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#  $A_{\infty}$-Deformations and their Derived Categories

Jasper van de Kreeke

(15 March 2023)

###### Abstract

Handling curved $A_{\infty}$-deformations is challenging and defining their
derived categories seems impossible. In this paper, we show how to welcome the
curvature and build derived categories despite the apparent difficulties. We
explicitly describe their $A_{\infty}$-structure by a deformed Kadeishvili
theorem.

This paper starts with a review of $A_{\infty}$-deformations for non-
specialists. We explain why $A_{\infty}$-deformations require curvature and
why curvature supposedly causes problems. We prove that infinitesimally curved
$A_{\infty}$-deformations are in fact non-problematic and provide the correct
definitions for their twisted completion and minimal model.

The classical Kadeishvili theorem collapses for curved deformations. We show
how to adapt it by iteratively optimizing the curvature and changing the
homological splitting infinitesimally. The final $A_{\infty}$-structure on the
minimal model is constructed in terms of trees.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on $A_{\infty}$-categories
  3. 3 Deformations of $A_{\infty}$-categories
    1. 3.1 The completed tensor product
    2. 3.2 Curved deformations
    3. 3.3 Twisted completion
    4. 3.4 Functors
  4. 4 The lens of DGLAs
    1. 4.1 The Hochschild DGLA
    2. 4.2 Gauge equivalence
    3. 4.3 $L_{\infty}$-algebras
    4. 4.4 Axioms on $A_{\infty}$-deformations
  5. 5 Derived categories of $A_{\infty}$-deformations
    1. 5.1 Twisted completion
    2. 5.2 Minimal models
    3. 5.3 Derived categories
    4. 5.4 The theory of uncurving
  6. 6 A deformed Kadeishvili theorem
    1. 6.1 Homological splittings
    2. 6.2 The classical Kadeishvili theorem
    3. 6.3 Deformed differentials
    4. 6.4 Optimizing curvature
    5. 6.5 Auxiliary minimal model procedure
    6. 6.6 The deformed Kadeishvili theorem
    7. 6.7 The $D=0$ case

##  1 Introduction

$A_{\infty}$-categories are a tool to store homotopical information in a
category. For instance, they are the standard method of homological mirror
symmetry to capture the higher products on Fukaya categories and categories of
coherent sheaves. From this background, mathematicians have also introduced
the notion of $A_{\infty}$-deformations. A prominent example is Seidel’s
relative Fukaya category [19], a deformation of the ordinary Fukaya category.
As Seidel observed, the relative Fukaya category must be defined as a so-
called curved $A_{\infty}$-deformation.

Many constructions are available for $A_{\infty}$-categories. For instance, an
$A_{\infty}$-category $\mathcal{C}$ has a twisted completion
$\operatorname{Tw}\mathcal{C}$ and minimal model
$\operatorname{H}\mathcal{C}$. The derived category of $\mathcal{C}$ is the
$A_{\infty}$-category $\operatorname{H}\operatorname{Tw}\mathcal{C}$, the
minimal model of the twisted completion of $\mathcal{C}$. A crucial necessity
for defining derived categories is that the differential of $\mathcal{C}$
squares to zero.

$A_{\infty}$-category $\mathcal{C}$Deformed $A_{\infty}$-category
$\mathcal{C}_{q}$Derived category
$\operatorname{H}\operatorname{Tw}\mathcal{C}$deformationderivedequivalent

At first sight, it may seem impossible to fill this square and define a
derived category for $\mathcal{C}_{q}$ which is a deformation of
$\operatorname{H}\operatorname{Tw}\mathcal{C}$. The problem is that the
deformed differential of $\mathcal{C}_{q}$ need not square to zero because of
the curvature. In the present paper, we solve the dilemma and show how to
correctly define the twisted completion $\operatorname{Tw}\mathcal{C}_{q}$ and
minimal model $\operatorname{H}\mathcal{C}_{q}$ of any curved
$A_{\infty}$-deformation. We define the derived category of $\mathcal{C}_{q}$
as the minimal model of the twisted completion, effectively filling the
square:

$A_{\infty}$-category $\mathcal{C}$Deformed $A_{\infty}$-category
$\mathcal{C}_{q}$Derived category
$\operatorname{H}\operatorname{Tw}\mathcal{C}$Deformed derived category
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$deformationdeformationderivedequivalentderivedequivalent

The literature includes attempts to avoid this “curvature problem” [11], for
instance Lowen and Van den Bergh [15] investigate how to get rid of curvature
by passing to a Morita equivalent model of $\mathcal{C}$, and Positselski [18]
shows how to define an alternative notion of derived categories for curved
$A_{\infty}$-categories. However, the approaches do not lead to a notion of
derived category that is a deformation of
$\operatorname{H}\operatorname{Tw}\mathcal{C}$.

### Context

To the benefit of the reader, we summarize here the background of
$A_{\infty}$-theory. We comment on $A_{\infty}$-deformations and the problems
one encounters when trying to build their derived categories. We also explain
how the results of the present paper are used to conduct an explicit minimal
model calculation in [14].

####  $A_{\infty}$-categories and their deformations

$A_{\infty}$-categories are a type of category used to store homotopical
information. In comparison with the better-known dg categories, they allow for
more flexibility. While dg categories come with only two operations, the
differential and the composition, an $A_{\infty}$-category allows room for
higher products $\mu^{3},\mu^{4},\ldots$. The flexibility entails that many
different $A_{\infty}$-categories are equivalent to a given one, varying in
size and amount of nonzero higher products.

Deformation theory is a means of capturing all ways to algebraically or
geometrically deform objects. The language of $A_{\infty}$-categories makes it
possible to define immediately an algebraic deformation theory over a given
local ring $B$. The idea is to look for all ways to deform the
$A_{\infty}$-products $\mu^{1},\mu^{2},\ldots$ of $\mathcal{C}$ and allow
additional curvature $\mu^{0}$ in such a way that the products satisfy the
“curved $A_{\infty}$-relations” and reduce to the $A_{\infty}$-structure of
$\mathcal{C}$ once the maximal ideal of $B$ is divided out.

#### Minimal models and derived categories

Given an $A_{\infty}$-category $\mathcal{C}$ and an equivalent
$A_{\infty}$-category $\mathcal{D}$, one also says that $\mathcal{D}$ is a
different model for $\mathcal{C}$. Switching to a different model of a given
$A_{\infty}$-category is often tedious because it requires computing the
higher products on the model. The Kadeishvili construction is an explicit
method to construct these higher products. The idea is originally due to
Kadeishvili, but has been worked out explicitly in [13, Chapter 6, 3.3.2].

Minimal models are a way to decrease the size of the hom spaces of an
$A_{\infty}$-category. An $A_{\infty}$-category is called minimal if its
differential $\mu^{1}$ vanishes. A minimal model for $\mathcal{C}$ is by
definition any $A_{\infty}$-category $\operatorname{H}\mathcal{C}$ which is
quasi-isomorphic to $\mathcal{C}$ and whose differential
$\mu^{1}_{\operatorname{H}\mathcal{C}}$ vanishes. Thanks to the Kadeishvili
construction, every $A_{\infty}$-category clearly has a minimal model: We take
the cohomology of all the hom spaces and equip them with the
$A_{\infty}$-structure produced by the Kadeishvili construction.

Twisted completion is a way to form an $A_{\infty}$-category of formal chain
complexes of objects. The idea is to include arbitrary formal sums of shifted
objects together with an upper-triangular matrix which represents the
differential. Derived categories of $A_{\infty}$-categories are defined by
combining the twisted completion and minimal model construction. More
precisely, the derived category of $\mathcal{C}$ is defined as the
$A_{\infty}$-category $\operatorname{H}\operatorname{Tw}\mathcal{C}$.

#### The curvature problem and derived categories

In formal $A_{\infty}$-deformation theory, one regards infinitesimal
deformations of a given $A_{\infty}$-structure such that the
$A_{\infty}\,$-relations are preserved. An issue is that only deforming the
$A_{\infty}$-structure does not give a notion of deformations that is
invariant under quasi-equivalence of $A_{\infty}$-categories. In order to
obtain a notion invariant under quasi-equivalence, one needs to permit the
deformation to have curvature. More precisely, the curvature must be
infinitesimal in the sense that it lies in a multiple of the maximal ideal of
the local ring. Infinitesimal curvature is inevitable for a good notion of
$A_{\infty}$-deformations.

Dealing with curvature is however regarded as tedious, because the curvature
prevents the differential from squaring to zero. The presence of curvature is
often referred to as the “curvature problem”. A main question is how to gauge
away the curvature or otherwise how to deal with the remaining curvature. An
instance of the uncurving problem has been studied by Lowen and Van den Bergh
[15].

Unfortunately the deformation makes it difficult to speak about twisted
completions, minimal models and derived categories: Twisted complexes are
classically assumed to satisfy the Maurer-Cartan equation, a condition which
ensures that the category $\operatorname{Tw}\mathcal{C}$ becomes curvature-
free. Minimal models require that the differential squares to zero in order to
be able to take cohomology. Derived categories require both the twisted
complex and minimal model construction to be well-defined.

#### Application of the Kadeishvili construction

In [14], we work out an explicit example where the deformed
$A_{\infty}\,$-category is a deformed gentle algebra
$\mathcal{C}_{q}=\operatorname{Tw}\operatorname{Gtl}_{q}Q$. We provide an
explicit matching of “result components” coming from the minimal model
computation for $\operatorname{H}\mathcal{C}_{q}$ and relative Fukaya disks of
the surface $|Q|$. This entails working out the meaning of all complicated
Kadeishvili trees. This way, we prove that
$\operatorname{H}\operatorname{Tw}\operatorname{Gtl}_{q}Q$ agrees with the
relative Fukaya category of $|Q|$ with respect to the set of punctures
$Q_{0}$.

out$h_{1}$$h_{2}$$h_{3}$$h_{4}$$h_{5}$$h_{6}$[14]$h_{6}$$h_{5}$$h_{4}$$h_{3}$$h_{2}$$h_{1}$$h_{q}\mu^{2}_{q}$$h_{q}\mu^{2}_{q}$$h_{q}\mu^{2}_{q}$$h_{q}\mu^{2}_{q}$$\pi_{q}\mu^{2}_{q}$

### Results

The aim of the present paper is to show that deformations of
$A_{\infty}$-categories have derived categories. The first step is to define
the twisted completion and minimal model of deformed $A_{\infty}$-categories
in an abstract way. This abstract method entails first forming the twisted
completion or minimal model of the $A_{\infty}$-category without deformation
and afterwards pushing the deformation onto it. The second step is to describe
twisted completion and minimal model explicitly. For the twisted completion,
the construction is tricky but straight-forward. For the minimal model, the
construction is a fine-tuned variant of the Kadeishvili construction.

| Original | Deformed
---|---|---
Category | $\mathcal{C}$ | $\mathcal{C}_{q}$
Twisted completion | $\operatorname{Tw}\mathcal{C}$ | $\operatorname{Tw}\mathcal{C}_{q}$
Minimal model | $\operatorname{H}\mathcal{C}$ | $\operatorname{H}\mathcal{C}_{q}$
Derived category | $\operatorname{H}\operatorname{Tw}\mathcal{C}$ | $\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$

#### Abstract definition for twisted completion

The most conceptual way to define $\operatorname{Tw}\mathcal{C}_{q}$ is to
take the twisted completion $\operatorname{Tw}\mathcal{C}$ and induce the
deformation afterwards. More precisely, the first step is to form the twisted
completion $\operatorname{Tw}\mathcal{C}$ and regard the inclusion
$i:\mathcal{C}\to\operatorname{Tw}\mathcal{C}$. The second step is to observe
the existence of a noncanonical $L_{\infty}$-isomorphism of Hochschild DGLAs
$i_{*}:\operatorname{HC}(\mathcal{C})\to\operatorname{HC}(\operatorname{Tw}\mathcal{C})$
since $i$ is a derived equivalence. The deformation $\mathcal{C}_{q}$ is a
Maurer-Cartan element of $\operatorname{HC}(\mathcal{C})$ over the given local
ring. The third step is to push the Maurer-Cartan element via $i_{*}$ and
obtain a Maurer-Cartan element of
$\operatorname{HC}(\operatorname{Tw}\mathcal{C})$. This is the desired
deformation $\operatorname{Tw}\mathcal{C}_{q}$.

#### Explicit construction for twisted completion

The explicit way to define $\operatorname{Tw}\mathcal{C}_{q}$ is by gathering
the same objects as $\operatorname{Tw}\mathcal{C}$, namely those twisted
complexes satisfying the non-deformed Maurer-Cartan equation
$\mu^{1}(\delta)+\mu^{2}(\delta,\delta)+\ldots=0$. This is not the same as
gathering twisted complexes with twisted differential $\delta$ satisfying the
deformed Maurer-Cartan equation $\mu^{0}_{q}+\mu^{1}_{q}(\delta)+\ldots=0$. In
contrast to $\operatorname{Tw}\mathcal{C}$, the category
$\operatorname{Tw}\mathcal{C}_{q}$ inherits infinitesimal curvature, stemming
precisely from the infinitesimal failure of the twisted differentials to
satisfy the deformed Maurer-Cartan equation.

#### Abstract definition for the minimal model

The most conceptual way to define $\operatorname{H}\mathcal{C}_{q}$ consists
of taking any minimal model $\operatorname{H}\mathcal{C}$ and inducing the
deformation $\mathcal{C}_{q}$ onto $\operatorname{H}\mathcal{C}$ via any
quasi-isomorphism $\pi:\mathcal{C}\to\operatorname{H}\mathcal{C}$. This
abstract approach means that the “minimal model”
$\operatorname{H}\mathcal{C}_{q}$ may have infinitesimal curvature as well as
a residual infinitesimal differential. This is not the same as taking
cohomology of the hom complexes
$(\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y),\mu^{1}_{q})$. In fact, the
deformed differential of an $A_{\infty}$-deformation need not even square to
zero because of the curvature.

#### Explicit construction for the minimal model

Minimal models of $A_{\infty}$-categories can classically be computed by means
of homological splittings and Kadeishvili trees. In our deformed Kadeishvili
theorem, we show that this method carries over to the deformed case. The
starting point is an $A_{\infty}$-category $\mathcal{C}$ together with an
deformation $\mathcal{C}_{q}$. The difficulties encountered in constructing
the minimal model are the presence of curvature $\mu^{0}_{q}$, the fact that
the deformed differential $\mu^{1}_{q}$ does not square to zero and the fact
that $\mu^{1}_{q}$ is not compatible with the homological splitting chosen for
$\mathcal{C}$. In section 6, we show how to adapt the Kadeishvili construction
to these special circumstances.

Our construction can be summarized as follows:

  1. 1.

Choose a homological splitting for $\mathcal{C}$.

  2. 2.

Perform an infinitesimal change on the homological splitting in order to adapt
it to $\mu^{1}_{q}$.

  3. 3.

Gauge away part of the deformation’s curvature.

  4. 4.

Repeat steps 2 and 3 indefinitely. Take the limit of this process.

  5. 5.

Calculate the deformed codifferential $h_{q}$ and projection $\pi_{q}$.

  6. 6.

Define the structure of $\operatorname{H}\mathcal{C}_{q}$ by sums over
deformed Kadeishvili trees.

### Relation to the literature

The theory of $A_{\infty}$-deformations presented in section 3 and 4 is known
to experts. In the literature the topic is however under-represented, to the
extent that it is hard to pinpoint a single location where
$A_{\infty}$-deformations are defined. In what follows, we summarize three
articles which gave rise to the understanding of the theory presented in this
paper.

#### Barmeier and Wang

In [1], Barmeier and Wang prove the power of $L_{\infty}$-morphisms in
deformation theory of ordinary algebras. They depart from a quiver algebra
with relations $A=\mathbb{C}Q/I$, where the ideal $I$ is supposed to come from
a reduction system. In order to classify all deformations of $A$, they replace
the Hochschild DGLA $\operatorname{HC}(A)$ by a quasi-isomorphic
$L_{\infty}$-algebra $L(A)$.

To define the Hochschild DGLA, we typically use the bar resolution. Barmeier
and Wang demonstrate that this resolution can be substituted by any other one,
in particular the very simple bimodule resolution $P_{\bullet}$ of Chouhy-
Solotar, see [1, Section 4.2]. The cochain map between resolutions immediately
gives a quasi-isomorphism of complexes

$\operatorname{Hom}_{A\otimes
A^{\mathsf{opp}}}\left(\bigoplus\limits_{i\in\mathbb{N}}A\otimes
A^{\otimes_{i}}\otimes
A,A\right)\xrightarrow{\sim}\operatorname{Hom}_{A\otimes
A^{\mathsf{opp}}}(P_{\bullet},A).$ (1.1)

The left-hand side already being a DGLA, the $L_{\infty}$-structure transfer
theorem induces an $L_{\infty}$-structure on the right-hand side such that
(1.1) becomes a quasi-isomorphism of $L_{\infty}$-algebras. Barmeier and Wang
then compute part of the $L_{\infty}$-structure on the right-hand side, just
enough to classify all of its Maurer-Cartan elements. All deformations of $A$
can then be written down explicitly.

Barmeier and Wang’s realization that $L_{\infty}$-quasi-isomorphisms transport
Maurer-Cartan elements has greatly helped us shape the curved
$A_{\infty}$-deformation theory of section 4. There, we make constant use of
the fact that quasi-equivalences of $A_{\infty}$-categories induces
$L_{\infty}$-quasi-isomorphisms of their Hochschild DGLAs:

$F:\mathcal{C}\xrightarrow{\sim}\mathcal{D}\quad\rightsquigarrow\quad
F_{*}:\operatorname{HC}(\mathcal{C})\xrightarrow{\sim}\operatorname{HC}(\mathcal{D}).$

Correspondingly, their sets of Maurer-Cartan elements over any deformation
base $B$ match:

$\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{C}),B)\xrightarrow{\sim}\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{D}),B).$

This is the main principle that lets us push deformations to and fro between
different categories. For example, it lets us seamlessly reduce an
$A_{\infty}$-category to a skeleton of non-isomorphic objects without changing
its deformation theory. It enable us to prove that if an object
$X\in\mathcal{C}_{q}$ is uncurvable, then all objects $Y\in\mathcal{C}_{q}$
quasi-isomorphic to $X$ are uncurvable as well.

#### Keller

In [9], Keller proves the $L_{\infty}$-invariance of the Hochschild DGLA under
Morita equivalence of dg-categories. Keller actually regards the Hochschild
DGLA as a $B_{\infty}$-algebra which is even stronger than the notion of
$L_{\infty}$-algebra.

Let $A$ and $B$ be Morita equivalent dg algebras, with Morita equivalence
provided by the $A$-$B$-bimodule $M$. Then Keller’s core argument for
invariance is as follows: Embed $A$ and $B$ into the triangular dg algebra

$D\coloneqq\begin{pmatrix}A&M\\\ 0&B\end{pmatrix}.$

Both embeddings $A,B\subseteq D$ turn out to be Morita equivalences. Keller
exploits the natural restriction maps
$\operatorname{HC}(D)\to\operatorname{HC}(A)$ and
$\operatorname{HC}(D)\to\operatorname{HC}(B)$, which automatically respect the
$B_{\infty}$-structure. Departing from the knowledge that Hochschild
cohomology of dg algebras is invariant under Morita equivalences as graded
vector space, Keller concludes that both
$\operatorname{HC}(D)\to\operatorname{HC}(A)$ and
$\operatorname{HC}(D)\to\operatorname{HC}(B)$ are $B_{\infty}$-quasi-
isomorphisms. Correspondingly, $\operatorname{HC}(A)$ and
$\operatorname{HC}(B)$ are $B_{\infty}$-quasi-isomorphic.

Unfortunately, Keller’s result is currently restricted to the case dg
algebras. To the present paper, this means that we simply assume as axioms
that the theory extends to all $A_{\infty}$-categories. The heavy use of
$A_{\infty}$-deformations in our paper shows how imperative it is to
rigorously extend Keller’s paper to the $A_{\infty}$-context. According to
private communication with Keller, these results can likely be obtained by the
same triangular construction together with appropriate $A_{\infty}$-bimodule
theory.

#### Lowen and Van den Bergh

In [15], Lowen and Van den Bergh explain how to remove curvature from
$A_{\infty}$-deformations of dg categories. They depart from a dg algebra $A$
together with an infinitesimally curved $A_{\infty}$-deformation $A_{q}$ over
$\mathbb{C}\llbracket q\rrbracket$. Lowen and Van den Bergh observe that a
category $\operatorname{Tw}(A_{q})$ of twisted complexes over $A_{q}$ can be
formed even with infinitesimal entries below the diagonal, just as in our
Definition 5.1. Interpret $A_{q}$ as an $A_{\infty}$-deformation with a single
object. Then the core observation of Lowen and Van den Bergh is that the
following twisted complex has vanishing curvature:

$X\coloneqq\left(A\oplus A[1],\begin{pmatrix}0&\mu^{0}_{q}/q\\\
q\mathrm{id}_{A}&0\end{pmatrix}\right)\in\operatorname{Tw}(A_{q}).$ (1.2)

This means that
$C_{q}\coloneqq\operatorname{End}_{\operatorname{Tw}(A_{q})}(X,X)$ is a
curvature-free deformed $A_{\infty}$-algebra. It is interesting to note that
its special fiber $C$ at $q=0$ is actually a dg algebra. Indeed, the higher
products $\mu^{\geq 3}$ on $C_{q}$ are given by embracing $\mu_{A_{q}}$ with
the matrix entries $\mu^{0}_{q}/q$ and $q\mathrm{id}_{A}$:

$\mu^{k\geq 3}_{C_{q}}(a_{k},\ldots,a_{1})=\sum\mu^{\geq
3}_{\operatorname{Add}A_{q}}(\delta,\ldots,\delta,a_{k},\ldots,\delta,\ldots,\delta,a_{1},\delta,\ldots,\delta).$

Restricting this sum to $q=0$ yields only higher products $\mu^{\geq 3}$ of
$A$. Since $A$ is a dg algebra, we deduce $\mu^{k\geq 3}_{C}=0$. In
conclusion, $C$ is a dg algebra as well and $C_{q}$ is a curvature-free
$A_{\infty}$-deformation of $C$.

Lowen and Van den Bergh prove that $A$ and $C$ are in fact related by Morita
equivalence. This costs substantial effort and uses the assumption that the
curvature $\mu^{0}_{A_{q}}$ is nilpotent in the cohomology of $A$. The result
is however that $A$ and $C$ are Morita equivalent, and moreover that $C_{q}$
is the deformation of $C$ corresponding to the deformation $A_{q}$ of $A$
along this Morita equivalence:

dg algebra A$\rightsquigarrow$dg algebra $C$curved $A_{\infty}$-deformation
$A_{q}$$\rightsquigarrow$uncurved $A_{\infty}$-deformation $C_{q}$

The work of Lowen and Van den Bergh illustrates that curvature is essential in
the notion of $A_{\infty}$-deformations, but not an invariant on its own.
Curvature can sometimes be removed by changing to a derived equivalent
$A_{\infty}$-category. Our uncurving construction for band objects in [14]
provides an example where a mere gauge transformation suffices to remove
curvature.

Uncurving is a crucial contribution to our deformed Kadeishvili construction.
The presence of curvature namely allows the differential of an
$A_{\infty}$-deformation to be behave in an arbitrary way. In contrast to the
classical Kadeishvili construction, the differential fails to structure the
hom spaces into cohomology, image and remainder term. By iteratively gauging
the deformation, we optimizing the curvature value until in the limit the
differential structures the hom spaces sufficiently. Only after this uncurving
procedure can the deformed minimal model be read off from Kadeishvili trees.

As aftermath of our paper, we conclude that there is theoretically no
hindrance to forming a derived category of a curved $A_{\infty}$-deformation:
By section 5.3, a derived category
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ exists even for
infinitesimally curved deformations. The statement of Lowen and Van den Bergh
that a curved $A_{\infty}$-deformation has no classical derived category
remains true, but our paper contends that the study of deformations profits
greatly from permitting also these “non-classical derived categories”
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$.

### Structure of the paper

In section 2, we recall $A_{\infty}$-categories. In section 3, we recall
$A_{\infty}$-deformations, their twisted completion and functors. In section
4, we interpret $A_{\infty}$-deformations via deformation theory and DGLAs. In
section 5, we explain the abstract definitions of twisted completion and
minimal models and comment on uncurving theory. In section 6, we construct our
deformed Kadeishvili theorem.

For the purposes of section 4, 5.2 and 5.4, we have to assume axioms on the
invariance of the Hochschild DGLA under $A_{\infty}$-quasi-equivalences. We
summarize these axioms in Convention 4.1.

### Acknowledgements

This paper is part of the author’s PhD thesis supervised by Raf Bocklandt.
Crucial insights were gained at the 2020 Summer Camp on Derived Categories,
Stability Conditions and Deformations, which the author organized in his
garden in Berlin. The author thanks all participants for sharing their
knowledge, in particular Severin Barmeier for advice and expertise on
deformation theory through the lens of $L_{\infty}$-algebras. The author’s PhD
project was supported by the NWO grant “Algebraic methods and structures in
the theory of Frobenius manifolds and their applications” (TOP1.17.012).

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

The notion of $A_{\infty}$-category is now widely used as a homological-
algebraic tool to study symplectic and algebraic geometry. Its relation to dg
categories, triangulated categories and $\infty$-categories can be described
as follows:

  * •

DG categories are more rigid than $A_{\infty}$-categories. DG quasi-
isomorphisms cannot be quasi-inverted on the dg level, while they always have
a quasi-inverse if one interprets the dg structures as
$A_{\infty}$-categories. In particular, zigzags of dg quasi-isomorphisms can
be resolved into a single $A_{\infty}$-quasi-isomorphism. DG structures are
very easy to work with, for instance homotopy colimits are easy to calculate
due to the well-known Tabuada model structure [20, 10].

  * •

Triangulated categories are weaker than $A_{\infty}$-categories. They do not
remember any of the “higher structure” that $A_{\infty}$-categories contain.
The higher structure on an $A_{\infty}$-category makes the entire category
reconstructible from a subcategory of generators. There is an $A_{\infty}$
notion of derived category
$\operatorname{D}\mathcal{C}\coloneqq\operatorname{H}\operatorname{Tw}\mathcal{C}$,
whose “lower structure” $(\operatorname{D}\mathcal{C})^{0}$ is a triangulated
category [12].

  * •

Stable $\infty$-categories also remember the homotopy information, similar to
$A_{\infty}$-categories. A priori they lack the linearity over a base field,
which can be added afterwards. In fact, there is a direct correspondence
between $k$-linear stable $\infty$-categories and $A_{\infty}$-categories over
$k$, given in one direction by taking the derived category [6, 7].

We work over an algebraically closed field of characteristic zero and always
write $\mathbb{C}$. Let us now recall the definition of
$A_{\infty}$-categories:

######  Definition 2.1.

A ($\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded, strictly unital)
$A_{\infty}$-category $\mathcal{C}$ consists of a collection of objects
together with $\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded hom spaces
$\operatorname{Hom}(X,Y)$, distinguished identity morphisms
$\mathrm{id}_{X}\in\operatorname{Hom}^{0}(X,X)$ for all $X\in\mathcal{C}$,
together with multilinear higher products

$\mu^{k}:\operatorname{Hom}(X_{k},X_{k+1})\times\ldots\times\operatorname{Hom}(X_{1},X_{2})\to\operatorname{Hom}(X_{1},X_{k+1}),\quad
k\geq 1$

of degree $2-k$ such that the $A_{\infty}$-relations and strict unitality
axioms hold: For all compatible morphisms $a_{1},\ldots,a_{k}$ we have

$\displaystyle\sum_{0\leq n<m\leq
k}(-1)^{\|a_{n}\|+\ldots+\|a_{1}\|}\mu(a_{k},\ldots,\mu(a_{m},\ldots,a_{n+1}),a_{n},\ldots,a_{1})=0,$
$\displaystyle\mu^{2}(a,\mathrm{id}_{X})=a,~{}\mu^{2}(\mathrm{id}_{Y},a)=(-1)^{|a|}a,~{}\mu^{\geq
3}(\ldots,\mathrm{id}_{X},\ldots)=0.$

######  Remark 2.1.

Let us write down the first few $A_{\infty}$-relations. In particular, we note
that if $\mu^{1}$ or $\mu^{3}$ vanishes then $\mu^{2}$ is graded associative:

$\displaystyle\mu^{1}(\mu^{1}(a))$ $\displaystyle=0,$
$\displaystyle(-1)^{\|b\|}\mu^{2}(\mu^{1}(a),b)+\mu^{2}(a,\mu^{1}(b))+\mu^{1}(\mu^{2}(a,b))$
$\displaystyle=0,$
$\displaystyle(-1)^{\|c\|}\mu^{2}(\mu^{2}(a,b),c)+\mu^{2}(a,\mu^{2}(b,c))+\mu^{1}(\mu^{3}(a,b,c))\phantom{XYZ}$
$\displaystyle+(-1)^{\|b\|+\|c\|}\mu^{3}(\mu^{1}(a),b,c)+(-1)^{\|c\|}\mu^{3}(a,\mu^{1}(b),c)+\mu^{3}(a,b,\mu^{1}(c))$
$\displaystyle=0.$

Recall that a functor $F:\mathcal{C}\to\mathcal{D}$ of $A_{\infty}$-categories
is a map which intertwines the products of $\mathcal{C}$ and $\mathcal{D}$:

######  Definition 2.2.

Let $\mathcal{C}$ and $\mathcal{D}$ be $A_{\infty}$-categories. A functor
$F:\mathcal{C}\to\mathcal{D}$ consists of a map
$F:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$ together
with for every $k\geq 1$ a degree $1-k$ multilinear map

$F^{k}:\operatorname{Hom}_{\mathcal{C}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{C}}(FX_{1},FX_{k+1})$

such that the $A_{\infty}$-functor relations hold:

$\sum_{0\leq j<i\leq
k}(-1)^{\|a_{j}\|+\ldots+\|a_{1}\|}F(a_{k},\ldots,a_{i+1},\mu(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})\\\
=\sum_{\begin{subarray}{c}l\geq 0\\\ 1=j_{1}<\ldots<j_{l}\leq
k\end{subarray}}\mu(F(a_{k},\ldots,a_{j_{l}}),\ldots,F(\ldots,a_{j_{2}}),F(\ldots,a_{j_{1}})).$

When $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{E}$ are
$A_{\infty}$-functors, then their composition $GF$ is given on objects by
$G\circ F:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{E})$ and
on morphisms by

$(GF)(a_{k},\ldots,a_{1})=\sum G(F(a_{k},\ldots),\ldots,F(\ldots,a_{1})).$

There are several notions for functors between $A_{\infty}$-categories to be
equivalences, which we recall as follows:

######  Definition 2.3.

Let $\mathcal{C}$ and $\mathcal{D}$ be two $A_{\infty}$-categories and
$F:\mathcal{C}\to\mathcal{D}$ be a functor. Then $F$ is

  * •

an isomorphism if it is an isomorphism on object level and
$F^{1}:\operatorname{Hom}(X,Y)\to\operatorname{Hom}(FX,FY)$ is an isomorphism
for every $X,Y\in\mathcal{C}$,

  * •

a quasi-isomorphism if the induced functor
$\operatorname{H}F:\operatorname{H}\mathcal{C}\to\operatorname{H}\mathcal{D}$
is an isomorphism,

  * •

a quasi-equivalence if
$(\operatorname{H}F)^{1}:\operatorname{Hom}_{\operatorname{H}\mathcal{C}}(X,Y)\to\operatorname{Hom}_{\operatorname{H}\mathcal{D}}(FX,FY)$
is an isomorphism for every $X,Y\in\mathcal{C}$ and if $\operatorname{H}F$
reaches every object in $\mathcal{D}$ up to quasi-isomorphism,

  * •

a derived equivalence if the induced functor
$\operatorname{H}\operatorname{Tw}F:\operatorname{H}\operatorname{Tw}\mathcal{C}\to\operatorname{H}\operatorname{Tw}\mathcal{D}$
is an equivalence.

There are several notions for $A_{\infty}$-categories to be equivalent,
without explicit reference to a functor:

######  Definition 2.4.

Let $\mathcal{C}$ and $\mathcal{D}$ be two $A_{\infty}$-categories. Then
$\mathcal{C}$ and $\mathcal{D}$ are

  * •

isomorphic if there exists an isomorphism $F:\mathcal{C}\to\mathcal{D}$,

  * •

quasi-isomorphic if there exists a quasi-isomorphism
$F:\mathcal{C}\to\mathcal{D}$,

  * •

quasi-equivalent if there exists a quasi-equivalence
$F:\mathcal{C}\to\mathcal{D}$,

  * •

derived equivalent if $\operatorname{Tw}\mathcal{C}$ and
$\operatorname{Tw}\mathcal{D}$ are quasi-equivalent.

######  Remark 2.2.

No functor in either direction is required for two $A_{\infty}$-categories to
be derived equivalent. The contrast with quasi-isomorphisms is due to the lack
of functors $\operatorname{Tw}\mathcal{C}\to\mathcal{C}$, in contrast to the
existence of natural functors $\operatorname{H}\mathcal{C}\to\mathcal{C}$ and
$\mathcal{C}\to\operatorname{H}\mathcal{C}$. There are many more equivalent
and equally esthetic ways to define every of the above notions, so we have
only presented a selection.

Let us recall minimal models of $A_{\infty}$-categories. These models are
called minimal because the differential on all hom spaces vanish. In section
6, we comment extensively on minimal models. Minimal models of
$A_{\infty}$-category are not uniquely defined. Instead we fix the following
parlance:

######  Definition 2.5.

An $A_{\infty}$-category is minimal if its differential vanishes. Let
$\mathcal{C}$ be an $A_{\infty}$-category. Then a minimal model for
$\mathcal{C}$ is any minimal $A_{\infty}$-category
$\operatorname{H}\mathcal{C}$ such that $\mathcal{C}$ and
$\operatorname{H}\mathcal{C}$ are quasi-isomorphic.

If $\mathcal{C}$ is a minimal $A_{\infty}$-category, let us regard its degree-
zero part $\mathcal{C}^{0}$. It has the same objects as $\mathcal{C}$ but only
including the degree-zero part of the hom spaces. Then the product composition
$a\circ b\coloneqq(-1)^{|b|}\mu^{2}(a,b)$ is associative on $\mathcal{C}^{0}$,
rendering $\mathcal{C}^{0}$ an ordinary $\mathbb{C}$-linear category. Two
objects $X,Y\in\mathcal{C}$ are quasi-isomorphic if they are isomorphic in
$(\operatorname{H}\mathcal{C})^{0}$.

As an auxiliary construction, we recall the additive completion of
$A_{\infty}$-categories. The idea is to form a category whose objects are
formal sums of objects. The hom spaces and $A_{\infty}$-products are extended
accordingly. We fix the definition and notation as follows:

######  Definition 2.6.

Let $\mathcal{C}$ be an $A_{\infty}$ category with product
$\mu_{\mathcal{C}}$. The additive completion $\operatorname{Add}\mathcal{C}$
of $\mathcal{C}$ is the category of formal sums of shifted objects of
$\mathcal{C}$:

$A_{1}[k_{1}]\oplus\ldots\oplus A_{n}[k_{n}].$

The hom space between two such objects $X=\bigoplus A_{i}[k_{i}]$ and
$Y=\bigoplus B_{i}[m_{i}]$ is

$\operatorname{Hom}_{\operatorname{Add}\mathcal{C}}(X,Y)=\bigoplus_{i,j}\operatorname{Hom}_{\mathcal{C}}(A_{i},B_{j})[m_{j}-k_{i}].$

Here $[-]$ denotes the right-shift. The products on
$\operatorname{Add}\mathcal{C}$ are given by multilinear extensions of

$\mu_{\operatorname{Add}\mathcal{C}}^{k}(a_{k},\ldots,a_{1})=(-1)^{\sum_{j<i}\|a_{i}\|l_{j}}\mu_{\mathcal{C}}^{k}(a_{k},\ldots,a_{1}).$

Here each $a_{i}$ lies in some
$\operatorname{Hom}(X_{i}[k_{i}],X_{i+1}[k_{i+1}])$. The integer $l_{i}$
denotes the difference $k_{i+1}-k_{i}$ between the shifts and the degree
$\|a_{i}\|$ is the degree of $a_{i}$ as element of
$\operatorname{Hom}_{\mathcal{C}}(X_{i},X_{i+1})$.

Next we recall the twisted completion $\operatorname{Tw}\mathcal{C}$ of an
$A_{\infty}$-category $\mathcal{C}$. The objects of this category are virtual
chain complexes of objects of $\mathcal{C}$. The usage of the twisted
completion construction is entirely analogous to the usage of the category of
complexes over an abelian category. We recall the definition and notation as
follows:

######  Definition 2.7.

A twisted complex in $\mathcal{C}$ is an object
$X\in\operatorname{Add}\mathcal{C}$ together with a morphism
$\delta\in\operatorname{Hom}^{1}_{\operatorname{Add}\mathcal{C}}(X,X)$ of
degree $1$ such that $\delta$ is strictly upper triangular and satisfies the
Maurer-Cartan equation:

$\operatorname{MC}(\delta)\coloneqq\mu^{1}(\delta)+\mu^{2}(\delta,\delta)+\ldots=0.$

We may refer to the morphism $\delta$ as the twisted differential. Note that
the upper triangularity ensures that this sum is well-defined. The twisted
completion of $\mathcal{C}$ is the $A_{\infty}$-category
$\operatorname{Tw}\mathcal{C}$ whose objects are twisted complexes. Its hom
spaces are the same as for the additive completion:

$\operatorname{Hom}_{\operatorname{Tw}\mathcal{C}}(X,Y)=\operatorname{Hom}_{\operatorname{Add}\mathcal{C}}(X,Y).$

The products on $\operatorname{Tw}\mathcal{C}$ of $\mathcal{C}$ are given by
embracing with $\delta$’s:

$\mu_{\operatorname{Tw}\mathcal{C}}^{k}(a_{k},\ldots,a_{1})=\sum_{n_{0},\ldots,n_{k}\geq
0}\mu_{\operatorname{Add}\mathcal{C}}(\underbrace{\delta,\ldots,\delta}_{n_{k}},a_{k},\ldots,a_{1},\underbrace{\delta,\ldots,\delta}_{n_{0}}).$

The notion of a curved $A_{\infty}$-category is a generalization of ordinary
$A_{\infty}$-category. In contrast to an ordinary $A_{\infty}$-category, a
curved $A_{\infty}$-category also has an element $\mu^{0}_{X}$ of degree $2$
associated with every object $X\in\mathcal{C}$. The required curved
$A_{\infty}$-relations are the same as those of an $A_{\infty}$-category, in
particular allowing $\mu^{0}_{X}$ to appear as inner $\mu$. The first two
curved $A_{\infty}$-relations read

$\mu^{1}(\mu^{0})=0,\quad\mu^{1}(\mu^{1}(a))+(-1)^{\|a\|}\mu^{2}(\mu^{0},a)+\mu^{2}(a,\mu^{0})=0.$

Curved $A_{\infty}$-categories are however ill-behaved: They do not have a
notion of derived category. One may form the twisted completion, but if one
enforces the Maurer-Cartan equation $\mu^{0}+\mu^{1}(\delta)+\ldots=0$, very
few objects remain and the construction is not functorial. If one instead
discards the Maurer-Cartan requirement for twisted complexes, one obtains a
category $\operatorname{Tw}\mathcal{C}$ with curvature, the curvature given by
the Maurer-Cartan formula $\mu^{0}+\mu^{1}(\delta)+\ldots$. This category
however has no minimal model $\operatorname{H}\operatorname{Tw}\mathcal{C}$,
because curvature prevents us from bringing $\mu^{1}$ to zero. In short,
curved $A_{\infty}$-categories are not useful, except for the purpose of
matrix factorizations.

We would like to stress that infinitesimal curvature however does not hurt. In
section 3.2, we recall what infinitesimal curvature entails and we explain in
section 5 that infinitesimally curved $A_{\infty}$-categories do have derived
categories.

##  3 Deformations of $A_{\infty}$-categories

In this section, we recollect material on $A_{\infty}$-deformations. Theory on
$A_{\infty}$-deformations is scarce and implicit in the literature, so we
provide a rigorous definition here that includes the infinitesimally curved
case. We explain why curvature is inevitable if one wants to obtain a
deformation theory invariant under quasi-equivalences.

In section 3.1, we recall the completed tensor products of the form
$B\widehat{\otimes}X$, where $B$ is a local algebra. In section 3.2, we recall
curved deformations of $A_{\infty}$-categories. In section 3.3, we define
their twisted completion. In section 3.4, we recall functors between
$A_{\infty}$-categories and between deformations of $A_{\infty}$-categories.

The material in the entire section is not original. For instance, Seidel
already mentioned curved $A_{\infty}$-deformations in [19] and Lowen and Van
den Bergh considered the “curvature problem” for formal deformations in [15].

In this section, we will be comparing different deformations with another. We
should set some terminology right before we get started. A “deformation base”
$B$ will be a complete local Noetherian unital $\mathbb{C}$-algebra with
residue field $B/\mathfrak{m}=\mathbb{C}$. Every deformation $\mathcal{C}_{q}$
over $B$ can be interpreted as object of two different universes: either as
deformation of $\mathcal{C}$, or as a deformation of any
$A_{\infty}$-category. Let us depict this pictorially:

deformations of $\mathcal{C}$deformations of any
category$\mathcal{C}_{q}^{\prime}$$\mathcal{C}_{q}$$\mathcal{C}_{q}$$\mathcal{C}_{q}^{\prime}$$\mathcal{D}_{q}$

This perspective makes a difference: The correct notion for two deformations
$\mathcal{C}_{q},\mathcal{C}_{q}^{\prime}$ to be similar on the left side is
to be gauge equivalent, while the correct notion for two deformations
$\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ to be similar on the right side is to
be quasi-equivalent. In the former case, the leading term of the functor
connecting the two is supposed to be the identity, in the latter case the
leading term can be any quasi-equivalence of $A_{\infty}$-categories.

To distinguish the two perspectives, we will often use the terminology
$A_{\infty}$-deformation to refer to the left side, and deformed
$A_{\infty}$-category to refer to the right side.

###  3.1 The completed tensor product

In this section, we recall the completed tensor products of the form
$B\widehat{\otimes}X$, where $B$ is a local algebra and $X$ is a vector space.
This serves as a preparation for section 3.2 where we define
$A_{\infty}$-deformations.

Throughout this paper, we deform over local rings like $\mathbb{C}\llbracket
q\rrbracket$. There are a few more conditions we put on the local ring: In
order to work with the $A_{\infty}$-formalism we need the ring to be a
$\mathbb{C}$-algebra. In order to speak of a special fiber, or algebraically
of an $\mathfrak{m}$-adic leading term, we need to require that its residue
field is $\mathbb{C}$ itself. As is customary in deformation theory, we shall
also require that the ring be complete and Noetherian. We have decided to give
the type of local rings a name in this paper:

######  Definition 3.1.

A deformation base is a complete local Noetherian unital $\mathbb{C}$-algebra
$B$ with residue field $B/\mathfrak{m}=\mathbb{C}$. The maximal ideal is
always denoted $\mathfrak{m}$.

######  Remark 3.1.

By the Cohen structure theorem, every deformation base is of the form
$\mathbb{C}\llbracket x_{1},\ldots,x_{n}\rrbracket/I$ with $I$ denoting some
ideal.

The idea behind formal $A_{\infty}$-deformations is to tensor the hom spaces
with a deformation base. One then looks at $A_{\infty}$-structures on the
enlarged collection of hom spaces which reduce to the original
$A_{\infty}$-structure once the maximal ideal is divided out. The construction
of these tensored hom spaces makes use of the completed tensor product, which
we now recall.

######  Definition 3.2.

Let $B$ be a deformation base and $X$ a vector space. Then the completed
tensor product $B\widehat{\otimes}X$ is the $B$-module limit

$B\widehat{\otimes}X=\lim(\ldots\to B/\mathfrak{m}^{1}\otimes X\to
B/\mathfrak{m}^{0}\otimes X),$

For simplicity, we write $\mathfrak{m}^{k}X$ to denote the infinitesimal part
$\mathfrak{m}^{k}X=\mathfrak{m}^{k}\widehat{\otimes}X\subseteq
B\widehat{\otimes}X$.

######  Remark 3.2.

In case $B=\mathbb{C}\llbracket q\rrbracket$, the completed tensor product
$B\widehat{\otimes}X$ equals the even more well-known space $X\llbracket
q\rrbracket$. The space $X\llbracket q\rrbracket$ consists of formal
$X$-valued power series in one variable. This way $X\llbracket q\rrbracket$
becomes naturally a $\mathbb{C}\llbracket q\rrbracket$-module. The space is
different from $\mathbb{C}\llbracket q\rrbracket\otimes X$. In fact, elements
of $\mathbb{C}\llbracket q\rrbracket\otimes X$ are only those power series
which can be written as a finite sum of pure tensors $a\otimes x$. Simply
speaking, in $\mathbb{C}\llbracket q\rrbracket\otimes X$, the power series
entries are divided into finitely many partitions in which all coefficients
are interrelated. Meanwhile in $X\llbracket q\rrbracket$ any entries can be
chosen at random. However if $X$ is finite-dimensional, then $X\llbracket
q\rrbracket=\mathbb{C}\llbracket q\rrbracket\otimes X$.

There are two popular ways of defining formal deformations of an associative
algebra $A$. The first definition asks for a product $\mu_{q}:A\otimes A\to
B\widehat{\otimes}A$ and the second definition asks for a product
$\mu_{q}:(B\widehat{\otimes}A)\otimes(B\widehat{\otimes}A)\to
B\widehat{\otimes}A$. The first definition immediately gives rise to a Maurer-
Cartan element, while in the second definition associativity is formulated
more naturally. In what follows, we shall explain briefly why both are
equivalent.

######  Definition 3.3.

Let $B$ be a deformation base. Then the $\mathfrak{m}$-adic topology on $B$ is
the topology on $B$ generated by open neighborhoods
$x+\mathfrak{m}^{k}\subseteq B$ for $x\in B$ and $k\in\mathbb{N}$. The
$\mathfrak{m}$-adic topology on $B\widehat{\otimes}X$ is generated by the open
neighborhoods $x+\mathfrak{m}^{k}\widehat{\otimes}X\subseteq
B\widehat{\otimes}X$ for every $x\in B\widehat{\otimes}X$ and
$k\in\mathbb{N}\,$. A map $\varphi:B\widehat{\otimes}X\to B\widehat{\otimes}Y$
is continuous if it is continuous with respect to the $\mathfrak{m}$-adic
topologies. A map
$\varphi:(B\widehat{\otimes}X_{k})\otimes\ldots\otimes(B\widehat{\otimes}X_{1})\to
B\widehat{\otimes}Y$ is continuous if for every $1\leq i\leq k$ and every
sequence of elements $x_{1},\ldots,\hat{x}_{i},\ldots,x_{k}$ the map

$\mu(x_{k},\ldots,-,\ldots,x_{1}):B\widehat{\otimes}X_{i}\to
B\widehat{\otimes}Y$

is continuous.

######  Remark 3.3.

It is well-known that the $\mathfrak{m}$-adic topology turns
$B\widehat{\otimes}X$ into a sequential Hausdorff space. It is also well-known
that $B\widehat{\otimes}X$ can simultaneously be interpreted as limit and
completion. More precisely, $B\widehat{\otimes}X$ is the completion of
$B\otimes X$ with respect to the so-called $\mathfrak{m}$-adic metric on
$B\otimes X$. For convenience, we may from time to time use expressions like
$x=\mathcal{O}(\mathfrak{m}^{k})$ to indicate $x\in\mathfrak{m}^{k}X$.

######  Remark 3.4.

Every element in $B\widehat{\otimes}X$ can be written as a series
$\sum_{i=0}^{\infty}m_{i}x_{i}$. Here $m_{i}$ is a sequence of elements
$m_{i}\in\mathfrak{m}^{\to\infty}$ and $x_{i}$ is a sequence of elements
$x_{i}\in X$. We have used the notation $m_{i}\in\mathfrak{m}^{\to\infty}$ to
indicate that $m_{i}\in\mathfrak{m}^{k_{i}}$ for some sequence
$(k_{i})\subseteq\mathbb{N}$ with $k_{i}\to\infty$.

######  Remark 3.5.

Let $\varphi:B\widehat{\otimes}X\to B\widehat{\otimes}Y$ be a $B$-linear map.
If $\varphi(\mathfrak{m}^{k}X)\subseteq\mathfrak{m}^{k}Y$, then $\varphi$ is
continuous.

Whenever $X\to B\widehat{\otimes}Y$ is a linear map, we can extend it uniquely
to a $B$-linear continuous map $B\widehat{\otimes}X\to B\widehat{\otimes}Y$.
Conversely, we can restrict any $B$-linear map $B\widehat{\otimes}X\to
B\widehat{\otimes}Y$ to a linear map $X\to B\widehat{\otimes}Y$. Restriction
and extension are in fact inverse to each other, providing a one-to-one
correspondence:

######  Lemma 3.1.

Let $B$ be a deformation base and $X,Y$ be vector spaces. Then a $B$-linear
map $B\widehat{\otimes}X\to B\widehat{\otimes}Y$ is automatically continuous.
There is a one-to-one correspondence between:

  * •

linear maps $X\to B\widehat{\otimes}Y$,

  * •

$B$-linear maps $B\widehat{\otimes}X\to B\widehat{\otimes}Y$,

  * •

$B~{}$-linear continuous maps $B\widehat{\otimes}X\to B\widehat{\otimes}Y$.

###### Proof.

The proof consists of three parts: First, we show that every $B$-linear map
$B\widehat{\otimes}X\to B\widehat{\otimes}Y$ is continuous. Second, we recall
how to extend a map $X\to B\widehat{\otimes}Y$ to $B\widehat{\otimes}X\to
B\widehat{\otimes}Y$. Third, we comment on the one-to-one aspect of the claim.

For the first step, we show that any $B$-linear map
$\varphi:B\widehat{\otimes}X\to B\widehat{\otimes}Y$ is continuous. Let
$k\in\mathbb{N}$. We claim that
$\varphi(\mathfrak{m}^{k}X)\subseteq\mathfrak{m}^{k}Y$. The idea is to exploit
the Cohen structure theorem. Write $B=\mathbb{C}\llbracket
q_{1},\ldots,q_{n}\rrbracket/I$, and regard the maximal ideal
$\mathfrak{m}=(q_{1},\ldots,q_{n})$. With this in mind, we can write any
element $x\in\mathfrak{m}^{k}X$ as a series

$x=\sum_{i=0}^{\infty}m_{i}\tilde{m}_{i}y_{i}.$

Here $m_{i}$ is a monomial of degree $k$ in the variables
$q_{1},\ldots,q_{n}$, the letter $\tilde{m}_{i}$ denotes a sequence
$\tilde{m}_{i}\in\mathfrak{m}^{\to\infty}$, and $y_{i}\in Y$. We conclude

$x=\sum_{\begin{subarray}{c}\text{monomials }M\\\ \text{ of degree
}k\end{subarray}}M\sum_{\begin{subarray}{c}i\geq 0\\\
m_{i}=M\end{subarray}}\tilde{m}_{i}y_{i}.$

The outer sum is finite. For every monomial $M$ of degree $k$, the inner sum
is an element $x_{M}\in B\widehat{\otimes}X$. We get that

$\varphi(x)=\sum_{\begin{subarray}{c}\text{monomials }M\\\ \text{ of degree
}k\end{subarray}}M\varphi(x_{M}).$

We conclude that $\varphi(x)\in\mathfrak{m}^{k}Y$. This shows
$\varphi(\mathfrak{m}^{k}X)\subseteq\mathfrak{m}^{k}Y$. In particular,
$\varphi$ is continuous.

For the second step, denote by $\pi_{i}$ the projection maps
$\pi_{i}:B\widehat{\otimes}X\to B/\mathfrak{m}^{i}\otimes X$ or
$B\widehat{\otimes}Y\to B/\mathfrak{m}^{i}\otimes Y$ and by $\pi_{ij}$ the
projection maps $\pi_{ij}:B/\mathfrak{m}^{i}\otimes Y\to
B/\mathfrak{m}^{j}\otimes Y$ for $i>j$. Note that
$(\pi_{ij}\otimes\operatorname{Id}_{Y})\circ\pi_{i}=\pi_{j}$ for $i>j$.

Let now $F:X\to B\widehat{\otimes}Y$ be a linear map. In order to define a map
$\hat{F}:B\widehat{\otimes}X\to B\widehat{\otimes}Y$, we shall build maps
$\hat{F}_{i}:B\widehat{\otimes}X\to B/\mathfrak{m}^{i}\otimes Y$ and then use
the universal property to combine them into $\hat{F}$.

Let us now construct auxiliary maps $F_{i}$ and the maps $\hat{F}_{i}$. Define
$F_{i}:B/\mathfrak{m}^{i}\otimes X\to B/\mathfrak{m}^{i}\otimes Y$ as the
$B/\mathfrak{m}^{i}$-linear extension of $\pi_{i}F:X\to
B/\mathfrak{m}^{i}\otimes Y$. Then let $\hat{F}_{i}:B\widehat{\otimes}X\to
B/\mathfrak{m}^{i}\otimes Y$ be the composition $F_{i}\pi_{i}$. More directly,
we could write $\hat{F}_{i}(z)=F(\pi_{i}(z))$, where on the right-hand side
$F$ is interpreted $B/\mathfrak{m}^{i}$-linearly.

We claim that
$\hat{F}_{j}=(\pi_{ij}\otimes\operatorname{Id}_{Y})\circ\hat{F}_{i}$ for
$i>j$. Indeed,

$\pi_{ij}(\hat{F}_{i}(z))=(\pi_{ij}\otimes\operatorname{Id}_{Y})(F(\pi_{i}(z))=F((\pi_{ij}\otimes\operatorname{Id}_{X})(\pi_{i}(z)))=F(\pi_{j}(z))=\hat{F}_{j}(z).$

This proves that the family of maps $\\{\hat{F}_{i}\\}_{i\in\mathbb{N}}$ is
compatible with the projections $\pi_{ij}\otimes\operatorname{Id}_{Y}$. By the
universal property of $B\widehat{\otimes}Y$, this family of maps factors
through the limit $B\widehat{\otimes}Y$, yielding a $B$-linear map

$\hat{F}:B\widehat{\otimes}X\to B\widehat{\otimes}Y,\text{ with
}\pi_{i}\hat{F}=\hat{F}_{i}.$

The map $\hat{F}$ can also be written explicitly as follows: Let
$x=\sum_{i=0}^{\infty}m_{i}x_{i}$ be an element of $B\widehat{\otimes}X$, with
$m_{i}\in\mathfrak{m}^{\to\infty}$ and $x_{i}\in X$. Then $\hat{F}(\sum
m_{i}x_{i})=\sum m_{i}F(x_{i})$.

As third step of the proof, we comment on the one-to-one aspect in the claim.
The only remaining statement to explain is that for a given map $F:X\to
B\widehat{\otimes}Y$, there is only one single $B$-linear continuous extension
to a map $B\widehat{\otimes}X\to B\widehat{\otimes}Y$. But this is obvious,
since $B$-linearity already determines the value on $B\otimes X$ and
continuity then determines the value on all of $B\widehat{\otimes}X$. This
finishes the proof. ∎

We shall provide a few more standard utilities. Recall that a map
$\varphi:X\to Y$ of topological spaces is an embedding if it is a
homeomorphism onto its image, the image being equipped with subspace topology.
The leading term of a $B$-linear map
$\varphi:B\widehat{\otimes}X\to\,B\widehat{\otimes}Y$ is the map
$\varphi_{0}:X\to Y$ given by the composition $\varphi_{0}=\pi\varphi|_{X}$,
where $\pi:B\widehat{\otimes}Y\to Y$ denotes the standard projection.

######  Lemma 3.2.

Let $X,Y$ be vector spaces and $\varphi:B\widehat{\otimes}X\to
B\widehat{\otimes}Y$ be $B$-linear with injective leading term. Then $\varphi$
is an embedding and its image is closed. If the leading term is surjective,
then $\varphi$ is an isomorphism.

###### Proof.

Before we dive into the proof, we start with an observation regarding the
leading term. Let $\varphi_{0}:X\to Y$ be the leading term of $\varphi$ and
define for every $i\in\mathbb{N}$ the map

$\varphi_{i}:\mathfrak{m}^{i}/\mathfrak{m}^{i+1}\otimes
X\to\mathfrak{m}^{i}/\mathfrak{m}^{i+1}\otimes Y$ (3.1)

induced from $\varphi$. Our observation is that $\varphi_{i}$ is in fact equal
to $\mathrm{id}_{\mathfrak{m}^{i}/\mathfrak{m}^{i+1}}\otimes\varphi_{0}$.

We are now ready to start the proof. First we show that $\varphi$ is
injective. Second we show that $\varphi^{-1}:\operatorname{Im}(\varphi)\to
B\widehat{\otimes}X$ is continuous. Third we show that
$\operatorname{Im}(\varphi)$ is closed. Fourth we prove $\varphi$ surjective
if $\varphi_{0}$ is surjective.

For the first part of the proof, we show that $\varphi$ injective. Let $x\in
B\widehat{\otimes}X$ with $\varphi(x)=0$. By induction, we show that
$x\in\mathfrak{m}^{i}X$ for every $i\in\mathbb{N}$. For $i=0$ this is clear.
As induction hypothesis, assume that $x\in\mathfrak{m}^{i}X$. Then
$\varphi_{i}([x])=[\varphi(x)]=0$. Since $\varphi_{i}$ is injective, we obtain
$x\in\mathfrak{m}^{i+1}X$. This finishes the induction. In consequence we have
$x\in\mathfrak{m}^{i}X$ for every $i\in\mathbb{N}$ and therefore $x=0$. This
proves $\varphi$ injective.

For the second part of the proof, we show that $\varphi^{-1}$ preserves the
filtration by $\mathfrak{m}$. In other words, we show that
$\varphi(x)\in\mathfrak{m}^{i}Y$ implies $x\in\mathfrak{m}^{i}X$. We do this
by induction over $i\in\mathbb{N}\,$. For $i=0$ this is clear. Assume this
statements holds for a certain $i\in\mathbb{N}$. Now let
$\varphi(x)\in\mathfrak{m}^{i+1}Y$. Then $x\in\mathfrak{m}^{i}X$ by induction
hypothesis. We get $\varphi_{i}([x])=0$, since
$\varphi(x)\in\mathfrak{m}^{i+1}Y$. Since $\varphi_{i}$ is injective, we get
$x\in\mathfrak{m}^{i+1}X$. This finishes the induction. Finally, we have shown
$\varphi^{-1}(\mathfrak{m}^{i}Y)\subseteq\mathfrak{m}^{i}X$. This renders
$\varphi^{-1}:\varphi(B\widehat{\otimes}X)\to B\widehat{\otimes}X$ continuous.

For the third part of the proof, we check that $\operatorname{Im}(\varphi)$ is
closed. Since $B\widehat{\otimes}Y$ is a sequential space, it suffices to
check that any sequence $\varphi(x_{n})\subseteq\operatorname{Im}(\varphi)$
converging in $B\widehat{\otimes}Y$ converges in $\operatorname{Im}(\varphi)$.
Pick a sequence $(x_{n})\subseteq B\widehat{\otimes}X$ and assume
$\varphi(x_{n})\to y\in B\widehat{\otimes}Y$. This makes
$\varphi(x_{n})-y\in\mathfrak{m}^{\to\infty}Y$. We have
$\varphi(x_{n})-\varphi(x_{m})\in\mathfrak{m}^{\to\infty}Y$ and by continuity
of $\varphi^{-1}$ we get $x_{n}-x_{m}\in\mathfrak{m}^{\to\infty}$. In
particuar, $x_{n}$ is a Cauchy sequence and converges to some $x\in
B\widehat{\otimes}X$. We get $\varphi(x)=\lim\varphi(x_{n})=y$ and therefore
$y\in\operatorname{Im}(\varphi)$. This proves $\operatorname{Im}(\varphi)$
closed.

For the fourth part of the proof, consider $y\in B\widehat{\otimes}Y$. We
construct inductively a sequence $(x_{n})$ with
$x_{n}\in\mathfrak{m}^{n}\widehat{\otimes}X$ such that
$\varphi(\sum_{i=0}^{n}x_{i})=\pi_{n}(y)$. For $i=0$, let $x_{0}\in X$ be
defined as $x_{0}=\varphi_{0}^{-1}(\pi_{0}(y))$. Now assume the sequence has
been constructed for indices until $n\in\mathbb{N}$. Put $z\coloneqq
y-\varphi(\sum^{n}x_{i})\in\mathfrak{m}^{n}Y$. Since $\varphi_{0}$ is an
isomorphism, the map $\varphi_{n}$ from (3.1) is an isomorphism as well. Put

$x_{n+1}^{\prime}\coloneqq\varphi_{n+1}^{-1}(z)\in\mathfrak{m}^{n+1}/\mathfrak{m}^{n+2}X.$

Let $x_{n+1}\in\mathfrak{m}^{n+1}X$ be any lift of $x_{n+1}^{\prime}$. Then we
have

$\varphi\left(\sum_{i=0}^{n+1}x_{i}\right)=y-z+\varphi(x_{n+1})=y+\mathcal{O}(\mathfrak{m}^{n+2}).$

This finishes the inductive construction of the sequence $(x_{i})$. Finally,
the series $x=\sum x_{i}$ converges and $\varphi(x)=y$. This proves $\varphi$
surjective. ∎

The pathway to defining $A_{\infty}$-deformations is now clear: When
$\mathcal{C}$ is an $A_{\infty}\,$-category, a deformation of $\mathcal{C}$
will always be modeled on the collection of enlarged hom spaces
$\\{B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)\\}_{X,Y\in\mathcal{C}}$.
Any $B$-multilinear product on these hom spaces is automatically continuous.
Similarly, functors of $A_{\infty}$-deformations will be defined as maps
between tensor products of the enlarged hom spaces and will be automatically
continuous as well.

###  3.2 Curved deformations

In this section, we give a quick definition of curved deformations of
$A_{\infty}$-categories. We have already used this notion in [4]. The notion
is not surprising and known to experts. Despite being uncomfortable to work
with, curvature comes naturally via the Hochschild complex and is necessary if
one aims at deformation theory invariant under quasi-equivalences. The
curvature in $A_{\infty}$-deformations will always be infinitesimal, which we
consider harmless in contrast with the different notion of curved
$A_{\infty}$-categories [5].

Let us recall the setup of Hochschild cohomology in the classical case. Let
$A$ be an associative algebra. As the reader probably knows, Hochschild
cohomology $\operatorname{HH}^{2}(A)$ captures the associative deformations of
the algebra over $\mathbb{C}[\varepsilon]/(\varepsilon^{2})$. More precisely,
a 2-cochain $\nu\in\operatorname{HC}^{2}(A)$ is a cocycle if and only if
$\mu+\varepsilon\nu$ is an associative product on $A\oplus A\varepsilon$ (note
$\varepsilon^{2}=0$). Classical Hochschild cohomology in other degrees than 2
helps characterize the deformation problem, but provides no actual
deformations.

Since Hochschild’s original definition in 1946, Hochschild cohomology has been
generalized to $A_{\infty}$-categories. The trick is to use the same formula
for the differential, with two adaptations: include also higher products
$\mu^{k}$ instead of only $\mu^{2}$, and use grading induced from the shift
$\mathcal{C}[1]$. We will make this precise in section 4.1.

What should an $A_{\infty}$-deformation be then? The naive answer would be to
allow deformations of all products $\mu^{k}$. This is however too
shortsighted: We contend that for a notion of $A_{\infty}$-deformations whose
infinitesimal deformations are classified by a cochain complex (or better a
DGLA), the $A_{\infty}$-Hochschild complex is the natural choice. Its Maurer-
Cartan elements consist however not only of deformations to the products
$\mu^{k}$ with $k\geq 1$, but also introduce curvature $\mu^{0}$.

We are now ready to define $A_{\infty}$-deformations precisely.

######  Definition 3.4.

Let $\mathcal{C}$ be an $A_{\infty}$ category with products $\mu$ and $B$ a
deformation base. An (infinitesimally curved) deformation $\mathcal{C}_{q}$ of
$\mathcal{C}$ consists of

  * •

The same objects as $\mathcal{C}$,

  * •

Hom spaces
$\operatorname{Hom}_{\mathcal{C}_{q}}(X,Y)=B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X,Y)$
for $X,Y\in\mathcal{C}$,

  * •

$B$-multilinear products of degree $2-k$

$\mu_{q}^{k}:\operatorname{Hom}_{\mathcal{C}_{q}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{k+1}),~{}k\geq
1$

  * •

Curvature of degree $2$ for every object $X\in\mathcal{C}$

$\mu_{q,X}^{0}\in\mathfrak{m}\operatorname{Hom}_{\mathcal{C}_{q}}(X,X),$

such that $\mu_{q}$ reduces to $\mu$ once the maximal ideal
$\mathfrak{m}\subseteq R$ is divided out, and $\mu_{q}$ satisfies the curved
$A_{\infty}$ ($cA_{\infty}$) relations

$\sum_{k\geq l\geq m\geq
0}(-1)^{\|a_{m}\|+\ldots+\|a_{1}\|}\mu_{q}(a_{k},\ldots,\mu_{q}(a_{l},\ldots),a_{m},\ldots,a_{1})=0.$

The deformation is unital if the deformed higher products still satisfy the
unitality axioms

$\mu^{2}_{q}(a,\mathrm{id}_{X})=a,~{}\mu^{2}_{q}(\mathrm{id}_{Y},a)=(-1)^{|a|}a,~{}\mu^{\geq
3}_{q}(\ldots,\mathrm{id}_{X},\ldots)=0.$

We use the terms $A_{\infty}$-deformation and deformation of an
$A_{\infty}$-category interchangeably. Whenever we speak of deformations of
$A_{\infty}$-categories in this paper, they are allowed to be
(infinitesimally) curved.

######  Remark 3.6.

As explained in section 3.1, the product $\mu^{k}_{q}$ is automatically
$\mathfrak{m}$-adically continuous. Spelling out this continuity requirement,
the datum of the map $\mu^{k}$ is equivalent to the datum of merely
multilinear maps

$\mu_{q}:\operatorname{Hom}_{\mathcal{C}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{2})~{}\to~{}B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{k+1}).$

There is a notion for two objects to be quasi-isomorphic in an
$A_{\infty}$-deformation. We provide here an ad-hoc definition which seems odd
at first, but we will encounter evidence in section 5 and 6 which supports
correctness of the definition.

######  Definition 3.5.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Let $X,Y\in\mathcal{C}_{q}$ be two objects. Then $X$ and $Y$ are
quasi-isomorphic if they are quasi-isomorphic in $\mathcal{C}$.

It sometimes comes handy to work with deformations that include more objects
than $\mathcal{C}$ does. We fix terminology as follows:

######  Definition 3.6.

Let $\mathcal{C}$ be an $A_{\infty}$-category. Let $O$ be an arbitrary set of
objects and $F:O\to\operatorname{Ob}(\mathcal{C})$ a map. Then the object-
cloned version $F^{*}\mathcal{C}$ of $\mathcal{C}$ is the
$A_{\infty}\,$-category given by object set $O$, hom spaces

$\operatorname{Hom}_{F^{*}\mathcal{C}}(X,Y)=\operatorname{Hom}_{\mathcal{C}}(F(X),F(Y)),\quad
X,Y\in O,$

and products simply given by the same composition as in $\mathcal{C}$.

An $A_{\infty}$-deformation of $\mathcal{C}$ always gives an induced
deformation of $F^{*}\mathcal{C}$. This provides a map from deformations of
$\mathcal{C}$ to deformations of $F^{*}\mathcal{C}$. In case $F$ is
surjective, the categories $\mathcal{C}$ and $F^{*}\mathcal{C}$ are equivalent
and as we elaborate in section 4 actually have the same deformation theory.
For this reason, we fix the following terminology:

######  Definition 3.7.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $B$ a deformation base. Let
$O$ be an arbitrary set and $F:O\to\operatorname{Ob}\mathcal{C}$ a map. An
object-cloning deformation is a deformation $\mathcal{D}_{q}$ of
$\mathcal{D}=F^{*}\mathcal{C}$. The object-cloning deformation is essentially
surjective if $F:\operatorname{Ob}\mathcal{D}\to\operatorname{Ob}\mathcal{C}$
reaches all objects of $\mathcal{C}$ up to isomorphism.

###  3.3 Twisted completion

In this section, we define the twisted completion of a deformed
$A_{\infty}$-category. The starting point is an $A_{\infty}$-category
$\mathcal{C}$ together with a deformation $\mathcal{C}_{q}$. The aim is to
define a deformation $\operatorname{Tw}\mathcal{C}_{q}$ of
$\operatorname{Tw}\mathcal{C}$ which incorporates the deformation given by
$\mathcal{C}_{q}$. More formally, the aim is to push the deformation
$\mathcal{C}_{q}$ onto $\operatorname{Tw}\mathcal{C}$ along the inclusion
$\mathcal{C}\to\operatorname{Tw}\mathcal{C}$ which is a derived equivalence.
Since we only explain the pushforward of deformations in section 4, we give
here a direct and explicit definition of $\operatorname{Tw}\mathcal{C}_{q}$.

We start by defining additive completions for deformed
$A_{\infty}$-categories. Let $\mathcal{C}$ be an $A_{\infty}$-category and
$\mathcal{C}_{q}$ a deformation. The aim is to define an additive completion
$\operatorname{Add}\mathcal{C}_{q}$ in such a way that it is a deformation of
$\operatorname{Add}\mathcal{C}$. This is straight-forward:

######  Definition 3.8.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Then the additive completion $\operatorname{Add}\mathcal{C}_{q}$
is the deformation of $\operatorname{Add}\mathcal{C}$ given by the following
deformed product:

$\mu_{\operatorname{Add}\mathcal{C}_{q}}^{k}(\alpha_{k},\ldots,\alpha_{1})=\sum(-1)^{\sum_{j<i}\|\alpha_{i}\|l_{j}}\mu_{\mathcal{C}_{q}}^{k}(\alpha_{k},\ldots,\alpha_{1}),$

with the same sign convention as in the non-deformed case.

######  Remark 3.7.

The only difference between $\mu_{\operatorname{Add}\mathcal{C}}$ and
$\mu_{\operatorname{Add}\mathcal{C}_{q}}$ lies in using the non-deformed
product $\mu_{\mathcal{C}}$ for the former and the deformed product
$\mu_{\mathcal{C}_{q}}$ for the latter.

We now extend the twisted completion construction to the case of
$A_{\infty}$-deformations. Let $\mathcal{C}$ be an $A_{\infty}$-category and
$\mathcal{C}_{q}$ a deformation. The aim is to define a twisted completion
$\operatorname{Tw}\mathcal{C}_{q}$ in such a way that it is a deformation of
$\operatorname{Tw}\mathcal{C}$. In particular, the category
$\operatorname{Tw}\mathcal{C}_{q}$ should have the same objects as
$\operatorname{Tw}\mathcal{C}$. With this in mind, we define:

######  Definition 3.9.

Let $\mathcal{C}$ be an $A_{\infty}$ category with products
$\mu_{\mathcal{C}}$ and $\mathcal{C}_{q}$ a deformation with products
$\mu_{\mathcal{C}_{q}}$. Then the twisted completion
$\operatorname{Tw}\mathcal{C}_{q}$ is the (possibly curved) deformation of
$\operatorname{Tw}\mathcal{C}$ given by the deformed products

$\mu_{\operatorname{Tw}\mathcal{C}_{q}}^{k}(\alpha_{k},\ldots,\alpha_{1})=\sum_{n_{0},\ldots,n_{k}\geq
0}\mu_{\operatorname{Add}\mathcal{C}_{q}}(\underbrace{\delta,\ldots,\delta}_{n_{k}},\alpha_{k},\ldots,\alpha_{1},\underbrace{\delta,\ldots,\delta}_{n_{0}}).$

######  Remark 3.8.

As one may have expected, the product $\mu_{\operatorname{Tw}\mathcal{C}_{q}}$
now simply uses $\mu_{\mathcal{C}_{q}}$ instead of $\mu_{\mathcal{C}}$. The
set of objects of $\operatorname{Tw}\mathcal{C}_{q}$ may however be
surprising: The objects of $\operatorname{Tw}\mathcal{C}_{q}$ are not formed
with twisted differentials
$\delta\in\operatorname{Hom}^{1}_{\mathcal{C}_{q}}(X,X)$. Instead, the objects
of $\operatorname{Tw}\mathcal{C}_{q}$ are twisted complexes
$(X,\delta)\in\operatorname{Tw}\mathcal{C}$, in other words, their
$\delta$-differential must lie in $\operatorname{Hom}^{1}_{\mathcal{C}}(X,X)$.
It is easily checked that $\mu_{\operatorname{Tw}\mathcal{C}_{q}}$ satisfies
the (curved) $A_{\infty}$-axioms, rendering $\operatorname{Tw}\mathcal{C}_{q}$
a genuine $A_{\infty}$-deformation of $\operatorname{Tw}\mathcal{C}$.

######  Remark 3.9.

Denoting the twisted completion of $\mathcal{C}_{q}$ by
$\operatorname{Tw}\mathcal{C}_{q}$ constitutes a slight abuse of notation:
“$\operatorname{Tw}\mathcal{C}_{q}$” suggests that twisted complexes are to be
taken with the $\delta$-differential formed from values in $\mathcal{C}_{q}$,
which is not the case. A more proper notation for the twisted completion of
$\mathcal{C}_{q}$ would have been $(\operatorname{Tw}\mathcal{C})_{q}$, which
we however found too complicated.

######  Remark 3.10.

Typical objects in $\operatorname{Tw}\mathcal{C}_{q}$ have curvature. Indeed,
according to Definition 3.9, an object
$(X,\delta)\in\operatorname{Tw}\mathcal{C}_{q}$ has curvature

$\mu^{0}_{(X,\delta)}=\mu^{0}_{X,\operatorname{Add}\mathcal{C}_{q}}+\mu^{1}_{\operatorname{Add}\mathcal{C}_{q}}(\delta)+\mu^{2}_{\operatorname{Add}\mathcal{C}_{q}}(\delta,\delta)+\ldots.$

The curvature $\mu^{0}_{X,\operatorname{Add}\mathcal{C}_{q}}$ can be spelled
out more concretely as the sum of the curvatures of all constituents $A_{i}$
of $X$. The differential $\mu^{1}_{\operatorname{Add}\mathcal{C}_{q}}(\delta)$
is concretely the sum of the deformed differentials
$\mu^{1}_{\mathcal{C}_{q}}$ applied to each entry of $\delta$ as a matrix.
Whatever the value of $\mu^{0}_{(X,\delta)}$ adds up to, the reader can see
that it does typically not vanish because the twisted differential
$\delta\in\operatorname{Hom}^{1}(X,X)$ only satisfies the Maurer-Cartan
equation with respect to the non-deformed differential
$\mu^{1}_{\mathcal{C}}$.

######  Remark 3.11.

A popular way to form twisted completions of curved categories is to pick only
curvature-free twisted complexes. This might be the way to luck in case of
curved $A_{\infty}$-categories, because a curved twisted completion cannot be
passed to the minimal model. In contrast, for $A_{\infty}$-deformations,
curvature on $\operatorname{Tw}\mathcal{C}_{q}$ is not problematic at all.
Picking all twisted complexes of $\mathcal{C}$ is in fact necessary in order
to render $\operatorname{Tw}\mathcal{C}_{q}$ a deformation of
$\operatorname{Tw}\mathcal{C}$.

###  3.4 Functors

In this section, we define the notion of functors between deformed
$A_{\infty}$-categories. These functors serve as a framework for gauge
equivalences, quasi-equivalences and pushforwards of
$A_{\infty}$-deformations. This class of functors is presumably known to
experts. Our sign conventions are those of [8] and [2, Section
3.1.4/symplectic].

Based on the classical definition of $A_{\infty}$-functors which we recalled
in Definition 2.2, we are now ready to explain the natural extension to the
deformed case. We fix the definition as follows:

######  Definition 3.10.

Let $\mathcal{C},\mathcal{D}$ be two $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ deformations. A functor of deformed
$A_{\infty}$-categories consists of a map
$F_{q}:\operatorname{Ob}(\mathcal{C})\to\operatorname{Ob}(\mathcal{D})$
together with for every $k\geq 1$ a $B$-multilinear degree $1-k$ map

$F^{k}_{q}:\operatorname{Hom}_{\mathcal{C}_{q}}(X_{k},X_{k+1})\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}_{q}}(X_{1},X_{2})\to\operatorname{Hom}_{\mathcal{D}_{q}}(F_{q}X_{1},F_{q}X_{k+1})$

and infinitesimal curvature
$F^{0}_{q,X}\in\mathfrak{m}\operatorname{Hom}^{1}_{\mathcal{D}}(F_{q}X,F_{q}X)$
for every $X\in\mathcal{C}$, such that the curved $A_{\infty}$-functor
relations hold:

$\sum_{0\leq j\leq i\leq
k}(-1)^{\|a_{j}\|+\ldots+\|a_{1}\|}F_{q}(a_{k},\ldots,a_{i+1},\mu_{q}(a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1})\\\
=\sum_{\begin{subarray}{c}l\geq 0\\\ 1=j_{1}\leq\ldots\leq j_{l}\leq
k\end{subarray}}\mu_{q}(F_{q}(a_{k},\ldots,a_{j_{l}}),\ldots,F_{q}(\ldots,a_{j_{2}}),F_{q}(\ldots,a_{j_{1}})).$

######  Remark 3.12.

A functor $F_{q}$ of deformed $A_{\infty}$-categories consists of maps between
hom spaces which are allowed to have deformed (nonconstant) terms themselves.
As we have seen in section 3.1, the maps $F_{q}^{k}$ are automatically
continuous. Apart from the components $F_{q}^{\geq 1}$, the functor is allowed
to have infinitesimal curvature $F^{0}_{q}$. This curvature is a feature of
the deformed world where infinitesimal curvature is not only welcome, but is
necessary. The first two curved $A_{\infty}$-functor relations read

$\displaystyle F^{0}_{q}+F^{1}_{q}(\mu^{0}_{\mathcal{C}_{q},X})$
$\displaystyle=\mu^{1}_{\mathcal{D}_{q}}(F^{0}_{q,X}),$ $\displaystyle
F^{1}_{q}(\mu^{1}_{\mathcal{C}_{q}}(a))+(-1)^{\|a\|}F^{2}_{q}(\mu^{0}_{\mathcal{C}_{q},Y},a)+F^{2}_{q}(a,\mu^{0}_{\mathcal{C}_{q},X})$
$\displaystyle=\mu^{1}_{\mathcal{D}_{q}}(F^{1}_{q}(a))+\mu^{2}_{\mathcal{D}_{q}}(F^{0}_{q,Y},F^{1}_{q}(a))$
$\displaystyle\quad+\mu^{2}_{\mathcal{D}_{q}}(F^{1}_{q}(a),F^{0}_{q,X}),\quad\forall~{}a:X\to
Y.$

We can interpret a functor $F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ as an
extension of an ordinary $A_{\infty}$-functor $\mathcal{C}\to\mathcal{D}$.
Indeed, when forgetting the terms of $F=\\{F_{q}^{k}\\}_{k\geq 0}$ in higher
$\mathfrak{m}$-adic order, we get a collection of maps $F=\\{F^{k}\\}_{k\geq
1}$. Since $F_{q}$ satisfies the curved $A_{\infty}$-functor relations, its
reduction $F$ satisfies the ordinary $A_{\infty}$-functor relations. We fix
this terminology as follows:

######  Definition 3.11.

Let $F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ be a functor of deformed
$A_{\infty}$-categories. Then its leading term is the functor
$F:\mathcal{C}\to\mathcal{D}$ obtained by dividing out the maximal ideal
$\mathfrak{m}$.

Let us now define various notions of equivalences between deformed
$A_{\infty}$-categories. The aim is to transfer the notions of equivalences
between $A_{\infty}$-categories recalled in Definition 2.3 and 2.4 to the
world of deformed $A_{\infty}$-categories. The main idea is to declare a
functor $F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ an isomorphism if its
leading term $F:\mathcal{C}\to\mathcal{D}$ is an isomorphism. Note that this
definition is not vacuous: It still requires the functor $F_{q}$ to satisfy
the (curved) $A_{\infty}$-relations, but displays the isomorphism property as
a side issue which is dealt with on the leading term part.

######  Definition 3.12.

Let $\mathcal{C},\mathcal{D}$ be $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ deformations. Let
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ be a functor of deformed
$A_{\infty}$-categories and denote by $F:\mathcal{C}\to\mathcal{D}$ its
leading term. Then $F_{q}$ is

  * •

an isomorphism if $F$ is an isomorphism,

  * •

a quasi-isomorphism if $F$ is a quasi-isomorphism,

  * •

a quasi-equivalence if $F$ is a quasi-equivalence,

  * •

a derived equivalence if $F$ is a derived equivalence.

Two deformed $A_{\infty}$-categories $\mathcal{C}_{q}$ and $\mathcal{D}_{q}$
are

  * •

isomorphic if there is an isomorphism
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$,

  * •

quasi-isomorphic if there is a quasi-isomorphism
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$,

  * •

quasi-equivalent if there is a quasi-equivalence
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$,

  * •

derived equivalent if $\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{Tw}\mathcal{D}_{q}$ are quasi-equivalent.

We explain in Lemma 4.1 why relations such as quasi-equivalence and derived
equivalence are equivalence relations among deformed $A_{\infty}$-categories.
Among the $A_{\infty}$-deformations of one single category $\mathcal{C}$,
there is one further notion of equivalence, known as gauge equivalence:

######  Definition 3.13.

Let $\mathcal{C}$ be an $A_{\infty}$-category and
$\mathcal{C}_{q},\mathcal{C}_{q}^{\prime}$ be deformations. Then a gauge
equivalence between $\mathcal{C}_{q}$ and $\mathcal{C}_{q}^{\prime}$ is a
functor $F_{q}:\mathcal{C}_{q}\to\mathcal{C}_{q}^{\prime}$ of deformed
$A_{\infty}$-categories whose leading term $F:\mathcal{C}\to\mathcal{C}$ is
the identity.

We elaborate gauge equivalence further in the context of the Hochschild DGLA
in section 4.2.

##  4 The lens of DGLAs

In this section, we show how to abstractly transfer $A_{\infty}$-deformations
from one category onto another.

In section 4.1, we recall how to view $A_{\infty}$-deformations as Maurer-
Cartan elements via the so-called Hochschild DGLA. In section 4.2, we recall
the notion of gauge equivalence of Maurer-Cartan elements and explain its
application in the specific case of $A_{\infty}$-deformations. In section 4.3,
we recall the generalization of DGLAs known as $L_{\infty}$-algebras. In
section 4.4, we explain how to push forward deformations between quasi-
equivalent categories.

For the interpretation of gauge equivalences and pushforwards of deformations
in terms of $L_{\infty}$-theory we will not provide any proof. Rather, we
state in section 4.4 a collection of axioms on the Hochschild DGLA which we
will just assume in this paper.

The material in this section is not original. The Hochschild DGLA is classical
at least in the case of ordinary algebras. Keller proved its invariance under
quasi-equivalence in the dg case in [9].

Instead, this section serves to support a very specific viewpoint on
$A_{\infty}$-deformations. While there is widespread belief that curvature is
a hindrance, we want to demonstrate here that it is at least possible to live
peacefully with infinitesimal curvature. Let us present which section supports
which claim: From section 3.3 we learn that the twisted completion of an
$A_{\infty}$-deformation can be formed even under the presence of
infinitesimal curvature. This construction is entirely natural in the sense
that the deformed twisted completion is a deformation of the twisted
completion of the non-deformed $A_{\infty}$-category. From section 4.1 we
learn that infinitesimal curvature is inevitable for a good notion of
$A_{\infty}$-deformations. The core summary is that curvature does not hurt
for constructions with $A_{\infty}$-categories because it enters the picture
automatically when a deformation is induced from one category to another.

###  4.1 The Hochschild DGLA

In this section, we recall how to view $A_{\infty}$-deformations from the DGLA
point of view. The material is all known to experts and nicely shows how
curvature enters the picture. Useful references include [16, Chapter V] and
[21].

Before we recall the concept of DGLA and Maurer-Cartan elements, let us
summarize the philosophy: The aim is to capture every deformation problem by a
DGLA. The solutions of the deformation problem should then correspond to
Maurer-Cartan elements of the DGLA. This empowers the mathematician to use the
force of DGLA theory. Standard questions in DGLAs include: to find a quasi-
isomorphism $F:L\to L^{\prime}$ between two DGLAs, or to classify all Maurer-
Cartan elements of $L$ up to gauge equivalence.

deformation problemDGLA1\. reformulation2\. algebraic power3\. harvest

We are now ready to recall the notion of DGLAs and Maurer-Cartan elements.

######  Definition 4.1.

A DG Lie algebra (DGLA) is a $\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded
vector space $L$ together with

  * •

a differential $d:L^{i}\to L^{i+1}$,

  * •

a bracket $[-,-]:L\times L\to L$ of degree zero,

satisfying skew-symmetry, Leibniz rule and Jacobi identity:

$\displaystyle[a,b]=(-1)^{|a||b|+1}[b,a],$ $\displaystyle
d([a,b])=[da,b]+(-1)^{|a|}[a,db],$
$\displaystyle(-1)^{|a||c|}[a,[b,c]]+(-1)^{|b||a|}[b,[c,a]]+(-1)^{|c||b|}[c,[a,b]]=0.$

For example, the bracket $[-,-]$ is commutative on odd elements. With this
consideration, we recall the definition of Maurer-Cartan elements:

######  Definition 4.2.

Let $L$ be a DGLA and $B$ a deformation base. Regard the tensored DGLA
$B\widehat{\otimes}L$ with differential $d$ and bracket $[-,-]$ simply
extended continuously. A Maurer-Cartan element of $L$ over $B$ is an element
$\nu\in B\widehat{\otimes}L^{1}$ which satisfies the Maurer-Cartan equation

$d\nu+\frac{1}{2}[\nu,\nu]=0.$

The set of Maurer-Cartan elements of $L$ over $B$ is denoted
$\operatorname{MC}(L,B)$. In case the DGLA $L$ is
$\mathbb{Z}/2\mathbb{Z}$-graded, a Maurer-Cartan element is supposed to lie in
$B\widehat{\otimes}L^{\operatorname{odd}}$.

In the rest of this section, we will make sense of these definitions in the
case of the so-called Hochschild DGLA:

######  Definition 4.3.

Let $\mathcal{C}$ be a $\mathbb{Z}$\- or $\mathbb{Z}/2\mathbb{Z}$-graded
$A_{\infty}$-category. Then its Hochschild complex
$\operatorname{HC}(\mathcal{C})$ is the graded vector space

$\operatorname{HC}(\mathcal{C})=\prod_{\begin{subarray}{c}X_{1},\ldots,X_{k+1}\in\mathcal{C}\\\
k\geq
0\end{subarray}}\operatorname{Hom}\big{(}\operatorname{Hom}_{\mathcal{C}}(X_{k},X_{k+1})[1]\otimes\ldots\otimes\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{2})[1],~{}\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{k+1})[1]\big{)}.$

Here $[1]$ denotes the left-shift and $\|a\|=|a|-1$ denotes the reduced degree
of a morphism $a\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$. The grading
$\|\cdot\|$ on $\operatorname{HC}(\mathcal{C})$ is the one induced from the
shifted degrees of the hom spaces of $\mathcal{C}$. In other words, we have

$\|\eta(a_{k},\ldots,a_{1})\|=\|\eta\|+\|a_{k}\|+\ldots+\|a_{1}\|,\quad\eta\in\operatorname{HC}(\mathcal{C}).$

For $\eta,\omega\in\operatorname{HC}(\mathcal{C})$, the Gerstenhaber product
$\mu\cdot\omega\in\operatorname{HC}(\mathcal{C})$ is defined as

$(\eta\cdot\omega)(a_{k},\ldots,a_{1})=\sum(-1)^{(\|a_{l}\|+\ldots+\|a_{1}\|)\|\omega\|}\eta(a_{k},\ldots,\omega(\ldots),a_{l},\ldots,a_{1}).$

The Hochschild DGLA is the following $\mathbb{Z}$\- or
$\mathbb{Z}/2\mathbb{Z}$-graded DGLA structure on
$\operatorname{HC}(\mathcal{C})$: The bracket on
$\operatorname{HC}(\mathcal{C})$ is the Gerstenhaber bracket

$[\eta,\omega]=\eta\cdot\omega-(-1)^{\|\omega\|\|\eta\|}\omega\cdot\eta.$

The differential on $\operatorname{HC}(\mathcal{C})$ consists of commuting
with the product $\mu_{\mathcal{C}}\in\operatorname{HC}^{1}(\mathcal{C})$:

$d\nu=[\mu_{\mathcal{C}},\nu].$

######  Remark 4.1.

It is not hard to check that $\operatorname{HC}(\mathcal{C})$ is indeed a
DGLA. The reader who wishes to perform the computation is advised to write all
double brackets in terms of Gerstenhaber products and use the associator
relation

$\displaystyle(a\cdot b)\cdot c-a\cdot(b\cdot c)$
$\displaystyle=\sum(-1)^{\|b\|(\|a_{1}\|+\ldots+\|a_{i}\|)+\|c\|(\|a_{1}\|+\ldots+\|a_{j}\|)+\|b\|\|c\|}a(\ldots,b(\ldots),a_{i},\ldots,c(\ldots),a_{j},\ldots)$
$\displaystyle+\sum(-1)^{\|b\|(\|a_{1}\|+\ldots+\|a_{j}\|)+\|c\|(\|a_{1}\|+\ldots+\|a_{i}\|)}a(\ldots,c(\ldots),a_{i},\ldots,b(\ldots),a_{j},\ldots)).$

The sum in these formulas runs over all ways to insert $b$ and $c$ into $a$.
Despite the way the formulas are written, it is not necessary that any
elements $a_{i}$ or $a_{j}$ actually lie in between or behind $b$ and $c$.
This is merely an artifact needed to define the sign right: the sign shall be
the total reduced degree of all elements coming after $b$ or after $c$,
respectively.

######  Remark 4.2.

In the terminology of the Hochschild DGLA $\operatorname{HC}(\mathcal{C})$, we
can interpret $A_{\infty}$-deformations of $\mathcal{C}$ precisely as Maurer-
Cartan elements of $\operatorname{HC}(\mathcal{C})$. We start by observing
that the $A_{\infty}$-product $\mu_{\mathcal{C}}$ amounts to an element
$\mu_{\mathcal{C}}\in\operatorname{HC}^{1}(\mathcal{C})$ and the
$A_{\infty}$-relations translate to
$\mu_{\mathcal{C}}\cdot\mu_{\mathcal{C}}=0$.

Now let $\mathcal{C}_{q}$ be a deformation of $\mathcal{C}$. Then the curved
$A_{\infty}$-relations for $\mu_{\mathcal{C}_{q}}$ translate to
$\mu_{\mathcal{C}_{q}}\cdot\mu_{\mathcal{C}_{q}}=0$. Decompose
$\mu_{\mathcal{C}_{q}}=\mu_{\mathcal{C}}+\nu$ as non-deformed part
$\mu_{\mathcal{C}}$ plus deformation
$\nu\in\mathfrak{m}\operatorname{HC}^{1}(\mathcal{C})$. Given that
$\mu_{\mathcal{C}}$ already satisfies the $A_{\infty}$-relation
$\mu_{\mathcal{C}}\cdot\mu_{\mathcal{C}}=0$, the element $\nu$ itself
satisfies the Maurer-Cartan equation

$0=(\mu_{\mathcal{C}}+\nu)\cdot(\mu_{\mathcal{C}}+\nu)=d\nu+[\nu,\nu]/2.$

Conversely, pick a Maurer-Cartan element
$\nu\in\operatorname{MC}(\operatorname{HC}(\mathcal{C}),B)$. According to
Lemma 3.1, the element $\mu_{\mathcal{C}}+\nu$ extends in a multilinear and
$\mathfrak{m}$-adically continuous way to a collection of mappings

$\mu_{\mathcal{C}_{q}}^{k\geq
0}:(B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X_{k},X_{k+1}))\otimes\ldots\otimes(B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{2}))\to
B\widehat{\otimes}\operatorname{Hom}_{\mathcal{C}}(X_{1},X_{k+1}).$

The Maurer-Cartan identity for $\nu$ makes that $\mu_{\mathcal{C}_{q}}$
satisfies the curved $A_{\infty}$-relations.

######  Remark 4.3.

In case $L$ is a $\mathbb{Z}/2\mathbb{Z}$-graded DGLA, then Maurer-Cartan
elements of $L$ are by definition odd elements
$\nu\in\mathfrak{m}L^{\operatorname{odd}}$ with $d\nu+[\nu,\nu]/2=0$. For
example, our deformation $\operatorname{Gtl}_{q}Q$ is only a
$\mathbb{Z}/2\mathbb{Z}$-graded deformation of $\operatorname{Gtl}Q$. In the
context of deformations, we have to view both $\operatorname{Gtl}Q$ and its
Hochschild DGLA $\operatorname{HC}(\operatorname{Gtl}Q)$ as
$\mathbb{Z}/2\mathbb{Z}$-graded.

######  Remark 4.4.

For ordinary algebras, which are concentrated in degree zero and have
vanishing higher products, the Hochschild cohomology is typically defined
without the shifts. This results in a grading difference of $1$ from what we
present here. For example, the center of the algebra is the classical zeroth
Hochschild cohomology. In our $A_{\infty}$-setting, this cohomology will
rather be found in degree $-1$.

###  4.2 Gauge equivalence

In this section, we recall the notion of gauge equivalence. By virtue of
algebraic deformation theory, we have two ways of defining this equivalence:
via an explicit definition and via the Hochschild DGLA. Both notions are
defined here in parallel. Useful references are [16] and [21].

Gauge equivalenceby functor
$F:\mathcal{C}_{q}\to\mathcal{C}_{q}^{\prime}$Gauge equivalenceas MC elements
$\mu_{\mathcal{C}_{q}}\sim\mu_{\mathcal{C}_{q}^{\prime}}$

Recall from Definition 3.13 that a gauge equivalence between two deformations
$\mathcal{C}_{q}$, $\mathcal{C}_{q}^{\prime}$ of an $A_{\infty}$-category
$\mathcal{C}$ consists of a functor
$F:\mathcal{C}_{q}\to\mathcal{C}_{q}^{\prime}$ whose leading term is the
identity.

######  Remark 4.5.

The idea behind Definition 3.13 is that the set of automorphisms
$\mathcal{C}_{q}\to\mathcal{C}_{q}^{\prime}$ as deformed
$A_{\infty}$-categories is rather large. The leading term of an automorphism
can be any automorphism of $\mathcal{C}$, which is not interesting from the
perspective of deformation theory. Therefore one restricts to those functors
whose leading term is the identity on $\mathcal{C}$.

After the success of the notion of gauge equivalence throughout mathematics
and physics, a definition has also been captured in the abstract DGLA
approach. The idea here is that the gauging functor $F_{q}$ can be viewed as
an element of a “gauge group”. The infinitesimal action of this gauge group
can be described purely in terms of the DGLA:

######  Definition 4.4.

Let $L$ be a DGLA and $B$ a deformation base. Then there is a group action by
$\exp(\mathfrak{m}L^{0})$ on $B\widehat{\otimes}L^{1}$ with infinitesimal
generator

$\varphi.\nu=d\varphi+[\nu,\varphi]\in
B\widehat{\otimes}L^{1},\quad\nu\in\mathfrak{m}L^{0}.$

The group action preserves the set of Maurer-Cartan elements
$\operatorname{MC}(L;B)$. Two Maurer-Cartan elements
$\nu,\nu^{\prime}\in\operatorname{MC}(L;B)$ are gauge-equivalent if they lie
in the same orbit. The set of Maurer-Cartan elements up to gauge-equivalence
is denoted $\overline{\operatorname{MC}}(L;B)$.

######  Remark 4.6.

In case $L$ is a $\mathbb{Z}/2\mathbb{Z}$-graded DGLA, then the gauge group is
$\exp(\mathfrak{m}L^{\operatorname{even}})$. This group acts on Maurer-Cartan
elements of $L$, which are by definition odd elements
$\nu\in\mathfrak{m}L^{\operatorname{odd}}$ with $d\nu+[\nu,\nu]/2=0$.

Let us compare the infinitesimal generators of the action, at the non-deformed
product $\mu_{\mathcal{C}}$: An “infinitesimal functor”
$\operatorname{Id}+\varepsilon F$ pushes the non-deformed product
$\mu_{\mathcal{C}}$ to some $\mu^{\prime}$ such that
$(\operatorname{Id}+\varepsilon
F)\cdot\mu_{\mathcal{C}}=\mu^{\prime}\circ(\operatorname{Id}+\varepsilon F)$.
Here $\circ$ denotes functor composition, in contrast to the Gerstenhaber
product “$\cdot$”. Setting $\varepsilon^{2}=0$, we read off

$\mu^{\prime}=\mu_{\mathcal{C}}+\varepsilon[\mu_{\mathcal{C}},F].$

Meanwhile, the trivial deformation $\mu_{\mathcal{C}}$ corresponds to the zero
Maurer-Cartan element in $\operatorname{MC}(\operatorname{HC}(\mathcal{C}))$.
Gauging it by $\varepsilon F$ under $\varepsilon^{2}=0$ gives the element

$(\varepsilon F).0=0+d(\varepsilon F)+[0,\varepsilon
F]=\varepsilon[\mu_{\mathcal{C}},F].$

This element corresponds to the deformation
$\mu_{\mathcal{C}}+\varepsilon[\mu_{\mathcal{C}},F]$. We see that gauging a
deformation by a gauge functor $\operatorname{Id}+\varepsilon F$ is the same
as gauging its corresponding Maurer-Cartan element in the Hochschild DGLA:

Infinitesimal functor$\operatorname{Id}+\varepsilon
F$$\mu^{\prime}=\mu_{\mathcal{C}}+\varepsilon[\mu_{\mathcal{C}},F]$Infinitesimal
DGLA gauge$\varepsilon F$$\mu^{\prime}=\mu_{\mathcal{C}}+(d(\varepsilon
F)+[0,\varepsilon F])$

More precisely, two deformations are gauge equivalent in the sense of
Definition 3.13 if and only if their corresponding Maurer-Cartan elements are
gauge equivalent in the sense of Definition 4.4.

######  Remark 4.7.

$A_{\infty}$-Hochschild cohomology of $\mathcal{C}$ is defined as the
cohomology of $\operatorname{HC}(\mathcal{C})$, merely regarded as a cochain
complex instead of DGLA. This way, Hochschild cohomology is the linear
approximation of Maurer-Cartan elements up to gauge equivalence.

###  4.3 $L_{\infty}$-algebras

In this section we recall the formalism of $L_{\infty}$-algebras. In our
context of $A_{\infty}$-categories, we namely want to transport deformations
from one category to another, so that one needs morphisms between their
Hochschild DGLAs. The world of DGLAs and their morphisms is quite restrictive,
but there is a more flexible version of DGLAs known as $L_{\infty}$-algebras.
We finish this section with a definition and brief discussion of the
$L_{\infty}$-theory. We follow version 3 of [1]. The reader be cautioned that
version 4 of that paper has changed signs.

######  Definition 4.5.

An $L_{\infty}$-algebra is a graded vector space $L$ together with multilinear
maps

$l_{k}:\underbrace{L\times\ldots\times L}_{k\text{ times}}\to L$

of degree $2-k$ which are graded skew-symmetric and satisfy the higher Jacobi
identities:

$\displaystyle
l_{k}(x_{s(1)},\ldots,x_{s(k)})=\chi(s)l_{k}(x_{1},\ldots,x_{k}),$
$\displaystyle\sum_{\begin{subarray}{c}i+j=k+1\\\ i,j\geq
1\end{subarray}}\sum_{s\in
S_{i,k-i}}(-1)^{i(n-i)}\chi(s)l_{j}(l_{i}(x_{s(1)},\ldots,x_{s(i)}),x_{s(i+1)},\ldots,x_{s(k)})=0.$

Here $S_{i,k-i}$ denotes the set of shuffles, i.e. $s\in S_{k}$ with
$s(1)<\ldots<s(i)$ and $s(i+1)<\ldots<s(k)$. The sign $\chi(s)$ is the product
of the signum of $s$ and the Koszul sign of $s$. The Koszul sign of a
transposition $(i~{}j)$ is $(-1)^{|x_{i}||x_{j}|}$, and this rule is extended
multiplicatively to arbitrary permutations.

With this sign convention, a DGLA as in Definition 4.1 is simply an
$L_{\infty}$-algebra without higher products. In particular, the Hochschild
DGLA can automatically be regarded as an $L_{\infty}$-algebra.

Morphisms between $L_{\infty}$-algebras are indeed more flexible than between
DGLAs: $L_{\infty}$-morphisms are allowed to have higher components, just like
$A_{\infty}$-functors allow for higher components.

######  Definition 4.6.

A morphism of $L_{\infty}$-algebras $\varphi:L\to L^{\prime}$ is given by
multilinear maps

$\varphi^{k}:L\times\ldots\times L\to L$

of degree $1-k$ for all $k\geq 1$ such that
$\varphi(x_{s(1)},\ldots,x_{s(k)})=\chi(s)\varphi(x_{1},\ldots,x_{k})$ for any
$s\in S_{k}$ and

$\displaystyle\sum_{\begin{subarray}{c}i+j=k+1\\\ i,j\geq
1\end{subarray}}\sum_{s\in
S_{i,n-i}}(-1)^{i(k-i)}\chi(s)\varphi(l(x_{s(1)},\ldots,x_{s(i)}),x_{s(i+1)},\ldots,x_{s(k)})$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq k\\\
i_{1}+\ldots+i_{r}=k\end{subarray}}\sum_{t}(-1)^{u}\chi(t)l^{\prime}_{r}(\varphi(x_{t(1)},\ldots,x_{t(i_{1})}),\ldots,\varphi(x_{t(i_{1}+\ldots+i_{r-1}+1)},\ldots,x_{t(k)})),$

where $t$ runs over all $(i_{1},\ldots,i_{r})$-shuffles for which

$t(i_{1}+\ldots+i_{l-1}+1)<t(i_{1}+\ldots+i_{l}+1).$

and $u=(r-1)(i_{1}-1)+\ldots+2(i_{r-2}-1)+(i_{r-1}-1)$. A morphism
$\varphi:L\to L^{\prime}$ of $L_{\infty}$-algebras is a quasi-isomorphism if
$\varphi^{1}$ is a quasi-isomorphism of complexes.

######  Definition 4.7.

Let $L$ be an $L_{\infty}$-algebra and $B$ a deformation base. Then a Maurer-
Cartan element is an element $x\in\mathfrak{m}\widehat{\otimes}L^{1}$
satisfying the Maurer-Cartan equation

$\sum_{k\geq 1}\frac{l_{k}(x,\ldots,x)}{k!}=0.$

We write $\operatorname{MC}(L,B)$ for the set of Maurer-Cartan elements of $L$
over $B$.

In contrast to the DGLA case, there is no gauge group acting on the Maurer-
Cartan elements. Instead, one should regard an equivalence relation of
“homotopy”. All we should assume here is that the notion of homotopy exists
and gives rise to a quotient set $\overline{\operatorname{MC}}(L,B)$, just as
in the DGLA case.

###  4.4 Axioms on $A_{\infty}$-deformations

There is a slight gap in our treatment of $A_{\infty}$-deformations: We cannot
prove invariance of the Hochschild DGLA under quasi-equivalences. While
derived invariance is known in the dg case due to [9], according to private
communication with Keller there is no literature available for the
$A_{\infty}$-case. We do not fill the gap here. Instead, we state a small set
of axioms in this section which we will simply assume for the purpose of
section 5.4 and 5.2.

The motivation for our axioms is that we need to be able to push deformations
from one category to another. Assume two $A_{\infty}$-categories $\mathcal{C}$
and $\mathcal{D}$ are quasi-equivalent by means of a quasi-equivalence
$F:\mathcal{C}\to\mathcal{D}$. Intuition says that a deformation
$\mathcal{C}_{q}$ of $\mathcal{C}$ can be “pushed” via $F$ to a deformation
$\mathcal{D}_{q}=F_{*}(\mathcal{C}_{q})$ of $\mathcal{D}$ such that
$\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ still quasi-equivalent to each other
by a deformation of the functor $F$. We formalize this notion as follows:

######  Definition 4.8.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ be a
deformation. Let $F:\mathcal{C}\to\mathcal{D}$ be a quasi-equivalence. Then we
call any deformation $\mathcal{D}_{q}$ of $\mathcal{D}$ a naive pushforward of
the deformation $\mathcal{C}_{q}$ along $F$ if there exists a functor
$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ with leading term $F$.

######  Remark 4.8.

If forming Hochschild DGLAs were functorial, pushforwards would be easy.
Indeed, a quasi-equivalence $\mathcal{C}\to\mathcal{D}$ would ideally induce a
quasi-isomorphism of $L_{\infty}$-algebras
$\operatorname{HC}(\mathcal{C})\to\operatorname{HC}(\mathcal{D})$ and
therefore a bijection
$\overline{\operatorname{MC}}(\mathcal{C},B)\to\overline{\operatorname{MC}}(\mathcal{D},B)$
of Maurer-Cartan elements. The naive pushforward of $\mathcal{C}_{q}$ would
then simply be obtained as the image under this map of the Maurer-Cartan
elemement defined by $\mathcal{C}_{q}$. It is an awkward fact of algebra that
however neither the Hochschild DGLA nor Hochschild cohomology is functorial.
For instance, even the center $\operatorname{HH}^{0}(A)=Z(A)$ of an algebra
$A$ is not functorial a property. Two quasi-equivalent $A_{\infty}$-categories
however have quasi-isomorphic Hochschild DGLAs. At least, this is a folklore
statement, with actual proof only available by Keller in the dg case [9].

This definition of naive pushforwards raises many detail questions. For
instance, let $G:\mathcal{D}\to\mathcal{E}$ be yet another quasi-equivalence.
Then we are interested in the question whether the double pushforward of
$\mathcal{C}_{q}$ along $F$ and $G$ is gauge-equivalent to the single
pushforward of $\mathcal{C}_{q}$ along $GF$. We shall provide answer to this
question in the form of axioms:

######  Convention 4.1.

We assume the following axioms regarding the Hochschild DGLA: Let $B$ be a
fixed deformation base. Then:

  * (A1)

Two $A_{\infty}$-deformations $\mathcal{C}_{q},\mathcal{C}_{q}^{\prime}$ over
$B$ are gauge equivalent if and only if they are gauge equivalent as Maurer-
Cartan elements of $\operatorname{HC}(\mathcal{C})$.

  * (A2)

Let $F:\mathcal{C}\to\mathcal{D}$ be a quasi-equivalence of
$A_{\infty}$-categories. Then there exists a quasi-isomorphism of
$L_{\infty}$-algebras
$F_{*}:\operatorname{HC}(\mathcal{C})\to\operatorname{HC}(\mathcal{D})$.

  * (A3)

We call $F_{*}$ the pushforward of $F$. The pushforward is noncanonical.
However its induced map
$(F_{*})^{\operatorname{MC}}:\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{C}),B)\to\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{D}),B)$
has the following property: $\mathcal{D}_{q}$ is a naive pushforward of
$\mathcal{C}_{q}$ along $F$ if and only if
$\mu_{\mathcal{D}_{q}}=F_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})$.

  * (A4)

Let $\mathcal{C}\subseteq\mathcal{D}$ be a subcategory such that the inclusion
$i:\mathcal{C}\to\mathcal{D}$ is a quasi-equivalence. Then
$i_{*}^{\operatorname{MC}}(\mu_{\mathcal{D}_{q}}|_{\mathcal{C}})=\mu_{\mathcal{D}_{q}}$.

  * (A5)

Push-forward is functorial on Maurer-Cartan elements:
$(GF)_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})=G_{*}^{\operatorname{MC}}(F_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})$.

######  Remark 4.9.

There is a slight abuse of the notation in Convention 4.1. Where we have
written $F_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})$, we actually mean
the Maurer-Cartan element $\mu_{\mathcal{C}_{q}}-\mu_{\mathcal{C}}$ instead of
$\mu_{\mathcal{C}_{q}}$. In fact, the element
$\mu_{\mathcal{C}_{q}}\in\,B\widehat{\otimes}\operatorname{HC}^{1}(\mathcal{C})$
is not a Maurer-Cartan element itself. In similar abuse of notation, we may
occasionally write $F_{*}^{\operatorname{MC}}(\mathcal{C}_{q})$ instead of
$F_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})$.

As an application of Convention 4.1, we show here that quasi-equivalence of
deformed $A_{\infty}$-categories is an equivalence relation. By quasi-
equivalence of two deformed $A_{\infty}$-categories, we refer to the relation
defined in Definition 3.12. A proof without assuming the axioms would either
need to deal with complicated explicit constructions, or make use of an
$\infty$-categorical level. In other words, assuming the axioms seems to be a
healthy midway for the scope of the paper.

######  Lemma 4.1.

Quasi-equivalence of deformed $A_{\infty}$-categories is an equivalence
relation. Even stronger, for every quasi-equivalence
$F:\mathcal{C}\to\mathcal{D}$ there exists a quasi-equivalence
$G:\mathcal{D}\to\mathcal{C}$ such that $G_{*}F_{*}=\mathrm{id}$. Derived
equivalence is an equivalence relation as well.

###### Proof.

We need to prove reflexivity, transitivity and symmetry of the quasi-
equivalence relation. The first two properties are easy: The identity functor
obviously provides a quasi-equivalence from any deformation $\mathcal{C}_{q}$
to itself. And if $F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$ and
$G_{q}:\mathcal{D}_{q}\to\mathcal{E}_{q}$ are quasi-equivalences, then the
composition $G_{q}F_{g}:\mathcal{C}_{q}\to\mathcal{E}_{q}$ is a quasi-
equivalence as well. We conclude that only symmetry remains to be proven.

The proof of symmetry consists of five steps: First, we define a set of “good”
quasi-equivalences $F$ for which there exists a quasi-equivalence $G$ with
$G_{*}F_{*}=\mathrm{id}$. The second, third and fourth step establish basic
properties of this “good” set. In the fifth step, we show that those
properties already make every quasi-equivalence lie in $\mathcal{F}$.

As step 1, let us recall our context. We are interested in the set of quasi-
equivalences $F:\mathcal{C}\to\mathcal{D}$ such that there exists a quasi-
equivalence $G:\mathcal{D}\to\mathcal{C}$ with $G_{*}F_{*}=\mathrm{id}$.
Denote this set by $\mathcal{F}$:

$\mathcal{F}\coloneqq\\{F:\mathcal{C}\to\mathcal{D}\text{ q.e.}~{}|~{}\exists
G:\mathcal{D}\to\mathcal{C}\text{ q.e.}:G_{*}F_{*}=\mathrm{id}\\}.$

Our aim is to show that any quasi-equivalence lies in $\mathcal{F}$. Steps 2,
3, 4 are devoted to proving several properties of $\mathcal{F}$.

As step 2, we prove the property

$F\in\mathcal{F}\text{ with }G\text{ q.e. such that
}G_{*}F_{*}=\mathrm{id}\quad\Longrightarrow\quad F_{*}G_{*}=\mathrm{id}\text{
and }G\in\mathcal{F}.$

Indeed, pick $F$ as on the left-hand side. Since both $F$ and $G$ are quasi-
equivalences, both pushforwards $F_{*}$ and $G_{*}$ are bijections. Together
with $G_{*}F_{*}=\mathrm{id}$, we conclude that $G_{*}$ and $F_{*}$ are simply
inverses to each other. In other words, $F_{*}G_{*}=\mathrm{id}$ holds as
well. We conclude that $G\in\mathcal{F}$.

As step 3, we prove for composable quasi-equivalences $F,G$ the property

$F,G\in\mathcal{F}\quad\Longleftrightarrow\quad GF\in\mathcal{F}.$

One should think of this as a strong version of the two-out-of-three property.
To prove it, pick $F,G\in\mathcal{F}$ with $F^{\prime}_{*}F_{*}=\mathrm{id}$
and $G^{\prime}_{*}G_{*}=\mathrm{id}$. We get
$(F^{\prime}G^{\prime})_{*}(GF)_{*}=F^{\prime}_{*}F_{*}=\mathrm{id}$, which
renders $GF\in\mathcal{F}$. Conversely assume $GF\in\mathcal{F}$ with
$H_{*}(GF)_{*}=\mathrm{id}$. Then $(HG)_{*}F_{*}=\mathrm{id}$, hence
$F\in\mathcal{F}$. Step 2 implies $F_{*}(HG)_{*}=\mathrm{id}$. In other words
$(FH)_{*}G_{*}=\mathrm{id}$, hence $G\in\mathcal{F}$. We conclude that both
$F$ and $G$ lie in $\mathcal{F}$, as desired.

As step 4, we show that the following lie in $\mathcal{F}$:

$\displaystyle\text{quasi-equivalences with a one-sided inverse},$
$\displaystyle\text{inclusions of skeletal subcategories},$
$\displaystyle\text{inclusion
}i:\operatorname{H}\mathcal{C}\to\mathcal{C}\text{ and projection
}\pi:\mathcal{C}\to\operatorname{H}\mathcal{C}.$

To this end, assume $F$ and $G$ are quasi-equivalences with
$FG=\operatorname{Id}$. Combining $\operatorname{Id}\in\mathcal{F}$ with step
3, we deduce $F,G\in\mathcal{F}$. In particular, isomorphisms fall under this
regime. Inclusions of skeletal subcategories also fall under this regime,
since a skeletal subcategory $S\subseteq\mathcal{C}$ produces a quasi-
equivalence $\mathcal{C}\to S$ which reduces to the identity on $S$. Now
regard a category $\mathcal{C}$ and its minimal model
$\operatorname{H}\mathcal{C}$. The minimal model is not unique, but every
minimal model comes with quasi-isomorphisms
$i:\operatorname{H}\mathcal{C}\to\mathcal{C}$ and
$\pi:\mathcal{C}\to\operatorname{H}\mathcal{C}$. Regard the map $\pi
i:\operatorname{H}\mathcal{C}\to\operatorname{H}\mathcal{C}$. Since both $i$
and $\pi$ are quasi-isomorphisms, $\pi i$ is a quasi-isomorphism as well.
Moreover, $\operatorname{H}\mathcal{C}$ is already a minimal category, hence
$\pi i$ is an isomorphism. We conclude that $\pi i\in\mathcal{F}$. By step 3,
we deduce that both $\pi$ and $i$ lie in $\mathcal{F}$. This finishes step 4.

As final step 5, we prove that any quasi-equivalence lies in $\mathcal{F}$. To
this end, let $F:\mathcal{C}\to\mathcal{D}$ be any quasi-equivalence. Our
strategy is to build a diagram to whose arrows we can apply step 3 and 4 to
deduce that $F$ also lies in $\mathcal{F}$. In order to write down the
diagram, pick minimal models $\operatorname{H}\mathcal{C}$ and
$\operatorname{H}\mathcal{D}$, together with inclusion map
$i_{\mathcal{C}}:\operatorname{H}\mathcal{C}\to\mathcal{C}$ and projection
$\pi_{\mathcal{D}}:\mathcal{D}\to\operatorname{H}\mathcal{D}$. Define
$F^{\prime}\coloneqq\pi_{\mathcal{D}}Fi_{\mathcal{C}}$. This gives a diagram,
commutative by definition,

$\mathcal{C}$$\mathcal{D}$$\operatorname{H}\mathcal{C}$$\operatorname{H}\mathcal{D}$$F$$i_{\mathcal{C}}$$\pi_{\mathcal{D}}$$F^{\prime}$

Choose a skeletal subcategory
$S_{\mathcal{C}}\subseteq\operatorname{H}\mathcal{C}$ and set
$S_{\mathcal{D}}\coloneqq F^{\prime}(S_{\mathcal{C}})$. Then
$S_{\mathcal{D}}\subseteq\operatorname{H}\mathcal{D}$ is a skeletal
subcategory as well: Any object $X\in\operatorname{H}\mathcal{D}$ is
isomorphic to some $F^{\prime}(Y)$, and $Y$ in turn is isomorphic to some
$Z\in S_{\mathcal{C}}$, hence $X\cong F^{\prime}(Z)\in S_{\mathcal{D}}$.
Moreover if $Y,Z\in S_{\mathcal{C}}$ and $F^{\prime}(Y)\cong F^{\prime}(Z)$,
then $Y\cong Z$. This implies $Y=Z$ because $S_{\mathcal{C}}$ is a skeleton.
In total, we conclude that
$S_{\mathcal{D}}\subseteq\operatorname{H}\mathcal{D}$ is a skeletal
subcategory, and we obtain a restricted quasi-equivalence
$F^{\prime\prime}:S_{\mathcal{C}}\to S_{\mathcal{D}}$. Putting everything
together, we have the commutative diagram

$\mathcal{C}$$\mathcal{D}$$\operatorname{H}\mathcal{C}$$\operatorname{H}\mathcal{D}$$S$$S^{\prime}$$F$$i_{\mathcal{C}}$$\pi_{\mathcal{D}}$$F^{\prime}$$I$$J$$F^{\prime\prime}$

Here we denoted by $I$ and $J$ the inclusion of the full subcategories $S$ and
$S^{\prime}$ into $\operatorname{H}\mathcal{C}$ resp.
$\operatorname{H}\mathcal{D}$. The top and bottom square are strictly
commutative by definition of $F^{\prime}$ and $F^{\prime\prime}$.

We now count everything together: $F^{\prime\prime}$ is an isomorphism and $J$
is an inclusion of a skeletal subcategory. By step 4, we get
$J,F^{\prime\prime}\in\mathcal{F}$. By step 3, we get
$JF^{\prime\prime}\in\mathcal{F}$. By commutativity of the bottom of the
diagram we have $F^{\prime}I=JF^{\prime\prime}$. By step 3, this implies
$F^{\prime}\in\mathcal{F}$. By commutativity of the top of the diagram we have
$\pi_{\mathcal{D}}Fi_{\mathcal{C}}=F^{\prime}\in\mathcal{F}$. A double
application of step 3 renders $F\in\mathcal{F}$. Since $F$ was arbitrary, this
shows that all quasi-equivalences lie in $\mathcal{F}$. In other words, for
every $F:\mathcal{C}\to\mathcal{D}$ there exists a
$G:\mathcal{D}\to\mathcal{C}$ such that $G_{*}F_{*}=\mathrm{id}$. In
particular, this shows quasi-equivalence is an equivalence relation.

Let us now explain why derived equivalence is an equivalence relation as well.
Indeed, $\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ derived equivalent according
to Definition 3.12 if $\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{Tw}\mathcal{D}_{q}$ are quasi-equivalent. Since we have just
shown that quasi-equivalence is an equivalence relation, we conclude that
derived equivalence is an equivalence relation. This finishes the proof. ∎

######  Remark 4.10.

Pushing deformations from one category to another is not only possible via
quasi-equivalences. We can also push forward deformations from one category to
a derived equivalent category. Namely, let
$F:\operatorname{Tw}\mathcal{C}\to\operatorname{Tw}\mathcal{D}$ be a quasi-
equivalence and let $\mathcal{C}_{q}$ be a deformation of $\mathcal{C}$. Then
the twisted completion $\operatorname{Tw}\mathcal{C}_{q}$ from Definition 3.9
is canonically a deformation of $\operatorname{Tw}\mathcal{C}$. The
pushforward $F_{*}^{\operatorname{MC}}$ on Maurer-Cartan elements now
transports this deformation $\operatorname{Tw}\mathcal{C}_{q}$ to a
deformation
$F_{*}^{\operatorname{MC}}(\mu_{\operatorname{Tw}\mathcal{C}_{q}})$ of
$\operatorname{Tw}\mathcal{D}$. Restricting this deformation to
$\mathcal{D}\subseteq\operatorname{Tw}\mathcal{D}$ gives a deformation of
$\mathcal{D}$. Via the route of twisted completion, pushforward and
restriction, the deformation $\mathcal{D}_{q}$ corresponds to the deformation
$\mathcal{C}_{q}$. We may equally call $\mathcal{D}_{q}$ the pushforward of
$\mathcal{C}_{q}$ along $F$.

##  5 Derived categories of $A_{\infty}$-deformations

In this section, we define and prove existence of derived categories of
$A_{\infty}$-deformations. In section 5.1, we revisit the notion of twisted
completion from section 3.3. In section 5.2, we define minimal models for
$A_{\infty}$-deformations, in such a way that the minimal model
$\operatorname{H}\mathcal{C}_{q}$ of $\mathcal{C}_{q}$ is a deformation of
$\operatorname{H}\mathcal{C}$. In section 5.3, we combine both steps and
define the derived category of $\mathcal{C}_{q}$ to be
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$, the minimal model of the
twisted completion. In section 5.4 we comment on uncurving of
$A_{\infty}$-deformations and on uncurvable objects.

###  5.1 Twisted completion

In this section, we revisit the twisted completion construction for deformed
$A_{\infty}$-categories from an abstract perspective. While in section 3.3 we
already defined $\operatorname{Tw}\mathcal{C}_{q}$ in an ad-hoc way, in the
present section we have the additional abstract operations of section 4
available. We explain that $\operatorname{Tw}\mathcal{C}_{q}$ is the
pushforward of the deformation $\mathcal{C}_{q}$ onto
$\operatorname{Tw}\mathcal{C}$ along the derived equivalence
$\mathcal{C}\to\operatorname{Tw}\mathcal{C}$. This section is meant to
illustrate that the story of $\operatorname{Tw}\mathcal{C}_{q}$ is well-
rounded and $\operatorname{Tw}\mathcal{C}_{q}$ is correctly defined by
Definition 3.9.

To start with, let us recall the additive completion and twisted completion
from Definition 3.9. The additive completion
$\operatorname{Add}\mathcal{C}_{q}$ is the deformation of
$\operatorname{Add}\mathcal{C}$ whose deformed products are simply induced
from those of $\mathcal{C}_{q}$. The twisted completion
$\operatorname{Tw}\mathcal{C}_{q}$ is the deformation of
$\operatorname{Tw}\mathcal{C}_{q}$ whose deformed products are defined by
embracing the deformed products of $\mathcal{C}_{q}$ with the $\delta$-matrix.
Note that the objects of $\operatorname{Tw}\mathcal{C}_{q}$ are by definition
the same as those of $\operatorname{Tw}\mathcal{C}$, namely twisted complexes
$(A_{1}[k_{1}]\oplus\ldots\oplus A_{n}[k_{n}],\delta)$ where $\delta$
satisfies the Maurer-Cartan equation with respect to $\mu_{\mathcal{C}}$. The
category $\operatorname{Tw}\mathcal{C}_{q}$ is a curved deformation and its
curvature is given by the failure of $\delta$ to satisfy the Maurer-Cartan
equation with respect to $\mu_{\mathcal{C}_{q}}$.

With the help of Definition 3.12, we can put the definition of
$\operatorname{Tw}\mathcal{C}_{q}$ into a better framing. The easiest way to
make this precise is to note that the inclusion
$\mathcal{C}\to\,\operatorname{Tw}\mathcal{C}$ is trivially a derived
equivalence. The twisted completion $\operatorname{Tw}\mathcal{C}_{q}$
satisfies precisely the deformed analog of this classical statement:

######  Lemma 5.1.

The inclusion $\mathcal{C}_{q}\subseteq\operatorname{Tw}\mathcal{C}_{q}$ is a
derived equivalence of deformed $A_{\infty}$-categories.

###### Proof.

Due to our definition of quasi-equivalences of deformed
$A_{\infty}$-categories, this statement is trivial. Indeed, regard the
inclusion functor $F_{q}:\mathcal{C}_{q}\to\operatorname{Tw}\mathcal{C}_{q}$.
This is by definition a functor of deformed $A_{\infty}$-categories. Its
leading term is the standard inclusion
$F:\mathcal{C}\to\operatorname{Tw}\mathcal{C}$. Ultimately, the functor $F$ is
a derived equivalence and we conclude that $F_{q}$ is a derived equivalence
according to Definition 3.12. ∎

In this section, let us go beyond the definition of
$\operatorname{Tw}\mathcal{C}_{q}$ and define a few more utilities. It is
possible to define a variant $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ of
the twisted completion of $A_{\infty}$-deformations. It is not a deformation
of $\operatorname{Tw}\mathcal{C}$, but rather an object-cloning deformation in
the sense of Definition 3.7.

######  Definition 5.1.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. The objects of $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ are be
pairs

$(X,\delta=\delta_{0}+\delta^{\prime}),\quad
X\in\operatorname{Add}\mathcal{C},\quad\delta_{0}\in\operatorname{Hom}_{\mathcal{C}}^{1}(X,X),\quad\delta^{\prime}\in\mathfrak{m}\operatorname{Hom}_{\mathcal{C}}^{1}(X,X).$

Only the leading part $\delta_{0}$ is required to be upper triangular and
satisfy the Maurer-Cartan equation with respect to $\mu_{\mathcal{C}}$. The
hom spaces and higher products of $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$
shall be defined by embracing with $\delta$ as in Definition 3.9.

######  Remark 5.1.

In an object
$(X,\delta_{0}+\delta^{\prime})\in\operatorname{Tw}^{\prime}\mathcal{C}_{q}$,
the infinitesimal part $\delta^{\prime}$ can also lie below the diagonal. The
category $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ is clearly a deformed
$A_{\infty}$-category. When dividing out the maximal ideal we do not exactly
reach $\operatorname{Tw}\mathcal{C}$ though, but a version of
$\operatorname{Tw}\mathcal{C}$ with lots of isomorphic objects, one copy for
every choice of infinitesimal terms being added in the $\delta$ matrix. This
makes $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ an essentially surjective
object-cloning deformation of $\operatorname{Tw}\mathcal{C}$ along the cloning
map
$O:\operatorname{Ob}(\operatorname{Tw}^{\prime}\mathcal{C}_{q})\to\operatorname{Ob}(\operatorname{Tw}\mathcal{C})$
given by $(X,\delta_{0}+\delta^{\prime})\mapsto(X,\delta_{0})$.

######  Remark 5.2.

Let us check that products of this deformed $A_{\infty}$-category
$\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ are still well-defined: Regard a
sequence of $k$ compatible entries of $\delta$. Then at least $k-d$ of them
are infinitesimal, where $d$ is the dimension of the $\delta$-matrix. We
conclude that a term

$\mu_{q}(\underbrace{\delta,\ldots,\delta}_{k_{n+1}\text{
times}},a_{n},\underbrace{\delta,\ldots,\delta}_{k_{n}\text{
times}},\ldots,a_{1},\underbrace{\delta,\ldots,\delta}_{k_{1}\text{ times}})$

lies in the $K\coloneqq(k_{1}+\ldots+k_{n+1}-(n+1)d)$-th power of the maximal
ideal $\mathfrak{m}$. When the number $k_{1}+\ldots+k_{n+1}$ of total $\delta$
insertions goes to infinity, this exponent $K$ tends towards infinity as well.
This renders all products in $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ well-
defined.

The category $\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ allows us to define
twisted complexes more liberally than in $\operatorname{Tw}\mathcal{C}_{q}$.
Nevertheless both categories are essentially equal:

######  Lemma 5.2.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Then $\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{Tw}^{\prime}\mathcal{C}_{q}$ are quasi-equivalent as deformed
$A_{\infty}$-categories. Moreover, let
$S\coloneqq\\{(X_{i},\delta_{i})\\}_{i=1,\ldots,n}$ be a collection of objects
in $\operatorname{Tw}\mathcal{C}_{q}$ and
$\delta_{i}^{\prime}\in\mathfrak{m}\operatorname{End}^{1}(X_{i})$ be
infinitesimal terms. Then the category

$S^{\prime}=\\{(X_{i},\delta_{i}+\delta_{i}^{\prime})\\}_{i=1,\ldots,n}\subseteq\operatorname{Tw}^{\prime}\mathcal{C}_{q}$

is gauge equivalent to $S$.

###### Proof.

For the first part, regard the inclusion
$\operatorname{Tw}\mathcal{C}_{q}\subseteq\operatorname{Tw}^{\prime}\mathcal{C}_{q}$.
Upon dividing out the maximal ideal $\mathfrak{m}$, this inclusion reduces to
the inclusion of $\operatorname{Tw}\mathcal{C}$ into a version of
$\operatorname{Tw}\mathcal{C}$ with lots of copies of objects. We conclude
that the inclusion
$\operatorname{Tw}\mathcal{C}_{q}\subseteq\operatorname{Tw}^{\prime}\mathcal{C}_{q}$
is a quasi-equivalence of deformed $A_{\infty}$-categories.

For the second part, build the functor

$F_{q}:S^{\prime}\to S,\quad
F^{0}_{q,(X_{i},\delta_{i}+\delta_{i}^{\prime})}=\delta_{i}^{\prime},\quad
F_{q}^{1}=\operatorname{Id},\quad F_{q}^{\geq 2}=0.$

This functor is the identity on objects and indeed satisfies the curved
$A_{\infty}$-functor relations:

$\displaystyle\sum\mu_{S}(\underbrace{F_{q}^{0},\ldots,F_{q}^{0}}_{\geq
0\text{ times}},F_{q}^{1}(a_{k}),$
$\displaystyle\underbrace{F_{q}^{0},\ldots,F_{q}^{0}}_{\geq 0\text{
times}},\ldots,F^{1}(a_{1}),\underbrace{F_{q}^{0},\ldots,F_{q}^{0}}_{\geq
0\text{ times}})$
$\displaystyle=\sum\mu_{\operatorname{Add}\mathcal{C}_{q}}(\underbrace{\delta_{i},\delta_{i}^{\prime},\ldots}_{\text{any
mix}},a_{k},\ldots,a_{1},\underbrace{\delta_{i},\delta_{i}^{\prime},\ldots}_{\text{any
mix}})$
$\displaystyle=\sum\mu_{\operatorname{Add}\mathcal{C}_{q}}(\underbrace{\delta,\ldots,\delta}_{\geq
0\text{ times}},a_{k},\underbrace{\delta,\ldots,\delta}_{\geq 0\text{
times}},\ldots,a_{1},\underbrace{\delta,\ldots,\delta}_{\geq 0\text{ times}})$
$\displaystyle=F_{q}^{1}(\mu_{S^{\prime}}(a_{k},\ldots,a_{1}).$

This shows that $F_{q}$ is a gauge-equivalence $S^{\prime}\to S$. ∎

###  5.2 Minimal models

In this section we define a notion of minimal model for deformed
$A_{\infty}$-categories. The definition might differ from the reader’s
expectation. Indeed, a minimal model according to our definition need not have
vanishing differential. We finish the section by explaining why every
$A_{\infty}$-deformation has a minimal model.

In Definition 5.2, we define the notion of minimal model for
$A_{\infty}$-deformations. The aim is to have a definition which is compatible
with the classical notion and make minimal models exist for any deformation.

######  Definition 5.2.

Let $\mathcal{C}$ and $\mathcal{D}$ be $A_{\infty}$-categories and
$\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ deformations. Then $\mathcal{D}_{q}$
is a minimal model for $\mathcal{C}_{q}$ if $\mathcal{D}$ is a minimal
category and there is a functor of deformed $A_{\infty}$-categories

$F_{q}:\mathcal{C}_{q}\to\mathcal{D}_{q}$

whose leading term is a quasi-isomorphism $F:\mathcal{C}\to\mathcal{D}$. We
denote any minimal model of $\mathcal{C}_{q}$ by
$\operatorname{H}\mathcal{C}_{q}$.

Let us discuss Definition 5.2. Classical minimal models of
$A_{\infty}$-categories have vanishing curvature $\mu^{0}$ and differential
$\mu^{1}$ by definition. This is not the case anymore for minimal models of
deformations. Instead, we require that $\mathcal{D}$ itself is minimal, while
$\mathcal{D}_{q}$ is allowed to have both curvature and nonvanishing
differential. Of course, the curvature and differential of $\mathcal{D}_{q}$
are both infinitesimal, since $\mathcal{D}_{q}$ is a deformation of the
minimal $A_{\infty}$-category $\mathcal{D}$:

$\displaystyle\mu^{0}_{\mathcal{D}_{q},X}$
$\displaystyle\in\mathfrak{m}\operatorname{Hom}_{\mathcal{D}}(X,X),\quad\forall
X\in\mathcal{D},$ $\displaystyle\mu^{1}_{\mathcal{D}_{q}}(x)$
$\displaystyle\in\mathfrak{m}\operatorname{Hom}_{\mathcal{D}}(X,Y),\quad\forall
X,Y\in\mathcal{D},~{}x\in\operatorname{Hom}_{\mathcal{D}}(X,Y).$

At this point, writing $\operatorname{H}\mathcal{C}_{q}$ for a minimal model
constitutes abuse of notation, since we have not explained their existence.
This is done in Corollary 5.1.

######  Remark 5.3.

An alternative definition is obtained by requiring that the map
$\mathcal{C}\to\mathcal{D}$ induced by $F$ be only a quasi-equivalence. The
difference in the two notions is merely cosmetic. While in the version of
Definition 5.2 the objects of $\mathcal{C}$ and $\mathcal{D}$ are required to
match, a definition requiring quasi-equivalence allows for additional bloat:
One may add any amount of quasi-isomorphic objects to both $\mathcal{C}$ and
$\mathcal{D}$.

There are several equivalent ways of characterizing minimal models. Recall the
notion of quasi-isomorphism from Definition 3.12.

######  Lemma 5.3.

Let $\mathcal{C},\mathcal{D}$ be $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ deformations. The following statements are
equivalent:

  1. 1.

$\mathcal{D}_{q}$ is a minimal model for $\mathcal{C}_{q}$.

  2. 2.

$\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ are quasi-isomorphic, and
$\mathcal{D}$ is minimal.

  3. 3.

$\mathcal{C}_{q}$ and $\mathcal{D}_{q}$ are quasi-isomorphic, and
$\mathcal{D}$ is a minimal model for $\mathcal{C}$.

  4. 4.

There is a quasi-isomorphism $F:\mathcal{C}\to\mathcal{D}$ such that
$F_{*}^{\operatorname{MC}}(\mu_{\mathcal{C}_{q}})=\mu_{\mathcal{D}_{q}}$ and
$\mathcal{D}$ is minimal.

Here
$F_{*}^{\operatorname{MC}}:\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{C}),B)\to\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{D}),B)$
denotes the push-forward map of Maurer-Cartan elements along $F$.

###### Proof.

This is a simple consequence of the axioms stated in Convention 4.1. ∎

A consequence of Lemma 5.3 is that minimal models always exist:

######  Corollary 5.1.

Let $\mathcal{C}$ be an $A_{\infty}$-category. Then any deformation
$\mathcal{C}_{q}$ has a minimal model.

###### Proof.

Pick any (classical) minimal model $\mathcal{D}$ for $\mathcal{C}$. Then
$\mathcal{D}$ is a minimal category and there is an $A_{\infty}$-quasi-
isomorphism $F:\mathcal{C}\to\mathcal{D}$. Now pick the push-forward

$\mu_{\mathcal{D}_{q}}\coloneqq
F_{*}(\mu_{\mathcal{C}_{q}})\in\overline{\operatorname{MC}}(\operatorname{HC}(\mathcal{D}),B).$

This Maurer-Cartan element defines a deformation $\mathcal{D}_{q}$ of
$\mathcal{D}$. This satisfies statement 4 of Lemma 5.3. ∎

###  5.3 Derived categories

In this section, we define derived categories of $A_{\infty}$-deformations.
The starting point is an $A_{\infty}$-category $\mathcal{C}$ and a deformation
$\mathcal{C}_{q}$. In section 3.3 we have already defined the twisted
completion $\operatorname{Tw}\mathcal{C}_{q}$. In section 5.2 we have
introduced the notion of minimal models for $A_{\infty}$-deformations. By
applying it to $\operatorname{Tw}\mathcal{C}_{q}$, we obtain the derived
category
$\operatorname{D}\mathcal{C}_{q}\coloneqq\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$.

######  Definition 5.3.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Then its derived category is the $A_{\infty}$-deformation
$\operatorname{D}\mathcal{C}_{q}\coloneqq\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$
of $\operatorname{H}\operatorname{Tw}\mathcal{C}$.

Derived categories are not unique. For instance, any deformation of
$\operatorname{H}\operatorname{Tw}\mathcal{C}$ which is gauge equivalent to a
given model $\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ is a derived
category for $\mathcal{C}$ as well.

######  Lemma 5.4.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Then the derived category
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ is defined uniquely up to
quasi-isomorphism.

###### Proof.

According to our definition, the twisted completion
$\operatorname{Tw}\mathcal{C}_{q}$ is defined uniquely. However, the minimal
model $\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ is not unique.
Instead, minimal models are only defined up to quasi-isomorphism. This
finishes the proof. ∎

######  Lemma 5.5.

Let $\mathcal{C},\mathcal{D}$ be $A_{\infty}$-categories and
$\mathcal{C}_{q},\mathcal{D}_{q}$ deformations. If $\mathcal{C}_{q}$ and
$\mathcal{D}_{q}$ are derived equivalent, then
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}$ are quasi-equivalent.

###### Proof.

By Definition 3.12, the deformed $A_{\infty}$-categories
$\operatorname{Tw}\mathcal{C}_{q}$ and $\operatorname{Tw}\mathcal{D}_{q}$ are
quasi-equivalent. By definition
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{Tw}\mathcal{C}_{q}$ are quasi-isomorphic. Similarly
$\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}$ and
$\operatorname{Tw}\mathcal{D}_{q}$ are quasi-isomorphic. By Lemma 4.1, quasi-
equivalence is an equivalence relation and hence
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$ and
$\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}$ are quasi-equivalent. ∎

The proof above is inconvenient in that it seems to give no clue about an
actual quasi-equivalence
$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}\to\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}$.
However we can construct such a functor by properly inverting and chaining the
given information. Indeed, the minimal models by definition come with quasi-
isomorphisms
$\operatorname{Tw}\mathcal{C}_{q}\to\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$
and
$\operatorname{Tw}\mathcal{D}_{q}\to\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}$.
As we see in the proof of Lemma 4.1, the quasi-isomorphism
$\operatorname{Tw}\mathcal{C}_{q}\to\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}$
has a kind of quasi-inverse. By chaining up all functors, we now obtain an
explicit quasi-equivalence

$\operatorname{H}\operatorname{Tw}\mathcal{C}_{q}\to\operatorname{Tw}\mathcal{C}_{q}\to\operatorname{Tw}\mathcal{D}_{q}\to\operatorname{H}\operatorname{Tw}\mathcal{D}_{q}.$

###  5.4 The theory of uncurving

In this section, we recollect uncurving theory for $A_{\infty}$-deformations.
The starting point is an arbitrary deformed $A_{\infty}$-category with
curvature. We are interested in the question which objects become curvature-
free once we apply a gauge functor to the category. In the present section, we
explain this question and define terminology. We prove elementary properties.
Uncurving has been studied in the literature, for instance under the name of
the “curvature problem” in [15].

It is our aim to explain in how far a gauge equivalence can change the
curvature of a deformation. As a starting point, let $\mathcal{C}$ be an
$A_{\infty}$ category. Let $\mathcal{C}_{q}$ and $\mathcal{C}_{q}^{\prime}$ be
two deformations of $\mathcal{C}$ connected by a gauge equivalence
$F:\mathcal{C}_{q}\to\mathcal{C}_{q}^{\prime}$. The curvature of
$\mathcal{C}_{q}$ and $\mathcal{C}_{q}^{\prime}$ are related by the zeroth
(curved) $A_{\infty}$-functor relation:

$F^{1}(\mu^{0}_{\mathcal{C}_{q}})=\mu^{0}_{\mathcal{C}_{q}^{\prime}}+\mu^{1}(F^{0})+\mu^{2}(F^{0},F^{0})+\ldots~{}.$

The map $F^{1}$ has leading term the identity and is then a linear isomorphism
by Lemma 3.2. This means that the curvature $\mu^{0}_{\mathcal{C}_{q}}$
depends only on $F^{0}$ and $F^{1}$, and not on the higher components $F^{\geq
2}$. If we approximate $F^{1}$ as the identity, we conclude that uncurving
essentially depends only on the choice of $F^{0}$. This gives rise to the
following definition:

######  Definition 5.4.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Let $X\in\mathcal{C}$. Then $X$ is uncurvable if there exists an
$S\in\mathfrak{m}\operatorname{End}^{1}(X)$ such that

$\mu^{0}_{X}+\mu^{1}_{\mathcal{C}_{q}}(S)+\mu^{2}_{\mathcal{C}_{q}}(S,S)+\ldots=0.$

We will now prove several basic properties regarding uncurvable objects.

######  Lemma 5.6.

Let $F:\mathcal{C}_{q}\to\mathcal{D}_{q}$ be a functor of deformed
$A_{\infty}$-categories. If $X\in\mathcal{C}_{q}$ is uncurvable, then so is
$F(X)$.

###### Proof.

Let $S\in\mathfrak{m}\operatorname{End}^{1}(X)$ be the uncurving morphism,
that is

$\mu^{0}_{X}+\mu^{1}(S)+\mu^{2}(S,S)+\ldots=0.$

Now set

$T:=F^{0}_{X}+F^{1}(S)+F^{2}(S,S)+\ldots\in\mathfrak{m}\operatorname{End}^{1}(F(X)).$

We claim that $T$ is an uncurving morphism for $F(X)$. Indeed,

$\displaystyle\mu^{0}_{F(X)}+\mu^{1}(T)+\mu^{2}(T,T)+\ldots$
$\displaystyle=\big{(}\mu^{0}_{F(X)}+\mu^{1}(F^{0}_{X})+\mu^{2}(F^{0}_{X},F^{0}_{X})+\ldots\big{)}$
$\displaystyle+\big{(}\mu^{1}(F^{1}(S))+\mu^{2}(F^{1}(S),F^{0}_{X})+\mu^{2}(F^{0}_{X},F^{1}(S))+\ldots\big{)}$
$\displaystyle+\big{(}\mu^{1}(F^{2}(S,S))+\mu^{2}(F^{1}(S),F^{1}(S))+\mu^{2}(F^{2}(S,S),F^{0}_{X})+\ldots\big{)}+\ldots$

We apply the curved $A_{\infty}$ rule to these terms and continue

$\displaystyle=F^{1}\big{(}\mu^{0}_{X}+\mu^{1}(S)+\mu^{2}(S,S)+\ldots\big{)}+F^{2}\big{(}\mu^{0}_{X}+\mu^{1}(S)+\mu^{2}(S,S)+\ldots,S\big{)}$
$\displaystyle\quad+F^{2}\big{(}S,\mu^{0}_{X}+\mu^{1}(S)+\mu^{2}(S,S)+\ldots\big{)}+\ldots$
$\displaystyle=F^{1}(0)+F^{2}(0,S)+F^{2}(S,0)+F^{3}(0,S,S)+\ldots=0.$

This shows that $T$ is an uncurving morphism for $F(X)$. ∎

######  Lemma 5.7.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Assume $\mathcal{C}$ is minimal, $X\cong Y$ in $\mathcal{C}$ and
$X$ is uncurvable. Then so is $Y$.

###### Proof.

Regard the embedding

$\\{Y\\}_{q}\to\\{X,Y\\}_{q},$

where both sides are defined as full subcategories of $\mathcal{C}_{q}$. The
embedding is an equivalence, since it reduces to the inclusion
$\\{Y\\}\to\\{X,Y\\}$ which is essentially surjective in cohomology. By Lemma
4.1, there is a quasi-equivalence in opposite direction

$\\{X,Y\\}_{q}\to\\{Y\\}_{q}.$

In particular, it maps $X$ to $Y$. Since $X$ is uncurvable, an application of
Lemma 5.6 shows that $Y$ is uncurvable. ∎

The next lemma concerns uncurving of objects in minimal models. As a
preparation let us recall that a minimal model
$\operatorname{H}\mathcal{C}_{q}$ comes with a choice of quasi-isomorphism
$F_{q}:\mathcal{C}_{q}\to\operatorname{H}\mathcal{C}_{q}$. Correspondingly,
the objects of $\mathcal{C}_{q}$ and $\operatorname{H}\mathcal{C}_{q}$ are
identified via $F_{q}$.

######  Lemma 5.8.

Let $\mathcal{C}$ be an $A_{\infty}$-category and $\mathcal{C}_{q}$ a
deformation. Then an object $X$ is uncurvable in $\mathcal{C}_{q}$ if and only
if it is uncurvable in $\operatorname{H}\mathcal{C}_{q}$.

###### Proof.

Let $F_{q}:\mathcal{C}_{q}\to\operatorname{H}\mathcal{C}_{q}$ be a quasi-
isomorphism and let $X\in\mathcal{C}_{q}$. We need to show that $X$ is
uncurvable if and only if $F_{q}(X)$ is uncurvable. By Lemma 5.6, $F_{q}(X)$
is clearly uncurvable if $X$ is uncurvable. For the other direction, regard
the restriction $F_{q}|_{\\{X\\}}:\\{X\\}_{q}\to\\{F_{q}(X)\\}_{q}$. It is a
quasi-isomorphism and by Lemma 4.1 there exists a quasi-isomorphism
$\\{F_{q}(X)\\}_{q}\to\\{X\\}_{q}$ in opposite direction. Now if $F_{q}(X)$ is