# Enriched Koszul duality for dg categories

J. Holstein  Department of Mathematics
Universität Hamburg
20146 Hamburg
Germany<EMAIL_ADDRESS>and  A. Lazarev  Department of
Mathematics and Statistics
Lancaster University
Lancaster LA1 4YF
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

It is well-known that the category of small dg categories $\mathsf{dgCat}$,
though it is monoidal, does not form a monoidal model category. In this paper
we construct a monoidal model structure on the category of pointed curved
coalgebras ${\mathsf{ptdCoa}^{*}}$ and show that the Quillen equivalence
relating it to $\mathsf{dgCat}$ is monoidal. We also show that
$\mathsf{dgCat}$ is a ${\mathsf{ptdCoa}^{*}}$-enriched model category. As a
consequence, the homotopy category of $\mathsf{dgCat}$ is closed monoidal and
is equivalent as a closed monoidal category to the homotopy category of
${\mathsf{ptdCoa}^{*}}$. In particular, this gives a conceptual construction
of a derived internal hom in $\mathsf{dgCat}$. As an application we obtain a
new description of simplicial mapping spaces in $\mathsf{dgCat}$ and a
calculation of their homotopy groups in terms of Hochschild cohomology groups,
reproducing and slightly generalizing well-known results of Toën. Comparing
our approach to Toën’s, we also obtain a description of the core of Lurie’s dg
nerve in terms of the ordinary nerve of a discrete category.

###### Key words and phrases:

DG categories, coalgebras, monoidal categories, model categories, enriched
categories, bar-construction, cobar-construction

###### 2020 Mathematics Subject Classification:

18N40, 18D20, 18M70

This work was partially supported by EPSRC grant EP/T029455/1

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Definitions, notation and conventions
    3. 1.3 Recalling categorical Koszul duality
  2. 2 Quivers, bicomodules and resolutions
  3. 3 Convolution
  4. 4 Closed monoidal structure on coalgebras
  5. 5 A monoidal model category
  6. 6 Coalgebras and dg categories
  7. 7 Hochschild cohomology

##  1\. Introduction

The category $\mathsf{dgCat}$ of small differential graded (dg) categories has
an internal homotopy theory, underpinned by a Quillen model structure
constructed by Tabuada [18] whose weak equivalences are quasi-equivalences.
There is also a closed monoidal structure on $\mathsf{dgCat}$, however since
the tensor product of two dg categories is not a Quillen bifunctor, e.g. the
tensor product of two cofibrant dg categories is not itself cofibrant, the two
structures are not compatible. Thus, the internal hom in $\mathsf{dgCat}$ does
not determine an internal hom in the homotopy category $Ho(\mathsf{dgCat})$,
even though the tensor product does lift to $Ho(\mathsf{dgCat})$.
Nevertheless, Toën showed in [20] that $Ho(\mathsf{dgCat})$ does have a closed
monoidal structure, cf. also [5] for an alternative approach. The resulting
derived internal hom is constructed using bimodules of special kind (‘quasi-
representable functors’). Borrowing the terminology of algebraic geometry,
this result shows that all functors between dg-categories are of Fourier-Mukai
type, cf. [4] regarding this point of view. For the Morita model structure on
$\mathsf{dgCat}$ Tabuada gave a construction of internal homs by considering a
category of localizing pairs [19].

Looking at this issue from a different angle, Kontsevich suggested that the
category of unital $A_{\infty}$-functors between two dg categories can be
taken as the derived internal hom between them. A complete proof of this
statement was obtained only recently, in [3].

The purpose of this paper is to provide another, more structured version of
the derived hom for dg categories. Our starting point is categorical Koszul
duality developed in [10] and which states, roughly, that the category of
(small) dg categories is Quillen equivalent to the category of coalgebras of a
special kind. At the same time it is known that the category of coalgebras
(unlike that of algebras) does possess an internal hom making it a closed
monoidal category (this point of view and its various ramifications are
explained in [1]). One can hope, therefore, that the internal hom in
coalgebras is homotopically better behaved than that in dg categories, and
admits a straightforward lift to homotopy categories. Provided that Koszul
duality is compatible with monoidal structures on coalgebras, this would give
a derived internal hom for dg categories.

The programme thus outlined is carried out in the present paper. More
precisely, it establishes a closed monoidal structure on the category
${\mathsf{ptdCoa}^{*}}$ of pointed curved coalgebras and shows that this
structure is compatible with the model structure on ${\mathsf{ptdCoa}^{*}}$
and the Koszul adjunction to dg categories. Kontsevich’s characterization of
the derived internal hom in $\mathsf{dgCat}$ is an immediate consequence of
these results.

Note that there is a close analogy between dg-categories and
$\infty$-categories: just like in dg-categories, simplicial categories do not
possess a well-behaved internal hom, and this problem is resolved by replacing
simplicial categories by simplicial sets with the Joyal model structure which
does possess a monoidal model structure and thus, a well-behaved internal hom.
In fact, this is more than an analogy; there exists a direct relationship
between the differential graded and simplicial pictures, it is explained in
Remark 5.8 below.

Recall that a coalgebra $C$ is pointed if its coradical is a direct sum of
copies of the ground field $\mathbf{k}$ that we fix throughout the paper. A
pointed curved coalgebra is a curved coalgebra that is pointed and has a
splitting of the coradical satisfying some compatibilities which we recall
below, see Definition 1.3. We denote by ${\mathsf{ptdCoa}^{*}}$ the category
of pointed curved coalgebras equipped with a final object.

According to [10] there is a natural model structure on
${\mathsf{ptdCoa}^{*}}$ and a Quillen equivalence
$\Omega:{\mathsf{ptdCoa}^{*}}\rightleftarrows\mathsf{dgCat}^{\prime}:\mathrm{B}$
between pointed curved coalgebras and dg categories (with the Dwyer-Kan model
structure). Here $\mathsf{dgCat}^{\prime}$ is an equivalent model for
$\mathsf{dgCat}$ that will be defined below. $\Omega$ and $\mathrm{B}$ are
suitable versions of the cobar and bar construction.

In this setting we prove the following results:

######  Theorem 5.4.

There is a closed monoidal model structure on ${\mathsf{ptdCoa}^{*}}$.

Moreover the cobar construction is quasi-strong monoidal in the sense that it
induces a strong monoidal functor on homotopy categories (Lemma 5.1), and it
induces an equivalence of monoidal category $Ho({\mathsf{ptdCoa}^{*}})\cong
Ho(\mathsf{dgCat})$ (Corollary 6.1).

The compatibility between ${\mathsf{ptdCoa}^{*}}$ and $\mathsf{dgCat}$ goes
beyond the homotopy categories. While $\mathsf{dgCat}$ is not a monoidal model
category, it is enriched, tensored and cotensored (powered) over
${\mathsf{ptdCoa}^{*}}$ and this structure is compatible with the model
structures of $\mathsf{dgCat}^{\prime}$ and ${\mathsf{ptdCoa}^{*}}$. We say
for short $\mathsf{dgCat}$ is a ${\mathsf{ptdCoa}^{*}}$-enriched model
category.

######  Theorem 6.2.

The category $\mathsf{dgCat}^{\prime}$ is a ${\mathsf{ptdCoa}^{*}}$-enriched
model category.

Our results have immediate consequences for describing internal homs in
$Ho(\mathsf{dgCat})$.

######  Corollary 6.5.

The internal hom in $Ho(\mathsf{dgCat})$ may be computed as (a small
modification of) the Maurer-Cartan category of a convolution category
$R\operatorname{\underline{Hom}}(D,D^{\prime})\simeq\overline{\mathsf{MC}}\\{BD,D^{\prime}\\}$.

Unravelling definitions, this internal hom is identified with the category of
unital $A_{\infty}$-functors as proposed by Kontsevich.

We also obtain a new description of mapping spaces in $\mathsf{dgCat}$, that
differs from the classical description in [20].

######  Theorem 6.7.

Given two dg categories $D,D^{\prime}$ the mapping space
$\operatorname{Map}(D,D^{\prime})$ is weakly equivalent to the core of
$\operatorname{N_{dg}}R\operatorname{\underline{Hom}}(D,D^{\prime})$ where
$\operatorname{N_{dg}}$ denotes Lurie’s dg nerve.

Combining this with Toen’s characterization of the mapping space, we obtain a
new description of the core of the dg nerve $\operatorname{N_{dg}}$ in terms
of the _ordinary_ nerve of the category of weak equivalences (Corollary 6.9
below).

By defining a Morita model structure on ${\mathsf{ptdCoa}^{*}}$ we may apply
our techniques also to the Morita model structure on $\mathsf{dgCat}$ and
compute internal homs similarly.

As an application, we compute the homotopy groups of simplicial mapping spaces
between dg categories in terms of Hochschild cohomology, reproducing and
slightly generalizing the well-known result of Toën [20].

######  Theorem 7.4.

Let $G:D\to D^{\prime}$ be a functor of dg categories. We then have
$\mathsf{HH}^{0}(D,D^{\prime})^{\times}\cong\pi_{1}(\operatorname{Map}(D,D^{\prime}),G)$
and
$\mathsf{HH}^{i}(D,D^{\prime})\cong\pi_{1-i}(\operatorname{Map}(D,D^{\prime}),G)$
for $i<0$.

Finally, we mention that analogous results hold for ordinary dg Koszul duality
(i.e. for the Quillen equivalence between augmented dg algebras and
conilpotent dg coalgebras). In this context, the monoidal structure on
coalgebras is given by a smash-product (as opposed to a tensor product) and it
lacks a monoidal unit, which brings about specific subtleties. This theory is
being developed in [9].

###  1.1. Outline

We first recall some concepts and definitions as well as the main result of
[10] in the remainder of this section. Then Section 2 provides some auxiliary
results on resolving dg categories by free dg categories and pointed curved
coalgebras by cofree ones. In Section 3 we construct the convolution category
structure on homs from a pointed curved coalgebra to a dg category, see
Definition 3.2. This is the main ingredient in constructing the MC category
$\overline{\mathsf{MC}}\\{C,D\\}$ (Definition 4.2) and the closed monoidal
structure on pointed curved coalgebras in Section 4. The compatibility with
the model structure on ${\mathsf{ptdCoa}^{*}}$ is provided in Section 5.
Section 6 then compares the monoidal structures on ${\mathsf{ptdCoa}^{*}}$ and
$\mathsf{dgCat}^{\prime}$ and shows that dg categories form a
${\mathsf{ptdCoa}^{*}}$-enriched model category. We show the same is true for
the Morita model structure on $\mathsf{dgCat}^{\prime}$ if we adjust the model
structure on ${\mathsf{ptdCoa}^{*}}$. We conclude with applications to
Hochschild cohomology in Section 7.

###  1.2. Definitions, notation and conventions

We refer to [10] for a more detailed overview of the background material
described below. The symbol $\mathbf{k}$ stands for a fixed ground field;
$\mathbf{k}$-linear Hom sets and Hom complexes will be denoted by
$\operatorname{Hom}$ while internal hom objects will be denoted by
$\operatorname{\underline{Hom}}$. The modifier ‘$\mathbf{k}$-linear’ will
usually be omitted later on as no other ground rings or fields will be
considered (e.g. an ‘algebra’ will stand for a ‘$\mathbf{k}$-algebra’ etc.)
Mapping spaces in model categories will be denoted as
$\operatorname{Map}(-,-)$.

Let $\mathsf{dgCat}$ be the category of _differential graded (dg) categories_
, considered as a model category with its Dwyer-Kan(-Bergner-Tabuada) model
structure where weak equivalences are given by quasi-equivalences [18]. (When
considering the Morita model structure on $\mathsf{dgCat}$ at the end of
Section 6 we will make this explicit.)

We will mainly consider $\mathsf{dgCat}^{\prime}$, the full subcategory of
$\mathsf{dgCat}$ consisting of dg categories without zero objects together
with the dg category $\mathbb{0}$ with one object and a single zero morphism.
The reason for this minor modification is that $\mathsf{dgCat}^{\prime}$ fits
better with categorical Koszul duality of [10] than $\mathsf{dgCat}$. The
category $\mathsf{dgCat}^{\prime}$ inherits the Dwyer-Kan model structure and
the associated $\infty$-categories of $\mathsf{dgCat}$ and
$\mathsf{dgCat}^{\prime}$ are equivalent.

There is a natural right Quillen functor from $\mathsf{dgCat}$ to
$\infty$-categories. We write $\mathsf{qCat}$ for simplicial sets with the
Joyal model structure, then the _dg nerve_
$\operatorname{N_{dg}}:\mathsf{dgCat}\to\mathsf{qCat}$ is constructed [15,
Section 1.3.1]; it was further analyzed in [17, Section 4] and [10, Section
4]. It may be restricted to $\mathsf{dgCat}^{\prime}$.

The monoidal structure on $\mathsf{dgCat}$ may easily be extended to
$\mathsf{dgCat}^{\prime}$ with one adjustment: We need to define
$\mathbb{0}\otimes D$ to be $\mathbb{0}$ for all dg categories $D$ (the
standard definition would be a dg category with the same objects as $D$ but
only zero morphisms and this is not an object of $\mathsf{dgCat}^{\prime}$).
The incluson $\mathsf{dgCat}^{\prime}\to\mathsf{dgCat}$ is lax monoidal and
induces a strong monoidal functor on homotopy categories. We call such
functors _quasi-strong monoidal_.

A _curved category_ is a graded category equipped with a degree 1 map $d$ on
all morphism spaces satisfying the Leibniz rule, and a degree 2 curvature
endomorphism $h_{X}$ for each object $X$ such that for $f:X\to Y$ we have
$d^{2}f=h_{Y}f-fh_{X}$.

Let $D$ be a dg category with the set of objects $\operatorname{Ob}D$ and
consider the coalgebra $D_{0}:=\mathbf{k}[\operatorname{Ob}D]$ spanned by
grouplike elements, one for every object in $D$. Then $D$ can be viewed it as
a monoid in bicomodules over the cosemisimple coalgebra $D_{0}$. This is an
example of a _semialgebra_.

In particular the space of morphisms between two objects $d,d^{\prime}$ is
given by the cotensor product
$\mathbf{k}_{d}\boxempty_{D_{0}}D\boxempty_{D_{0}}\mathbf{k}_{d^{\prime}}$
where $\mathbf{k}_{d}$ is the 1-dimensional comodule whose coaction map is
induced by the inclusion of $d$ into $\operatorname{Ob}D$.

Let $C$ be a coalgebra. We denote by $C_{0}$ is its coradical, i.e. its
maximal cosemisimple subcoalgebra and set $\overline{C}:=C/C_{0}$. We say $C$
is a _pointed coalgebra_ if $C_{0}$ is a direct sum of copies of the ground
field. Such coalgebras are also called cocomplete augmented cocategories. We
then denote the set of grouplike elements by $\operatorname{Ob}C$ so that
$C_{0}\cong\mathbf{k}[\operatorname{Ob}C]$.

A pointed coalgebra $C$ is _split_ if it is equipped with a section
$\epsilon:C\to C_{0}$ of the inclusion $C_{0}\to C$.

We will now consider curved coalgebras.

As in [10] many statements become easier when moving from the category of
coalgebras to the opposite category of _pseudocompact algebras_ by taking
continuous duals. Recall that a pseudocompact algebra is a topological algebra
that is the projective limit of (discrete) finite-dimensional algebras.

Recall the following definition, analogous to [16, Section 3.1].

######  Definition 1.1.

A _curved pseudocompact algebra_ $A=(A,d,h)$ is a graded pseudocompact algebra
supplied with a derivation $d:A\to A$ (a differential) of degree 1 and an
element $h\in A^{2}$ called the _curvature_ of $A$, such that $d^{2}(x)=[h,x]$
and $d(h)=0$ for any $x\in A$.

A _curved morphism_ between two curved pseudocompact algebras $A\to B$ is a
pair $(f,b)$ where $f:A\to B$ is a map of graded algebras of degree zero and
$b\in B^{1}$ so that:

  1. (1)

$f(d_{A}x)=d_{B}f(x)+[b,f(x)]$;

  2. (2)

$f(h_{A})=h_{B}+d_{B}(b)+b^{2}$.

Two such morphisms $(f,b)$ and $(g,c)$ are composed as
$(g,c)\circ(f,b)=(g\circ f,c+g(b))$. In particular, every map $(f,b)$ can be
decomposed $(f,b)=(\operatorname{id},b)\circ(f,0)$.

######  Definition 1.2.

A _curved coalgebra_ is a coalgebra $C$ equipped with an odd coderivation $d$
and a homogeneous linear function $h:C\to\mathbf{k}$ of degree 2, called the
_curvature_ , such that its dual $(C^{*},d^{*},h^{*})$ is a curved
pseudocompact algebra.

A morphism of curved dg coalgebras from $(C,d_{C},h_{C})$ to $(D,d_{D},h_{D})$
is given by the data $(f,a)$ where $f:C\to D$ is a morphism of graded
coalgebras and $a:C\to\mathbf{k}$ is a linear map of degree 1, such that
$(f^{*},a^{*})$ is a curved morphisms $D^{*}\to C^{*}$. The composition rule
is $(g,b)\circ(f,a)=(g\circ f,b\circ f+a)$.

######  Definition 1.3.

A _pointed curved coalgebra_ is a tuple
$(C,\Delta_{\mathbf{k}},\epsilon_{\mathbf{k}},d,h_{\mathbf{k}},\epsilon_{C})$
such that

  * •

$(C,\Delta_{\mathbf{k}},\epsilon_{\mathbf{k}},d,h_{\mathbf{k}})$ is a curved
coalgebra (over $\mathbf{k}$)

  * •

the restriction of $d$ to the coradical $C_{0}\hookrightarrow C$ is zero,

  * •

$\epsilon_{C}:C\to C_{0}$ is a a coalgebra map compatible with the
differential $d$, which is left inverse to $i:C_{0}\hookrightarrow C$.

We will often write simply $C$ or $(C,\epsilon_{C})$ for
$(C,\Delta_{\mathbf{k}},\epsilon_{\mathbf{k}},d,h_{\mathbf{k}},\epsilon_{C})$
when it does not cause confusion.

The map $\epsilon$ together with the comultiplication induce the structure of
a $C_{0}$-bicomodule on $C$. It follows from coassociativity that
$\Delta_{\mathbf{k}}$ factors as $C\xrightarrow{\Delta}C\boxempty_{C_{0}}C\to
C\otimes C$. Thus, $\Delta:C\to C\boxempty_{C_{0}}C$ and $\epsilon$ exhibit
$C$ as a comonoid in $C_{0}$-bicomodules. The inclusion $C_{0}\hookrightarrow
C$ provides a coaugmentation of this comonoid. The differential $d$ is
compatible with the $C_{0}$-bicomodule structure and the comonoid structure
given by $\Delta$ and $\epsilon$. Note also that there is automatically a
curvature $h$ with values in $C_{0}$, obtained by factorizing the curvature
$h_{\mathbf{k}}:C\to\mathbf{k}$ as
$\epsilon_{\mathbf{k}}\circ(h_{\mathbf{k}}\otimes\operatorname{id}_{C_{0}})\circ\rho_{C}:C\to
C\otimes C_{0}\to\mathbf{k}\otimes C_{0}\to\mathbf{k}$, where $\rho_{C}$ is
the right coaction. We define $h$ as
$(h_{\mathbf{k}}\otimes\operatorname{id}_{C_{0}})\circ\rho_{C}$.

We note that in particular the zero vector space $0$ with coradical $0$ and
all defining maps equal to the zero map is a pointed curved coalgebra which we
will also denote by $0$.

######  Definition 1.4.

A morphism $(f,a):(C,\epsilon)\to(D,\delta)$ of pointed curved coalgebras
consists of

  * •

a morphism $(f,a_{\mathbf{k}})$ of curved coalgebras

  * •

a factorization of $a_{\mathbf{k}}$ as the composition
$C\xrightarrow{a}D_{0}\to\mathbf{k}$

such that

  * •

$\delta\circ f=f\circ\epsilon$,

  * •

$f$ and $a$ are $D_{0}$-bicomodule maps,

where the $D_{0}$-bicomodule structure on $C$ is induced by the map
$f:C_{0}\to D_{0}$ on coradicals induced by $f$ (this is a slight abuse of
notation, but it will be clear from context what the domain of $f$ is).

The composition is then defined as

$(g,b)\circ(f,a)=(g\circ f,b\circ f+g\circ a).$

The category of pointed curved coalgebras will be denoted by
${\mathsf{ptdCoa}}$. The same category together with a final object $*$ will
be denoted by ${\mathsf{ptdCoa}^{*}}$.

Given a pair of grouplike elements $x,y$ in a $C\in{\mathsf{ptdCoa}^{*}}$ we
write $C(x,y)$ for the complex
$\mathbf{k}_{x}\boxempty_{C_{0}}C\boxempty_{C_{0}}\mathbf{k}_{y}$, analogous
to the hom space in a category.

###  1.3. Recalling categorical Koszul duality

In [10] we proved that ${\mathsf{ptdCoa}^{*}}$ and $\mathsf{dgCat}^{\prime}$
are Quillen equivalent using the bar and cobar constructions.

Given $C\in{\mathsf{ptdCoa}^{*}}$ the cobar construction $\Omega C$ is defined
to have underlying graded category given by the tensor monoid
$T_{C_{0}}\bar{C}[-1]$ in $C_{0}$-bicomodules and differential induced by
differential, comultiplication and curvature.

Similarly, given a dg category $D$ viewed as a monoid in $D_{0}$-bicomodules
the bar construction is defined as the tensor coalgebra $T_{D_{0}}\bar{D}[1]$
where $D\cong D_{0}\oplus\bar{D}$ is a choice of splitting of $D$ as a
bicomodule. The differential and curvature are induced by differential and
composition in $D$. Note that the splitting is not canonical and this is a
source of substantial technical difficulties (also present in the one-object
situation), for the details we refer to Section 3 of [10].

The adjunction is naturally written as

$\operatorname{Hom}(C,\mathrm{B}D)\cong\mathsf{MC}(\\{\overline{C},D\\})\cong\operatorname{Hom}(\Omega
C,D)$

where the middle term is the set of Maurer-Cartan (MC) elements in the reduced
convolution category $\\{\overline{C},D\\}$ which will be discussed in detail
below, see Definitions 3.2 and 4.2.

We note the following special cases. Let $\mathbb{0}$ denote the dg category
with one object and only the zero morphism and $\mathbb{\emptyset}$ the empty
dg category. Recall furthermore the inital object $0$ and the final object $*$
in ${\mathsf{ptdCoa}^{*}}$. Then $\mathrm{B}\mathbb{\emptyset}=0$ and $\Omega
0=\mathbb{\emptyset}$ by unravelling the definitions. Furthermore we define
$\mathrm{B}\mathbb{0}=*$ and $\Omega*=\mathbb{0}$.

##  2\. Quivers, bicomodules and resolutions

We recall the category of _graded $\mathbf{k}$-quivers_ following Keller [12].
A graded $\mathbf{k}$-quiver $V$ consists of a set of vertices or _objects_
$\operatorname{Ob}V$ and a graded $\mathbf{k}$-module $V(x,y)$ of _arrows_ for
every pair of objects $x,y$.

A morphism $V\to W$ consists of a map on objects
$f:\operatorname{Ob}V\to\operatorname{Ob}W$ and for each pair
$x,y\in\operatorname{Ob}V$ a morphism of arrows $V(x,y)\to W(fx,fy)$.

The category of graded quivers will be denoted by ${\mathsf{grQuiv}}$.

One sees that the data of a graded quiver is equivalent to a pair $(N,C)$
where $C$ is a coalgebra of the form $\oplus\mathbf{k}$ and $N$ is a graded
bicomodule $N$ over it. One defines the correspondence by
$N=\oplus_{x,y}V(x,y)$ and $C=\mathbf{k}[\operatorname{Ob}V]$.

######  Lemma 2.1.

There is a closed monoidal structure on ${\mathsf{grQuiv}}$, with
$\operatorname{Ob}(V\otimes W)=\operatorname{Ob}(V)\times\operatorname{Ob}(W)$
and $(V\otimes W)(x,y)=V(x)\otimes V(y)$. The internal hom is given by
$\operatorname{Ob}\operatorname{\underline{Hom}}(V,W)=\operatorname{Hom}(\operatorname{Ob}V,\operatorname{Ob}W)$
and
$\operatorname{Hom}(f,g)=\oplus_{x,y\in\operatorname{Ob}V}\operatorname{Hom}(V(x,y),W(fx,gy))$.

###### Proof.

Immediate from unravelling definitions. ∎

We also consider the category of _augmented graded $\mathbf{k}$-quivers_
${\mathsf{grQuiv}}^{\mathrm{aug}}$ whose objects are graded quivers $V$
together with a factorization
$\mathbf{k}[\operatorname{Ob}V]\xrightarrow{\eta}V\xrightarrow{\epsilon}\mathbf{k}[\operatorname{Ob}V]$
of the identity on $\mathbf{k}[\operatorname{Ob}V]$, which is defined as the
quiver with $V(x,y)=\mathbf{k}\delta_{x,y}$. Morphisms of augmented quivers
are morphisms of quivers compatible with the augmentation maps $\eta$ and
$\epsilon$.

######  Remark 2.2.

The tensor product defined as above of two augmented quivers is naturally
augmented and ${\mathsf{grQuiv}}^{\mathrm{aug}}$ is a closed monoidal
category. The internal hom is somewhat delicate, it is described in detail in
Section Section 5.1 of [12].

We now show that aribtrary objects in ${\mathsf{ptdCoa}^{*}}$ can be resolved
by cofree coalgebras, and in fact by bar constructions of dg categories. This
will be important later.

Let $\mathsf{ptdgrCo}$ denote the category of _pointed graded coalgebras_ ,
i.e. graded coalgebras $C$ with coradical $C_{0}$ of the form
$\oplus\mathbf{k}$ that are coaugmented comonoids over $C_{0}$.

######  Lemma 2.3.

There is a forgetful-cofree adjunction
$U:\mathsf{ptdgrCo}\rightleftarrows{\mathsf{grQuiv}}^{\mathrm{aug}}:G$, where
the cofree functor sends a quiver $Q$ to the tensor coalgebra over
$Q_{0}=\mathbf{k}[\operatorname{Ob}Q]$.

###### Proof.

For any fixed pair of objects $C,Q$ we have the cofree-forgetful adjunction of
coalgebras in $Q_{0}$-bicomodules, thus we find
$\operatorname{Hom}(UC,Q)=\oplus_{f:C_{0}\to
Q_{0}}\operatorname{Hom}_{Q_{0}}(f_{*}UC,Q)$ but as $f$ commutes with $U$ this
is

$\oplus_{f:C_{0}\to
Q_{0}}\operatorname{Hom}_{Q_{0}}(Uf_{*}C,Q)\cong\oplus_{f:C_{0}\to
Q_{0}}\operatorname{Hom}_{Q_{0}}(f_{*}C,T_{Q_{0}}Q)\cong\operatorname{Hom}(C,GQ)$

and this bijection is natural. ∎

Let ${\mathsf{ptdCoa}^{\mathrm{str}}}$ denote the category of strict pointed
curved coalgebras, with the same objects as ${\mathsf{ptdCoa}^{*}}$ but
morphisms only the strict, i.e. uncurved, morphisms
$f:(C,d,h)\to(C^{\prime},d^{\prime},h^{\prime})$ given by $f:C\to C^{\prime}$
compatible with differential and curvature.

######  Lemma 2.4.

There is a comonadic adjunction
$V:{\mathsf{ptdCoa}^{\mathrm{str}}}\rightleftarrows\mathsf{ptdgrCo}:H$ with
left adjoint $V$ given by the functor forgetting differential and curvature.

###### Proof.

For simplicity we formulate the proof for pseudocompact algebras, which is
contravariantly equivalent by dualizing. Thus we claim that there is a monadic
adjunction
$H^{*}:\mathsf{ptdgrpcAlg}\rightleftarrows\mathsf{ptdcupcAlg^{\textrm{str}}}:V^{*}$
with the right adjoint $V^{*}$ forgetting differential and curvature.

Given a pointed graded pseudocompact algebra $A$ we define $H^{*}A$ to be
freely generated by an element $h$ in degree $2$ as well as differentials $da$
in degree $|a|+1$ for every $a\in A$ satisfying the rules for derivations and
compatibility with the bimodule structure. Then $d^{2}(a)\coloneqq[h,a]$ and
this is easily seen to be a left adjoint of $V^{*}$.

It is clear that $V^{*}$ reflects isomorphisms, thus the adjunction is monadic
by the Barr-Beck theorem if we can show $V^{*}$ preserves certain
coequalizers. In fact, $V^{*}$ preserves all coequalizers as coequalizers in
the strict curved category are given by coequalizers in the graded category
equipped with the induced differential and curvature.

Dualizing we see that $V\dashv H$ is comonadic. ∎

######  Lemma 2.5.

Any pointed curved coalgebra $C$ is the equalizer of a diagram of cofree
pointed curved coalgebras, i.e. pointed curved coalgebras of the form $HGV$
for a graded augmented quiver $V$.

###### Proof.

We consider $C$ as an object in ${\mathsf{ptdCoa}^{\mathrm{str}}}$. From the
composition of the comonadic adjunctions
${\mathsf{ptdCoa}^{\mathrm{str}}}\rightleftarrows\mathsf{ptdgrCo}$ from Lemma
2.4 and $\mathsf{ptdgrCo}\rightleftarrows{\mathsf{grQuiv}}$ from Lemma 2.3 we
obtain a comonad $K=HGUV$ on ${\mathsf{ptdCoa}^{\mathrm{str}}}$ and
$C=\operatorname{eq}(KC\rightrightarrows KKC)$.

Note that a composition of comonadic functors is not necessarily comonadic,
but it is if the first functor ($V$ in our case) satisfies the crude
monadicity theorem (which is true as $V$ preserves all equalizers), see
Theorem 3.5.1 in [2]. We conclude by noting that this diagram is also an
equalizer diagram in ${\mathsf{ptdCoa}^{*}}$, which follows from the
construction of equalizers in curved coalgebras (or equivalently coequalizers
in curved pseudocompact algebras) in the proof of Lemma 3.30 in [10]. ∎

######  Lemma 2.6.

Any cofree coalgebra arises as the bar construction of a dg category.

###### Proof.

Let $V$ be a graded augmented quiver and write
$V_{0}=\mathbf{k}[\operatorname{Ob}V]$ for the quiver given by a copy of the
ground field at every object and $\bar{V}=V/V_{0}$ where
$\operatorname{Ob}\bar{V}=\operatorname{Ob}V$ and the quotient is taken for
each pair of objcts.

Then we define a differential graded quiver
$V^{\prime}=V_{0}\oplus\bar{V}[-1]\oplus\bar{V}\oplus V_{0}[1]$ with
differential given by the identity on $V_{0}[1]$ and $\bar{V}$ respectively.
We note that there is a map of quivers $V_{0}\to V^{\prime}$, but the natural
projection is not compatible with the differential, so this is not an
augmentation.

Then define the dg category $D$ by setting all composition zero except that
each $V_{0}(x,x)\cong\mathbf{k}$ is the unit at
$x\in\operatorname{Ob}V^{\prime}$.

It now follows from the definitions that $\mathrm{B}D\cong HGV$. ∎

######  Corollary 2.7.

Every object of ${\mathsf{ptdCoa}^{*}}$ is of the form
$\operatorname{eq}(\mathrm{B}D_{1}\rightrightarrows\mathrm{B}D_{2})$ for
suitable dg categories $D_{1}$ and $D_{2}$.

###### Proof.

For a pointed curved coalgebra this is immediate from combining Lemma 2.5 and
Lemma 2.6. The final object $*$ is by definition the bar construction of the
dg category with one object and only the zero morphism. ∎

A similar result holds for dg categories:

######  Lemma 2.8.

Any small dg category $D$ is of the form $\operatorname{coeq}(\Omega
C_{1}\rightrightarrows\Omega C_{2})$ for pointed curved coalgebras
$C_{1},C_{2}$.

###### Proof.

We again construct a composition of monadic adjunctions. Forgetting the
differential provides an adjunction
$U:\mathsf{dgCat}\rightleftarrows\mathsf{grCat}:D$, which satisfies the crude
monadicity theorem, as $U$ preserves all coequalizers. (Coequalizers of
differential graded categories can be computed on the underlying graded
category and then equipped with a suitable differential.)

There is also a monadic adjunction from graded categories to graded reflexive
quivers (quivers $V$ equipped with a unit $\mathbf{k}[\operatorname{Ob}V]\to
V$), see [22].

Together these give a monad $T$ on $\mathsf{dgCat}$ and thus any dg category
is the coequalizer of free dg categories.

It remains to observe that any free dg category on a reflexive quiver $V$ is
the cobar construction of a coalgebra $C$ obtained by equipping $V$ with the
zero comultiplication on $V/V_{0}$. ∎

##  3\. Convolution

We now introduce a convolution structure on the maps between a pointed curved
coalgebra and a dg category. We will need the construction for not necessarily
counital coalgebras, so let us define a _noncounital pointed graded coalgebra_
to be a pair $(C_{0},C)$ where $C_{0}$ is a coalgebra of the form
$\oplus\mathbf{k}$ and $C$ is a graded $C_{0}$ bicomodule which has a
coassociative comultiplication $C\to C\boxempty_{C_{0}}C$, which is not
necessarily counital.

######  Definition 3.1.

A _noncounital pointed curved coalgebra_ is a noncounital pointed graded
coalgebra that moreover has a differential $d$ and a curvature $h:C\to C_{0}$
such that the square of the differential is given by the coaction of the
curvature.

In particular any pointed curved coalgebra $C$ is noncounital pointed curved
coalgebra by forgetting the counit, and the quotient $C/C_{0}$ is also a
noncounital pointed curved coalgebra.

Given a dg category $D$ and a possibly noncounital pointed curved coalgebra
$C$ we will now construct a convolution category $\\{C,D\\}$ as follows. Note
that unless $C$ is counital this is a non-unital category, i.e. a category
without units.

The set of objects of $\\{C,D\\}$ is given by
$\operatorname{Hom}(C_{0},D_{0})\cong\operatorname{Hom}_{\mathbf{Set}}(\operatorname{Ob}C,\operatorname{Ob}D)$.
Given two objects $f,g$ in $\operatorname{Hom}(D_{0},C_{0})$ we make $C$ into
a $D_{0}$-bicomodule $(f,g)_{*}C$ via the composition

$C\to C_{0}\otimes C\otimes
C_{0}\xrightarrow{f\otimes\operatorname{id}_{C}\otimes g}D_{0}\otimes C\otimes
D_{0}.$

Note that for $(f,f)_{*}C$ we will write $f_{*}C$.

Then we define

$\operatorname{Hom}_{\\{C,D\\}}(f,g)=\operatorname{Hom}_{D_{0}}((f,g)_{*}{C},D).$

We note this agrees with the internal hom of quivers from Lemma 2.1.

To define the composition note that for fixed $f,h$ the comultiplication
$\Delta:C\to C\boxempty_{C_{0}}C$ induces a map
$(f,h)_{*}C\to(f,g)_{*}C\boxempty_{D_{0}}(g,h)_{*}C$ of $D_{0}$-bicomodules
for each $g$, put differently
$(f,h)_{*}(C\boxempty_{C_{0}}C)\subset(f,g)_{*}C\boxempty_{D_{0}}(g,h)_{*}C$
(by definition of the cotensor product).

The composition
$\operatorname{Hom}_{\\{C,D\\}}(f,g)\otimes\operatorname{Hom}_{\\{C,D\\}}(g,h)\to\operatorname{Hom}_{\\{C,D\\}}(f,h)$,
i.e.
$\operatorname{Hom}_{D_{0}}((f,g)_{*}C,D)\otimes\operatorname{Hom}_{D_{0}}((g,h)_{*}C,D)\to\operatorname{Hom}_{D_{0}}((f,h)_{*}C,D)$
is then defined by convolution:

$\phi\circ\psi:C\xrightarrow{\Delta}C\boxempty_{C_{0}}C\hookrightarrow
C\boxempty_{D_{0}}C\xrightarrow{\phi\otimes\psi}D\boxempty_{D_{0}}D\xrightarrow{\mu}D$

or, spelling out the $D_{0}$-bicomodule structures:

$(f,h)_{*}C\xrightarrow{\Delta}(f,h)_{*}(C\boxempty_{C_{0}}C)\hookrightarrow(f,g)_{*}C\boxempty_{D_{0}}(g,h)_{*}C\xrightarrow{\phi\otimes\psi}D\boxempty_{D_{0}}D\xrightarrow{\mu}D$

The composition is associative as $\Delta$ and $\mu$ are (co)associative. Thus
$\\{C,D\\}$ is a (nonunital) monoid in
$\mathbf{k}[\operatorname{Hom}(C_{0},D_{0})]$-bicomodules.

The differential on $C$ and $D$ induces a natural differential on $\\{C,D\\}$.
Let $C$ have curvature $h_{C}:C\to C_{0}$ and let $\eta:D_{0}\to D$ be the
unit map of $D$. Then at an object $f\mapsto\eta\circ f\circ
h_{C}:\operatorname{Hom}(C_{0},D_{0})\to\operatorname{Hom}(C,D)$ defines a
curvature on the convolution category, compatible with the differential.

######  Definition 3.2.

Given a dg category $D$ and a possibly noncounital pointed curved coalgebra
$C$ the _convolution category_ $\\{C,D\\}$ is the possibly non-unital curved
category of maps from $C$ to $D$, with objects
$\operatorname{Hom}(C_{0},D_{0})$, morphisms
$\operatorname{Hom}_{\\{C,D\\}}(f,g)=\operatorname{Hom}_{D_{0}}((f,g)_{*}{C},D)$
and composition given by the convolution product as defined above.

If $C$ has a counit $\epsilon:C\to C_{0}$ and $D$ has a unit $D_{0}\to D$ then
every object $f$ has a unit $\eta\circ f\circ\epsilon$ and $\\{C,D\\}$ is a
(unital) curved category.

We note that we may also define a curved convolution monoid if $D$ is a curved
category as long as $C$ is counital, then the curvature will aquire an
additional summand $f\mapsto h_{D}\circ f\circ\epsilon_{C}$.

######  Lemma 3.3.

Let $C,C^{\prime}$ be (possibly noncounital) pointed graded coalgebras and $D$
a graded category. Then there is an isomorphism of (possibly nonunital) graded
convolution categories $\\{C,\\{C^{\prime},D\\}\\}\cong\\{C\otimes
C^{\prime},D\\}$.

###### Proof.

The statement is true on the underlying bicomodules by Lemma 2.1. Next we
compare the convolution products, which essentially follows by naturality of
the coproduct: If $f^{\\#},g^{\\#}:C\to\operatorname{Hom}(C^{\prime},D)$ are
adjoint to $f,g:C\otimes C^{\prime}\to D$ we apply the two compositions to
$c\otimes c^{\prime}$. We obtain
$f^{\\#}(c^{(1)})(c^{\prime(1)}).g^{\\#}(c^{(2)})(c^{\prime(2)})$ and
$f(c^{(1)}\otimes c^{\prime(1)}).g(c^{(2)}\otimes c^{\prime(2)})$ in Sweedler
notation, which of course agree. ∎

The curved version of the lemma needs an extra assumption as
$\\{C^{\prime},D\\}$ will be curved in general.

######  Corollary 3.4.

Let $C,C^{\prime}$ be (possibly noncounital) pointed curved coalgebras and $D$
a curved category. Then there is an isomorphism of (possibly nonunital) curved
convolution categories $\\{C,\\{C^{\prime},D\\}\\}\cong\\{C\otimes
C^{\prime},D\\}$ in either of the following three cases:

  * •

$C$ is counital, $D$ has no curvature,

  * •

$C^{\prime}$ and $D$ have no curvature,

  * •

$C$ and $C^{\prime}$ are both counital.

###### Proof.

The conditions of the corollary are exactly those needed to define the
curvature on both sides. It is then straightforward to check that the induced
differential and curvature are compatible with the adjunction and agree on
both sides. ∎

##  4\. Closed monoidal structure on coalgebras

In order to define a closed monoidal structure on coalgebras we need to
introduce the Maurer-Cartan (MC) category of a curved category.

######  Definition 4.1.

Given a curved category $D$ we construct its _dg MC category_
$\mathsf{MC_{dg}}(D)$ whose objects are pairs $(X,\xi)$ with $X$ an object of
$D$ and $\xi$ an MC element in $\operatorname{End}(X)$, i.e $\xi$ satisfies
$d\xi+\xi^{2}+h_{X}=0$ where $h_{X}$ is the curvature of $D$ at $X$.

The hom spaces are given by the complex of morphisms between twisted elements:

$\operatorname{Hom}_{MC}((X,\xi),(X^{\prime},\xi^{\prime}))=\operatorname{Hom}_{D}(X,X^{\prime})^{[\xi,\xi^{\prime}]}\coloneqq(\operatorname{Hom}_{D}(X,X^{\prime}),d^{[\xi,\xi^{\prime}]})$

where $d^{[\xi,\xi^{\prime}]}(f)=df+\xi^{\prime}f-(-1)^{|f|}f\xi$.

We also write $\operatorname{End}_{D}(X)^{\xi}$ for
$\operatorname{Hom}_{D}(X)^{[\xi,\xi]}$.

Note that even if $D$ is curved $\mathsf{MC_{dg}}(D)$ is naturally a dg
category. We denote by $\mathsf{MC}(D)$ the set of objects of
$\mathsf{MC_{dg}}(D)$.

######  Definition 4.2.

Given $C\in{\mathsf{ptdCoa}^{*}}$ and $D\in\mathsf{dgCat}$ we define the
Maurer-Cartan category $\overline{\mathsf{MC}}\\{C,D\\}$ to be the full
subcategory of $\mathsf{MC_{dg}}(\\{C,D\\})$ whose objects are the objects of
the nonunital dg category $\mathsf{MC_{dg}}(\\{\overline{C},D\\})$.

Thus, an object of $\overline{\mathsf{MC}}\\{C,D\\}$ consists of a map
$f:\operatorname{Ob}C\to\operatorname{Ob}D$ together with a Maurer-Cartan
element in
$\xi\in\operatorname{End}_{\mathsf{MC_{dg}}(\\{\overline{C},D\\})}(f)=\operatorname{Hom}_{D_{0}}(f_{*}\overline{C},D)$.

######  Remark 4.3.

Given a pointed curved coalgebra $C$ and a dg category $D$ definition 4.2
categorifies the set of MC elements in maps from $\overline{C}$ to $D$ which
mediates the Koszul adjunction $\Omega\dashv\mathrm{B}$. This leads naturally
to consider the reduced convolution category $\\{\overline{C},D\\}$, however
as this is a non-unital category we need to consider morphisms coming from the
larger (unital) category $\\{C,D\\}$.

The category of pointed curved coalgebras admits a symmetric monoidal
structure given by the ordinary tensor product. Indeed, if $C,C^{\prime}$ are
objects in ${\mathsf{ptdCoa}^{*}}$ with curvature functions $h:C\to C_{0}$ and
$h^{\prime}:C^{\prime}\to C^{\prime}_{0}$ and counits $\epsilon:C\to C_{0}$
and $\epsilon^{\prime}:C^{\prime}\to C^{\prime}_{0}$ then $C\otimes
C^{\prime}$ is likewise pointed curved with the coradical
$\mathbf{k}[\operatorname{Ob}C]\otimes\mathbf{k}[\operatorname{Ob}C^{\prime}]$
and curvature $h\otimes\epsilon^{\prime}+\epsilon\otimes h^{\prime}:C\otimes
C^{\prime}\to C_{0}\otimes C_{0}^{\prime}$. Furthermore we define
$C\otimes*=*\otimes C\coloneqq*$.

We are now ready for our first main result.

######  Theorem 4.4.

The tensor product defines a closed monoidal structure on
${\mathsf{ptdCoa}^{*}}$ where the internal hom
$\operatorname{\underline{Hom}}(C,C^{\prime})$ is defined as
$\operatorname{\underline{Hom}}(C,\mathrm{B}D)=\mathrm{B}\overline{\mathsf{MC}}\\{C,D\\}$
whenever $C^{\prime}=\mathrm{B}D$. We also define
$\operatorname{\underline{Hom}}(C,*)=*$ and
$\operatorname{\underline{Hom}}(*,C)=0$ if $C\neq*$.

Note that from this definition
$\operatorname{\underline{Hom}}(C,0)=\mathrm{B}\overline{\mathsf{MC}}\\{C,\emptyset\\}=\mathrm{B}\emptyset=0$
and
$\operatorname{\underline{Hom}}(0,\mathrm{B}D)=\mathrm{B}\overline{\mathsf{MC}}\\{0,D\\}=\mathrm{B}\mathbb{0}=*$.
From the latter it follows (see the proof of Theorem 5.4 below) that
$\operatorname{\underline{Hom}}(0,C)=*$ for all $C$.

######  Remark 4.5.

It is easy to see that the tensor product makes ${\mathsf{ptdCoa}^{*}}$ into a
monoidal category and one may check that $\otimes$ commutes with all colimits.
As ${\mathsf{ptdCoa}^{*}}$ is locally presentable, the existence of an
internal hom as a right adjoint to the bifunctor $\otimes$ then follows. The
explicit description that we give is more complicated but turns out to be very
fruitful.

We prove some lemmas before turning to the proof of Theorem 4.4.

######  Lemma 4.6.

Given $C,C^{\prime}\in{\mathsf{ptdCoa}^{*}}$ and $D\in\mathsf{dgCat}$ there is
a decomposition

$\mathsf{MC}(\\{\overline{C\otimes
C^{\prime}},D\\})\cong\coprod_{f\in\operatorname{Hom}(C_{0}\otimes
C^{\prime}_{0},D)}\coprod_{\phi\in\mathsf{MC}(\operatorname{End}_{\\{C_{0}\otimes\overline{C^{\prime}},D\\}}(f))}\mathsf{MC}(\operatorname{End}_{\\{\overline{C}\otimes
C^{\prime},D\\}}(f)^{\phi})$

where $(f,\phi)$ is an object in
$\mathsf{MC}(\\{C_{0}\otimes\overline{C^{\prime}},D\\})$ and the twist by
$\phi$ is induced by the natural action of
$\\{C_{0}\otimes\overline{C^{\prime}},D\\}$ on $\\{\overline{C}\otimes
C^{\prime},D\\}$.

###### Proof.

The decomposition $\overline{C\otimes C^{\prime}}\cong
C_{0}\otimes\overline{C^{\prime}}\oplus\overline{C}\otimes C^{\prime}$ induces
a bicomodule decomposition of $\\{\overline{C\otimes C^{\prime}},D\\}$ and the
product decomposes as
$(\phi,\psi).(\phi^{\prime},\psi^{\prime})=(\phi.\phi^{\prime},\phi.\psi^{\prime}+\psi.\phi^{\prime}+\psi.\psi^{\prime})$,
i.e. $\\{C_{0}\otimes\overline{C^{\prime}},D\\}$ is a subsemialgebra and
$\\{\overline{C}\otimes C^{\prime},D\\}$ is an ideal of the convolution
category. In particular the second summand has an action by the first.

Thus an MC element in $\\{\overline{C\otimes C^{\prime}},D\\}$ is given by an
object $f\in\operatorname{Hom}(C_{0}\otimes C_{0}^{\prime},D_{0})$ together
with MC elements
$\phi\in\operatorname{End}_{\\{C_{0}\otimes\overline{C^{\prime}},D\\}}(f)$ and
$\psi\in\operatorname{End}_{\\{\overline{C}\otimes
C^{\prime},D\\}}(f)^{\phi}$. ∎

The following lemma is the heart of constructing the closed monoidal
structure. The proof looks quite complicated, but it consists mostly in
unravelling notations and book-keeping in order to reduce the result to the
usual tensor hom adjunction.

######  Lemma 4.7.

Given $C,C^{\prime}\in{\mathsf{ptdCoa}^{*}}$ and $D\in\mathsf{dgCat}$ there is
a natural isomorphism in ${\mathsf{ptdCoa}^{*}}$:

(4.1) $\operatorname{Hom}(C\otimes
C^{\prime},\mathrm{B}D)\cong\operatorname{Hom}(C,\operatorname{\underline{Hom}}(C^{\prime},\mathrm{B}D))$

###### Proof.

To make the argument clearer we first consider the case where $C,C^{\prime}$
and $D$ have one object. In this case we may discard the object
$f:\operatorname{Ob}C\times\operatorname{Ob}C^{\prime}\to\operatorname{Ob}D$
in our considerations.

By the Koszul adjunction the left hand side of (4.1) is the set of MC elements
in the convolution algebra $\\{\overline{C\otimes C^{\prime}},D\\}$. There is
a decomposition of vector spaces $\\{\overline{C\otimes
C^{\prime}},D\\}\cong\\{\overline{C}\otimes
C^{\prime},D\\}\times\\{\overline{\mathbf{k}\otimes C^{\prime}},D\\}$ with
$\\{\overline{\mathbf{k}\otimes C^{\prime}},D\\}$ a subalgebra. From Lemma 4.6
we obtain

$\displaystyle\mathsf{MC}(\\{\overline{C\otimes C^{\prime}},D\\})$
$\displaystyle\cong$
$\displaystyle\coprod_{\phi\in\mathsf{MC}(\\{\mathbf{k}\otimes\overline{C^{\prime}},D\\})}\mathsf{MC}(\\{\overline{C}\otimes
C^{\prime},D\\}^{\phi})$ $\displaystyle\cong$
$\displaystyle\coprod_{\phi\in\mathsf{MC}(\\{\mathbf{k}\otimes\overline{C^{\prime}},D\\})}\mathsf{MC}(\\{\overline{C},\\{C^{\prime},D\\}^{\phi^{\prime}}\\})$

where the second line follows from Lemma 3.4. To be precise, Lemma 3.4 only
identifies the underlying graded algebras. It remains to compare the
differential and curvature on both sides. Here $\phi^{\prime}$ is defined as
the image of $\phi$ in $\\{C^{\prime},D\\}$ under the map induced by
$C^{\prime}\to\overline{C^{\prime}}$. Then $\phi^{\prime}$ is MC and thus the
twisted convolution algebra $\\{C^{\prime},D\\}^{\phi^{\prime}}$ is a dg
algebra and $\\{\overline{C},\\{C^{\prime},D\\}^{\phi^{\prime}}\\}$ is a well-
defined curved algebra. Unravelling definitions matches up differentials and
curvatures in the two different convolution algebras and thus the MC elements
agree.

Thus we write an element of the left hand side of (4.1) as a pair
$(\phi,\psi)$ with
$\phi\in\mathsf{MC}(\\{\mathbf{k}\otimes\overline{C^{\prime}},D\\})$ and
$\psi\in\mathsf{MC}(\\{\overline{C},\\{C^{\prime},D\\}^{\phi^{\prime}})$.

On the right hand side of (4.1) we have MC elements in
$\\{\overline{C},\overline{\mathsf{MC}}\\{C^{\prime},D\\}\\}$, i.e. pairs
$(X,\eta)$ where
$X\in\operatorname{Ob}\\{\overline{C},\overline{\mathsf{MC}}\\{C^{\prime},D\\})\\}$
and $\eta\in\operatorname{End}(X)$ is MC. Thus $X$ is an MC element of
$\\{\overline{C^{\prime}},D\\}$, equivalent to $\phi$ on the LHS. The MC
element $\eta$ lives in
$\operatorname{End}(X)\cong\operatorname{Hom}(\overline{C},\operatorname{\underline{Hom}}(C^{\prime},D)^{X^{\prime}})$,
writing $X^{\prime}$ for the image of $X$. Thus after identifying $X$ and
$\phi$ we may identify $\eta$ with $\psi$.

We consider arbitrary objects next. Again we decompose the data of an element
of the left hand side of (4.1) first. An object of
$\operatorname{\underline{Hom}}(\overline{C\otimes C^{\prime}},BD)$ is an MC
element of $\\{\overline{C\otimes C^{\prime}},D\\}$, so it consists of an
object $f$, given by a map
$f:\operatorname{Ob}C\times\operatorname{Ob}C^{\prime}\to\operatorname{Ob}D$,
and a MC element in the endomorphism algebra of $f$, i.e.
$\Phi\in\operatorname{Hom}_{D_{0}}(f_{*}\overline{C\otimes C^{\prime}},D)$
satisfying $d\Phi+\Phi^{2}+h=0$ where $h$ is the curvature induced by the
curvatures of $C$ and $C^{\prime}$.

As above we use Lemma 4.6 to identify $\Phi$ with a pair

$\phi\in\mathsf{MC}(\operatorname{Hom}_{D_{0}}(f_{*}(C_{0}\otimes\overline{C^{\prime}}),D),\quad\psi\in\mathsf{MC}(\operatorname{Hom}_{D_{0}}(f_{*}({\overline{C}}\otimes
C^{\prime}),D)^{\phi})$

and consider $\phi$ and $\psi$ separately. Here we spelled out the
endomorphisms of $f$ as bicomodule maps,
$\operatorname{End}_{\\{C_{0}\otimes\overline{C^{\prime}},D\\}}(f)\coloneqq\operatorname{Hom}_{D_{0}}(f_{*}(C_{0}\otimes\overline{C^{\prime}}),D)$.

Using the definition of comodule maps we find:

$\displaystyle\phi$ $\displaystyle\in$
$\displaystyle\operatorname{Hom}_{D_{0}}(f_{*}(C_{0}\otimes\overline{C^{\prime}}),D)$
$\displaystyle\cong$
$\displaystyle\bigoplus_{d_{1},d_{2}\in\operatorname{Ob}D}\bigoplus_{\begin{subarray}{c}c_{1},c_{1}^{\prime}\\\
f(c_{1},c^{\prime}_{1})=d_{1}\end{subarray}}\bigoplus_{\begin{subarray}{c}c_{2},c^{\prime}_{2}\\\
f(c_{2},c_{2}^{\prime})=d_{2}\end{subarray}}\operatorname{Hom}(C_{0}(c_{1},c_{2})\otimes\overline{C^{\prime}}(c_{1}^{\prime},c_{2}^{\prime}),D(d_{1},d_{2}))$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c\in\operatorname{Ob}C}\bigoplus_{d_{1},d_{2}\in\operatorname{Ob}D}\bigoplus_{\begin{subarray}{c}c_{1}^{\prime},c_{2}^{\prime}\in\operatorname{Ob}C^{\prime}\\\
f(c,c^{\prime}_{1})=d_{1},f(c,c_{2}^{\prime})=d_{2}\end{subarray}}\operatorname{Hom}(\overline{C^{\prime}}(c_{1}^{\prime},c_{2}^{\prime}),D(d_{1},d_{2}))$

where we used $C_{0}(c_{1},c_{2})=\mathbf{k}.\delta_{c_{1},c_{2}}$. The MC
condition says that each $\phi(c)$ is an MC element in
$\operatorname{Hom}_{D_{0}}(f^{\sharp}(c)_{*}\overline{C^{\prime}},D)$. We
also have

$\displaystyle\psi$ $\displaystyle\in$
$\displaystyle\operatorname{Hom}_{D_{0}}(f_{*}({\overline{C}}\otimes
C^{\prime}),D)^{\phi}$ $\displaystyle=$
$\displaystyle\bigoplus_{d_{1},d_{2}\in\operatorname{Ob}D}\bigoplus_{\begin{subarray}{c}c_{1},c_{2}\in\operatorname{Ob}C,\
c_{1}^{\prime},c_{2}^{\prime}\in\operatorname{Ob}C^{\prime}\\\
f(c_{1},c^{\prime}_{1})=d_{1},\
f(c_{2},c_{2}^{\prime})=d_{2}\end{subarray}}\operatorname{Hom}(\overline{C}(c_{1},c_{2})\otimes
C^{\prime}(c_{1}^{\prime},c_{2}^{\prime}),D(d_{1},d_{2}))^{\phi}$
$\displaystyle=$
$\displaystyle\bigoplus_{d_{1},d_{2}\in\operatorname{Ob}D}\bigoplus_{\begin{subarray}{c}c_{1},c_{2}\in\operatorname{Ob}C,\
c_{1}^{\prime},c_{2}^{\prime}\in\operatorname{Ob}C^{\prime}\\\
f(c_{1},c^{\prime}_{1})=d_{1},\
f(c_{2},c_{2}^{\prime})=d_{2}\end{subarray}}\operatorname{Hom}\left(\overline{C}(c_{1},c_{2}),\operatorname{Hom}(C^{\prime}(c_{1}^{\prime},c_{2}^{\prime}),D(d_{1},d_{2}))^{\phi(c_{1}),\phi(c_{2})}\right)$

where we used Lemmas 3.4 and 4.6 again to identify the last two lines as
graded algebras and then observe that differential and curvature also match
up.

On the right hand side of (4.1) we unravel similarly. A MC element of
$\operatorname{\underline{Hom}}(C,\mathrm{B}\overline{\mathsf{MC}}\\{C^{\prime},D\\}))$
is given by an MC element of the convolution category
$\\{C,\overline{\mathsf{MC}}\\{C^{\prime},D\\})\\}$, i.e. an object
$\Xi:\operatorname{Ob}C\to\operatorname{Ob}(\overline{\mathsf{MC}}\\{C^{\prime},D\\})$
with an MC element $\eta$ in
$\operatorname{End}(\Xi)=\operatorname{Hom}_{M_{0}}(\Xi_{*}C,\overline{\mathsf{MC}}\\{C^{\prime},D\\})$
where $M_{0}$ is the coalgebra
$\mathbf{k}[\overline{\mathsf{MC}}\\{C^{\prime},D\\}]$.

Thus $\Xi$ sends any $c\in\operatorname{Ob}C$ to a pair
$(f^{\sharp}(c),\xi(c))$ with
$f^{\sharp}(c):\operatorname{Ob}C^{\prime}\to\operatorname{Ob}D$ an object and
$\xi(c)$ an MC element

$\xi(c)\in\operatorname{End}_{\overline{\mathsf{MC}}\\{C^{\prime},D\\}}(f^{\sharp})\cong\operatorname{Hom}_{D_{0}}({f^{\sharp}}_{*}\overline{C^{\prime}},D)$

Here we write
$f^{\sharp}:\operatorname{Ob}C\to\operatorname{Hom}(\operatorname{Ob}C^{\prime},\operatorname{Ob}D)$
as it may be seen as the adjoint of
$f:\operatorname{Ob}C\times\operatorname{Ob}C^{\prime}\to\operatorname{Ob}D$
from the LHS.

Summing over $c$ we find that $\xi$ is equivalent to a MC element in

$\displaystyle\bigoplus_{c\in\operatorname{Ob}C}\operatorname{End}_{\\{\overline{C^{\prime}},D\\}}(f^{\sharp}(c))$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c\in\operatorname{Ob}C}\operatorname{Hom}_{D_{0}}((f^{\sharp})_{*}\overline{C^{\prime}},D)$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c\in\operatorname{Ob}C}\bigoplus_{d_{1},d_{2}\in\operatorname{Ob}D}\bigoplus_{\begin{subarray}{c}c_{1}^{\prime},c_{2}^{\prime}\in\operatorname{Ob}C\\\
f^{\sharp}(c)(c^{\prime}_{i})=d_{i}^{\prime}\end{subarray}}\operatorname{Hom}(\overline{C^{\prime}}(c^{\prime}_{1},c_{2}^{\prime}),D(d_{1},d_{2}))$

which is exactly the same curved algebra that $\phi$ lives in.

For the final step we need to match up $\eta$ with $\psi$. By definition
$\eta$ is a MC element in

$\displaystyle\operatorname{Hom}_{D_{0}}(\Xi_{*}\overline{C},\overline{\mathsf{MC}}\\{C^{\prime},D\\})$
$\displaystyle\cong$
$\displaystyle\bigoplus_{\zeta_{1},\zeta_{2}\in\operatorname{Ob}\overline{\mathsf{MC}}\\{C^{\prime},D\\}}\bigoplus_{\begin{subarray}{c}c_{1},c_{2}\in\operatorname{Ob}C\\\
\Xi(c_{i})=\zeta_{i}\end{subarray}}\operatorname{Hom}(\overline{C}(c_{1},c_{2}),\overline{\mathsf{MC}}\\{C^{\prime},D\\}(\zeta_{1},\zeta_{2}))$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c_{1},c_{2}\in\operatorname{Ob}C}\operatorname{Hom}(\overline{C}(c_{1},c_{2}),\operatorname{Hom}_{\\{C^{\prime},D\\}}(f^{\sharp}(c_{1}),f^{\sharp}(c_{2}))^{\xi(c_{1}),\xi(c_{2})})$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c_{1},c_{2}\in\operatorname{Ob}C}\operatorname{Hom}(\overline{C}(c_{1},c_{2}),\operatorname{Hom}_{D_{0}}((f^{\sharp}(c_{1}),f^{\sharp}(c_{2}))_{*}{C^{\prime}},D)^{\xi(c_{1}),\xi(c_{2})})$
$\displaystyle\cong$
$\displaystyle\bigoplus_{c_{1},c_{2}\in\operatorname{Ob}C}\bigoplus_{\begin{subarray}{c}d_{1},d_{2}\in\operatorname{Ob}D\\\
c_{1}^{\prime},c_{2}^{\prime}\in\operatorname{Ob}C^{\prime}\\\
f^{\sharp}(c_{i})(c_{i}^{\prime})=d_{i}\end{subarray}}\operatorname{Hom}\left(\overline{C}(c_{1},c_{2}),\operatorname{Hom}({C^{\prime}}(c_{1}^{\prime},c_{2}^{\prime}),D(d_{1},d_{2}))^{\xi(c_{1}),\xi(c_{2})}\right)$

where we used that the hom spaces in $\overline{\mathsf{MC}}\\{C,D\\}$ are
defined in terms of the nonreduced coalgebra $C^{\prime}$.

Matching $\phi$ with $\xi$ this shows that $\psi$ and $\eta$ are MC elements
in isomorphic curved algebras and the construction of the bijection is
complete.

It remains to consider the special case that $C$ is $*$. Then the left hand
side is $\operatorname{\underline{Hom}}(*,\mathrm{B}D)$ which is $0$ unless
$D=\emptyset$ in which case it equals $*$. The right hand side is
$\operatorname{\underline{Hom}}(*,\operatorname{\underline{Hom}}(C^{\prime},\mathrm{B}D))$,
which is $0$ unless $\operatorname{\underline{Hom}}(C^{\prime},\mathrm{B}D)$
is $*$, which by definition is only possible if $D=\emptyset$. A similar
argument applies if $C^{\prime}=*$. As all objects here are initial or final
naturality is immediate. ∎

###### Proof of Theorem 4.4.

We observe first that the construction in Lemma 4.7 is functorial on the full
subcategory whose objects are bar constructions of dg categories. This is
immediate for the first variable, for the second variable it follows from the
Yoneda embedding: it suffices to construct a map
$\operatorname{Hom}(C,\operatorname{\underline{Hom}}(C^{\prime},BD))\to\operatorname{Hom}(C,\operatorname{\underline{Hom}}(C^{\prime},BD^{\prime}))$
for any map $\mathrm{B}D\to BD^{\prime}$. But such a map is equivalent to
$\operatorname{Hom}(C\otimes C^{\prime},BD)\to\operatorname{Hom}(C\otimes
C^{\prime},BD^{\prime})$, induced by functoriality of the ordinary hom.

Thus the theorem follows from Lemma 4.7 as we are able to rewrite an arbitrary
pointed curved coalgebra in terms of bar constructions. Indeed, using
Corollary 2.7 we write $C^{\prime}=\lim_{i}{\mathrm{B}D_{i}}$ and define

$\operatorname{\underline{Hom}}(C,C^{\prime})\coloneqq\lim_{i}\operatorname{\underline{Hom}}(C,\mathrm{B}D_{i})\cong\lim_{i}(\mathrm{B}\overline{\mathsf{MC}}\\{C,D_{i}\\}).$

This satisfies the adjointness isomorphism for fixed $C^{\prime}$ and thus is
functorial in maps
$\lim_{i}\mathrm{B}D_{i}\to\lim_{i}\mathrm{B}D^{\prime}_{i}$ by the Yoneda
lemma again. Naturality of adjointness follows. ∎

######  Remark 4.8.

Note that the compatibility with limits also holds for
$\operatorname{\underline{Hom}}(*,-)$ as
$\operatorname{\underline{Hom}}(*,\lim
C_{i})=\lim\operatorname{\underline{Hom}}(*,C_{i})$ is $0$ unless all $C_{i}$
equal *, in which case we get $\operatorname{\underline{Hom}}(*,*)=*$.

######  Corollary 4.9.

For $C,C^{\prime},C^{\prime\prime}\in{\mathsf{ptdCoa}^{*}}$ we have an
isomorphism of pointed curved coalgebras
$\operatorname{\underline{Hom}}(C,\operatorname{Hom}(C^{\prime},C^{\prime\prime}))\cong\operatorname{\underline{Hom}}(C\otimes
C^{\prime},C^{\prime\prime})$.

###### Proof.

This follows from Theorem 4.4 using the Yoneda Lemma. ∎

######  Remark 4.10.

Our construction may be compared to the internal hom of (cocomplete augmented)
cocategories in terms of the internal hom of augmented quivers that is
considered by Keller, see Theorem 5.3 in [12] and the following discussion.
Note that [12] considers uncurved cocategories and uses them to express
functors of augmented $A_{\infty}$ categories. As usual in curved Koszul
duality the introduction of curved coalgebras allows us to remove the
augmentation.

##  5\. A monoidal model category

We next show that ${\mathsf{ptdCoa}^{*}}$ is in fact a monoidal model category
with its model structure defined in [10].

We will first compare the monoidal structure on ${\mathsf{ptdCoa}^{*}}$ and
$\mathsf{dgCat}^{\prime}$.

Let $C$ and $C^{\prime}$ be pointed curved coalgebras. Let
$\xi_{C}\in\overline{\mathsf{MC}}\\{C,\Omega C\\}$ and
$\xi_{C^{\prime}}\in\overline{\mathsf{MC}}\\{C^{\prime},\Omega{C^{\prime}}\\}$
be the canonical MC elements corresponding to the identity maps $\Omega
C\to\Omega C$ and $\Omega C^{\prime}\to\Omega C^{\prime}$. We take their
images in the curved categories $\\{C,\Omega C\\}$ and $\\{C^{\prime},\Omega
C^{\prime}\\}$ induced by $C\to\overline{C}$ and
$C^{\prime}\to\overline{C^{\prime}}$. Then the element $\xi_{C}\otimes
1+1\otimes\xi_{C^{\prime}}$ is an MC element in the curved category
$\\{C,\Omega C\\}\otimes\\{C^{\prime},\Omega C^{\prime}\\}$. Consider the
natural map of curved semialgebras

$m:\\{C,\Omega C\\}\otimes\\{C^{\prime},\Omega C^{\prime}\\}\to\\{{C\otimes
C^{\prime}},\Omega C\otimes\Omega C^{\prime}\\}\to\\{(\overline{C\otimes
C^{\prime}},\Omega C\otimes\Omega C^{\prime}\\}$

where the second map is induced by the inclusion of the kernel of the counit.
Then $m(\xi_{C}\otimes 1+1\otimes\xi_{C^{\prime}})$ is an MC element in
$\operatorname{Hom}(\overline{C\otimes C^{\prime}},\Omega C\otimes\Omega
C^{\prime})$ and thus by Koszul duality, it determines a map of dg categories

$M:\Omega(C\otimes C^{\prime})\to\Omega(C)\otimes\Omega(C^{\prime}).$

######  Lemma 5.1.

The map $M$ defined above is a quasi-equivalence of dg categories. Thus the
cobar construction is quasi-strong monoidal.

###### Proof.

Viewing all categories as semialgebras it suffices to prove that $M$ induces a
quasi-isomorphism.

Assume first that the pointed curved coalgebras $C,C^{\prime}$ have no
curvature. Consider the cosimplicial complexes
$T_{C_{0}}C=\\{C^{\boxempty_{C_{0}}i}\\}_{i=0}^{\infty}$ and
$T_{C^{\prime}_{0}}C^{\prime}=\\{C^{\prime\boxempty_{C^{\prime}_{0}}i}\\}_{i=0}^{\infty}$.
The cosimplicial structures comes from considering the standard cosimplicial
resolution $\\{C^{\boxempty_{C_{0}}i}\\}_{i=2}^{\infty}$ and cotensoring on
the left and on the right with $C_{0}$. Thus the coface map come from the
comultiplication on $C$ (respectively $C^{\prime}$)and the comodule coaction
$C_{0}\to C$. The codegeneracy maps are induced by the counit.

Then $\Omega C$ and $\Omega C^{\prime}$ are the totalizations of the
normalized cochain complexes of these cosimplicial complexes : $\Omega(C)\cong
N(T_{C_{0}}C)$, $\Omega(C^{\prime})\cong N(T_{C_{0}^{\prime}}C^{\prime})$ and
$\Omega(C\otimes C^{\prime})\cong N({T_{C_{0}\otimes C^{\prime}_{0}}C\otimes
C^{\prime}})$. Next we observe that $T_{C_{0}\otimes C^{\prime}_{0}}C\otimes
C^{\prime}\cong T_{C_{0}}C\otimes T_{C^{\prime}_{0}}C^{\prime}$. We then claim
that we can identify the map $M:\Omega(C\otimes
C^{\prime})\to\Omega(C)\otimes\Omega(C^{\prime})$ with the dual Eilenberg-
Zilber map. As such it induces a quasi-isomorphism on total complexes, proving
the lemma in this special case.

To prove this claim we recall the dual Eilenberg-Zilber map $EZ^{*}:N(A\otimes
B)\to N(A)\otimes N(B)$ that sends $a\otimes b\in A_{n}\otimes B_{n}$ to

$\sum_{p+q=n}\sum_{(\lambda,\mu)\in
Sh(p,q)}\epsilon(\lambda,\mu)\lambda^{*}(a)\otimes\mu^{*}(b)$

where we sum over all shuffles and $\epsilon(\lambda,\mu)$ denotes the sign.
In the case at hand $A=T_{C_{0}}C$ and $B=T_{C_{0}^{\prime}}C^{\prime}$ and we
can check that $EZ^{*}$ is an algebra map.

Indeed, unravelling definitions we just have to match up shuffles on $n$
objects with pairs of shuffles on $k$ and on $n-k$ objects (with the correct
signs), but this is a classical computation, see §17 in [8].

Since $M$ is an algebra map by construction and on generators $C\otimes
C^{\prime}$ the maps $EZ^{*}$ and $M$ are easily seen to agree (both send
$c\otimes e$ to $c\otimes 1+1\otimes e$) we have shown the claim.

We turn to the general case where $C,C^{\prime}$ may have curvature. We note
that the coradical filtration on any pointed curved coalgebra has the property
that its associated graded has no curvature. Thus, consider the coradical
filtrations on $C$, $C^{\prime}$ and $C\otimes C^{\prime}$ and the induced
filtrations on $\Omega C\otimes\Omega C^{\prime}$ and $\Omega(C\otimes
C^{\prime})$. The map $M:\Omega(C\otimes
C^{\prime})\to\Omega(C)\otimes\Omega(C^{\prime})$ is compatible with these
filtrations, and thus it induces a map on associated graded. These are nothing
but the cobar constructions of the associated graded coalgebras. By the
special case proved above, this induced map is a quasi-isomorphism and so, $F$
is a quasi-isomorphism to begin with.

As usual we need to consider the special case $*\in{\mathsf{ptdCoa}^{*}}$, but
both $*$ and $\Omega(*)=\mathbb{0}$ are absorbing for the respective tensor
product. ∎

######  Remark 5.2.

The argument above using the Eilenber-Zilber map (albeit in the dual setting
of bar-constructions of algebras over a field) goes back to Cartan and
Eilenberg, [6, Chapter XI, 6]

######  Corollary 5.3.

The tensor product with an arbitary pointed coalgebra preserves all weak
equivalences.

###### Proof.

Let $E,C,C^{\prime}$ be pointed curved coalgebras and $C\to C^{\prime}$ a weak
equivalence. Then we claim $E\otimes C\to E\otimes C^{\prime}$ is also a weak
equivalence. By definition we need to check $\Omega(E\otimes
C)\simeq\Omega(E\otimes C^{\prime})$, or, by Lemma 5.1,
$\Omega(E)\otimes\Omega(C)\simeq\Omega(E)\otimes\Omega(C^{\prime})$.

Thus the result follows if tensoring with a cofibrant dg category preserves
quasi-equivalences. But this is readily verified, both quasi-full faitfulness
and quasi-essential surjectivity are easy to check. ∎

######  Theorem 5.4.

The category ${\mathsf{ptdCoa}^{*}}$ is a monoidal model category.

###### Proof.

Recall the model structure on pointed curved coalgebras from [10]. The
cofibrations are given by injections and a map $f:C\to C^{\prime}$ is a weak
equivalence exactly if $\Omega(f)$ is a quasi-equivalence.

It remains to show that this is compatible with the closed monoidal structure
from Theorem 4.4. The unit axiom follows from Corollary 5.3 as tensor product
preserves all weak equivalences.

It remains to check the pushout-product axiom, i.e. let $E\to E^{\prime}$ and
$C\to C^{\prime}$ be cofibrations in ${\mathsf{ptdCoa}^{*}}$. We first check
that $(E\otimes C^{\prime})\amalg_{E\otimes C}(E^{\prime}\otimes C)\to
E^{\prime}\otimes C^{\prime}$ is a cofibration. Indeed, rewriting the pushout
as a coequalizer it follows from the description of coequalizers in Lemma 3.30
of [10] that the coequalizer of any two parallel arrows with a cone has
underlying space the coequalizer in graded augmented quivers. This suffices to
show that the canonical map to $E^{\prime}\otimes C^{\prime}$ is injective,
and thus a cofibration.

Next we assume that $C\to C^{\prime}$ is, moreover, a weak equivalence. By the
first part (or by inspection) $E\otimes C\to E\otimes C^{\prime}$ and
$E\otimes C\to E^{\prime}\otimes C$ are cofibrations. As $\Omega$ preserves
cofibrations we see that $\Omega(E\otimes C^{\prime})\amalg_{\Omega(E\otimes
C)}\Omega(E^{\prime}\otimes C)$ is in fact a homotopy colimit, and weakly
equivalent to the homotopy colimit $(\Omega C\otimes\Omega
E^{\prime})\amalg^{h}_{\Omega C\otimes\Omega E}(\Omega C^{\prime}\otimes\Omega
E)$. But the latter is weakly equivalent to $\Omega C\otimes\Omega E^{\prime}$
as $\Omega C\otimes\Omega E\to\Omega C^{\prime}\otimes\Omega E$ is a weak
equivalence. Thus the canonical map to $\Omega(C^{\prime}\otimes
E^{\prime})\simeq\Omega C^{\prime}\otimes\Omega E^{\prime}$ is also a weak
equivalence. ∎

######  Remark 5.5.

It is well-known that $\mathsf{dgCat}$ is not a monoidal model category. As
$\Omega$ is only quasi-monoidal there is no contradiction to
${\mathsf{ptdCoa}^{*}}$ being a monoidal model category.

In order to consider mapping spaces, we recall an adjunction between
simplicial sets with the Joyal model structure (which we denote
$\mathsf{qCat}$ for quasi-categories) and pointed curved coalgebras. Define
$F(C):=\operatorname{Hom}_{{\mathsf{ptdCoa}^{*}}}(\tilde{C}_{*}(\Delta^{\bullet}),C)$
where $C$ is a a pointed curved coalgebra, $K$ is a simplicial set and
$\tilde{C}(K)$ is the twisted chain coalgebra of $K$ (the detailed definition
of is found in Section 4 of [10]). Then there is a Quillen adjunction
$\tilde{C}_{*}:\mathsf{qCat}\leftrightarrows:{\mathsf{ptdCoa}}:F$.

To consider mapping spaces we will be interested in the ’maximal subgroupoid’
of an $\infty$-category.

######  Definition 5.6.

We call the maximal Kan subset of a quasi-category $K$ the _core_ of $K$.

It is clear that the core is right adjoint to the inclusion of Kan complexes
into quasi-categories (weak Kan complexes).

######  Corollary 5.7.

Let $C,C^{\prime}$ be pointed curved coalgebras. Then their mapping space
$\operatorname{Map}(C,C^{\prime})$ in the model category
${\mathsf{ptdCoa}^{*}}$ is weakly equivalent to the core of the
$\infty$-category $F\operatorname{\underline{Hom}}(C,C^{\prime})$.

###### Proof.

We define a cosimplicial simplicial set $E^{\bullet}$ by letting $E^{n}$ be
the nerve of the groupoid with object set $[n]$ and one arrow connecting each
pair of objects. By Section 4.1 of [7] this is a Reedy cosimplicial resolution
of the point in the Joyal model structure i.e. a cosimplicial resolution of
the trivial $\infty$-category.

We claim that tensoring with $\tilde{C}_{*}(E^{\bullet})$ gives a cosimplicial
resolution in ${\mathsf{ptdCoa}^{*}}$. Indeed as $\tilde{C}_{*}$ and the
tensor product are left Quillen, they preserve colimits (and thus latching
objects) and cofibrations, and thus Reedy cofibrant objects.

We compute

$\displaystyle\operatorname{Map}(C,C^{\prime})$ $\displaystyle\cong$
$\displaystyle\operatorname{Hom}_{\mathsf{ptdCoa}^{*}}(C\otimes\tilde{C}_{*}(E^{\bullet}),C^{\prime})$
$\displaystyle\cong$
$\displaystyle\operatorname{Hom}_{\mathsf{ptdCoa}^{*}}(\tilde{C}_{*}(E^{\bullet}),\operatorname{\underline{Hom}}(C,C^{\prime}))$
$\displaystyle\cong$
$\displaystyle\operatorname{Hom}_{\mathsf{sSet}}(E^{\bullet},F\operatorname{\underline{Hom}}(C,C^{\prime}))$

by the closed monoidal structure of ${\mathsf{ptdCoa}^{*}}$. But the
expression in the final line above is, by construction, the core of
$F\operatorname{\underline{Hom}}(C,C^{\prime})$. (We may also view it as
$\operatorname{Map}_{\mathsf{qCat}}(*,F\operatorname{\underline{Hom}}(C,C^{\prime}))$,
which is known to be the core.) ∎

######  Remark 5.8.

We recall from [10, Section 4.2] the relation of Koszul duality and the
coherent nerve $\operatorname{N_{coh}}$ construction sending simplicial
categories to quasi-categories. There is the following diagram, where the
inner and outer square commute up to homotopy.

$\textstyle{\mathsf{qCat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{C}}$$\scriptstyle{\widetilde{C}_{*}}$$\textstyle{\mathsf{sCat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{N_{coh}}}$$\scriptstyle{G_{*}}$$\textstyle{{\mathsf{ptdCoa}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega}$$\scriptstyle{F}$$\textstyle{\mathsf{dgCat}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{B}}$$\scriptstyle{H}$

Here the horizontal arrows are $\infty$-equivalences and downward arrows are
induced by normalized chain functors, the upwards maps are right adjoints (on
the level of homotopy categories). We also showed in [10] that the dg nerve
$\operatorname{N_{dg}}$, the natural functor from $\mathsf{dgCat}^{\prime}$ to
$\mathsf{qCat}$ which is equivalent to $\operatorname{N_{coh}}\circ H$,
factors through $\mathrm{B}$.

On the left hand side of the above diagram we have simplicial sets with the
Joyal model structure and pointed curved coalgebras, which both possesss
monoidal model structures. In contrast, simplicial categories or dg categories
on the right hand side do not have monoidal model structures.

Note that, moreover, $\tilde{C}_{*}$ has a lax monoidal structure given by the
Eilenberg-Zilber map together with the isomorphism $\tilde{C}_{*}(K)\cong
C_{*}(K)$ of curved coalgebras. Thus $\tilde{C}_{*}$ also has a lax closed
structure. This closed structure is far from being quasi-strong. Indeed,
restricting to the subcategory of spaces in $\mathsf{qCat}$ it is clear that
the map
$\tilde{C}_{*}\operatorname{Map}(X,Y)\to\operatorname{\underline{Hom}}(\tilde{C}_{*}X,\tilde{C}_{*}Y)$
does not become an isomorphism in the homotopy category:
$\operatorname{\underline{Hom}}(\tilde{C}_{*}X,\tilde{C}_{*}Y)$ ignores the
homotopy cocommutative nature of chain coalgebras and so cannot faithfully
reflect maps between the corresponding spaces.

##  6\. Coalgebras and dg categories

######  Corollary 6.1.

There is an equivalence of closed monoidal model categories between
$Ho({\mathsf{ptdCoa}^{*}})$ and $Ho(\mathsf{dgCat})$.

###### Proof.

There is an equivalence of categories $Ho({\mathsf{ptdCoa}^{*}})$ and
$Ho(\mathsf{dgCat})$ by [10]. This equivalence is strong monoidal by Lemma 5.1
and thus also closed by general principles. ∎

Note that in the statement of the corollary we could replace
$\mathsf{dgCat}^{\prime}$ by the usual $\mathsf{dgCat}$ as the two have
equivalent homotopy categories (and the natural inclusion
$\mathsf{dgCat}^{\prime}\to\mathsf{dgCat}$ is quasi-strong monoidal).

In fact, before passing to homotopy categories there is a very strong relation
between the monoidal model category ${\mathsf{ptdCoa}^{*}}$ and the model
category $\mathsf{dgCat}$.

Recall that given a model category $\mathcal{M}$ and a monoidal model category
$\mathcal{C}$ we say $\mathcal{M}$ is a _$\mathcal{C}$ -enriched model
category_ [14, Definition A.3.1.5] if $\mathcal{M}$ is enriched in
$\mathcal{C}$, tensored and cotensored over $\mathcal{C}$ and moreover the
tensor action $\tilde{\otimes}:\mathcal{C}\times\mathcal{M}\to\mathcal{M}$ is
a left Quillen bifunctor (which ensures that enrichment and cotensor are also
compatible with the model structures). This is a natural generalisation of the
notion of simplicial model category.

######  Theorem 6.2.

The category $\mathsf{dgCat}^{\prime}$ is a ${\mathsf{ptdCoa}^{*}}$-enriched
model category.

###### Proof.

We define the external hom of a coalgebra $C$ and dg category $D$ as
$\overline{\operatorname{Hom}}(\Omega
C,D)\coloneqq\mathrm{B}\overline{\mathsf{MC}}\\{C,D\\}$ and extend to all dg
categories by writing a dg category as a colimit of cobar constructions using
Lemma 2.8.

We then define the cotensoring $D^{C}\coloneqq\overline{\mathsf{MC}}\\{C,D\\}$
and the tensoring by $C\tilde{\otimes}\Omega
C^{\prime}\coloneqq\Omega(C\otimes C^{\prime})$ for dg categories in the image
of $\Omega$, extended by colimits to all dg categories.

In both cases functoriality follows from the Yoneda lemma as in the proof of
Theorem 4.4. E.g. for fixed $C,D$ any map $\Omega C^{\prime}\to\Omega
C^{\prime\prime}$ induces a map $\operatorname{Hom}(\Omega
C^{\prime\prime},\overline{\mathsf{MC}}\\{C,D\\})\to\operatorname{Hom}(\Omega
C^{\prime},\overline{\mathsf{MC}}\\{C,D\\})$’ and thus
$\operatorname{Hom}(\Omega(C^{\prime\prime}\otimes
C),D)\to\operatorname{Hom}(\Omega(C^{\prime}\otimes C),D)$.

To show compatibility for the cotensoring we need to check that
${\operatorname{\overline{Hom}}}(D^{\prime},D^{C})\cong\operatorname{\underline{Hom}}(C,\overline{\operatorname{Hom}}(D^{\prime},D))$.
Assume first $D^{\prime}=\Omega C^{\prime}$. We have

$\displaystyle\operatorname{\underline{Hom}}(C,\operatorname{\overline{Hom}}(\Omega
C^{\prime},D))$ $\displaystyle\cong$
$\displaystyle\operatorname{\underline{Hom}}(C,\mathrm{B}\overline{\mathsf{MC}}\\{C^{\prime},D\\})$
$\displaystyle\cong$
$\displaystyle\operatorname{\underline{Hom}}(C,\operatorname{\underline{Hom}}(C^{\prime},\mathrm{B}D))$
$\displaystyle\cong$ $\displaystyle\operatorname{\underline{Hom}}(C\otimes
C^{\prime},\mathrm{B}D)$ $\displaystyle\cong$
$\displaystyle\operatorname{\underline{Hom}}(C^{\prime},\operatorname{\underline{Hom}}(C,\mathrm{B}D))$
$\displaystyle\cong$ $\displaystyle\operatorname{\underline{Hom}}(\Omega
C^{\prime},\overline{\mathsf{MC}}\\{C,D\\})$

and the last term is $\operatorname{\overline{Hom}}(D,D^{C}))$ by definition.
The general case follows by applying Lemma 2.8 and observing that the
construction on both sides is compatible with colimits as in the proof of
Theorem 4.4.

The compatibility condition of the tensor product is
$\overline{\operatorname{Hom}}(C\tilde{\otimes}D^{\prime},D)\cong\operatorname{\underline{Hom}}(C,\overline{\operatorname{Hom}}(D^{\prime},D))$.
Again it suffices to show this if $D^{\prime}=\Omega C^{\prime}$. Then we have

$\displaystyle\operatorname{\overline{Hom}}(C\tilde{\otimes}\Omega C,D)$
$\displaystyle\cong$
$\displaystyle\operatorname{\overline{Hom}}(\Omega(C\otimes C^{\prime},D))$
$\displaystyle\cong$ $\displaystyle\operatorname{\underline{Hom}}(C\otimes
C^{\prime},\mathrm{B}D)$ $\displaystyle\cong$
$\displaystyle\operatorname{\underline{Hom}}(C,\operatorname{\underline{Hom}}(C^{\prime},\mathrm{B}D))$
$\displaystyle\cong$
$\displaystyle\operatorname{\underline{Hom}}(C,\operatorname{\overline{Hom}}(\Omega
C^{\prime},D))$

completing the proof of compatibility. To obtain the second line we used that
$\operatorname{\underline{Hom}}(C,\mathrm{B}D)\cong
B\overline{\mathsf{MC}}\\{C,D\\}\cong\operatorname{\overline{Hom}}(\Omega
C,D)$ as pointed curved coalgebras by definition.

It remains to show that the enrichment is compatible with the model
structures, i.e. that $\tilde{\otimes}$ is a left Quillen bifunctor,
associating to a pair of cofibrations $f:C\to C^{\prime}$ and $g:D\to
D^{\prime}$ a cofibration

$f\boxempty
g:(C\tilde{\otimes}D^{\prime})\amalg_{C\tilde{\otimes}D}(C^{\prime}\tilde{\otimes}D)\to
C^{\prime}\tilde{\otimes}D^{\prime}$

that is acyclic if $f$ or $g$ is.

We show first that $f\boxempty g$ is a cofibration. By [11, Corollary 4.25] it
suffices to check this only in the case that $f$ and $g$ are generating
cofibrations. But the generating cofibrations in $\mathsf{dgCat}^{\prime}$ are
images of cofibrations under $\Omega$, as was shown in the proof of [10,
Proposition 3.33].

Thus we may assume $D\to D^{\prime}$ is of the form $\Omega(\tilde{g})$ for a
cofibration $\tilde{g}:E\to E^{\prime}$ and we consider

$\Omega(C\otimes E^{\prime})\amalg_{\Omega(C\otimes
E)}\Omega(C^{\prime}\otimes E)\to\Omega(C^{\prime}\otimes E^{\prime})$

which is a cofibration as it is an image under $\Omega$ of
$f\boxempty\tilde{g}$, which in turn is a cofibration by Theorem 5.4.

Similarly, to check the case one of $f$, $g$ is acyclic it suffices to check
on generating trivial cofibrations in $\mathsf{dgCat}$. But these also lie in
the image of $\Omega$, so the same argument applies. ∎

The following observation was used in the proof of Theorem 6.2, it is worth
re-stating:

######  Corollary 6.3.

The Koszul adjunction
$\Omega:{\mathsf{ptdCoa}^{*}}\rightleftarrows\mathsf{dgCat}:\mathrm{B}$ is
enriched in ${\mathsf{ptdCoa}^{*}}$, i.e. there is a natural isomorphism
$\operatorname{\underline{Hom}}(C,\mathrm{B}D)\cong\operatorname{\overline{Hom}}(\Omega
C,D)$ of pointed curved coalgebras enhancing the adjunction isomorphism.

######  Remark 6.4.

In particular, note that the MC elements constructed in an ad-hoc way to prove
categorical Koszul duality in [10] are just the objects of the MC category
$\overline{\mathsf{MC}}\\{C,D\\}$.

######  Corollary 6.5.

The internal hom in $Ho(\mathsf{dgCat})$ may be computed as

$R\operatorname{\underline{Hom}}(D,D^{\prime})\simeq\overline{\mathsf{MC}}\\{BD,D^{\prime}\\}.$

###### Proof.

On the level of homotopy categories, $L\Omega$ is a strong monoidal
equivalence of categories, thus it identifies internal homs. It follows that
$L\Omega(R\operatorname{\underline{Hom}}(\mathrm{B}D,\mathrm{B}D^{\prime}))$
is an internal hom in the homotopy category. Since $\mathrm{B}D^{\prime}$ is
always fibrant and all pointed curved coalgebras are cofibrant we may rewrite
the internal hom underived as
$\Omega(\operatorname{\underline{Hom}}(\mathrm{B}D,\mathrm{B}D^{\prime}))\simeq\overline{\mathsf{MC}}\\{\mathrm{B}D,D^{\prime}\\}$.
∎

######  Remark 6.6.

The objects in the category $\overline{\mathsf{MC}}\\{BD,D^{\prime}\\}$ are
unital $A_{\infty}$ functors $D\to D^{\prime}$, either by definition, or by
unravelling the relevant definitions formulated in terms of multilinear maps
cf. [3]. The morphisms in $\overline{\mathsf{MC}}\\{BD,D^{\prime}\\}$ are then
unital $A_{\infty}$ transformations between the corresponding $A_{\infty}$
functors. We obtain a characterization of the derived internal homs in
$\mathsf{dgCat}$ as the category of unital $A_{\infty}$ functors as proposed
by Kontsevich and proved rigorously in [3]. Note that [3] considers not
necessarily unital $A_{\infty}$ transformations between functors, but the two
notions agree. This was shown in [13, Lemma 8.2.1.3], it also follows
immediately from Lemma 7.2 below, the difference between unital and non-unital
transformations corresponds exactly to the difference between reduced and non-
reduced Hochschild cochains.

While the embedding of dg categories into $A_{\infty}$ categories thus
computes the correct internal hom object, it is worth recalling that the
category of $A_{\infty}$ categories does not have a sensible monoidal
structure.

######  Theorem 6.7.

Given two dg categories $D,D^{\prime}$ the mapping space
$\operatorname{Map}(D,D^{\prime})$ is weakly equivalent to the core of
$\operatorname{N_{dg}}R\operatorname{\underline{Hom}}(D,D^{\prime})$.

###### Proof.

By the main results of [10] we may compute $\operatorname{Map}(D,D^{\prime})$
as $\operatorname{Map}(\mathrm{B}D,\mathrm{B}D^{\prime})$ in
${\mathsf{ptdCoa}^{*}}$. By Theorem 4.4 and Corollary 5.7 this is given by the
core of

$F\mathrm{B}\overline{\mathsf{MC}}\\{\mathrm{B}D,D^{\prime}\\}\simeq\operatorname{N_{dg}}\overline{\mathsf{MC}}\\{\mathrm{B}D,D^{\prime}\\}\simeq\operatorname{N_{dg}}R\operatorname{\underline{Hom}}(D,D^{\prime}),$

where we used $\operatorname{N_{dg}}\simeq F\mathrm{B}$ from Theorem 4.16 in
[10] as well as Corollary 6.5. ∎

######  Remark 6.8.

The theorem may also be deduced directly from the existence of an internal hom
in $Ho(\mathsf{dgCat})$ and some other standard results. We denote the left
adjoint of the dg nerve by $L$ and note that $L(*)=\mathbf{k}$ (considered as
a dg category with one object). Then the core of
$\operatorname{N_{dg}}(R\operatorname{\underline{Hom}}(D,D^{\prime}))$ may be
computed as

$\operatorname{Map}_{\mathsf{qCat}}(*,\operatorname{N_{dg}}(R\operatorname{\underline{Hom}}(D,D^{\prime})))\simeq\operatorname{Map}_{\mathsf{dgCat}}(L*,R\operatorname{\underline{Hom}}(D,D^{\prime}))\simeq\operatorname{Map}_{\mathsf{dgCat}}(\mathbf{k}\otimes
D,D^{\prime})$

Here we use that the closed structure $\mathsf{dgCat}$ induces a weak
equivalence of mapping spaces even though it is not Quillen. This follows from
the Yoneda Lemma by taking hom out an arbitrary simplicial set and using the
simplicial enrichment on $\mathsf{dgCat}$ [20, Section 5].

We may compare Theorem 6.7 with Toën’s characterization of
$\operatorname{Map}(D,D^{\prime})$ as the (classical) nerve of the category of
weak equivalences in the category of right quasi-representable cofibrant
$D\otimes{D^{\prime}}^{\operatorname{op}}$-modules [20, Theorem 1.1]. In the
special case of $\operatorname{Map}(\mathbf{k},D)$ this gives the following
corollary:

######  Corollary 6.9.

For a dg category $D$ the core of $\operatorname{N_{dg}}(D)$ is equivalent to
the nerve of the 1-category of quasi-isomorphisms between cofibrant quasi-
representable $D^{\operatorname{op}}$-modules.

This corollary is already interesting in the case that the dg category has one
object. Then it says that the core of the simplicial Maurer-Cartan set of a dg
algebra $A$ (as considered in [10, Section 4.1]) is given by the $A$-component
of the nerve of the category of quasi-isomorphisms between cofibrant
$A$-modules.

We have considered the Dwyer-Kan model structure on $\mathsf{dgCat}$ so far,
but similar results apply for the Morita model structure. We denote by
$\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$ the left Bousfield localization of
$\mathsf{dgCat}^{\prime}$ at all _Morita equivalences_ , i.e. functors
inducing equivalences of derived categories.

######  Lemma 6.10.

There is a Morita model structure on ${\mathsf{ptdCoa}^{*}}$ whose weak
equivalences are all those maps whose cobar construction is a Morita
equivalence and whose cofibrations are the injections. It is Quillen
equivalent to $\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$.

We denote pointed curved coalgebras with the Morita model structure by
${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$.

###### Proof.

By [10, Proposition 3.33] ${\mathsf{ptdCoa}^{*}}$ is a left proper
combinatorial model category, thus the left Bousfield localization at Morita
equivalences exists.

The cobar construction on ${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$ defines a
left Quillen functor to $\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$, which
induces an equivalence on homotopy categories. Thus the Quillen equivalence
${\mathsf{ptdCoa}^{*}}\rightleftarrows\mathsf{dgCat}$ restricts to a Quillen
equivalence ∎

######  Lemma 6.11.

The tensor product makes ${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$ into a
monoidal model category whose homotopy category is monoidally equivalent to
$Ho(\mathsf{dgCat}^{\prime}_{\mathrm{Mor}})$.

###### Proof.

The closed monoidal structure is the one considered in Theorem 5.4.

We first note that Lemma 5.1 implies that $\Omega$ is quasi-strong monoidal
also for the Morita model structures. We then observe that tensor product of
dg categories preserves Morita equivalences. This is well-known, see Exercise
32 in [21]. ∎

######  Proposition 6.12.

The category $\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$ is
${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$-enriched model category.

###### Proof.

Given $f:C\to C^{\prime}$ in ${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$ and
$g:D\to D^{\prime}$ in $\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$ we consider

$f\boxempty
g:(C\tilde{\otimes}D^{\prime})\amalg_{C\tilde{\otimes}D}(C^{\prime}\tilde{\otimes}D)\to
C^{\prime}\tilde{\otimes}D^{\prime}$

As the cofibrations in $\mathsf{dgCat}$ and
$\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$ agree we see that $f\boxempty g$ is a
cofibration if $f$ and $g$ are, by the proof of Theorem 6.2.

We now have to check that $f\boxempty g$ is a Morita equivalence if $f$ or a
$g$ is a (Morita) acyclic cofibration. It would suffice to show that all
generating acyclic cofibrations in $\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$
lie in the image of $\Omega$. We expect this to be true, but to avoid
excessive computations we may use another argument.

By imitating the proof of Theorem 5.4, it suffices to show that the action
$\tilde{\otimes}$ preserves Morita equivalences in
${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$ and
$\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$. Morita equivalences will be denoted
by $\simeq$ in the following.

The proof uses the simple observation that for any dg category $D$ we have a
natural quasi-equivalence $\Omega\mathrm{B}D\xrightarrow{\sim}D$, thus by
Theorem 6.2 we have $C\tilde{\otimes}\Omega\mathrm{B}D\simeq
C\tilde{\otimes}D$ for any $C\in{\mathsf{ptdCoa}^{*}}$.

Let $C\xrightarrow{\sim}C^{\prime}$ be an acyclic cofibration in
${\mathsf{ptdCoa}^{*}_{\mathrm{Mor}}}$. As $\Omega$ is quasi-strong monoidal
and $\otimes$ preserves Morita equivalences, see Lemma 6.11, we have the
following zig-zag of Morita equivalences:

$C\tilde{\otimes}D\xleftarrow{\sim}C\tilde{\otimes}\Omega\mathrm{B}D\xleftarrow{\sim}\Omega
C\otimes\Omega\mathrm{B}D\xrightarrow{\sim}\Omega
C^{\prime}\otimes\Omega\mathrm{B}D\xrightarrow{\sim}C^{\prime}\otimes\Omega\mathrm{B}D\xrightarrow{\sim}C^{\prime}\tilde{\otimes}D^{\prime}$

and thus $C\tilde{\otimes}D\simeq C^{\prime}\tilde{\otimes}D$.

Similarly, let $D\to D^{\prime}$ be a Morita equivalence. As
$\Omega\mathrm{B}D\simeq D$ it follows from 2-out-of-3 that
$\Omega\mathrm{B}D\simeq\Omega\mathrm{B}D^{\prime}$. Putting this together
with the previous observations we get the following Morita equivalences

$C\tilde{\otimes}D\xleftarrow{\sim}C\tilde{\otimes}\Omega\mathrm{B}D\xleftarrow{\sim}\Omega
C\otimes\Omega\mathrm{B}D\xrightarrow{\sim}\Omega
C\otimes\Omega\mathrm{B}D^{\prime}\xrightarrow{\sim}C\tilde{\otimes}\Omega\mathrm{B}D^{\prime}\xrightarrow{\sim}C\tilde{\otimes}D^{\prime}.$

Thus $C\tilde{\otimes}D\simeq C\tilde{\otimes}D^{\prime}$ and the proposition
follows. ∎

######  Corollary 6.13.

The internal hom in $Ho(\mathsf{dgCat}^{\prime}_{\mathrm{Mor}})$ is computed
by

$R\operatorname{\underline{Hom}}(D,D^{\prime})\simeq\overline{\mathsf{MC}}\\{BD,D^{\prime}\\}$

whenever $D^{\prime}$ is Morita fibrant.

###### Proof.

As for Corollary 6.5, except that in order to ensure $\mathrm{B}D^{\prime}$ is
fibrant we need to assume $D^{\prime}$ is fibrant in
$\mathsf{dgCat}^{\prime}_{\mathrm{Mor}}$. ∎

##  7\. Hochschild cohomology

One key application of the construction of the derived internal hom of dg
categories is the computation of homotopy groups in terms of Hochschild
cohomology by Toën [20]. We recreate this computation in our setting in
somewhat greater generality.

######  Definition 7.1.

The Hochschild cochain complex of a dg category $D$ with coefficients in a
bimodule $M:D\otimes D^{\operatorname{op}}\to\mathsf{dgVect}$ is defined as

$C^{*}_{\mathsf{HH}}(D,M)\coloneqq R\operatorname{Hom}_{D\otimes
D^{\operatorname{op}}}(D,M).$

It is well-known that $C^{*}_{\mathsf{HH}}(D,M)$ may be computed by the
Hochschild complex of the dg category $D$ with coefficients in $M$. Writing
$D(d_{0},d_{1})$ for $\operatorname{Hom}_{D}(d_{0},d_{1})$ for better
legibility this complex is

$\prod
M(d_{0},d_{1})\to\prod\operatorname{Hom}(D(d_{0},d_{1}),M(d_{0},d_{1}))\to\prod\operatorname{Hom}(D(d_{1},d_{2})\otimes
D(d_{0},d_{1}),M(d_{0},d_{2}))\cdots\to,$

with a differential induced by the internal differentials, composition in $D$
and the action of $D$ on $M$. One may equivalently compute with the reduced
Hochschild complex replacing $D$ by $\overline{D}$ everywhere.

We specialise now to the case where $M$ is given by another dg category
$D^{\prime}$ with a pair of functors $F,G:D\to D^{\prime}$, i.e. we consider
the bimodule ${}_{F}D^{\prime}_{G}$, or by abuse of notation just
$D^{\prime}$, sending $d_{1}\otimes d_{0}$ to
$\operatorname{Hom}_{D^{\prime}}(F(d_{0}),G(d_{1}))$.

Any dg functor $F:D\to D^{\prime}$ gives rise to an object in
$R\operatorname{\underline{Hom}}(D,D^{\prime})$ in a natural way. We use this
to state the following lemma:

######  Lemma 7.2.

Let $F,G:D\to D^{\prime}$ be functors of dg categories, then Hochschild
cohomology of $D$ with coefficients in $D^{\prime}$ is given by

$C^{*}_{\mathsf{HH}}(D,{}_{F}D^{\prime}_{G})\cong\operatorname{Hom}_{R\operatorname{\underline{Hom}}(D,D^{\prime})}(F,G).$

###### Proof.

By definition
$\operatorname{\underline{Hom}}_{R\operatorname{Hom}(D,D^{\prime})}(F,G)\cong\operatorname{Hom}_{\overline{\mathsf{MC}}\\{\mathrm{B}D,D^{\prime}\\}}(F,G)$
is the twisted hom space
$\operatorname{Hom}_{\operatorname{\underline{Hom}}(\mathrm{B}D,D^{\prime})}(F,G)^{[\xi_{F},\xi_{G}]}$
where $\xi_{F},\xi_{G}$ are the MC elements in the convolution category
corresponding to $F,G:D\to D^{\prime}$. Unravelling definitions, we recognize
this as the reduced Hochschild complex which we may write as

$D^{\prime}\to\operatorname{Hom}_{D^{\prime}_{0}}(\overline{D},D^{\prime})\to\operatorname{Hom}_{D^{\prime}_{0}}(\overline{D}\boxempty_{D_{0}}\overline{D},D^{\prime})\to\cdots$

with differential induced by internal differentials, the composition in $D$
and action of $D$ on $D^{\prime}$ (the latter corresponding to the twist by
$\xi_{F}$ and $\xi_{G}$). ∎

If $F=G=\operatorname{id}_{D}$ this specializes to the well-known equivalence
$C^{*}_{\mathsf{HH}}(D)\cong\operatorname{End}_{R\operatorname{\underline{Hom}}(D,D)}(\operatorname{id})$.

######  Remark 7.3.

One can also directly relate
$\operatorname{Hom}_{R\operatorname{\underline{Hom}}(D,D^{\prime})}(F,G)$ to
$R\operatorname{Hom}_{D\otimes D^{\operatorname{op}}}(D,{}_{F}D^{\prime}_{G})$
without reference to any resolutions, as follows.

We use that the functor $F:D\to D^{\prime}$ gives rise to a
$D\otimes{D^{\prime}}^{\operatorname{op}}$-bimodule $D^{\prime}_{F}$ sending
$d,d^{\prime}\mapsto\operatorname{Hom}_{D^{\prime}}(d^{\prime},F(d))$. The
functor $\hat{F}=\operatorname{id}\otimes F^{\operatorname{op}}:D\otimes
D^{\operatorname{op}}\to D\otimes{D^{\prime}}^{\operatorname{op}}$ induces an
adjunction on module categories. Considering the $D\otimes
D^{\operatorname{op}}$-module $D$ and the
$D\otimes{D^{\prime}}^{\operatorname{op}}$-module $D^{\prime}_{G}:d\otimes
d^{\prime}\mapsto\operatorname{Hom}(d^{\prime},d)$ we have

$R\operatorname{Hom}_{D\otimes
D^{\operatorname{op}}}(D,\hat{F}^{*}(D^{\prime}_{G})\cong
R\operatorname{Hom}_{D\otimes{D^{\prime}}^{\operatorname{op}}}(\hat{F}_{!}D,D^{\prime}_{G})$

By definition $\hat{F}^{*}({}_{G}D^{\prime})$ is ${}_{F}D^{\prime}_{G}$ and
moreover one may check $\hat{F}_{!}D=D^{\prime}_{F}$, for example by rewriting
$\hat{F}_{!}$, which by definition is a left Kan extension, as a coend and
computing it.

The proof is completed by recalling from [20] that the functor category
$R\operatorname{\underline{Hom}}(D,D^{\prime})$ is weakly equivalent to a full
subcategory of (fibrant cofibrant)
$D\otimes{D^{\prime}}^{\operatorname{op}}$-modules, and in particular the map
$F\mapsto D^{\prime}_{F}$ induces weak equivalences of hom spaces. Thus we
have

$\operatorname{Hom}_{R\operatorname{\underline{Hom}}(D,D^{\prime})}(F,G)\simeq
R\operatorname{Hom}_{D\otimes{D^{\prime}}^{\operatorname{op}}}(D^{\prime}_{F},D^{\prime}_{G})\simeq
R\operatorname{Hom}_{D,D}(D,{}_{F}D^{\prime}_{G}).$

This is nothing but a categorical version of the classical formula
$\mathsf{HH}^{*}(A,B)\cong\operatorname{Ext}^{*}_{A\otimes
B^{\operatorname{op}}}(B,B)$ for algebras $A$ and $B$.

We now specialise to $F=G$ and consider the bimodule
$D^{\prime}={}_{G}D^{\prime}_{G}$.

######  Theorem 7.4.

Let $G:D\to D^{\prime}$ be a functor of dg categories. We then have
$\mathsf{HH}^{0}(D,D^{\prime})^{\times}\cong\pi_{1}(\operatorname{Map}(D,D^{\prime}),G)$
and
$\mathsf{HH}^{i}(D,D^{\prime})\cong\pi_{1-i}(\operatorname{Map}(D,D^{\prime}),G)$
for $i<0$.

###### Proof.

By Theorem 6.7 we have $\operatorname{Map}(D,D^{\prime})$ weakly equivalent to
the core of
$\operatorname{N_{dg}}R\operatorname{\underline{Hom}}(D,D^{\prime})$.

It now follows, cf. [15, Remark 1.3.1.12], that the Dold-Kan image of
$\operatorname{Hom}_{R\operatorname{\underline{Hom}}(D,D^{\prime})}(G,G)$ is
the infinity categorical mapping space from $G$ to $G$ in
$F\mathrm{B}\overline{\mathsf{MC}}\\{\mathrm{B}D,D^{\prime}\\}$. Taking
homotopy groups and using Lemma 7.2 we have
$\mathsf{HH}^{i}(D,D^{\prime})=\pi_{-i}(\Omega\operatorname{Map}(D,D^{\prime}),G)$
for $i>1$ as the higher homotopy groups of mapping spaces are unaffected by
taking the core. Finally $\mathsf{HH}^{0}(D,G)$ is given by
$\pi_{0}\operatorname{Map}_{F\Omega\overline{\mathsf{MC}}\\{BD,D\\}^{\prime}}(G,G)$
and taking units on both sides we obtain
$\mathsf{HH}^{0}(D,D^{\prime})^{*}\cong\pi_{0}\operatorname{Map}_{\operatorname{Map}(D,D^{\prime})}(G,G)=\pi_{1}(\operatorname{Map}(D,D^{\prime}),G)$.
∎

######  Corollary 7.5 ([20, Corollary 8.2]).

$\mathsf{HH}^{i}(D)\cong\pi_{1-i}(\operatorname{Map}(D,D),\operatorname{id})$
for $i<0$ and
$\mathsf{HH}^{0}(D)^{\times}\cong\pi_{1}(\operatorname{Map}(D,D),\operatorname{id}_{D})$.

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