categories, which we properly introduce in
Chapter 2 together with the necessary toolkit. In Chapter 3, we apply this new
technology to forge the intermediate notion of bounded derived category
$\mathsf{D^{b}}(\mathcal{A})$ associated to an $A_{\infty}$-category
$\mathcal{A}$. In Chapter 4, we enrich it to the split-closed derived category
$\mathsf{D}^{\pi}(\mathcal{A})$ appearing in the homological mirror
conjecture. Next, in Chapter 5, we move our focus to symplectic manifolds and
study Lagrangian Floer homology, integral to the definition of Fukaya category
$\mathscr{F}(X)$, which we provide in Chapter 6 along with the description of
its $A_{\infty}$-structure and special geometric features. Finally, in Chapter
7, we give a survey of the theory leading to the Homological Mirror Symmetry
Conjecture, briefly touching on its physical motivation and origins in mirror
symmetry, and aiming at a qualitative understanding of the B-side as well.

Acknowledgments. I would like to thank my supervisor, Will Merry, for
carefully reading through the various iterations of this thesis and, more
importantly, always seconding my ideas on how to best develop it, until its
publication.

##  Part I Homological Algebra

The first incarnation of $A_{\infty}$-categories were studied by Stasheff in
his Homotopy associativity of H-spaces, dating 1963 (see [Sta63]). In an
attempt to find useful tools for the study of topological spaces with group-
like features, he developed the notion of $A_{\infty}$-algebras: these are
graded vector spaces (properly, algebras) which are associative only up to a
system of higher order homotopies fulfilling certain relations (whence the
notation “$A_{\infty}$”); as such, they can be regarded as an enriched version
of differential graded algebras. Packing together this data, one speaks of
$A_{\infty}$-structure. The earliest example studied by Stasheff were loop
spaces, instances of the so-called $A_{\infty}$-spaces, which result from
endowing topological spaces with suitable $A_{\infty}$-structures.

Soon after their discovery, $A_{\infty}$-algebras became objects of interest
of many other illustrious mathematicians of that period, Kadeishvili among
these (particularly in [Kad80]), though always within the topological context,
specifically homotopy theory. This tendence didn’t change until the early 90s,
when $A_{\infty}$-structures started to exhibit their relevance towards
geometry and mathematical physics through works of the various Getzler–Jones
([GJ90]), Fukaya ([Fuk93]) and Stasheff himself. In particular, Fukaya’s take
sparked Kontsevich’s Homological Mirror Symmetry Conjecture of 1994 ([Kon94]),
which heavily relies on the $A_{\infty}$-structure of Fukaya categories in the
symplectic A-side.

Specifically, $A_{\infty}$-categories took the stage. Thinkable as
$A_{\infty}$-algebras with “several objects”, $A_{\infty}$-categories are
variations of the classical notion of categories. Albeit similar in aspect,
since consisting of objects and morphisms — the latters collected in graded
vector spaces (or modules), the hom-spaces — they are not categories in
general, because they lack units and violate associativity. Like for
$A_{\infty}$-algebras, associativity is supplanted by more complicated
equations involving higher order “composition maps” which generalize the
aforementioned homotopies, providing the final piece of data.

The goal of this first part is to give a thorough introduction to the
algebraic machinery of $A_{\infty}$-categories, gradually building towards the
item $\mathsf{D}^{\pi}(\mathcal{A})$ appearing on the A-side of the
Homological Mirror Symmetry Conjecture. We do this under the slogan: “the
larger the structure, the easier its description”. The precise plan is
articulated as follows:

  * •

In Chapter 1, we introduce $A_{\infty}$-algebras and $A_{\infty}$-modules in
order to acquire some familiarity with the type of algebraic structures in the
ensuing chapters.

  * •

In Chapter 2, we present the core ingredients of the subject without taking a
breath. We define $A_{\infty}$-categories and their cohomological
counterparts, $A_{\infty}$-functors between $A_{\infty}$-categories,
themselves related through pre-natural transformations. After a brief
technical digression in Hochschild cohomology, we give the notion of homotopic
$A_{\infty}$-functors and see how they take part in the fabrication of
$A_{\infty}$-categories. Then we move one step up, presenting
$A_{\infty}$-modules over $A_{\infty}$-categories, which offer a way of
studying the latters indirectly.
The basic vocabulary being set, we discuss the matter of units, with special
focus on cohomologically unital $A_{\infty}$-categories and quasi-equivalences
thereof, in fact the “best calibrated” sort of $A_{\infty}$-functors appearing
in most propositions. Finally, we define Yoneda embeddings, which are special
$A_{\infty}$-functors fundamental in bridging $A_{\infty}$-categories to their
respective category of $A_{\infty}$-modules.

  * •

In Chapter 3, we discuss the idea of triangulated categories, a major topic in
homological algebra, first axiomatized by Jean-Louis Verdier (in [Ver96]).
After a few preliminaries, we highlight some fundamental operations and
constructions for $A_{\infty}$-modules, particularly an abstract parent of the
well-known mapping cone from topology and some special diagrams called exact
triangles.
Shortly afterwards, we will see these are in fact part of the program to
extend triangulation to the $A_{\infty}$-world, ultimately resulting in the
$A_{\infty}$-category $Tw\mathcal{A}$ of twisted complexes. The latter is then
applied to produce a couple criteria of triangularity for
$A_{\infty}$-categories, finally proving that their cohomological categories
are triangulated in the classical sense. This culminates in the definition of
bounded derived category $\mathsf{D^{b}}(\mathcal{A})$.

  * •

In Chapter 4, we enlarge our construction to account for direct summands
forming objects of $\mathsf{D^{b}}(\mathcal{A})$. We explore the classical
notion of split images and split-closure, and later translate it to the
abstract realm of $A_{\infty}$-categories. We use the so obtained technology
to finally produce the split-closed derived category
$\mathsf{D}^{\pi}(\mathcal{A})$. At the end, we also discuss qualitatively the
twisting procedure, which links the algebra of Fukaya categories to their
subtended geometry.

References for Chapter 1 are [Kel01] and [Kel06]. Chapter 2 to 4 are based on
part I of the excellent book [Sei08] by Paul Seidel, currently one of the
major experts in the area. If not otherwise explicited, every notion must be
traced back to it. Also, at the beginning of each chapter we dedicate a
section to the necessary prerequisites from category theory and homological
algebra; especially when it comes to derived categories and functors, we will
refer to [GM03]. However, the advanced reader might skip these preliminaries
so to preserve a smoother reading experience.

###  1 $A_{\infty}$-algebras

####  1.1 Example of non-associativity: $A_{\infty}$-spaces

Let us begin our study of $A_{\infty}$-algebras from a topological example
which delivers most clearly the concept of non-associativity. Let $(X,x_{0})$
be a pointed topological space, $\Omega X\coloneqq\\{f:[0,1]\rightarrow X\mid
f\text{ continuous, }f(0)=f(1)=x_{0}\\}$ the space of loops in $X$ based at
$x_{0}\in X$. We can define a composition map by concatenating loops:
$m_{2}:\Omega X\times\Omega X\rightarrow\Omega
X,\;(f_{1},f_{2})\mapsto(f_{1}\ast f_{2})$, where

$(f_{1}\ast f_{2})(t)\coloneqq\begin{cases}f_{1}(2t)&$if $0\leq t\leq 1/2\\\
f_{2}(2t-1)&$if $1/2\leq t\leq 1\end{cases}\;,$

so that the resulting loop “spends half the time” along $f_{1}$, then half
along $f_{2}$. Obviously, $m_{2}$ is not associative, since
$m_{2}(m_{2}(f_{1},f_{2}),f_{3})\neq m_{2}(f_{1},m_{2}(f_{2},f_{3}))$ for a
third $f_{3}\in\Omega X$ (in the former concatenation only a “quarter of time”
is spent along $f_{1}$, against half in the latter).

However, we can modify the time distribution via a homotopy
$m_{3}:[0,1]\times(\Omega X)^{3}\rightarrow\Omega X$ which continuously
deforms the concatenation $(f_{1}\ast f_{2})\ast f_{3}$ into
$f_{1}\ast(f_{2}\ast f_{3})$. The same $m_{3}$ can be used to move between the
five possible concatenations of four loops: $(f_{1}\ast(f_{2}\ast f_{3}))\ast
f_{4}$, $((f_{1}\ast f_{2})\ast f_{3})\ast f_{4}$, $(f_{1}\ast
f_{2})\ast(f_{3}\ast f_{4})$, $f_{1}\ast(f_{2}\ast(f_{3}\ast f_{4}))$ and
$f_{1}\ast((f_{2}\ast f_{3})\ast f_{4})$. Drawing schematically these
configurations as vertices of a graph whose sides represent the path of
$m_{3}$, one ends up with the boundary of a pentagon. Call it $K_{4}$.

In turn, this can be extended to generic $(n-2)$-dimensional polytopes $K_{n}$
by iteratively defining homotopies $m_{n}:K_{n}\times(\Omega
X)^{n}\rightarrow\Omega X$ for $n>3$ (we set $K_{2}\coloneqq\\{*\\}$ and
$K_{3}\coloneqq[0,1]$; see Figure 1).

Figure 1: The first four associahedra $K_{2},K_{3},K_{4},K_{5}$. Each
concatenation at the vertices can be represented by a binary tree: for
example, $K_{2}$ consists of a single point $(f_{1}\ast f_{2})\in\Omega X$,
whose binary tree is trivial; $K_{3}$ has two vertices, $((f_{1}\ast
f_{2})\ast f_{3})$ and $(f_{1}\ast(f_{2}\ast f_{3}))$ respectively; the
concatenations forming $K_{4}$ are listed above. On the other hand, higher
dimensional cells parametrize trees with more nodal ramifications: for
example, the 1-cell of $K_{3}$ is labeled by a trifurcated tree, encoding
$f_{1}\ast f_{2}\ast f_{3}$.
[Source: http://pseudomonad.blogspot.com/2010/08/m-theory-lesson-345.html]

These $K_{n}$ are called associahedra, first constructed by Stasheff in
[Sta63]. He included them in his definition of $A_{\infty}$-spaces:
topological spaces $Y$ with a collection of compatible maps $m_{n}:K_{n}\times
Y^{n}\rightarrow Y$, for $n\geq 2$. As we have just seen, the loop space
$\Omega X$ is a prototypical example of $A_{\infty}$-space. Moreover, for any
such $Y$, the singular cochain complex $C^{\bullet}(Y)$ serves as our first
case of an $A_{\infty}$-algebra. Let us see why.

####  1.2 $A_{\infty}$-algebras and morphisms

Throughout this thesis, we will work over a fixed ground field $\mathbb{K}$
and assume that gradings are taken over $\mathbb{Z}$, unless otherwise stated.
We start with a basic review of graded vector spaces and complexes (cf.
[Kel06, section 2.1]).

######  Definition 1.1.

Let $V$ be a $\mathbb{Z}$-graded $\mathbb{K}$-vector space, thus of the form
$V=\bigoplus_{p\in\mathbb{Z}}V^{p}$. Its $n$-fold shift $V[n]$, for
$n\in\mathbb{Z}$, is the graded vector space with all indices shifted $n$
times downwards, so that $(V[n])^{p}=V^{p+n}$ for all $p\in\mathbb{Z}$. We
write $SV\coloneqq V[1]$ and call it the suspension of $V$.

Given another graded vector space $V^{\prime}$, a morphism of graded vector
spaces $f:V\rightarrow V^{\prime}[d]$ is a homogeneous graded
$\mathbb{K}$-linear map of degree $d=|f|\in\mathbb{Z}$ (so that a homogeneous
element $v\in V$ of degree $p=|v|$ will be mapped to a homogeneous element
$f(v)\in V^{\prime}$ of degree $p+d$; for practicality, we will often suppress
information about the degree of morphisms).

Graded vector spaces and their morphisms form the category
$\mathsf{GradVect}$.

When considering the tensor product $f\otimes g:V\otimes W\rightarrow
V^{\prime}\otimes W^{\prime}$ of two morphisms of graded vector spaces
$f:V\rightarrow V^{\prime}$ and $g:W\rightarrow W^{\prime}$, the output is
subject to the Koszul sign rule:

$(f\otimes g)(v\otimes w)=(-1)^{|g||v|}f(v)\otimes g(w)\;,$

for all homogeneous elements $v\in V,w\in W$.

######  Definition 1.2.

A graded vector space $V$ is a (cochain) complex when endowed with a
differential $d_{V}:V\rightarrow V[1]$, a homogeneous morphism of degree 1
such that $d_{V}\circ d_{V}=0$. We define $d_{SV}\coloneqq-d_{V}$, while

$d(f)\coloneqq d_{V^{\prime}}\circ f-(-1)^{|f|}f\circ d_{V}$

for a morphism of graded vector spaces $f:V\rightarrow V^{\prime}$. The latter
becomes a morphism of complexes (a cochain map) if and only if $d(f)=0$.

Complexes and their morphisms form the category $\mathsf{Ch}$. We can
therefore study its cohomological counterpart $H(\mathsf{Ch})$, whose objects
are graded vector spaces $H(V)=\bigoplus_{p\in\mathbb{Z}}H^{p}(V)$ and whose
morphisms are the induced graded morphisms (indeed
$H(\mathsf{Ch})\cong\mathsf{GradVect}$ as categories).

Finally, we recall that two morphisms of complexes $f$ and $f^{\prime}$ are
homotopic if and only if there exists a morphism of graded vector spaces $h$
such that $f-f^{\prime}=d(h)$. Homological algebra then tells us that the
induced maps in cohomology are the same: $H^{p}(f)=H^{p}(f^{\prime})$ for all
$p\in\mathbb{Z}$.

We proceed to define $A_{\infty}$-algebras, also known as strongly homotopy
associative algebras, for reasons that will become clear in Remark 1.5 below.
The main reference is [Kel01, sections 3–4].

######  Definition 1.3.

An $A_{\infty}$-algebra over $\mathbb{K}$ is a $\mathbb{Z}$-graded
$\mathbb{K}$-vector space

$A=\bigoplus_{p\in\mathbb{Z}}A^{p}$

endowed with graded morphisms $m_{n}:A^{\otimes n}\rightarrow A[2-n]$, for
$n\geq 1$, satisfying the equations

$\sum_{r+s+t=n}\mkern-12.0mu(-1)^{r+st}m_{r+1+t}(\textup{id}_{A}^{\otimes
r}\otimes m_{s}\otimes{\textup{id}_{A}^{\otimes t}})=0\;,$ (1)

as maps $A^{\otimes n}\rightarrow A$, for all $n\geq 1$.

$A$ is strictly unital if endowed with some $1_{A}\in A^{0}$ such that
$m_{1}(1_{A})=0,\;m_{2}(1_{A},a)=a=m_{2}(a,1_{A})$ for all $a\in A$ and
$m_{i}(...,1_{A},...)=0$ if $i>2$. (If it exists, $1_{A}$ is unique; we call
it strict unit.)

An $A_{\infty}$-subalgebra $B$ of $A$ is a graded subspace such that all
morphisms $m_{n}$ restrict to $B$.

Here we won’t concern ourselves too much about the signs appearing in (1),
since explicit inputs will modify it according to the Koszul rule from
Definition 1.1. We will be more precise when discussing
$A_{\infty}$-categories, where all formulae are presented in this fashion.

######  Remark 1.4.

If one wishes to make the degree of each morphism $m_{n}$ of Definition 1.3
independent from $n$, there is a canonical way to do it using suspensions:
define a bijective correspondence between the $m_{n}$’s and the graded maps
$b_{n}:(SA)^{\otimes n}\rightarrow SA$ by imposing commutativity of the
diagram

${A^{\otimes n}}$${A}$${(SA)^{\otimes
n}}$${SA}$$\scriptstyle{m_{n}}$$\scriptstyle{s^{\otimes
n}}$$\scriptstyle{s}$$\scriptstyle{b_{n}}$

for all $n\geq 1$. Here $s:A\rightarrow SA,\;a\in A^{p}\mapsto a\in(SA)^{p-1}$
has degree $-1$, so that each $b_{n}$ has $|b_{n}|=-(-1)\cdot n+(2-n)+(-1)=1$.
In particular, $b_{1}=-m_{1}$ and
$b_{2}(a_{1},a_{2})=(-1)^{|a_{1}|}m_{2}(a_{1},a_{2})$. One can prove that the
$m_{n}$’s give an $A_{\infty}$-algebra structure on $A$ if and only if

$\sum_{r+s+t=n}\mkern-12.0mub_{r+1+t}(\textup{id}_{A}^{\otimes r}\otimes
b_{s}\otimes{\textup{id}_{A}^{\otimes t}})=0$

for all $n\geq 1$, where we note that the signs in (1) have vanished.

Concretely, we will almost never work with suspensions, but one should account
for them when trying to put together the various sign conventions.

######  Remark 1.5.

The set of equations (1) yields the following consequences:

  * •

$m_{1}m_{1}=0$, so that $(A,m_{1})$ is a cochain complex (then, by definition,
so is $(SA,-m_{1})$).

  * •

$m_{1}m_{2}=m_{2}(m_{1}\otimes\textup{id}_{A}+\textup{id}_{A}\otimes m_{1})$
makes the differential $m_{1}$ a derivation with respect to the
multiplication222We refer to it as “multiplication” because, in the particular
situation when $A^{p}=0$ for all $p\neq 0$, $A=A^{0}$ has $m_{2}$ as only non-
vanishing map, of degree 0, hence thought of as multiplication for the
ordinary associative algebra $A$. $m_{2}$ (that is, compatible through the
graded Leibniz rule — as one would expect). Since
$(m_{1}\otimes\textup{id}_{A}+\textup{id}_{A}\otimes m_{1})$ is the
differential of $A^{\otimes 2}$, $m_{2}$ is a morphism of complexes.

  * •

$m_{2}(\textup{id}_{A}\otimes
m_{2}-m_{2}\otimes\textup{id}_{A})=m_{1}m_{3}+m_{3}(m_{1}\otimes\textup{id}_{A}\otimes\textup{id}_{A}+\textup{id}_{A}\otimes
m_{1}\otimes\textup{id}_{A}$ $+\textup{id}_{A}\otimes\textup{id}_{A}\otimes
m_{1})=d(m_{3})$ tells us that $m_{2}$ is associative only up to homotopy!
(whence the terminology “strongly homotopy associative” algebra). Therefore,
the corresponding cohomology $H(A)$ is an associative graded algebra with
respect to $H(m_{2})$ (its unit element, if it exists, is induced by the
cohomological unit of $A$).

  * •

if $m_{n}=0$ for all $n>2$, then $(A,m_{1},m_{2})$ turns into an
associative333Here $d(m_{3})=0$ vacuously. differential $\mathbb{Z}$-graded
algebra; conversely, any such algebra can be seen as particular instance of an
$A_{\infty}$-algebra.

The second bullet point exposes the nature of $A_{\infty}$-algebras as
algebras which are associative up to higher homotopies (and possibly lack
identities). Also, the third bullet point already foreshadows the advantage of
working at cohomological level, one of our guiding principles in the theory to
come.

######  Definition 1.6.

Given two $A_{\infty}$-algebras $A$ and $B$, with morphisms $m_{n}^{A}$
respectively $m_{n}^{B}$, a morphism of $A_{\infty}$-algebras $f:A\rightarrow
B$ is a sequence of graded maps $f_{n}:A^{\otimes n}\rightarrow B[1-n]$, for
$n\geq 1$, fulfilling the equations

$\sum_{r+s+t=n}\mkern-12.0mu(-1)^{r+st}f_{r+1+t}(\textup{id}_{A}^{\otimes
r}\otimes m_{s}^{A}\otimes{\textup{id}_{A}^{\otimes
t}})=\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq r\leq n\\\
s_{1}+...+s_{r}=n\end{subarray}}\mkern-24.0mu(-1)^{u}m_{r}^{B}(f_{s_{1}}\otimes...\otimes
f_{s_{r}})\;$ (2)

for all $n\geq 1$, where $u=\coloneqq\sum_{i=1}^{r-1}(r-i)(s_{i}-1)$.

Composition with a second morphism of $A_{\infty}$-algebras $g:B\rightarrow C$
is specified by the collection of graded maps

$(g\circ f)_{n}\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq r\leq n\\\
s_{1}+...+s_{r}=n\end{subarray}}\mkern-24.0mu(-1)^{u}g_{r}\circ(f_{s_{1}}\otimes...\otimes
f_{s_{r}})$

(plugging them into (2) shows that indeed $g\circ f:A\rightarrow C$ is a
morphism of $A_{\infty}$-algebras).

For $A=B$, it is easy to see that $f$ with $f_{1}\coloneqq\textup{id}_{A}$ and
$f_{n}\coloneqq 0$ otherwise corresponds to the identity morphism on $A$.

In particular, the first two equalities of (2),
$f_{1}m_{1}^{A}=m_{1}^{B}f_{1}$ and $f_{1}m_{2}^{A}-m_{2}^{B}(f_{1}\otimes
f_{1})=m_{1}^{B}f_{2}+f_{2}(m_{1}^{A}\otimes\textup{id}_{A}+\textup{id}_{A}\otimes
m_{1}^{A})=d(f_{2})$, tell us that $f_{1}$ is a morphism of complexes
commuting with $m_{2}$ only up to homotopy. Again, the signs in (2) can be
dispensed with upon use of suspensions (see Remark 1.4). Finally, we present a
few definitions that will reappear in their $A_{\infty}$-categorical
formulation.

######  Definition 1.7.

A morphism of $A_{\infty}$-algebras $f:A\rightarrow B$ is called a quasi-
isomorphism if $f_{1}$ is a quasi-isomorphism, that is, if it induces an
algebra isomorphism $H(f_{1}):H(A)\rightarrow H(B)$.

A morphism of strictly unital $A_{\infty}$-algebras
$f:(A,1_{A})\rightarrow(B,1_{B})$ is itself called strictly unital if
$f_{1}(1_{A})=1_{B}$ and $f_{i}(...,1_{A},...)=0$ for $i>1$.

####  1.3 $A_{\infty}$-modules and morphisms

######  Definition 1.8.

Let $A$ be an $A_{\infty}$-algebra, equipped with morphisms $m_{n}^{A}$ as in
Definition 1.3. An $A_{\infty}$-module over $A$ is a $\mathbb{Z}$-graded
$\mathbb{K}$-vector space

$M=\bigoplus_{p\in\mathbb{Z}}M^{p}$

endowed with maps $m_{n}^{M}:M\otimes A^{\otimes n-1}\rightarrow M[2-n]$, for
$n\geq 1$, fulfilling equations similar to (1):

$\begin{split}\sum_{r+s+t=n}\mkern-12.0mu(-1)^{r+st}m_{r+1+t}^{M}(\textup{id}_{M}\otimes\textup{id}_{A}^{\otimes
r-1}\otimes m_{s}^{A}\otimes\textup{id}_{A}^{\otimes t})&=0\qquad\text{if
}r>0,\\\
\sum_{s+t=n}\mkern-6.0mu(-1)^{st}m_{1+t}^{M}(m_{s}^{M}\otimes\textup{id}_{A}^{\otimes
t})&=0\qquad\text{if }r=0.\end{split}$ (3)

######  Definition 1.9.

Let $M$ and $N$ be $A_{\infty}$-modules over an $A_{\infty}$-algebra $A$, with
morphisms $m_{n}^{M}$ respectively $m_{n}^{N}$. A morphism of
$A_{\infty}$-modules $f:M\rightarrow N$ is a sequence of graded maps
$f_{n}:M\otimes A^{\otimes n-1}\rightarrow N[1-n]$, for $n\geq 1$, fulfilling
the equations

$\sum_{r+s+t=n}\mkern-12.0mu(-1)^{r+st}f_{r+1+t}(\textup{id}_{A}^{\otimes
r}\otimes m_{s}^{M}\otimes\textup{id}_{A}^{\otimes
t})=\mkern-6.0mu\sum_{s+t=n}\mkern-6.0mu(-1)^{(s+1)t}m_{1+t}^{N}(f_{s}\otimes\textup{id}_{A}^{\otimes
t})\;$ (4)

for all $n\geq 1$, where a distinction between the cases $r>0$ and $r=0$ as in
last definition must be made.

Composition with a second morphism of $A_{\infty}$-modules $g:N\rightarrow P$
is specified by a collection of graded maps

$(g\circ
f)_{n}\coloneqq\mkern-6.0mu\sum_{s+t=n}\mkern-6.0mu(-1)^{(s-1)t}f_{1+t}\circ(g_{s}\otimes\textup{id}_{A}^{\otimes
t})$

(which can be shown to satisfy equations (4)). The definitions of identity
morphism and quasi-isomorphism are identical to the $A_{\infty}$-algebra case.

$A_{\infty}$-modules together with morphisms of $A_{\infty}$-modules form a
category, which we denote by $\mathsf{C}_{\infty}(A)$.

Now that we have had a little taste of the kind of algebraic structures that
Part I deals with, let us move to the more general notion of
$A_{\infty}$-categories.

###  2 $A_{\infty}$-categories

####  2.1 Category theoretic preliminaries

Before studying $A_{\infty}$-categories, we summarize here some basic
definitions and results from category theory which will appear frequently and
play a key role in the proofs to come. Recall that we are working over some
fixed ground field $\mathbb{K}$.

######  Definition 2.1.

A category $\mathsf{C}$ is said to be:

  * •

linear and graded if all its Hom-sets are graded $\mathbb{K}$-vector spaces
and composition of compatible morphisms is a bilinear operation over
$\mathbb{K}$;

  * •

differential $\mathbb{Z}$-graded (dg) if each Hom-set is a $\mathbb{Z}$-graded
(cochain) complex, so that composition of morphisms is a cochain map and the
differential of each identity is zero (example: the category $\mathsf{Ch}$
from Definition 1.2);

  * •

small if both obj($\mathsf{C}$) and the collection of all Hom-sets are sets;

  * •

non-unital if possibly lacking some identity morphisms (if so, $\mathsf{C}$ is
not an actual category).

######  Definition 2.2.

Let $\mathsf{C}$ and $\mathsf{D}$ be categories. A functor
$\mathsf{F}:\mathsf{C}\rightarrow\mathsf{D}$ is:

  * •

an isomorphism if there exists a functor
$\mathsf{G}:\mathsf{D}\rightarrow\mathsf{C}$ such that
$\mathsf{G}\circ\mathsf{F}=\mathsf{Id_{C}}$ and
$\mathsf{F}\circ\mathsf{G}=\mathsf{Id_{D}}$ as functors; in this case
$\mathsf{C}$ and $\mathsf{D}$ are called isomorphic;

  * •

an equivalence if there exists a functor
$\mathsf{G}:\mathsf{D}\rightarrow\mathsf{C}$ (sometimes called quasi-inverse)
and natural isomorphisms
$\Phi:\mathsf{Id_{C}}\rightarrow\mathsf{G}\circ\mathsf{F}$ and
$\Psi:\mathsf{F}\circ\mathsf{G}\rightarrow\mathsf{Id_{D}}$ (meaning that each
$\Phi_{X}\in\textup{Hom}_{\mathsf{C}}\big{(}X,\mathsf{G}(\mathsf{F}(X))\big{)}$
for $X\in\textup{obj}(\mathsf{C})$ and each
$\Psi_{Y}\in\textup{Hom}_{\mathsf{D}}(\mathsf{F}(\mathsf{G}(Y)),Y)$ for
$Y\in\textup{obj}(\mathsf{D})$ is an isomorphism of objects); then
$\mathsf{C}$ and $\mathsf{D}$ are called equivalent;

  * •

full and faithful if
$\mathsf{F}_{X,Y}:\textup{Hom}_{\mathsf{C}}(X,Y)\rightarrow\textup{Hom}_{\mathsf{D}}(\mathsf{F}(X),\mathsf{F}(Y))$
is surjective respectively injective for all $X,Y\in$ obj$(\mathsf{C})$ (fully
faithful if both);

  * •

essentially surjective if for all $Y\in$ obj($\mathsf{D}$) there are an $X\in$
obj($\mathsf{C}$) and an isomorphism of objects $f\in$
Hom${}_{\mathsf{D}}(Y,\mathsf{F}(X))$;

  * •

an embedding if faithful and injective on objects (that is,
$\mathsf{F}(X_{1})=\mathsf{F}(X_{2})\in\textup{obj}(\mathsf{D})$ implies
$X_{1}=X_{2}\in\textup{obj}(\mathsf{C})$);

  * •

linear and graded if each $\mathsf{F}_{X,Y}$ as above is a
$\mathbb{K}$-linear, graded morphism between graded vector spaces (assuming
$\mathsf{C},\mathsf{D}$ are linear, graded);

  * •

(non-)unital if (not) carrying all identity morphisms to identity morphisms.

The following results are an easy consequence of above definitions (for
example, their proofs can be found in [GM03]).

######  Lemma 2.3.

A functor $\mathsf{F}:\mathsf{C}\rightarrow\mathsf{D}$ is:

  1. i.

an isomorphism if and only if it is bijective on objects and Hom-sets,

  2. ii.

an equivalence if and only if it is full, faithful and essentially surjective.

In particular, isomorphisms are equivalences, thus full and faithful.

[frame]Henceforth, we will assume that all our ($A_{\infty}$-)categories are
linear and small,
unless otherwise stated.

####  2.2 $A_{\infty}$-categories and $A_{\infty}$-functors

######  Definition 2.4.

A non-unital $A_{\infty}$-category $\mathcal{A}$ consists of the following
data:

  * –

a set of objects obj($\mathcal{A}$),

  * –

for any pair $X_{0},X_{1}\in$ obj($\mathcal{A}$), a $\mathbb{Z}$-graded
$\mathbb{K}$-vector space

$\textup{hom}_{\mathcal{A}}(X_{0},X_{1})=\bigoplus_{p\in\mathbb{Z}}\textup{hom}_{\mathcal{A}}^{p}(X_{0},X_{1})\;,$

whose elements we still call morphisms (homogeneous of degree $p$ when
belonging to $\textup{hom}_{\mathcal{A}}^{p}(X_{0},X_{1})$),

  * –

(multi)linear graded maps

$\mu_{\mathcal{A}}^{d}:\textup{hom}_{\mathcal{A}}(X_{d-1},X_{d})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{A}}(X_{0},X_{d})[2-d]$
(5)

for every $d\geq 1$ and $X_{0},...,X_{d}\in\textup{obj}(\mathcal{A})$, which
we refer to as composition maps. These must satisfy the quadratic
$A_{\infty}$-associativity equations

$\mkern-30.0mu\sum_{\begin{subarray}{c}1\leq m\leq d\\\ \quad 0\leq n\leq
d-m\end{subarray}}\mkern-24.0mu(-1)^{\dagger_{n}}\mu_{\mathcal{A}}^{d-m+1}(a_{d},...,a_{n+m+1},\underbrace{\mu_{\mathcal{A}}^{m}(a_{n+m},...,a_{n+1})}_{\in\textup{hom}_{\mathcal{A}}(X_{n},X_{n+m})},a_{n},...,a_{1})=0$
(6)

in $\textup{hom}_{\mathcal{A}}(X_{0},X_{d})$, where
$a_{i}\\!\in\\!\textup{hom}_{\mathcal{A}}(X_{i-1},X_{i})$ and
$\dagger_{n}\\!\coloneqq\\!(\sum_{i=1}^{n}|a_{i}|)-n$.444$\dagger_{n}$, along
with other variables regulating the signs of our expressions, will be
henceforth used as standard notation, without further comment.For $d=1,2$, the
first two such equations read
$\mu_{\mathcal{A}}^{1}(\mu_{\mathcal{A}}^{1}(a_{1}))=0$
and$\mu_{\mathcal{A}}^{1}(\mu_{\mathcal{A}}^{2}(a_{2},a_{1}))=(-1)^{|a_{1}|}\mu_{\mathcal{A}}^{2}(\mu_{\mathcal{A}}^{1}(a_{2}),a_{1})-\mu_{\mathcal{A}}^{2}(a_{2},\mu_{\mathcal{A}}^{1}(a_{1}))$.

An $A_{\infty}$-subcategory $\mathcal{B}\subset\mathcal{A}$ is an
$A_{\infty}$-category such that
$\textup{obj}(\mathcal{B})\subset\textup{obj}(\mathcal{A})$,
$\textup{hom}_{\mathcal{B}}(X_{0},X_{1})\subset\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$
are graded subspaces for all $X_{0},X_{1}\in\textup{obj}(\mathcal{B})$, and
the composition maps $\mu_{\mathcal{B}}^{d}$ have target spaces within
$\mathcal{B}$.

The labeling system in (5) and (6) can be somewhat misleading: it is an
attempt to reconcile various standard conventions. For consistency’s sake, we
will be faithful, here and in the future, to the notation adopted by Seidel in
[Sei08].

######  Remark 2.5.

Consider the case where $\text{obj}(\mathcal{A})\coloneqq\\{X\\}$. Then the
only possible graded vector space $A\coloneqq\textup{hom}_{\mathcal{A}}(X,X)$
endowed with graded maps $m_{n}\coloneqq\mu_{\mathcal{A}}^{n}$ is indeed an
$A_{\infty}$-algebra, since the equations (1) precisely coincide with those in
(6) (possibly up to a sign).

Moreover, an $A_{\infty}$-category $\mathcal{A}$ with
$\mu_{\mathcal{A}}^{d}=0$ if $d>2$ is a differential graded category, and vice
versa (for example, the dg category $\mathsf{Ch}$ can be considered an
$A_{\infty}$-category).

The $A_{\infty}$-associativity equations yield the exact same conclusions we
made in Remark 1.5 for $A_{\infty}$-algebras: non-unital
$A_{\infty}$-categories are not categories in general, since they may lack
identity morphisms and each composition map $\mu_{\mathcal{A}}^{2}$ is not
necessarily associative.

However, $\mu_{\mathcal{A}}^{1}\mu_{\mathcal{A}}^{1}=0$, so that each morphism
space $\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ is a cochain complex with
differential
$\mu_{\mathcal{A}}^{1}:\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{A}}(X_{0},X_{1})[1]$.
Hence, it makes sense to talk about cocycles (any
$a\in\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ such that
$\mu_{\mathcal{A}}^{1}(a)=0$), coboundaries (any $a=\mu_{\mathcal{A}}^{1}(b)$
for some $b\in\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$) and cohomology:

######  Definition 2.6.

Let $\mathcal{A}$ be a non-unital $A_{\infty}$-category. Its cohomological
category $H(\mathcal{A})$ is defined as follows:
$\textup{obj}(H(\mathcal{A}))\coloneqq\text{obj}(\mathcal{A})$, each pair
$X_{0},X_{1}\in\textup{obj}(\mathcal{A})$ yields a graded cohomology group

$\text{Hom}_{H(\mathcal{A})}(X_{0},X_{1})\coloneqq
H(\textup{hom}_{\mathcal{A}}(X_{0},X_{1}))=\bigoplus_{p\in\mathbb{Z}}H^{p}(\textup{hom}_{\mathcal{A}}(X_{0},X_{1}))\,,$
(7)

and the “usual” associative (!) composition map is555Observe the choice of
notation: whenever we speak of cohomological categories, we adopt the capital
“H” in “Hom” (against “hom” for $A_{\infty}$-categories) and the composition
symbol “$\circ$”.

$\displaystyle\circ:\;$
$\displaystyle\textup{Hom}_{H(\mathcal{A})}(X_{1},X_{2})\times\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{1})\rightarrow\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{2}),$
(8) $\displaystyle(\langle a_{2}\rangle,\langle a_{1}\rangle)\mapsto\langle
a_{2}\rangle\circ\langle
a_{1}\rangle\coloneqq(-1)^{|a_{1}|}\langle\mu_{\mathcal{A}}^{2}(a_{2},a_{1})\rangle\;.$

This is a non-unital ordinary linear graded category.

We designate by $H^{0}(\mathcal{A})\subset H(\mathcal{A})$ the zeroth
cohomological subcategory, having
$\textup{obj}(H^{0}(\mathcal{A}))\\!\coloneqq\\!\textup{obj}(H(\mathcal{A}))$
and
$\text{Hom}_{H^{0}(\mathcal{A})}(X_{0},X_{1})\\!\coloneqq\\!H^{0}(\textup{hom}_{\mathcal{A}}(X_{0},X_{1}))$.

######  Definition 2.7.

A non-unital $A_{\infty}$-functor
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ between two non-unital
$A_{\infty}$-categories $\mathcal{A}$ and $\mathcal{B}$ is a map
$\mathcal{F}:\textup{obj}(\mathcal{A})\rightarrow\textup{obj}(\mathcal{B})$
together with a sequence of (multi)linear graded maps

$\mkern-9.0mu\mathcal{F}^{d}:\textup{hom}_{\mathcal{A}}(X_{d-1},X_{d})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{B}}(\mathcal{F}(X_{0}),\mathcal{F}(X_{d}))[1-d]$
(9)

for every $d\geq 1$ and $X_{0},...,X_{d}\in$ obj($\mathcal{A}$), fulfilling
the equations

$\displaystyle\sum_{\begin{subarray}{c}1\leq r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0mu\mu_{\mathcal{B}}^{r}(\mathcal{F}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,\mathcal{F}^{s_{1}}(a_{s_{1}},...,a_{1}))=$
(10) $\displaystyle\;\sum_{\begin{subarray}{c}1\leq m\leq d\\\ 0\leq n\leq
d-m\end{subarray}}\mkern-18.0mu(-1)^{\dagger_{n}}\mathcal{F}^{d-m+1}(a_{d},...,a_{n+m+1},\mu_{\mathcal{A}}^{m}(a_{n+m},...,a_{n+1}),a_{n},...,a_{1})\;,$

in $\textup{hom}_{\mathcal{B}}(\mathcal{F}(X_{0}),\mathcal{F}(X_{d}))$, where
$a_{i}\in\textup{hom}_{\mathcal{A}}(X_{i-1},X_{i})$.

Given another non-unital $A_{\infty}$-functor
$\mathcal{G}:\mathcal{B}\rightarrow\mathcal{C}$, the (strictly associative)
composition $\mathcal{G}\circ\mathcal{F}:\mathcal{A}\rightarrow\mathcal{C}$ is
the $A_{\infty}$-functor specified by

$(\mathcal{G}\circ\mathcal{F})^{d}(a_{d},...,a_{1})\coloneqq\mkern-18.0mu\sum_{\begin{subarray}{c}1\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-18.0mu\mathcal{G}^{r}(\mathcal{F}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,\mathcal{F}^{s_{1}}(a_{s_{1}},...,a_{1}))$
(11)

in
$\textup{hom}_{\mathcal{C}}\big{(}\mathcal{G}(\mathcal{F}(X_{0})),\mathcal{G}(\mathcal{F}(X_{d}))\big{)}$
(one can show that these do satisfy the defining equations (10)). In
particular,
$(\mathcal{G}\circ\mathcal{F})^{1}(a_{1})=\mathcal{G}^{1}(\mathcal{F}^{1}(a_{1}))$
and
$(\mathcal{G}\circ\mathcal{F})^{2}(a_{2},a_{1})=\mathcal{G}^{1}(\mathcal{F}^{2}(a_{2},a_{1}))+\mathcal{G}^{2}(\mathcal{F}^{1}(a_{2}),\mathcal{F}^{1}(a_{1}))$.

If $\mathcal{A}=\mathcal{B}$ and $\mathcal{A}$ is unital, the identity
$A_{\infty}$-functor $\mathcal{I}d_{\mathcal{A}}$ is given by
$\mathcal{I}d_{\mathcal{A}}(X)\coloneqq X$,
$\mathcal{I}d_{\mathcal{A}}^{1}\coloneqq\textup{id}_{\textup{hom}_{\mathcal{A}}(X,X)}$
for $X\in\textup{obj}(\mathcal{A})$, and
$\mathcal{I}d_{\mathcal{A}}^{k}\coloneqq 0$ for all $k>1$.

As expected, an $A_{\infty}$-functor between one-object
$A_{\infty}$-categories is just an $A_{\infty}$-algebra morphism.

From (10) we deduce that the respective first order maps commute: for any
$a_{1}\in\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ holds
$\mu_{\mathcal{B}}^{1}(\mathcal{F}^{1}(a_{1}))\\!=\\!\mathcal{F}^{1}(\mu_{\mathcal{A}}^{1}(a_{1}))$.
The case $d=2$ produces instead
$\mu_{\mathcal{B}}^{1}(\mathcal{F}^{2}(a_{2},a_{1}))+\mu_{\mathcal{B}}^{2}(\mathcal{F}^{1}(a_{2}),\mathcal{F}^{1}(a_{1}))=\mathcal{F}^{1}(\mu_{\mathcal{A}}^{2}(a_{2},a_{1}))+\mathcal{F}^{2}(a_{2},\mathcal{A}^{1}(a_{1}))$
$+(-1)^{|a_{1}|-1}\mathcal{F}^{2}(\mu_{\mathcal{A}}^{1}(a_{2}),$ $a_{1})$.
These equations are compatible with what we obtained from Definition 1.6
(again, up to some signs).

######  Definition 2.8.

A non-unital $A_{\infty}$-functor
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ induces a cohomological
functor $H(\mathcal{F}):H(\mathcal{A})\rightarrow H(\mathcal{B})$ which maps
$X\in\textup{obj}(H(\mathcal{A}))\mapsto\mathcal{F}(X)\in\textup{obj}(H(\mathcal{B}))$
and

$\langle
a_{1}\rangle\in\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{1})\longmapsto\langle\mathcal{F}^{1}(a_{1})\rangle\in\textup{Hom}_{H(\mathcal{B})}(\mathcal{F}(X_{0}),\mathcal{F}(X_{1}))\;,$
(12)

for each $X_{0},X_{1}\in\textup{obj}(H(\mathcal{A}))$. This is an ordinary
non-unital linear graded functor. We call $\mathcal{F}$:

  * •

a quasi-isomorphism if $H(\mathcal{F})$ is an isomorphism;

  * •

cohomologically full and faithful if $H(\mathcal{F})$ is full and faithful;

  * •

cohomologically essentially surjective if $H(\mathcal{F})$ is essentially
surjective.

Note that $H(\mathcal{F}):H(\mathcal{A})\rightarrow H(\mathcal{B})$ itself
reduces to $H^{0}(\mathcal{F}):H^{0}(\mathcal{A})\rightarrow
H^{0}(\mathcal{B})$ on the zeroth cohomological categories.

As an immediate consequence of Lemma 2.3, quasi-isomorphisms are
cohomologically full and faithful $A_{\infty}$-functors.

There is a particular instance of $A_{\infty}$-functors which will become
useful in simplifying the setting of several proofs. The definition exploits
the following (non-trivial) fact: given an $A_{\infty}$-category $\mathcal{A}$
and a sequence of graded maps $(\Phi^{0},\Phi^{1},...)$ as in (9) such that
$\Phi^{1}:\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$
is a linear automorphism, recursive solving of the equations (10) with respect
to $\mu_{\mathcal{B}}$ produces a unique alternative $A_{\infty}$-structure
$\tilde{\mathcal{A}}$ for $\mathcal{A}$, given by
$\textup{obj}(\tilde{\mathcal{A}})\coloneqq\textup{obj}(\mathcal{A})$,
$\textup{hom}_{\tilde{\mathcal{A}}}(X_{0},X_{1})\coloneqq\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$
and $\mu_{\tilde{\mathcal{A}}}\coloneqq\mu_{\mathcal{B}}$. Then the
multilinear graded maps $\Phi^{d}$ fulfill (10), by construction.

######  Definition 2.9.

Let $\mathcal{A}$ be an $A_{\infty}$-category, with alternative
$A_{\infty}$-structure $\tilde{\mathcal{A}}$ constructed from a sequence of
graded maps $(\Phi^{0},\Phi^{1},...)$ as above. The associated non-unital
$A_{\infty}$-functor
$\Phi:\mathcal{A}\rightarrow\Phi_{*}\mathcal{A}\coloneqq\tilde{\mathcal{A}}$
is called formal diffeomorphism. By equation (12) and choice of $\Phi^{1}$, it
is in particular a quasi-isomorphism (hence cohomologically full and
faithful).

####  2.3 Pre-natural transformations and compositions

In standard category theory we can look at maps between functors, the natural
transformations. There is of course an adaptation also for
$A_{\infty}$-functors.

######  Definition 2.10.

Given two non-unital $A_{\infty}$-categories $\mathcal{A}$ and $\mathcal{B}$,
we define $\mathcal{Q}=nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$ to be the
non-unital $A_{\infty}$-category specified by:

  * –

a set obj($\mathcal{Q}$) consisting of non-unital $A_{\infty}$-functors
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$;

  * –

for any pair $\mathcal{F}_{0},\mathcal{F}_{1}\in$ obj($\mathcal{Q}$),
morphisms whose homogeneous parts
$T\in\textup{hom}_{\mathcal{Q}}^{g}(\mathcal{F}_{0},\mathcal{F}_{1})$ are
sequences $(T^{0},T^{1},...)$, where each $T^{d}$ for $d\geq 1$ is a
collection of (multi)linear graded maps

$\mkern-9.0mu\textup{hom}_{\mathcal{A}}(X_{d-1},X_{d})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\\!\rightarrow\\!\textup{hom}_{\mathcal{B}}(\mathcal{F}_{0}(X_{0}),\mathcal{F}_{1}(X_{d}))[g-d],$
(13)

where $X_{0},...,X_{d}\in\textup{obj}(\mathcal{A})$. Instead, $T^{0}$ must be
regarded as a collection $\\{T_{X}^{0}\equiv
T^{0}(X)\in\textup{hom}_{\mathcal{B}}^{g}(\mathcal{F}_{0}(X),\mathcal{F}_{1}(X))\mid
X\in\textup{obj}(\mathcal{A})\\}$ (in accordance with the standard definition
of natural transformation). $T$ is called pre-natural transformation of degree
$|T|=g$ from $\mathcal{F}_{0}$ to $\mathcal{F}_{1}$;

  * –

composition maps $\mu_{\mathcal{Q}}^{d}$ like those of (5), so that for
$T_{k}\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{k-1},\mathcal{F}_{k})$ the
output $S\coloneqq\mu_{\mathcal{Q}}^{d}(T_{d},...,T_{1})$
$\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{d})[2-d]$ is again
a sequence $(S^{0},S^{1},...)$ as in (13). The first two cases $d=1,2$
are:666Note: by definition, in (– ‣ 2.10) only $s_{i}$ can be zero, while in
(– ‣ 2.10) both $s_{i}$ and $s_{j}$ can.

$\displaystyle\mkern-12.0mu\mu_{\mathcal{Q}}^{1}$
$\displaystyle(T_{1})^{h}(a_{h},...,a_{1})\coloneqq\mkern-18.0mu\sum_{1\leq
r\leq h+1}\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq i\leq r\\\ \quad
s_{1}+...+s_{r}=h\end{subarray}}\mkern-30.0mu(-1)^{\ddagger_{i}^{1}}\mu_{\mathcal{B}}^{r}\Big{(}\mathcal{F}_{1}^{s_{r}}(a_{h},...,a_{h-s_{r}+1}),...,$
$\displaystyle\mathcal{F}_{1}^{s_{i+1}}(a_{h-...-s_{i+2}},...,a_{s_{1}+...+s_{i}+1}),T_{1}^{s_{i}}(a_{s_{1}+...+s_{i}},...,a_{s_{1}+...+s_{i-1}+1}),$
$\displaystyle\mathcal{F}_{0}^{s_{i-1}}(a_{s_{1}+...+s_{i-1}},...,a_{s_{1}+...+s_{i-2}+1}),...,\mathcal{F}_{0}^{s_{1}}(a_{s_{1}},...,a_{1})\Big{)}$
$\displaystyle-\mkern-30.0mu\sum_{\begin{subarray}{c}1\leq m\leq h\\\ \quad
0\leq n\leq
h-m\end{subarray}}\mkern-24.0mu(-1)^{\dagger_{n}+|T_{1}|-1}T_{1}^{h-m+1}(a_{h},...,\mu_{\mathcal{A}}^{m}(a_{n+m},...,a_{n+1}),...,a_{1})$
(14)

in
$\textup{hom}_{\mathcal{B}}(\mathcal{F}_{0}(X_{0}),\mathcal{F}_{1}(X_{h}))$,
and

$\displaystyle\mu_{\mathcal{Q}}^{2}$
$\displaystyle(T_{2},T_{1})^{h}(a_{h},...,a_{1})\coloneqq\mkern-18.0mu\sum_{1\leq
r\leq h+2}\mkern-12.0mu\sum_{\begin{subarray}{c}1\leq i<j\leq r\\\ \quad
s_{1}+...+s_{r}=h\end{subarray}}\mkern-30.0mu(-1)^{\ddagger_{i}^{1}+\ddagger_{j}^{2}}\mu_{\mathcal{B}}^{r}\Big{(}\mathcal{F}_{2}^{s_{r}}(a_{h},...,a_{h-s_{r}+1}),$
$\displaystyle...,\mathcal{F}_{2}^{s_{j+1}}(...),T_{2}^{s_{j}}(a_{s_{1}+...+s_{j}},...,a_{s_{1}+...+s_{j-1}+1}),\mathcal{F}_{1}^{s_{j-1}}(...),$
$\displaystyle...,\mathcal{F}_{1}^{s_{i+1}}(...),T_{1}^{s_{i}}(a_{s_{1}+...+s_{i}},...,a_{s_{1}+...+s_{i-1}+1}),\mathcal{F}_{0}^{s_{i-1}}(...),$
$\displaystyle...,\mathcal{F}_{0}^{s_{1}}(a_{s_{1}},...,a_{1})\Big{)}$ (15)

in
$\textup{hom}_{\mathcal{B}}(\mathcal{F}_{0}(X_{0}),\mathcal{F}_{2}(X_{h}))$,
where $h\geq 1$, $a_{k}\in\textup{hom}_{\mathcal{A}}(X_{k-1},X_{k})$ and
$\ddagger_{i}^{j}\coloneqq(|T_{j}|-1)\cdot\sum_{k=1}^{s_{1}+...+s_{i-1}}(|a_{k}|-1)$.

Higher order formulae for $\mu_{\mathcal{Q}}^{d}(T_{d},...,T_{1})$ can be
found for example in Fukaya’s work ([Fuk99, Definition 10.15]). Actual
computations are best tackled with help of some programming. Here, we limit
ourselves to give the first two terms of $\eqref{shit1}$:

$\mu_{\mathcal{Q}}^{1}(T)^{0}=\\{\mu_{\mathcal{Q}}^{1}(T)_{X}^{0}=\mu_{\mathcal{B}}^{1}(T_{X}^{0})\in\textup{hom}_{\mathcal{B}}(\mathcal{F}_{0}(X),\mathcal{F}_{1}(X))\mid
X\in\textup{obj}(\mathcal{A})\\}\,,$ (16)
$\displaystyle\mkern-46.0mu\mu_{\mathcal{Q}}^{1}(T)^{1}(a_{1})=\;$
$\displaystyle\mu_{\mathcal{B}}^{1}(T^{1}(a_{1}))+(-1)^{|T|-1}T^{1}(\mu_{\mathcal{A}}^{1}(a_{1}))+\mu_{\mathcal{B}}^{2}(\mathcal{F}_{1}^{1}(a_{1}),T_{X_{0}}^{0})$
$\displaystyle\\!+(-1)^{(|T|-1)(|a_{1}|-1)}\mu_{\mathcal{B}}^{2}(T_{X_{1}}^{0},\mathcal{F}_{0}^{1}(a_{1}))\;.$
(17)

######  Definition 2.11.

A pre-natural transformation which is closed with respect to the boundary
operator of $\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{1})$,
that is, any $T\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{1})$
with $\mu_{\mathcal{Q}}^{1}(T)^{d}=0$ for all $d\geq 0$, is called an
($A_{\infty}$-)natural transformation.

Consider a natural transformation
$T\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{1})$. By (2.3)
and (8), for all $a_{1}\in\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ the classes
$\langle T_{X}^{0}\rangle$ satisfy the naturality condition $\langle
T_{X_{1}}^{0}\rangle\circ\langle\mathcal{F}_{0}^{1}(a_{1})\rangle$
$=(-1)^{|a_{1}|\cdot|T|}\langle\mathcal{F}_{1}^{1}(a_{1})\rangle\circ\langle
T_{X_{0}}^{0}\rangle\in\textup{Hom}_{H(\mathcal{B})}(\mathcal{F}_{0}(X_{0}),\mathcal{F}_{1}(X_{1}))$.
Therefore, $H(T)\equiv H(T^{0})\coloneqq\\{\langle
T_{X}^{0}\rangle\in\textup{Hom}_{H(\mathcal{B})}(\mathcal{F}_{0}(X),\mathcal{F}_{1}(X))\mid
X\in\textup{obj}(\mathcal{A})\\}$ is a “usual” natural transformation, and we
may define:

######  Definition 2.12.

Let $\mathcal{Q}=nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$. Then
$\mathcal{H(Q)}=Nu\text{-}\\!Fun(H(\mathcal{A}),H(\mathcal{B}))$ denotes the
(non-unital) category whose objects are non-unital cohomological functors
$H(\mathcal{F}):H(\mathcal{A})\rightarrow H(\mathcal{B})$ and whose morphisms
are their natural trans-formations
$H(T)\\!\in\\!\textup{Hom}_{\mathcal{H(Q)}}(H(\mathcal{F}),$
$H(\mathcal{G}))$, defined as in the last paragraph. We obtain the non-unital
functor:

$\mathcal{H}:H(\mathcal{Q})\longrightarrow\mathcal{H(Q)},\;\begin{cases}\mathcal{F}\in\textup{obj}(\mathcal{Q})\longmapsto
H(\mathcal{F})\\\ \langle
T\rangle\in\textup{Hom}_{H(\mathcal{Q})}(\mathcal{F},\mathcal{G})\longmapsto
H(T)\end{cases}$ (18)

Two useful operations are left and right composition with some fixed
$A_{\infty}$-functor.

######  Definition 2.13.

Let $\mathcal{G}:\mathcal{A}\rightarrow\mathcal{B}$ be a non-unital
$A_{\infty}$-functor and $\mathcal{C}$ any non-unital $A_{\infty}$-category.
We define the non-unital $A_{\infty}$-functors of:

  * •

left composition
$\mathcal{L}_{\mathcal{G}}:nu\text{-}\\!fun(\mathcal{C},\mathcal{A})\rightarrow
nu\text{-}\\!fun(\mathcal{C},\mathcal{B})$, given by
$\mathcal{F}\mapsto\mathcal{G}\circ\mathcal{F}$ and
$\mathcal{L}_{\mathcal{G}}^{1}:\textup{hom}_{nu\text{-}\\!fun(\mathcal{C},\mathcal{A})}(\mathcal{F}_{0},\mathcal{F}_{1})\rightarrow\textup{hom}_{nu\text{-}\\!fun(\mathcal{C},\mathcal{B})}(\mathcal{G}\circ\mathcal{F}_{0},\mathcal{G}\circ\mathcal{F}_{1})$
specified by

$\displaystyle\mkern-12.0mu\mathcal{L}_{\mathcal{G}}^{1}$
$\displaystyle(T_{1})^{d}(a_{d},...,a_{1})\coloneqq\mkern-18.0mu\sum_{1\leq
r\leq d+1}\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq i\leq r\\\ \quad
s_{1}+...+s_{r}=d\end{subarray}}\mkern-30.0mu(-1)^{\ddagger_{i}^{1}}\mathcal{G}^{r}\Big{(}\mathcal{F}_{1}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,$
$\displaystyle\mathcal{F}_{1}^{s_{i+1}}(a_{d-...-s_{i+2}},...,a_{s_{1}+...+s_{i}+1}),T_{1}^{s_{i}}(a_{s_{1}+...+s_{i}},...,a_{s_{1}+...+s_{i-1}+1}),$
$\displaystyle\mathcal{F}_{0}^{s_{i-1}}(a_{s_{1}+...+s_{i-1}},...,a_{s_{1}+...+s_{i-2}+1}),...,\mathcal{F}_{0}^{s_{1}}(a_{s_{1}},...,a_{1})\Big{)}$
(19)

in
$\textup{hom}_{\mathcal{B}}\big{(}\mathcal{G}(\mathcal{F}_{0}(X_{0})),\mathcal{G}(\mathcal{F}_{1}(X_{d}))\big{)}$,
and with complicated higher order terms $\mathcal{L}_{\mathcal{G}}^{d}$;

  * •

right composition
$\mathcal{R}_{\mathcal{G}}:nu\text{-}\\!fun(\mathcal{B},\mathcal{C})\rightarrow
nu\text{-}\\!fun(\mathcal{A},\mathcal{C})$, given by
$\mathcal{F}\mapsto\mathcal{F}\circ\mathcal{G}$ and
$\mathcal{R}_{\mathcal{G}}^{1}:\textup{hom}_{nu\text{-}\\!fun(\mathcal{B},\mathcal{C})}(\mathcal{F}_{0},\mathcal{F}_{1})\rightarrow\textup{hom}_{nu\text{-}\\!fun(\mathcal{A},\mathcal{C})}(\mathcal{F}_{0}\circ\mathcal{G},\mathcal{F}_{1}\circ\mathcal{G})$
specified by

$\mkern-14.0mu\mathcal{R}_{\mathcal{G}}^{1}(T_{1})^{d}(a_{d},...,a_{1})\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}0\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0muT^{r}\Big{(}\mathcal{G}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,\mathcal{G}^{s_{1}}(a_{s_{1}},...,a_{1})\Big{)}$
(20)

in
$\textup{hom}_{\mathcal{C}}(\mathcal{F}_{0}(\mathcal{G}(X_{0})),\mathcal{F}_{1}(\mathcal{G}(X_{d}))$,
and $\mathcal{R}_{\mathcal{G}}^{d}\coloneqq 0$ for $d>1$.

Again, it is worth calculating the lower order terms:
$\mathcal{L}_{\mathcal{G}}^{1}(T)_{X}^{0}=\mathcal{G}^{1}(T_{X}^{0})$ and
$\mathcal{L}_{\mathcal{G}}^{1}(T)^{1}(a_{1})\\!=\\!\mathcal{G}^{1}(T^{1}(a_{1}))+\mathcal{G}^{2}(\mathcal{F}_{1}^{1}(a_{1}),T_{X_{0}}^{0})+(-1)^{(|T|-1)(|a_{1}|-1)}\mathcal{G}^{2}(T_{X_{1}}^{0},\mathcal{F}_{0}^{1}(a_{1}))$,
respectively $\mathcal{R}_{\mathcal{G}}^{1}(T)_{X}^{0}=T_{X}^{0}$ and
$\mathcal{R}_{\mathcal{G}}^{1}(T)^{1}(a_{1})=T^{1}(\mathcal{G}^{1}(a_{1}))$.

For compatible $A_{\infty}$-functors, left and right compositions themselves
can be composed:
$\mathcal{R}_{\mathcal{G}_{1}}\circ\mathcal{R}_{\mathcal{G}_{2}}=\mathcal{R}_{\mathcal{G}_{1}\circ\mathcal{G}_{2}}$,
$\mathcal{L}_{\mathcal{G}_{1}}\circ\mathcal{L}_{\mathcal{G}_{2}}=\mathcal{L}_{\mathcal{G}_{1}\circ\mathcal{G}_{2}}$
and
$\mathcal{L}_{\mathcal{G}_{1}}\circ\mathcal{R}_{\mathcal{G}_{2}}=\mathcal{R}_{\mathcal{G}_{2}}\circ\mathcal{L}_{\mathcal{G}_{1}}$.

####  2.4 Interlude: Hochschild cohomology

Hochschild cohomology is a useful tool for proving many key results by
comparison arguments. It is prominently used in deformation theory. However, a
full treatment would take us too far afield and hence we limit ourselves to
give the relevant definitions and main applications. A more rigorous take can
be found in [GM03, subsection III.7.3] and [Sei08, section (1f)].

######  Definition 2.14.

Let $\mathcal{A}$ be an $A_{\infty}$-category whose cohomological category
$H(\mathcal{A})$ is unital. We define the Hochschild cohomology group of
$\mathcal{A}$ to be the graded morphism space $HH(\mathcal{A})\coloneqq
H(\textup{hom}_{nu\text{-}\\!fun(\mathcal{A},\mathcal{A})}(\mathcal{I}d_{\mathcal{A}},\mathcal{I}d_{\mathcal{A}}))$.
It can be computed from the cochain complex $(CC(\mathcal{A}),d)$ whose
cochains $h=(h^{0},h^{1},...)$ $\in CC^{r}(\mathcal{A})$ are sequences of
(multi)linear graded maps

$h^{d}:\textup{hom}_{\mathcal{A}}(X_{d},X_{d+1})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{1},X_{2})\rightarrow\textup{hom}_{\mathcal{A}}(X_{1},X_{d+1})[r-d]$

for each tuple $(X_{1},...,X_{d+1})\subset\textup{obj}(\mathcal{A})$, with
differential acting on them as:

$\displaystyle(dh)^{d}(a_{d},...,a_{1})\coloneqq$
$\displaystyle\mkern-18.0mu\sum_{\quad 0\leq
i+j<d+1}\mkern-18.0mu(-1)^{(r+1)\dagger_{i}}\mu_{\mathcal{A}}^{d+1-j}(a_{d},...,h^{j}(a_{i+j},...,a_{i+1}),...,a_{1})\,+$
$\displaystyle\mkern-18.0mu\sum_{\quad 0<i+j\leq
d+1}\mkern-18.0mu(-1)^{r+1+\dagger_{i}}h^{d+1-j}(a_{d},...,\mu_{\mathcal{A}}^{j}(a_{i+j},...,a_{i+1}),...,a_{1})\;.$

In fact, one can be more explicit about the definition of Hochschild complex
in the bigraded case. Consider now two arbitrary non-unital
$A_{\infty}$-categories $\mathcal{A}$, $\mathcal{B}$, and
$\mathcal{Q}=nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$. Given two
$\mathcal{G}_{0}$, $\mathcal{G}_{1}\in\textup{obj}(\mathcal{Q})$, the vector
space $\textup{hom}_{\mathcal{Q}}(\mathcal{G}_{0},\mathcal{G}_{1})$ can be
decreasingly filtrated by allowing for the $r$-th term only those pre-natural
transformations $T$ with $T^{0}=...=T^{r-1}=0$. The spectral sequence
associated to this filtration starts with

$\displaystyle
E_{1}^{rs}=\mkern-12.0mu\prod_{X_{0},...,X_{r}}\mkern-12.0mu\textup{Hom}_{\mathbb{K}}^{s}\big{(}$
$\displaystyle\textup{Hom}_{H(\mathcal{A})}(X_{r-1},X_{r})\otimes...\otimes\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{1}),$
$\displaystyle\textup{Hom}_{H(\mathcal{B})}(\mathcal{G}_{0}(X_{0}),\mathcal{G}_{1}(X_{r}))\big{)}.$
(21)

Equation (– ‣ 2.10) for the boundary operators in $\mathcal{Q}$ then furnishes
the following differential $d\equiv d^{rs}$ on $E_{1}^{rs}$:

$\displaystyle(dT)(a_{r+1},...,a_{1})=\;$
$\displaystyle(-1)^{(r+s)(|a_{1}|-1)+1}T(a_{r+1},...,a_{2})H(\mathcal{G}_{0})(a_{1})\,+$
$\displaystyle(-1)^{r+s+\dagger_{r}}H(\mathcal{G}_{1})(a_{r+1})T(a_{r},...,a_{1})\,+$
$\displaystyle\mkern-28.0mu\sum_{\quad 0\leq n\leq
r-1}\mkern-18.0mu(-1)^{r+s-1+\dagger_{n+1}}T(a_{r+1},...,a_{n+2}a_{n+1},...,a_{1})$

(for $T$ and the $a_{i}$’s cohomological maps). Then we obtain a bigraded
complex $CC^{r+s}(H(\mathcal{A}),H(\mathcal{B}))^{s}\coloneqq E_{1}^{rs}$,
inducing the (bigraded) Hochschild cohomology
$HH^{r+s}(H(\mathcal{A}),H(\mathcal{B}))^{s}\coloneqq E_{2}^{rs}$ (in
particular, we have
$HH^{s}(H(\mathcal{A}),H(\mathcal{B}))^{s}=\textup{Hom}_{\mathcal{H}(\mathcal{Q})}^{s}(H(\mathcal{G}_{0}),H(\mathcal{G}_{1}))$).

For our purposes, here are the principal results whose proofs exploit
Hochschild cohomology:

######  Lemma 2.15.

Let $\mathcal{Q}=nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$,
$\mathcal{G}_{1},\mathcal{G}_{2},\mathcal{G}_{3}\\!\in\\!\textup{obj}(\mathcal{Q})$
and $T\\!\in\\!\textup{hom}_{\mathcal{Q}}(\mathcal{G}_{1},\mathcal{G}_{2})$ be
a natural transformation. Suppose that:

  * •

$\textup{Hom}_{H(\mathcal{B})}(\mathcal{G}_{2}(X),\mathcal{G}_{3}(X))\rightarrow\textup{Hom}_{H(\mathcal{B})}(\mathcal{G}_{1}(X),\mathcal{G}_{3}(X)),\;\langle
c\rangle\mapsto\langle c\rangle\circ\langle T_{X}^{0}\rangle$ is an
isomorphism for all $X\in\textup{obj}(\mathcal{A})$. Then
$\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}_{2},\mathcal{G}_{3})\\!\rightarrow\\!\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}_{1},\mathcal{G}_{3}),$
$\langle S\rangle\mapsto\langle S\rangle\circ\langle T\rangle$ is an
isomorphism too.

  * •

$\textup{Hom}_{H(\mathcal{B})}(\mathcal{G}_{3}(X),\mathcal{G}_{1}(X))\rightarrow\textup{Hom}_{H(\mathcal{B})}(\mathcal{G}_{3}(X),\mathcal{G}_{2}(X)),\;\langle
c\rangle\mapsto\langle T_{X}^{0}\rangle\circ\langle c\rangle$ is an
isomorphism for all $X\in\textup{obj}(\mathcal{A})$. Then
$\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}_{3},\mathcal{G}_{1})\\!\rightarrow\\!\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}_{3},\mathcal{G}_{2}),$
$\langle S\rangle\mapsto\langle T\rangle\circ\langle S\rangle$ is an
isomorphism too.

######  Lemma 2.16.

Let $\mathcal{C}$ be any non-unital $A_{\infty}$-category. Suppose
$\mathcal{G}\in nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$ is:

  * •

cohomologically full and faithful, then so is
$\mathcal{L}_{\mathcal{G}}\\!:\\!nu\text{-}\\!fun(\mathcal{C},\mathcal{A})\\!\rightarrow\\!nu\text{-}\\!fun(\mathcal{C},\mathcal{B})$;

  * •

a quasi-isomorphism, then so is
$\mathcal{R}_{\mathcal{G}}:nu\text{-}\\!fun(\mathcal{B},\mathcal{C})\rightarrow
nu\text{-}\\!fun(\mathcal{A},\mathcal{C})$.

####  2.5 Homological perturbation theory

As usual, let $\mathcal{Q}=nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$. Given
$\mathcal{F}_{0},\mathcal{F}_{1}\in\textup{obj}(\mathcal{Q})$, observe that
the pre-natural transformation
$D\coloneqq\mathcal{F}_{0}-\mathcal{F}_{1}\in\textup{hom}_{\mathcal{Q}}^{1}(\mathcal{F}_{0},\mathcal{F}_{1})$
defined by $D^{0}\coloneqq 0$ and
$D^{d}\coloneqq\mathcal{F}_{0}^{d}-\mathcal{F}_{1}^{d}$ for $d\geq 1$ is
actually a natural transformation (indeed, $\mu_{\mathcal{Q}}^{1}(D)=0$ can be
checked with help of (10) and (– ‣ 2.10)).

######  Definition 2.17.

Two non-unital $A_{\infty}$-functors
$\mathcal{F}_{0},\mathcal{F}_{1}\in\textup{obj}(\mathcal{Q})$ with
$\mathcal{F}_{0}(X)=\mathcal{F}_{1}(X)$ for all
$X\in\textup{obj}(\mathcal{A})$ are homotopic, denoted
$\mathcal{F}_{0}\simeq\mathcal{F}_{1}$, if there exists some
$T\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{F}_{0},\mathcal{F}_{1})$ with
$T^{0}=0$ such that
$\mathcal{F}_{0}-\mathcal{F}_{1}=\mu_{\mathcal{Q}}^{1}(T)$.

Being homotopic is a well-defined equivalence relation.

As one would suspect, homotopic $A_{\infty}$-functors induce the same map in
cohomology: $(H(\mathcal{F}_{0})-H(\mathcal{F}_{1}))\langle
c\rangle=H(\mu_{\mathcal{Q}}^{1}(T))\langle
c\rangle=\langle\mu_{\mathcal{Q}}^{1}(T)^{1}(c)\rangle=0$, while identity on
objects is readily checked. Moreover, if
$\mathcal{F}_{0}\simeq\mathcal{F}_{1}$ via $T$, so
$\mathcal{G}\circ\mathcal{F}_{0}\simeq\mathcal{G}\circ\mathcal{F}_{1}$ via
$\mathcal{L}_{\mathcal{G}}^{1}(T)$ respectively
$\mathcal{F}_{0}\circ\mathcal{G}\simeq\mathcal{F}_{1}\circ\mathcal{G}$ via
$\mathcal{R}_{\mathcal{G}}^{1}(T)$.

Here is our first important result.

######  Proposition 2.18 (Homological Perturbation Lemma).

Let $\mathcal{B}$ be a non-unital $A_{\infty}$-category. Suppose for each pair
$X_{0},X_{1}\in\textup{obj}(\mathcal{B})$ there is a diagram

${{\textup{hom}_{\mathcal{A}}(X_{0},X_{1})}}$${{\textup{hom}_{\mathcal{B}}(X_{0},X_{1})}}$$\scriptstyle{\mathcal{F}^{1}[0]}$$\scriptstyle{\mathcal{G}^{1}[0]}$$\scriptstyle{T^{1}[-1]}$
(22)

where $(\textup{hom}_{\mathcal{A}}(X_{0},X_{1}),\mu_{\mathcal{A}}^{1})$ is a
cochain complex (as in Definition 1.2), $\mathcal{F}^{1}$ and
$\mathcal{G}^{1}$ are degree $0$ chain maps, and $T^{1}$ is a degree $-1$
graded map fulfilling
$\mu_{\mathcal{B}}^{1}T^{1}+T^{1}\mu_{\mathcal{B}}^{1}=\mathcal{F}^{1}\mathcal{G}^{1}-\textup{id}$
as endomorphisms of $\textup{hom}_{\mathcal{B}}(X_{0},X_{1})$. Then there
exist:

  * –

a non-unital $A_{\infty}$-category $\mathcal{A}$ with
$\textup{obj}(\mathcal{A})=\textup{obj}(\mathcal{B})$,
$\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ as morphism spaces and associated
first order composition maps $\mu_{\mathcal{A}}^{1}$,

  * –

non-unital $A_{\infty}$-functors
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ and
$\mathcal{G}:\mathcal{B}\rightarrow\mathcal{A}$ which are identities on
objects and have first order terms $\mathcal{F}^{1}$ respectively
$\mathcal{G}^{1}$,

  * –

a homotopy
$T\in\textup{hom}_{nu\text{-}\\!fun(\mathcal{B},\mathcal{B})}^{0}(\mathcal{F}\circ\mathcal{G},\mathcal{I}d_{\mathcal{B}})$
with first order term $T^{1}$.

One can produce the desired formulae recursively. Indeed,

$\mathcal{F}^{d}(a_{d},...,a_{1})\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}2\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0muT^{1}\Big{(}\mu_{\mathcal{B}}^{r}(\mathcal{F}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,\mathcal{F}^{s_{1}}(a_{s_{1}},...,a_{1}))\Big{)}$
$\mu_{\mathcal{A}}^{d}(a_{d},...,a_{1})\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}2\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0mu\mathcal{G}^{1}\Big{(}\mu_{\mathcal{B}}^{r}(\mathcal{F}^{s_{r}}(a_{d},...,a_{d-s_{r}+1}),...,\mathcal{F}^{s_{1}}(a_{s_{1}},...,a_{1}))\Big{)}$

are seen to fulfill the defining equations (10) respectively (6) (ultimately
because $\mu_{\mathcal{B}}^{\bullet}$ is an $A_{\infty}$-structure on
$\mathcal{B}$). However, the proof leading to them is too technical, and we
would rather move on to the following application.

######  Lemma 2.19.

Any quasi-isomorphism $\mathcal{K}:\mathcal{B}\rightarrow\tilde{\mathcal{B}}$
of non-unital $A_{\infty}$-categories has an inverse up to homotopy. That is,
there exists some $\mathcal{J}:\tilde{\mathcal{B}}\rightarrow\mathcal{B}$ such
that $\mathcal{K}\circ\mathcal{J}\simeq\mathcal{I}d_{\tilde{\mathcal{B}}}$ and
$\mathcal{J}\circ\mathcal{K}\simeq\mathcal{I}d_{\mathcal{B}}$.

###### Proof.

We start by proving the following claim: for a suitable choice of complexes,
we can construct an $A_{\infty}$-category $\mathcal{A}$ as in Proposition 2.18
such that $\mathcal{F}$ is a quasi-isomorphism and
$\mathcal{G}\circ\mathcal{F}\simeq\mathcal{I}d_{\mathcal{A}}$ too.

Indeed, given two $X_{0},X_{1}\in\textup{obj}(\mathcal{B})$, we can decompose
the associated complex
$(\textup{hom}_{\mathcal{B}}(X_{0},X_{1}),\mu_{\mathcal{B}}^{1})$ as
$(\textup{hom}_{\mathcal{A}}(X_{0},X_{1}),0)\oplus(\textup{hom}_{\mathcal{B^{\prime}}}(X_{0},X_{1}),\mu_{\mathcal{B^{\prime}}}^{1})$
so that $\mathcal{F}^{1}$ and $\mathcal{G}^{1}$ are the inclusion respectively
projection of the first summand, while the second summand is acyclic. But
acyclicity is achieved if and only if there exists a contracting homotopy
$(T^{1})_{n}:\textup{hom}_{\mathcal{B^{\prime}}}^{n}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{B^{\prime}}}^{n-1}(X_{0},X_{1})$.
This provides the desired $T^{1}$.

The Perturbation Lemma applies to construct an $A_{\infty}$-category
$\mathcal{A}$ with vanishing boundary operator, as well as compatible
$A_{\infty}$-functors $\mathcal{F}$ and $\mathcal{B}$. Since $\mathcal{F}^{1}$
is the inclusion and
$\mathcal{F}\circ\mathcal{G}\simeq\mathcal{I}d_{\mathcal{B}}$, we see that
$H(\mathcal{G})$ is bijective on objects and morphisms, making $\mathcal{G}$
(thus also $\mathcal{F}$) a quasi-isomorphism. By Lemma 2.16,
$\mathcal{R}_{\mathcal{G}}$ is a quasi-isomorphism too, so that any

$H(\mathcal{R}_{\mathcal{G}})_{\mathcal{F}_{1},\mathcal{F}_{2}}:\textup{Hom}_{H(nu\text{-}\\!fun(\mathcal{A},\mathcal{A}))}(\mathcal{F}_{1},\mathcal{F}_{2})\longmapsto\textup{Hom}_{H(nu\text{-}\\!fun(\mathcal{B},\mathcal{A}))}(\mathcal{F}_{1}\circ\mathcal{G},\mathcal{F}_{2}\circ\mathcal{G})$

is a bijection, also when restricting to (classes of) pre-natural
transformations with vanishing zeroth order term. Choose
$\mathcal{F}_{1}\coloneqq\mathcal{I}d_{\mathcal{A}}$ and
$\mathcal{F}_{2}\coloneqq\mathcal{G}\circ\mathcal{F}$. As
$\mathcal{I}d_{\mathcal{B}}\simeq\mathcal{F}\circ\mathcal{G}$, also
$\mathcal{G}\simeq\mathcal{G}\circ\mathcal{F}\circ\mathcal{G}$, say via some
pre-natural transformation
$T\in\textup{hom}_{{nu\text{-}\\!fun}(\mathcal{B},\mathcal{A})}^{0}(\mathcal{G},\mathcal{G}\circ\mathcal{F}\circ\mathcal{G})$.
Then
$\mu_{\mathcal{Q}}^{1}(T)=\mathcal{G}-\mathcal{G}\circ\mathcal{F}\circ\mathcal{G}=\mathcal{R}_{\mathcal{G}}^{1}(\mathcal{I}d_{\mathcal{A}}-\mathcal{G}\circ\mathcal{F})$,
hence
$\langle\mu_{\mathcal{Q}}^{1}(T)\rangle=H(\mathcal{R}_{\mathcal{G}})_{\mathcal{I}d_{\mathcal{A}},\mathcal{G}\circ\mathcal{F}}\langle\mathcal{I}d_{\mathcal{A}}-\mathcal{G}\circ\mathcal{F}\rangle$,
and bijectivity of
$H(\mathcal{R}_{\mathcal{G}})_{\mathcal{I}d_{\mathcal{A}},\mathcal{G}\circ\mathcal{F}}$
provides a class mapping to
$\langle\mathcal{I}d_{\mathcal{A}}-\mathcal{G}\circ\mathcal{F}\rangle$. Any
representative thereof gives by construction a homotopy
$\mathcal{I}d_{\mathcal{A}}\simeq\mathcal{G}\circ\mathcal{F}$. This proves the
claim.

We use this result to construct the following diagram:

$\leavevmode\hbox to59.12pt{\vbox
to58.29pt{\pgfpicture\makeatletter\hbox{\hskip
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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where all arrows, including
$\mathcal{E}\coloneqq\tilde{\mathcal{G}}\circ\mathcal{K}\circ\mathcal{F}$, are
quasi-isomorphisms. But then $\mu_{A}^{1}=\mu_{\tilde{\mathcal{A}}}^{1}=0$ and
the equations (10) force
$\mathcal{E}:\mathcal{A}\rightarrow\tilde{\mathcal{A}}\equiv\mathcal{E}_{*}\mathcal{A}$
into a formal diffeomorphism, which is thus invertible. Therefore, the non-
unital $A_{\infty}$-functor
$\mathcal{J}\coloneqq\mathcal{F}\circ\mathcal{E}^{-1}\circ\tilde{\mathcal{G}}:\tilde{\mathcal{B}}\rightarrow\mathcal{B}$
is well defined and fulfills:
$\mathcal{K}\simeq\mathcal{I}d_{\tilde{\mathcal{B}}}\circ\mathcal{K}\circ\mathcal{I}d_{\mathcal{B}}\simeq\tilde{\mathcal{F}}\circ\tilde{\mathcal{G}}\circ\mathcal{K}\circ\mathcal{F}\circ\mathcal{G}=\tilde{\mathcal{F}}\circ\mathcal{E}\circ\mathcal{G}$,
so
$\mathcal{K}\circ\mathcal{J}\simeq\tilde{\mathcal{F}}\circ\mathcal{E}\circ\mathcal{I}d_{\mathcal{A}}\circ\mathcal{E}^{-1}\circ\tilde{\mathcal{G}}\simeq\mathcal{I}d_{\tilde{\mathcal{B}}}$
and
$\mathcal{J}\circ\mathcal{K}\simeq\mathcal{F}\circ\mathcal{E}^{-1}\circ\mathcal{I}d_{\tilde{\mathcal{A}}}\circ\mathcal{E}\circ\mathcal{G}\simeq\mathcal{I}d_{\mathcal{B}}$.
We conclude that $\mathcal{J}$ is a homotopy inverse of $\mathcal{K}$. ∎

####  2.6 $A_{\infty}$-modules over $A_{\infty}$-categories

Mirroring our definition for $A_{\infty}$-algebras, we extend the construction
of $A_{\infty}$-modules to $A_{\infty}$-categories.

Given any non-unital $A_{\infty}$-category $\mathcal{A}$, its opposite
$\mathcal{A}^{opp}$ is simply the non-unital $A_{\infty}$-category of same
objects, with
$\textup{hom}_{\mathcal{A}^{opp}}(X_{0},X_{1})\coloneqq\textup{hom}_{\mathcal{A}}(X_{1},X_{0})$
and
$\mu_{\mathcal{A}^{opp}}^{d}(a_{d},...,a_{1})\coloneqq(-1)^{\dagger_{d}}\mu_{\mathcal{A}}^{d}(a_{1},...,a_{d})$.
Also, $H(\mathcal{A}^{opp})=H(\mathcal{A})^{opp}$.

######  Definition 2.20.

Consider the $A_{\infty}$-category777cf. Remark 2.5. of complexes
$\mathsf{Ch}$ and any non-unital $A_{\infty}$-category $\mathcal{A}$. Then
$\mathsf{Q}=nu\text{-}mod(\mathcal{A})\coloneqq
nu\text{-}\\!fun(\mathcal{A}^{opp},\mathsf{Ch})$ is the non-unital
$A_{\infty}$-category of non-unital (right) $A_{\infty}$-modules over
$\mathcal{A}$, specified by the following data:

  * –

Objects $\mathcal{M}\in\textup{obj}(\mathsf{Q})$ are non-unital
$A_{\infty}$-functors $\mathcal{M}:\mathcal{A}^{opp}\rightarrow\mathsf{Ch}$,
thus collections of graded spaces
$\\{\mathcal{M}(X)=\bigoplus_{p\in\mathbb{Z}}\mathcal{M}(X)^{p}\in\textup{obj}(\mathsf{Ch})\mid
X\in\textup{obj}(\mathcal{A})\\}$ with (multi)linear graded maps of degree
$d-2$

$\mu_{\mathcal{M}}^{d}:\mathcal{M}(X_{d-1})\\!\otimes\\!\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1})\\!\otimes...\otimes\\!\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\mathcal{M}(X_{0})$
(23)

for all $d\geq 1$ and $X_{0},...,X_{d-1}\in\textup{obj}(\mathcal{A})$. In
particular, $\mu_{\mathcal{M}}^{1}$ serves as differential for the cochain
complex $\mathcal{M}(X_{0})$. The $\mu_{\mathcal{M}}^{d}$’s are subject to
associativity equations analogous to (10):

$\displaystyle\sum_{0\leq n\leq
d-1}\mkern-18.0mu(-1)^{\dagger_{n}}\mu_{\mathcal{M}}^{n+1}(\mu_{\mathcal{M}}^{d-n}(b,a_{d-1},...,a_{n+1}),a_{n},...,a_{1})\,+$
(24) $\displaystyle\\!\sum_{\begin{subarray}{c}1\leq m\leq d\\\ 0\leq
n<d-m\end{subarray}}\mkern-18.0mu(-1)^{\dagger_{n}}\mu_{\mathcal{M}}^{d-m+1}(b,a_{d-1},...,\mu_{\mathcal{A}}^{m}(a_{n+m},...,a_{n+1}),...,a_{1})=0\;.$

  * –

For any pair $\mathcal{M}_{0},\mathcal{M}_{1}\in\textup{obj}(\mathsf{Q})$,
morphisms $t\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$
consisting of sequences $(t^{1},t^{2},...)$ of (multi)linear graded maps

$\displaystyle
t^{d}:\;\mathcal{M}_{0}(X_{d-1})\otimes\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1}$
$\displaystyle)\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})$ (25)
$\displaystyle\mkern-15.0mu\longrightarrow\mathcal{M}_{1}(X_{0})[|t|-d+1]\;,$

for $d\geq 1$ and $X_{0},...,X_{d-1}\in\textup{obj}(\mathcal{A})$. Such $t$ is
called a pre-module homomorphism.

  * –

Composition maps $\mu_{\mathsf{Q}}^{d}$, so that for
$t_{k}\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{k-1},\mathcal{M}_{k})$ the
output
$s\coloneqq\mu_{\mathsf{Q}}^{d}(t_{d},...,t_{1})\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{d})[2-d]$
is again a sequence $(s^{1},s^{2},...)$ as above. The first two cases $d=1,2$
are:

$\displaystyle\mu_{\mathsf{Q}}^{1}$
$\displaystyle(t_{1})^{h}(b,a_{h-1},...,a_{1})\coloneqq\mkern-18.0mu\sum_{0\leq
n\leq
h-1}\mkern-18.0mu(-1)^{\natural_{n}}\mu_{\mathcal{M}_{1}}^{n+1}(t^{h-n}(b,a_{h-1},...,a_{n+1}),a_{n},...,a_{1})+$
$\displaystyle\mkern-24.0mu\sum_{\quad 0\leq n\leq
h-1}\mkern-18.0mu(-1)^{\natural_{n}}t^{n+1}(\mu_{\mathcal{M}_{0}}^{h-n}(b,a_{h-1},...,a_{n+1}),a_{n},...,a_{1})\,+$
$\displaystyle\mkern-27.0mu\sum_{\begin{subarray}{c}1\leq m\leq h\\\ \quad
0\leq
n<h-m\end{subarray}}\mkern-21.0mu(-1)^{\natural_{n}}t^{h-m+1}(b,a_{h-1},...,\mu_{\mathcal{A}}^{m}(a_{n+m},...,a_{n+1}),...,a_{1})\;,$
(26)

in $\mathcal{M}_{1}(X_{0})$, and

$\displaystyle\mkern-15.0mu\mu_{\mathsf{Q}}^{2}(t_{2},t_{1})^{h}(b,a_{h-1},...,a_{1})\coloneqq\mkern-24.0mu\sum_{0\leq
n\leq
h-1}\mkern-18.0mu(-1)^{\natural_{n}}t_{2}^{n+1}(t_{1}^{h-n}(b,a_{h-1},...),a_{n},...,a_{1}),$
(27)

in $\mathcal{M}_{2}(X_{0})$, where $h\geq 1$,
$a_{k}\in\textup{hom}_{\mathcal{A}}(X_{k-1},X_{k})$ and
$\natural_{n}\coloneqq|b|+|a_{h-1}|+...+|a_{n+1}|-(h-1-n)$. Meanwhile, for
$d>2$ one has $\mu_{\mathsf{Q}}^{d}\coloneqq 0$.

In the case of trivial $A_{\infty}$-categories, this data coincide with that
given in Definitions 1.8 and 1.9 (compare the defining equations (24) and (– ‣
2.20) with (3) respectively (4), observing that pre-module homomorphisms
simply reduce to morphisms of $A_{\infty}$-modules over
$A_{\infty}$-algebras).

Any pre-module homomorphism
$t\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$ identifies a
pre-natural transformation
$T\\!\in\textup{hom}_{nu\text{-}\\!fun(\mathcal{A}^{opp},\mathsf{Ch})}(\mathcal{M}_{0},\mathcal{M}_{1})$
by setting $T^{d-1}(a_{1},...,a_{d-1})(b)$
$\coloneqq(-1)^{\S}t^{d}(b,a_{d-1},...,a_{1})$, where
$\S\coloneqq(|T|-1)|b|+|T|(|T|-1)/2$.

The associativity equation for $d=1$ reads
$\mu_{\mathcal{M}}^{1}(\mu_{\mathcal{M}}^{1}(b))=0$ — as expected — while (– ‣
2.20) and (27) become

$\displaystyle\mu_{\mathsf{Q}}^{1}(t)^{1}(b)=(-1)^{|b|}\mu_{\mathcal{M}_{1}}^{1}(t^{1}(b))+(-1)^{|b|}t^{1}(\mu_{\mathcal{M}_{0}}^{1}(b))\;,$
$\displaystyle\mu_{\mathsf{Q}}^{2}(t_{2},t_{1})^{1}(b)=(-1)^{|b|}t_{2}^{1}(t_{1}^{1}(b))\;.$

######  Definition 2.21.

A pre-module homomorphism which is closed with respect to the boundary
operator of $\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$, that
is, any $t\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$ with
$\mu_{\mathsf{Q}}^{1}(t)^{d}=0$ for all $d\geq 0$, is called an
($A_{\infty}$-)module homomorphism.

Being an $A_{\infty}$-functor, $\mathcal{M}\in\textup{obj}(\mathsf{Q})$
induces a non-unital (contravariant) functor
$H(\mathcal{M}):H(\mathcal{A})^{opp}\rightarrow
H(\mathsf{Ch})\cong\mathsf{GradVect}$ to the category of graded vector spaces
(as described in Section 1.2).

######  Definition 2.22.

Let $\mathcal{M}\in\textup{obj}(\mathsf{Q})$ be an $A_{\infty}$-module. The
cohomological module
$H(\mathcal{M}):H(\mathcal{A})^{opp}\rightarrow\mathsf{GradVect}$ of
$\mathcal{M}$ is an $H(\mathcal{A})$-module consisting of a family
$\\{(H(\mathcal{M})(X),\partial_{X})\mid X\\!\in\textup{obj}(\mathcal{A})\\}$,
where $H(\mathcal{M})(X)\coloneqq
H(\mathcal{M}(X))\in\textup{obj}(\mathsf{GradVect})$ and
$\partial_{X}(b)\coloneqq(-1)^{|b|}\mu_{\mathcal{M}}^{1}(b)\in\mathcal{M}(X)[1]$
for $b\in\mathcal{M}(X)$, endowed with module multiplication

$H(\mathcal{M})(X_{1})\otimes\textup{Hom}_{H(\mathcal{A})}(X_{0},X_{1})\rightarrow
H(\mathcal{M})(X_{0}),\;\;\langle b\rangle\cdot\langle
a\rangle\coloneqq(-1)^{|a|}\langle\mu_{\mathcal{M}}^{2}(b,a)\rangle\;,$

(cf. equation (8)). Cohomological modules are the objects of the non-unital
category
$\mathcal{H}(\mathsf{Q})=Nu\text{-}\\!Fun(H(\mathcal{A})^{opp},\mathsf{GradVect})$
($=:\mathsf{Mod}^{0}(H(\mathcal{A}))$, if unital).

Moreover, any module homomorphism
$t\in\textup{hom}_{\mathsf{Q}}(\mathcal{M}_{0},\mathcal{M}_{1})$ induces an
ordinary module homorphism
$H(t)\in\textup{Hom}_{\mathcal{H}(\mathsf{Q})}(H(\mathcal{M}_{0}),H(\mathcal{M}_{1}))$,
namely a family of graded maps

$H(t)_{X}:H(\mathcal{M}_{0})(X)\rightarrow H(\mathcal{M}_{1})(X),\,\langle
b\rangle\mapsto\langle(-1)^{|b|}t^{1}(b)\rangle\;,$

for all $X\in\textup{obj}(\mathcal{A})$.

####  2.7 Strict unitality and cohomological unitality

As we have already observed at the beginning of Section 2.2,
$A_{\infty}$-categories generally lack identity morphisms, but — as one would
guess — it is desirable to work with them, as they simplify the overall theory
significantly. So, here we introduce two different notions of unitality
(generalizing those encountered with $A_{\infty}$-algebras) which turn out to
be equivalent.

######  Definition 2.23.

Let $\mathcal{A}$ be a non-unital $A_{\infty}$-category. We call it:

  * •

strictly unital if for each object $X$ there exists some
$e_{X}\in\textup{hom}_{\mathcal{A}}^{0}(X,X)$ (then unique), called a strict
unit, fulfilling the conditions:

$\displaystyle\mu_{\mathcal{A}}^{1}(e_{X})=0\;,$
$\displaystyle(-1)^{|a_{1}|}\mu_{\mathcal{A}}^{2}(e_{X_{1}},a_{1})=a_{1}=\mu_{\mathcal{A}}^{2}(a_{1},e_{X_{0}})\;,$
(28)
$\displaystyle\mu_{\mathcal{A}}^{d}(a_{d-1},...,a_{n+1},e_{X_{n}},a_{n},...,a_{1})=0\;,$

for $d>2$, $0\leq n<d$ and $a_{i}\in\textup{hom}_{\mathcal{A}}(X_{i-1},X_{i})$
(in particular, $e_{X}$ must be a cocycle);

  * •

c-unital (for cohomologically unital) if the cohomological category
$H(\mathcal{A})$ is unital (thus an “actual” linear graded category). We call
any $e_{X}\in\textup{hom}_{\mathcal{A}}^{0}(X,X)$ inducing the identity
morphism on $X$ in $H(\mathcal{A})$ a c-unit ($e_{X}$ must necessarily be a
degree 0 cocycle).

Strict unitality implies c-unitality, which can be proved by making use of the
middle notion of homotopy unitality: roughly, a homotopy unital
$A_{\infty}$-category is one endowed with additional composition maps
$\mu_{\mathcal{A}}^{d,(i_{d},...,i_{0})}$ of degree
$2-d-2\sum_{k=0}^{d}i_{k}$, for all $d+i_{0}+...+i_{d}>0$, coinciding with
those of Definition 2.4 when $i_{0}=...=i_{d}=0$, and fulfilling some modified
compatible $A_{\infty}$-associativity equations.

Actually, the converse implication is true as well (up to quasi-isomorphism
given by a formal diffeomorphism):

######  Lemma 2.24.

Let $\mathcal{A}$ be a c-unital $A_{\infty}$-category such that for all
$X\in\textup{obj}(\mathcal{A})$ either
$\textup{hom}_{\mathcal{A}}(X,X)=\\{0\\}$ or
$H(\textup{hom}_{\mathcal{A}}(X,X))$ is non-trivial. Then there exists a
formal diffeomorphism
$\Phi:\mathcal{A}\rightarrow\tilde{\mathcal{A}}=\Phi_{*}\mathcal{A}$ with
$\Phi_{X_{0},X_{1}}^{1}=\textup{id}_{\textup{hom}_{\mathcal{A}}(X_{0},X_{1})}$
and such that $\Phi_{*}\mathcal{A}$ is strictly unital.

###### Proof.

(Sketch) As in most cases, we lay out a recursive construction. First, for
each $X\in\textup{obj}(\mathcal{A})$ there exists by assumption a c-unit
$e_{X}\in\textup{hom}_{\mathcal{A}}^{0}(X,X)$, which is in particular a
cocycle. Set $\mu_{\tilde{\mathcal{A}}}^{1}\coloneqq\mu_{\mathcal{A}}^{1}$,
then clearly $\mu_{\tilde{\mathcal{A}}}^{1}(e_{X})=0$. Construct888Recall from
Definition 2.9 that $\Phi_{*}\mathcal{A}$ has same morphism spaces as
$\mathcal{A}$!
$\mu_{\tilde{\mathcal{A}}}^{2}:\textup{hom}_{\mathcal{A}}(X_{1},X_{2})\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{A}}(X_{0},X_{2})$
so as to represent the composition law (8) for $H(\mathcal{A})$ and satisfy
the second equation in (• ‣ 2.23). Let $\Phi^{1}\coloneqq\textup{id}$, and
define $\Phi^{2}$ to be a chain homotopy between $\mu_{\mathcal{A}}^{2}$ and
$\mu_{\tilde{\mathcal{A}}}^{2}$, on each morphism space.

The construction now becomes more involved, ultimately producing the desired
formal diffeomorphism $\Phi:\mathcal{A}\rightarrow\tilde{\mathcal{A}}$ from
composition maps:

$\displaystyle\mu_{\tilde{\mathcal{A}}}^{d}$
$\displaystyle(a_{d},...,a_{1})\coloneqq\mu_{\mathcal{A}}^{d}(a_{d},...,a_{1})-(-1)^{\gamma}\mu_{\mathcal{A}}^{2}(\mu_{\mathcal{A}}^{d}(a_{d},...,e_{X_{n+1}},a_{n+1},...,a_{2}),a_{1})$
$\displaystyle-(-1)^{\dagger_{n+1}}\sum_{j<n}(-1)^{\alpha_{j}}\mu_{\mathcal{A}}^{d}(a_{d},...,e_{X_{n+1}},a_{n+1},...,\mu_{\mathcal{A}}^{2}(a_{j+2},a_{j+1}),...,a_{1})$
$\displaystyle-(-1)^{\dagger_{n}}\sum_{i,j}(-1)^{\beta_{j}}\mu_{\mathcal{A}}^{d+2-i}(b_{d+1},...,\mu_{\mathcal{A}}^{i}(b_{j+i},...,b_{j+1}),b_{j},...,b_{1})\;,$

where $0\leq n<d$,
$(b_{1},...,b_{d+1})\coloneqq(a_{1},...,a_{n},e_{X_{n}},a_{n+1},...,a_{d})$,
$\gamma\coloneqq|a_{2}|+...+|a_{n+1}|-n$,
$\alpha_{j}\coloneqq|a_{1}|+...+|a_{j}|-j$,
$\beta_{j}\coloneqq|b_{1}|+...+|b_{j}|-j$ and the ranges for $i,j$ in last
summand are: $i=1$, $j\neq n$; $i=2$, $j>n$; $i=d$, $j=0$; $i=d+1$.

Now, one must check that such higher order $\mu_{\tilde{\mathcal{A}}}^{d}$’s
vanish when having an identity morphism among their entries. For the
strategical approach to this we refer to [Sei08, Lemma 2.1]. ∎

######  Definition 2.25.

Let $\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ be a functor of
$A_{\infty}$-categories. If $\mathcal{A}$ and $\mathcal{B}$ are:

  * •

strictly unital, then we call $\mathcal{F}$ a strictly unital
$A_{\infty}$-functor if $\mathcal{F}^{1}(e_{X})=e_{\mathcal{F}(X)}$ for all
$X\in\textup{obj}(\mathcal{A})$, and
$\mathcal{F}^{d}(a_{d-1},...,a_{n+1},e_{X_{n}},a_{n},...,a_{1})=0$ if $d>1$;

  * •

c-unital, then we call $\mathcal{F}$ a c-unital $A_{\infty}$-functor if
$H(\mathcal{F})$ is unital (thus an “actual” linear graded functor).

We put emphasis on c-unitality, as this is a characteristic of Fukaya
categories, which we will introduce in the second part of this thesis. For
now, let us investigate unitality in functor categories.

######  Proposition 2.26.

Let $\mathcal{A}$ be an $A_{\infty}$-category, $\mathcal{C}$ any non-unital
$A_{\infty}$-category. Then:

  1. i.

if $\mathcal{A}$ is strictly unital, then so is
$\mathcal{Q}=nu\text{-}\\!fun(\mathcal{C},\mathcal{A})$;

  2. ii.

if $\mathcal{A}$ is c-unital, then so is $\mathcal{Q}$.

In the latter case, if
$E_{\mathcal{F}}\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{F},\mathcal{F})$ is
a c-unit, thus representing
$\textup{id}_{\mathcal{F}}\in\textup{Hom}_{H(\mathcal{Q})}^{0}(\mathcal{F},\mathcal{F})$,
then each $E_{\mathcal{F}}^{0}(X)\equiv
E_{\mathcal{F},X}^{0}\in\textup{hom}_{\mathcal{A}}^{0}(\mathcal{F}(X),\mathcal{F}(X))$
for $X\in\textup{obj}(\mathcal{C})$ is a c-unit, thus representing
$\textup{id}_{\mathcal{F}(X)}\in\textup{Hom}_{H(\mathcal{A})}^{0}(\mathcal{F}(X),\mathcal{F}(X))$.

###### Proof.

  1. i.

We check that strict units for $\mathcal{Q}$ are all the pre-natural
transformations
$E_{\mathcal{F}}\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{F},\mathcal{F})$
given by $E_{\mathcal{F}}^{0}(X)\coloneqq
e_{\mathcal{F}(X)}\in\textup{hom}_{\mathcal{A}}^{0}(\mathcal{F}(X),\mathcal{F}(X))$
(the strict unit of $\mathcal{F}(X)$) and $E_{\mathcal{F}}^{d}\coloneqq 0$ if
$d>0$. For example, substitution in (16) gives
$\mu_{\mathcal{Q}}^{1}(E_{\mathcal{F}})_{X}^{0}=\mu_{\mathcal{A}}^{1}(e_{\mathcal{F}(X)})=0$,
while (2.3) yields
$\mu_{\mathcal{Q}}^{1}(E_{\mathcal{F}})^{1}(a)=0+0+\mu_{\mathcal{A}}^{2}(\mathcal{F}^{1}(a),e_{\mathcal{F}(X_{0})})+(-1)^{1-|a|}\mu_{\mathcal{A}}^{2}(e_{\mathcal{F}(X_{1})},\mathcal{F}^{1}(a))=\mathcal{F}^{1}(a)+(-1)^{1-|a|+|a|}$
$\cdot\mathcal{F}^{1}(a)=0$. Indeed, the equations (– ‣ 2.10) show that
$\mu_{\mathcal{Q}}^{1}(E_{\mathcal{F}})^{d}=0$ for all higher $d$, hence
$\mu_{\mathcal{Q}}^{1}(E_{\mathcal{F}})=0$ (in particular, $E_{\mathcal{F}}$
is a natural transformation). By (– ‣ 2.10), the second requirement for strict
unitality is also met, while the third asks for higher degree
$\mu_{\mathcal{Q}}^{d}$’s (see again [Fuk99]).

  2. ii.

We reduce this case to the strictly unital one from above. Thereto, consider a
formal diffeomorphism $\Phi:\mathcal{A}\rightarrow\tilde{\mathcal{A}}$ with
$\Phi^{1}=\textup{id}$ on morphism spaces as given by Lemma 2.24, so that
$\tilde{\mathcal{A}}$ is strictly unital (then so is
$\tilde{\mathcal{Q}}\coloneqq
nu\text{-}\\!fun(\mathcal{C},\tilde{\mathcal{A}})$, by part i.). By
definition, $\Phi$ is a quasi-isomorphism, and hence so is its left
composition $\mathcal{L}_{\Phi}:\mathcal{Q}\rightarrow\tilde{\mathcal{Q}}$ (by
Lemma 2.16). For each $\mathcal{F}\in\textup{obj}(\mathcal{Q})$, this enables
us to find some
$E_{\mathcal{F}}\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{F},\mathcal{F})$
such that $H(\mathcal{L}_{\Phi})\langle
E_{\mathcal{F}}\rangle=\langle\mathcal{L}_{\Phi}^{1}(E_{\mathcal{F}})\rangle$
is the identity morphism $\langle
E_{\tilde{\mathcal{F}}}\rangle\in\textup{Hom}_{H(\tilde{\mathcal{Q}})}^{0}(\tilde{\mathcal{F}},\tilde{\mathcal{F}})$
induced by the unique strict unit for
$\tilde{\mathcal{F}}\coloneqq\mathcal{L}_{\Phi}(\mathcal{F})\in\textup{obj}(\tilde{\mathcal{Q}})$.
Functoriality of the isomorphism $H(\mathcal{L}_{\Phi})$ guarantees that
$\langle E_{\mathcal{F}}\rangle$ itself is an identity morphism in
$H(\mathcal{Q})$, the unique such for $\mathcal{F}$, which proves that
$H(\mathcal{Q})$ is unital, thus $\mathcal{Q}$ c-unital.

Now, the functor $\mathcal{H}$ of (18) maps $\langle
E_{\mathcal{F}}\rangle=\textup{id}_{\mathcal{F}}$ to the identity natural
transformation
$H(E_{\mathcal{F}})\in\textup{Hom}_{\mathcal{H}(\mathcal{Q})}(H(\mathcal{F}),H(\mathcal{F}))$
in $\mathcal{H}(\mathcal{Q})=Nu\text{-}\\!Fun(H(\mathcal{C}),H(\mathcal{A}))$.
The definition of $\Phi$ (which makes $H(\mathcal{F})=H(\tilde{\mathcal{F}})$
and $H(\mathcal{A})\cong H(\tilde{\mathcal{A}})$, thus $H(\mathcal{Q})\cong
H(\tilde{\mathcal{Q}})$) allows us to construct the commutative diagram

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which in turn gives $H(E_{\mathcal{F}})=\mathcal{H}(\langle
E_{\mathcal{F}}\rangle)=\mathcal{H}\circ H(\mathcal{L}_{\Phi})\langle
E_{\mathcal{F}}\rangle=\mathcal{H}(\langle\tilde{E}_{\mathcal{F}}\rangle)=H(E_{\tilde{\mathcal{F}}})$
as identity natural transformations, so that each $\langle
E_{\mathcal{F}}^{0}(X)\rangle=\langle E_{\tilde{\mathcal{F}}}^{0}(X)\rangle$
for $X\in\textup{obj}(\mathcal{C})$ (cf. Definition 2.12) must be the identity
$\textup{id}_{\mathcal{F}(X)}$ in $H^{0}(\mathcal{A})$. ∎

######  Lemma 2.27.

Let $\mathcal{G}:\mathcal{A}\rightarrow\mathcal{B}$ be a c-unital
$A_{\infty}$-functor. Then for any non-unital $A_{\infty}$-category
$\mathcal{C}$,
$\mathcal{L}_{\mathcal{G}}:nu\text{-}\\!fun(\mathcal{C},\mathcal{A})\rightarrow
nu\text{-}\\!fun(\mathcal{C},\mathcal{B})$ is c-unital. A similar result holds
for right composition.

###### Proof.

For
$E_{\mathcal{F}}\in\textup{hom}_{nu\text{-}\\!fun(\mathcal{C},\mathcal{A})}^{0}(\mathcal{F},\mathcal{F})$
a c-unit, we know by Proposition 2.26 that $\langle
E_{\mathcal{F}}^{0}(X)\rangle=\textup{id}_{\mathcal{F}(X)}$ in
$H(\mathcal{A})$ for each $X\in\textup{obj}(\mathcal{C})$. Then
$\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})_{X}^{0}\rangle=\langle\mathcal{G}^{1}(E_{\mathcal{F}}^{0}(X))\rangle=H(\mathcal{G})\langle
E_{\mathcal{F}}^{0}(X)\rangle$ (using (• ‣ 2.13) for $d=0$) shows that also
$\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})_{X}^{0}$
$\in\textup{hom}_{\mathcal{B}}^{0}\big{(}\mathcal{G}(\mathcal{F}(X)),\mathcal{G}(\mathcal{F}(X))\big{)}$
is a c-unit (as $\mathcal{G}$ is c-unital, $H(\mathcal{G})$ will preserve
identity morphisms).

Now we apply Lemma 2.15 to the following data: $\mathcal{Q}\coloneqq
nu\text{-}\\!fun(\mathcal{C},\mathcal{B})$,
$\mathcal{G}_{1}=\mathcal{G}_{2}=\mathcal{G}_{3}\coloneqq\mathcal{G}\circ\mathcal{F}$,
$T\coloneqq\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\in\textup{hom}_{\mathcal{Q}}^{0}(\mathcal{G}\circ\mathcal{F},\mathcal{G}\circ\mathcal{F})$,
where the argument from last paragraph ensures that right composition with
each $\langle T_{X}^{0}\rangle=\textup{id}_{\mathcal{G}(\mathcal{F}(X))}$ is
indeed an automorphism of
$\textup{Hom}_{H(\mathcal{B})}\big{(}\mathcal{G}(\mathcal{F}(X)),\mathcal{G}(\mathcal{F}(X))\big{)}$.
Consequently, right composition with
$\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle$ is an
automorphism of
$\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}\circ\mathcal{F},\mathcal{G}\circ\mathcal{F})$.

But $\mathcal{Q}$ is c-unital by Proposition 2.26, and thus $H(\mathcal{Q})$
possesses an identity $\textup{id}_{\mathcal{G}\circ\mathcal{F}}$ and
whichever $\langle
S\rangle\in\textup{Hom}_{H(\mathcal{Q})}(\mathcal{G}\circ\mathcal{F},\mathcal{G}\circ\mathcal{F})$
maps to it under right composition with
$\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle$ is an inverse
for $\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle$. Moreover,
by functoriality of $H(\mathcal{L}_{\mathcal{G}})$ and equation (8),
$\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle\circ\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle=\langle\mathcal{L}_{\mathcal{G}}^{1}(\mu_{\mathcal{Q}}^{2}(E_{\mathcal{F}},E_{\mathcal{F}}))\rangle=\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle$
is idempotent. The only idempotent invertible map is the identity, whence we
conclude that $H(\mathcal{L}_{\mathcal{G}}):\langle
E_{\mathcal{F}}\rangle\mapsto\langle\mathcal{L}_{\mathcal{G}}^{1}(E_{\mathcal{F}})\rangle$
preserves identities, that is, $\mathcal{L}_{\mathcal{G}}$ is c-unital. ∎

######  Lemma 2.28.

Let $\mathcal{A}$ be a c-unital $A_{\infty}$-category, $\mathcal{C}$ any non-
unital one. If
$\mathcal{F}_{0},\mathcal{F}_{1}:\mathcal{C}\rightarrow\mathcal{A}$ are
homotopic non-unital $A_{\infty}$-functors, then
$\mathcal{F}_{0}\cong\mathcal{F}_{1}$ as objects of
$H^{0}(nu\text{-}\\!fun(\mathcal{C},\mathcal{A}))$.

###### Proof.

As in the proof of Proposition 2.26, we pass to the strictly unital image
$\tilde{\mathcal{A}}=\Phi_{*}\mathcal{A}$ via a suitable formal diffeomorphism
$\Phi$, recalling that $\mathcal{F}_{0}\simeq\mathcal{F}_{1}$ implies
$\tilde{\mathcal{F}}_{0}\simeq\tilde{\mathcal{F}}_{1}$, where
$\tilde{\mathcal{F}}_{i}\coloneqq\Phi\circ\mathcal{F}_{i}$.

Set again $\mathcal{Q}=nu\text{-}\\!fun(\mathcal{C},\mathcal{A})$ and
$\tilde{\mathcal{Q}}=nu\text{-}\\!fun(\mathcal{C},\tilde{\mathcal{A}})$. Let
$T\in\textup{hom}_{\tilde{\mathcal{Q}}}^{0}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$
have $T^{0}=0$, and define
$S\in\textup{hom}_{\tilde{\mathcal{Q}}}^{0}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$
such that
$S_{X}^{0}=e_{\tilde{\mathcal{F}}_{0}(X)}=e_{\tilde{\mathcal{F}}_{1}(X)}$999Recall
that two $A_{\infty}$-functors can be homotopic only when coinciding on
objects! for all $X\in\textup{obj}(\mathcal{C})$ (then
$\mu_{\tilde{\mathcal{Q}}}^{1}(S)_{X}^{0}=\mu_{\tilde{\mathcal{A}}}^{1}(e_{\tilde{\mathcal{F}}_{0}(X)})=0$)
and $S^{d}=T^{d}$ if $d>0$. The equations (– ‣ 2.10) show that $T$ is a
homotopy from $\tilde{\mathcal{F}}_{0}$ to $\tilde{\mathcal{F}}_{1}$ if and
only if $S$ is a natural transformation, as (2.3) readily suggests:
$\mu_{\tilde{\mathcal{Q}}}^{1}(S)^{1}(a)=\mu_{\tilde{\mathcal{A}}}^{1}(S^{1}(a))-S^{1}(\mu_{\mathcal{C}}^{1}(a))+\mu_{\tilde{\mathcal{A}}}^{2}(\tilde{\mathcal{F}}_{1}^{1}(a),S_{X_{0}}^{0})+(-1)^{1-|a|}\mu_{\tilde{\mathcal{A}}}^{2}(S_{X_{1}}^{0},\tilde{\mathcal{F}}_{0}^{1}(a))=\mu_{\tilde{\mathcal{A}}}^{1}(S^{1}(a))-S^{1}(\mu_{\mathcal{C}}^{1}(a))$
$+\tilde{\mathcal{F}}_{1}^{1}(a)-\tilde{\mathcal{F}}_{0}^{1}(a)=0$ if and only
if
$\mu_{\tilde{\mathcal{Q}}}^{1}(T)=\tilde{\mathcal{F}}_{0}-\tilde{\mathcal{F}}_{1}$.

Now, apply Lemma 2.15 to the data:
$\mathcal{G}_{1}\coloneqq\tilde{\mathcal{F}}_{0}$,
$\mathcal{G}_{2}=\mathcal{G}_{3}\coloneqq\tilde{\mathcal{F}}_{1}$ and natural
transformation
$S\in\textup{hom}_{\tilde{\mathcal{Q}}}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$,
so that $\langle c\rangle\mapsto\langle c\rangle\circ\langle
S_{X}^{0}\rangle=\langle\mu_{\tilde{\mathcal{A}}}(c,e_{\tilde{\mathcal{F}}_{0}(X)})\rangle=\langle
c\rangle$ is indeed an isomorphism. Then
$\textup{Hom}_{H(\tilde{\mathcal{Q}})}^{0}(\tilde{\mathcal{F}}_{1},\tilde{\mathcal{F}}_{1})\rightarrow\textup{Hom}_{H(\tilde{\mathcal{Q}})}^{0}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$
given by right composition with $\langle S\rangle$ is an isomorphism.
$\tilde{\mathcal{Q}}$ being strictly unital (since $\tilde{\mathcal{A}}$ is),
it is also c-unital, so that we can find an identity morphism
$\textup{id}_{\tilde{\mathcal{F}}_{1}}\in\textup{Hom}_{H(\tilde{\mathcal{Q}})}^{0}(\tilde{\mathcal{F}}_{1},\tilde{\mathcal{F}}_{1})$.
Then
$\tilde{\mathcal{F}}_{0}\cong\tilde{\mathcal{F}}_{1}\in\textup{obj}(H^{0}(\tilde{\mathcal{Q}}))$
via the isomorphism $\textup{id}_{\tilde{\mathcal{F}}_{1}}\circ\langle
S\rangle:\tilde{\mathcal{F}}_{0}\rightarrow\tilde{\mathcal{F}}_{1}$.

Finally, $\Phi$ being a quasi-isomorphism, it is also cohomologically full and
faithful, hence so is
$\mathcal{L}_{\Phi}:\mathcal{Q}\rightarrow\tilde{\mathcal{Q}}$ by Lemma 2.16.
This implies that $H(\mathcal{L}_{\Phi})_{\mathcal{F}_{0},\mathcal{F}_{1}}$ is
bijective, and hence preserves isomorphisms. Therefore,
$\mathcal{F}_{0}\cong\mathcal{F}_{1}\in\textup{obj}(H^{0}(\mathcal{Q}))$, as
desired. ∎

####  2.8 Quasi-equivalences

######  Definition 2.29.

For $\mathcal{A}$ and $\mathcal{B}$ c-unital $A_{\infty}$-categories, let
$fun(\mathcal{A},\mathcal{B})$ denote the full $A_{\infty}$-subcategory of
$nu\text{-}\\!fun(\mathcal{A},\mathcal{B})$ consisting of c-unital
$A_{\infty}$-functors.

Any $\mathcal{F}\in\textup{obj}(fun(\mathcal{A},\mathcal{B}))$ is called
quasi-equivalence if $H(\mathcal{F})$ is an equivalence. In particular, by
Lemma 2.3, we have the following chain of implications for $\mathcal{F}$:

[frame]quasi-isomorphism $\implies$ quasi-equivalence $\implies$
cohomologically full, faithful
$\mkern 338.0mu\implies$ c-unital

In the following, given any c-unital $A_{\infty}$-category $\mathcal{A}$,
write $\tilde{\mathcal{A}}\subset\mathcal{A}$ for the full c-unital
$A_{\infty}$-subcategory making the (c-unital) inclusion
$\mathcal{I}:\tilde{\mathcal{A}}\rightarrow\mathcal{A}$ a quasi-equivalence.

######  Lemma 2.30.

Given another c-unital $A_{\infty}$-category $\mathcal{B}$, the restriction
map
$\mathcal{Q}=fun(\mathcal{A},\mathcal{B})\rightarrow\tilde{\mathcal{Q}}=fun(\tilde{\mathcal{A}},\mathcal{B})$
specified by
$\mathcal{F}\mapsto\tilde{\mathcal{F}}\equiv\mathcal{F}|_{\tilde{\mathcal{A}}}$
and
$T\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{1})\mapsto\tilde{T}\in\textup{hom}_{\tilde{\mathcal{Q}}}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$
is a quasi-equivalence.

###### Proof.

(Sketch) Arguing with Hochschild cohomology, one can prove that
$T\in\textup{hom}_{\mathcal{Q}}(\mathcal{F}_{0},\mathcal{F}_{1})\mapsto\tilde{T}\in\textup{hom}_{\tilde{\mathcal{Q}}}(\tilde{\mathcal{F}}_{0},\tilde{\mathcal{F}}_{1})$
is a quasi-isomorphism, thus cohomologically full and faithful, and that any
c-unital $\tilde{\mathcal{F}}:\tilde{\mathcal{A}}\rightarrow\mathcal{B}$ can
be extended uniquely to an $\mathcal{F}\in\textup{obj}(\mathcal{Q})$, making
the restriction cohomologically essentially surjective. By Lemma 2.3, these
two conditions imply quasi-equivalence. ∎

The extension procedure alluded to in last proof, when applied to the identity
$\mathcal{I}d_{\tilde{\mathcal{A}}}\in
fun(\tilde{\mathcal{A}},\tilde{\mathcal{A}})$, produces a c-unital
$A_{\infty}$-functor $\mathcal{P}:\mathcal{A}\rightarrow\tilde{\mathcal{A}}$
such that
$\mathcal{P}|_{\tilde{\mathcal{A}}}=\mathcal{P}\circ\mathcal{I}=\mathcal{I}d_{\tilde{\mathcal{A}}}$
and which is a quasi-equivalence (as $\mathcal{I}$ is). Actually,
$\mathcal{I}\circ\mathcal{P}\cong{\mathcal{I}d_{\mathcal{A}}}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{A},\mathcal{A}))\big{)}$,
because $\mathcal{L}_{\mathcal{P}}$ is cohomologically full and faithful (by
Lemma 2.16), and thus

$H(\mathcal{L}_{\mathcal{P}}):\textup{Hom}_{H(fun(\mathcal{A},\mathcal{A}))}^{0}(\mathcal{I}\circ\mathcal{P},\mathcal{I}d_{\mathcal{A}})\rightarrow\textup{Hom}_{H(fun(\mathcal{A},\tilde{\mathcal{A}}))}^{0}(\overbrace{\mathcal{L}_{\mathcal{P}}(\mathcal{I}\circ\mathcal{P})}^{=\mathcal{L}_{\mathcal{P}}(\mathcal{I}d_{\mathcal{A}})},\mathcal{L}_{\mathcal{P}}(\mathcal{I}d_{\mathcal{A}}))$

a bijection mapping the identity of
$\mathcal{L}_{\mathcal{P}}(\mathcal{I}d_{\mathcal{A}})$ back to the desired
isomorphism.

These maps take part in proving the following very important result, which
explains why we care about quasi-equivalences. We will use it repeatedly
throughout next section.

######  Theorem 2.31.

Let $\mathcal{A},\mathcal{B}$ be c-unital $A_{\infty}$-categories,
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$ a quasi-equivalence. Then
there exists a quasi-equivalence
$\mathcal{G}:\mathcal{B}\rightarrow\mathcal{A}$ such that

$\mathcal{G}\circ\mathcal{F}\cong\mathcal{I}d_{\mathcal{A}}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{A},\mathcal{A}))\big{)}\,,\quad\mathcal{F}\circ\mathcal{G}\cong\mathcal{I}d_{\mathcal{B}}\in\textup{obj}\big{(}H^{0}(fun(\mathcal{B},\mathcal{B}))\big{)}\,.$

###### Proof.

Denote by $\tilde{\mathcal{A}}\subset\mathcal{A}$ and
$\tilde{\mathcal{B}}\subset\mathcal{B}$ the full c-unital
$A_{\infty}$-subcategories making the inclusions
$\mathcal{I}_{\mathcal{A}}:\tilde{\mathcal{A}}\rightarrow\mathcal{A}$
respectively
$\mathcal{I}_{\mathcal{B}}:\tilde{\mathcal{B}}\rightarrow\mathcal{B}$ quasi-
equivalences, invertible up to isomorphism via $\mathcal{P}_{\mathcal{A}}$ and
$\mathcal{P}_{\mathcal{B}}$, by the argument given in last paragraph. Writing
$\tilde{\mathcal{F}}\coloneqq{\mathcal{P}_{\mathcal{B}}}\circ\mathcal{F}\circ{\mathcal{I}_{\mathcal{A}}}:\tilde{\mathcal{A}}\rightarrow\tilde{\mathcal{B}}$,
we see that $\tilde{\mathcal{F}}$ is actually a quasi-isomorphism, and hence
by Lemma 2.19 has a homotopy inverse
$\tilde{\mathcal{G}}:\tilde{\mathcal{B}}\rightarrow\tilde{\mathcal{A}}$, so
that
$\tilde{\mathcal{F}}\circ\tilde{\mathcal{G}}\simeq\mathcal{I}d_{\tilde{\mathcal{B}}}$
and
$\tilde{\mathcal{G}}\circ\tilde{\mathcal{F}}\simeq\mathcal{I}d_{\tilde{\mathcal{A}}}$.
Now, Lemma 2.28 yields
$\tilde{\mathcal{F}}\circ\tilde{\mathcal{G}}\cong\mathcal{I}d_{\tilde{\mathcal{B}}}$
in $H^{0}(fun(\tilde{\mathcal{B}},\tilde{\mathcal{B}}))$ and
$\tilde{\mathcal{G}}\circ\tilde{\mathcal{F}}\cong\mathcal{I}d_{\tilde{\mathcal{A}}}$
in $H^{0}(fun(\tilde{\mathcal{A}},\tilde{\mathcal{A}}))$. ∎

######  Corollary 2.32.

Let $\mathcal{C}$ be any c-unital $A_{\infty}$-category. If
$\mathcal{G}:\mathcal{A}\rightarrow\mathcal{B}$ is a quasi-equivalence, then
so is its right composition
$\mathcal{R}_{\mathcal{G}}:fun(\mathcal{B},\mathcal{C})\rightarrow
fun(\mathcal{A},\mathcal{C})$ (and its left
$\mathcal{L}_{\mathcal{G}}:fun(\mathcal{C},\mathcal{A})\rightarrow
fun(\mathcal{C},\mathcal{B})$).

###### Proof.

By above, we have two cases which imply the general result: either
$\mathcal{G}$ is a quasi-isomorphism or it is the inclusion of a full
$A_{\infty}$-subcategory $\tilde{\mathcal{B}}\coloneqq\mathcal{A}$ (so that
the restriction of Lemma 2.30 is actually $\mathcal{R}_{\mathcal{G}}$). In
both cases, $\mathcal{R}_{\mathcal{G}}$ is cohomologically full and faithful
(by Lemmas 2.16 respectively 2.30).

Now, by Theorem 2.31 we can find a quasi-equivalence
$\mathcal{F}:\mathcal{B}\rightarrow\mathcal{A}$ such that
$\mathcal{F}\circ\mathcal{G}\cong\mathcal{I}d_{\mathcal{A}}$ as objects in
cohomology, and Lemma 2.27 ensures that
$\mathcal{L}_{\mathcal{E}}:fun(\mathcal{A},\mathcal{A})\rightarrow
fun(\mathcal{A},\mathcal{C})$ is well defined for any
$\mathcal{E}\in\textup{obj}(fun(\mathcal{A},\mathcal{C}))$. We have
$\mathcal{R}_{\mathcal{G}}(\mathcal{E}\circ\mathcal{F})=\mathcal{L}_{\mathcal{E}}(\mathcal{F}\circ\mathcal{G})\cong\mathcal{L}_{\mathcal{E}}(\mathcal{I}d_{\mathcal{A}})=\mathcal{E}$
in $H^{0}(fun(\mathcal{A},\mathcal{C}))$. Hence $\mathcal{R}_{\mathcal{G}}$ is
cohomologically essentially surjective. Together with fullness and
faithfulness, this makes it a quasi-equivalence (the argument for
$\mathcal{L}_{\mathcal{G}}$ is similar). ∎

Finally, the notion of unitality for $A_{\infty}$-modules.

######  Remark 2.33.

Notice that the dg category $\mathsf{Ch}$ is strictly unital: for any complex
$(V,d_{V})\in\textup{obj}(\mathsf{Ch})$, the unique strict unit is given by
$e_{V}\coloneqq\textup{id}_{V}\in\textup{hom}_{\mathsf{Ch}}^{0}(V,V)$. Indeed,
by Definition 1.2:
$\mu_{\mathsf{Ch}}^{1}(\textup{id}_{V})=d(\textup{id}_{V})=d_{V}\circ\textup{id}_{V}-(-1)^{0}\textup{id}_{V}\circ
d_{V}=0$,
$(-1)^{|f|}\mu_{\mathsf{Ch}}^{2}(\textup{id}_{W},f)=(-1)^{2|f|}\textup{id}_{W}\circ
f=f=f\circ\textup{id}_{V}=\mu_{\mathsf{Ch}}^{2}(f,\textup{id}_{V})$ for all
$f\in$ $\textup{Hom}_{\mathsf{Ch}}(V,W)$, and readily
$\mu_{\mathsf{Ch}}^{d}=0$ if $d>2$.

By last Remark and Proposition 2.26, for any non-unital $A_{\infty}$-category
$\mathcal{A}$,
$\mathsf{Q}=nu\text{-}mod(\mathcal{A})=nu\text{-}\\!fun(\mathcal{A}^{opp},\mathsf{Ch})$
is strictly unital. The strict unit associated to
$\mathcal{M}\in\textup{obj}(\mathsf{Q})$ is the module homomorphism
$t\coloneqq
e_{\mathcal{M}}\in\textup{hom}_{\mathsf{Q}}^{0}(\mathcal{M},\mathcal{M})$
given by

$e_{\mathcal{M}}^{1}(b)\coloneqq(-1)^{|b|}b\in\mathcal{M}(X_{0})\qquad\text{and}\qquad
e_{\mathcal{M}}^{d}\coloneqq 0\quad\text{ if }d>1\,.$ (29)

We verify this in dimension 1 with the help of equations (– ‣ 2.20) and (27):
$\mu_{\mathsf{Q}}^{1}(t)^{1}(b)$
$=(-1)^{|b|}\mu_{\mathcal{M}}^{1}(t^{1}(b))+(-1)^{|b|}t^{1}(\mu_{\mathcal{M}}^{1}(b))=\mu_{\mathcal{M}}^{1}(b)+(-1)^{2|b|+1}\mu_{\mathcal{M}}^{1}(b)=0$
and
$(-1)^{|t_{1}|}\mu_{\mathsf{Q}}^{2}(t,t_{1})^{1}(b)=(-1)^{|t_{1}|+|b|}t^{1}(t_{1}^{1}(b))=t_{1}^{1}(b)=\mu_{\mathsf{Q}}^{2}(t_{1},t)^{1}(b)$.

######  Definition 2.34.

Let $\mathcal{A}$ be a c-unital $A_{\infty}$-category. An $A_{\infty}$-module
$\mathcal{M}\in\textup{obj}(\mathsf{Q})$ is c-unital if the
$H(\mathcal{A})$-module $H(\mathcal{M})$ is unital, that is, if c-units
$e_{X}\in\textup{hom}_{\mathcal{A}}^{0}(X,X)$ yield maps
$\mu_{\mathcal{M}}^{2}(\;\cdot\;,e_{X}):\mathcal{M}(X)\rightarrow\mathcal{M}(X)$
inducing the identity in cohomology.

Denote by $mod(\mathcal{A})\\!\subset\\!nu\text{-}mod(\mathcal{A})$ the full
subcategory of c-unital $A_{\infty}$-modules.

####  2.9 The Yoneda embedding

Before moving to triangulation of $A_{\infty}$-categories, we discuss an
important $A_{\infty}$-functor which will play a major role in the coming
constructions.

######  Definition 2.35.

The Yoneda embedding for the non-unital $A_{\infty}$-category $\mathcal{A}$ is
the non-unital $A_{\infty}$-functor
$\Upsilon\equiv\Upsilon_{\mathcal{A}}:\mathcal{A}\rightarrow\mathsf{Q}=nu\text{-}mod(\mathcal{A})$
which:

  * –

maps $Y\in\textup{obj}(\mathcal{A})$ to
$\mathcal{Y}\in\textup{obj}(\mathsf{Q})$ (called Yoneda module of $Y$) defined
by $\mathcal{Y}(X)\coloneqq\textup{hom}_{\mathcal{A}}(X,Y)$ and
$\mu_{\mathcal{Y}}^{d}\coloneqq\mu_{\mathcal{A}}^{d}:\textup{hom}_{\mathcal{A}}(X_{d-1},Y)\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\textup{hom}_{\mathcal{A}}(X_{0},Y)$
for $d\geq 1$ and $X,X_{0},...,X_{d-1}\in\textup{obj}(\mathcal{A})$,

  * –

consists of a sequence of graded maps $\Upsilon^{d}$ as in (9), whose first
order term $\Upsilon^{1}$ (of degree 0) maps
$c_{1}\\!\in\\!\textup{hom}_{\mathcal{A}}(Y_{0},Y_{1})$ to the pre-module
homorphism
$\Upsilon^{1}(c_{1})\\!\in\\!\textup{hom}_{\mathsf{Q}}(\mathcal{Y}_{0},\mathcal{Y}_{1})$
given by

$\displaystyle\Upsilon^{1}(c_{1})^{d}\\!:\;$
$\displaystyle\mathcal{Y}_{0}(X_{d-1})\otimes\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1})\otimes...\otimes\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\rightarrow\mathcal{Y}_{1}(X_{0})$
$\displaystyle(b,a_{d-1},...,a_{1})\mapsto\mu_{\mathcal{A}}^{d+1}(c_{1},b,a_{d-1},...,a_{1}),$
(30)

The higher order $\Upsilon^{d}$ are defined similarly:
$\Upsilon^{2}(c_{2},c_{1})^{d}(b,a_{d-1},...,a_{1})\\!\coloneqq\mu_{\mathcal{A}}^{d+2}(c_{2},c_{1},b,a_{d-1},...,a_{1})\in\mathcal{Y}_{2}$
for $c_{2}\in\textup{hom}_{\mathcal{A}}(Y_{1},Y_{2})$, and so on.

If moreover $\mathcal{A}$ is c-unital, then by Definition 2.34 the Yoneda
embedding is actually an $A_{\infty}$-functor $\Upsilon:\mathcal{A}\rightarrow
mod(\mathcal{A})$.

One can show that for a c-unital $A_{\infty}$-category $\mathcal{A}$,
$\Upsilon_{\mathcal{A}}$ is cohomologically full and faithful (the proof
involves Hochschild cohomology; see [Sei08, section (2g)]). Furthermore, for
any $A_{\infty}$-module $\mathcal{M}$ and $Y\in\textup{obj}(\mathcal{A})$, we
have a natural chain map

$\displaystyle\lambda=\lambda_{\mathcal{M}}:\mathcal{M}(Y)\rightarrow\textup{hom}_{\mathsf{Q}}(\mathcal{Y},\mathcal{M}),$
$\displaystyle\lambda(c)^{d}(b,a_{d-1},...,a_{1})\coloneqq\mu_{\mathcal{M}}^{d+1}(c,b,a_{d-1},...,a_{1})$
(31)

extending $\Upsilon^{1}$. Actually, $\lambda(c)$ is a module homomorphism as
soon as $c\in\mathcal{M}(Y)$ is a cocycle, by equation (– ‣ 2.20). Moreover,
equation (27) shows that:

$\displaystyle\langle\lambda_{\mathcal{M}}(\mu_{\mathcal{M}}^{2}(c,b))\rangle$
$\displaystyle=\langle\mu_{\mathsf{Q}}^{2}(\lambda_{\mathcal{M}}(c),\Upsilon^{1}(b))\rangle\in\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Y}_{0},\mathcal{M})\;,$
(32)
$\displaystyle\langle\mu_{\mathsf{Q}}^{2}(t,\lambda_{\mathcal{M}_{0}}(\tilde{c}))\rangle$
$\displaystyle=\langle\lambda_{\mathcal{M}_{1}}(t^{1}(\tilde{c}))\rangle\in\textup{Hom}_{H(\mathsf{Q})}(\mathcal{Y},\mathcal{M}_{1})\;,$

for any $\langle c\rangle\in H(\mathcal{M}(Y_{1}))$, $\langle
b\rangle\in\textup{Hom}_{H(\mathcal{A})}(Y_{0},Y_{1})$, $\langle
t\rangle\in\textup{Hom}_{H(\mathsf{Q})}(\mathcal{M}_{0},\mathcal{M}_{1})$ and
$\langle\tilde{c}\rangle\in H(\mathcal{M}_{0}(Y))$.

In particular, the extension of the classical Yoneda Lemma holds in the
c-unital case (see [Sei08, Lemma 2.12]).

######  Lemma 2.36 (Yoneda Lemma).

Let $\mathcal{A}$ be a c-unital $A_{\infty}$-category,
$\mathsf{Q}=mod(\mathcal{A})$. Then the module homomorphism
$\lambda_{\mathcal{M}}$ from (2.9) is actually a quasi-isomorphism, for any
$\mathcal{M}\in\textup{obj}(\mathsf{Q})$ and $Y\in\textup{obj}(\mathcal{A})$.

There is a way to relate Yoneda embeddings of distinct $A_{\infty}$-categories
$\mathcal{A}$, $\mathcal{B}$. It makes use of the following construction.

######  Definition 2.37.

A non-unital $A_{\infty}$-functor
$\mathcal{G}:\mathcal{A}\rightarrow\mathcal{B}$ induces a non-unital (dg)
pullback functor
$\mathcal{G}^{*}\coloneqq\mathcal{R}_{\mathcal{G}^{opp}}:nu\text{-}mod(\mathcal{B})\rightarrow
nu\text{-}mod(\mathcal{A})$, defined as follows:

  * –

$\mathcal{M}\in\textup{obj}(nu\text{-}mod(\mathcal{B}))$ is mapped to the
module $\mathcal{G}^{*}\mathcal{M}\in\textup{obj}(nu\text{-}mod(\mathcal{A}))$
given by
$(\mathcal{G}^{*}\mathcal{M})(X)\coloneqq\mathcal{M}(\mathcal{G}(X))\in\textup{obj}(\mathsf{Ch})$
for $X\in\textup{obj}(\mathcal{A})$, with associated

$\mu_{\mathcal{G}^{*}\\!\mathcal{M}}^{d}(b,a_{d-1},...,a_{1})\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0mu\mu_{\mathcal{M}}^{r}(b,\mathcal{G}^{s_{r}}(a_{d-1},...,a_{d-s_{r}}),...,\mathcal{G}^{s_{1}}(a_{s_{1}},...,a_{1}))\;,$

  * –

$t\in\textup{hom}_{nu\text{-}mod(\mathcal{B})}(\mathcal{M}_{0},\mathcal{M}_{1})$
is mapped to the pre-module homomorphism
$\mathcal{G}^{*}t\in\textup{hom}_{nu\text{-}mod(\mathcal{A})}(\mathcal{G}^{*}\mathcal{M}_{0},\mathcal{G}^{*}\mathcal{M}_{1})$
given by

$(\mathcal{G}^{*}t)^{d}(b,a_{d-1},...,a_{1})\coloneqq\mkern-24.0mu\sum_{\begin{subarray}{c}1\leq
r\leq d\\\
s_{1}+...+s_{r}=d\end{subarray}}\mkern-24.0mut^{r}(b,\mathcal{G}^{s_{r}}(a_{d-1},...,a_{d-s_{r}}),...,\mathcal{G}^{s_{1}}(a_{s_{1}},...,a_{1}))\;.$

If $\mathcal{G}$ is c-unital, then so is $\mathcal{G}^{*}$ (by Lemma 2.27),
resulting in a strictly unital (dg) functor $mod(\mathcal{B})\rightarrow
mod(\mathcal{A})$. If moreover $\mathcal{G}$ is a quasi-equivalence, then so
is $\mathcal{G}^{*}$.

Consider now a non-unital $A_{\infty}$-functor
$\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}$. Then the respective Yoneda
embeddings $\Upsilon_{\mathcal{A}}$ and $\Upsilon_{\mathcal{B}}$ are related
via the natural transformation
$T:\Upsilon_{\mathcal{A}}\rightarrow\mathcal{F}^{*}\circ\Upsilon_{\mathcal{B}}\circ\mathcal{F}$
in $nu\text{-}\\!fun(\mathcal{A},\mathsf{Q})$, whose
$T^{1}:\textup{hom}_{\mathcal{A}}(Y_{0},Y_{1})\rightarrow\textup{hom}_{\mathsf{Q}}(\Upsilon_{\mathcal{A}}(Y_{0}),\mathcal{F}^{*}\Upsilon_{\mathcal{B}}\mathcal{F}(Y_{1}))$
assigns to each morphism in $\mathcal{A}$ a module homorphism in $\mathsf{Q}$,
itself a collection of graded maps

$\displaystyle(\Upsilon_{\mathcal{A}}(Y))(X_{d-1})\\!\otimes\\!\textup{hom}_{\mathcal{A}}(X_{d-2},X_{d-1})\\!\otimes\\!...\\!\otimes\\!\textup{hom}_{\mathcal{A}}(X_{0},X_{1})\\!\rightarrow\\!\overbrace{(\mathcal{F}^{*}\Upsilon_{\mathcal{B}}\mathcal{F}(Y))(X_{0})}^{=\textup{hom}_{\mathcal{B}}(\mathcal{F}(X_{0}),\mathcal{F}(Y))},$
$\displaystyle(b,a_{d-1},...,a_{1})\mapsto(-1)^{\dagger_{d-1}+|b|}\mathcal{F}^{d}(b,a_{d-1},...,a_{1})\;.$
(33)

######  Remark 2.38.

If we assume that $\mathcal{A}$, $\mathcal{B}$ are c-unital and $\mathcal{F}$
is cohomologically full and faithful, it turns out that $T^{1}$ is an
isomorphism of c-unital $A_{\infty}$-modules. Therefore, Lemma 2.15 implies
that $T$ is an isomorphism in the respective cohomological category, producing
commutativity of the diagram

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