Most of the work for the construction of this dg-category has already been
done, and we only have to put it together. We define a graph $G$ as follows:

  * •

$\operatorname{Obj}(G)=\mathcal{O}$.

  * •

For all $x,y\in\mathcal{O}$, we have that $G(x,y)=F(x,y)$, the representing
object of $F_{(x,y)}$.

We define then $T=L(G)$ as being the free category constructed from $G$.

Let us now prove that the morphism $\operatorname{Sing}(T)\to F$ is a dg-Segal
epimorphism in $\mathbf{dg-Segal}_{\mathcal{O}}$, or equivalently, that its
projection on $\mathbf{dg-Segal}$ is a dg-Segal epimorphism.

By the construction of $T=L(G)$, we have that for all $x,y\in\mathcal{O}$,

$L(G)(x,y)=\bigoplus_{m\in\mathbb{N}}\bigoplus_{(x_{1},x_{2},\ldots,x_{m})\in\mathcal{O}^{m}}(G(x,x_{1})\otimes\ldots\otimes
G(x_{m},y)),$

and in particular, $G(x,y)=F(x,y)$ is a factor in $T(x,y)$. That means that
for all $E\in\mathbf{C(k)}$ there exists an inclusion
$\operatorname{Map}(E,F(x,y))=F_{(x,y)}(E)\to\operatorname{Map}(E,T(x,y))$. It
is easy to see, using the definition of homotopy fiber, that for all dg-
category $T^{\prime}$,
$\operatorname{Sing}(T^{\prime})_{(x,y)}=\operatorname{Map}(-,T^{\prime}(x,y))$.
Putting it all together, we have then that the morphism
$\operatorname{Sing}(T)_{(x,y)}=\operatorname{Map}(-,T(x,y))\to
F_{(x,y)}=\operatorname{Map}(-,F(x,y))$ is a split epimorphism and we have
proven our result.

The morphism $\operatorname{Sing}(T)\to F$ is a dg-Segal epimorphism and we
have finished.
∎

Let us tackle now the hypercover result.

######  Theorem 4.6.8.

Let $F\in\mathbf{dg-Segal}$ be a functor that satisfies the dg-Segal
conditions and let $\mathcal{O}=\pi_{0}(F(k))$. Then there exists a simplicial
object $T_{*}$ in $\mathbf{dg-cat}_{\mathcal{O}}$ such that
$F_{*}=\operatorname{Sing}(T_{*})$ and a morphism $F_{*}\to F$ such that
$F_{*}\to F$ is a dg-Segal hypercover in $\mathbf{dg-Segal}_{\mathcal{O}}$.

###### Proof.

Again, this construction is almost identical to that of Theorem 4.4.6, and as
such we won’t be going into much detail. We will construct our hypercover by
induction, by proving that for all $n\in\mathbb{N}$ there exists an
$n$-truncated dg-Segal hypercover with fixed objects of $F$ where every level
is in the image of $\operatorname{Sing}$.

  * •

$n=0$ is true by Lemma 4.6.7. There exists a dg-category with fixed objects
$T\in\mathbf{dg-cat}_{\mathcal{O}}$ such that $\operatorname{Sing}(T)\to F$ is
a dg-Segal epimorphism, and that creates a 0-truncated dg-Segal hypercover.

  * •

$n\in\mathbb{N}$. By induction hypothesis there exists an $n$-truncated dg-
Segal hypercover of $F$, named $\operatorname{Sing}(T_{*})\to F$. Let us
construct an $(n+1)$-truncated hypercover of $F$, that we will call
$\operatorname{Sing}(T_{*})\to F$ too. As the construction doesn’t change the
first $n$ terms, there is no ambiguity in the notation.

We define
$V_{*}=\operatorname{sk}_{n+1}(\operatorname{cosk}_{n}\operatorname{Sing}(T_{*}))$.
This simplicial set is $(n+1)$-truncated, but the term $n+1$ is not
necessarily in the image of $\operatorname{Sing}$. By Lemma 4.6.7 again, there
exists a free dg-category $A$ with objects $\mathcal{O}$ and a morphism
$\operatorname{Sing}(A)\to V_{n+1}$ that is a dg-Segal epimorphism. We define
the $(n+1)$-truncated dg-Segal hypercover to be $\operatorname{Sing}(T_{i})$
for all $i\leq n$ and

$\operatorname{Sing}(T_{n+1})=\operatorname{Sing}(A)\coprod\operatorname{Sing}(T_{i})=\operatorname{Sing}(A\coprod
T_{i})$

for $n+1$. This is possible because $\operatorname{Sing}$ commutes with finite
coproducts.

With the same morphisms as in Theorem 4.4.6, we have a simplicial object in
$\mathbf{dg-Segal}_{\mathcal{O}}$. We now just have to prove that for all
$i\leq n+1$,
$\operatorname{Sing}(T_{i})\to\operatorname{Sing}(T_{*})^{\partial\Delta^{i}}$
is a dg-Segal epimorphism, which by construction gets instantly reduced to
proving that

$\operatorname{Sing}(T_{n+1})=\operatorname{Sing}(A)\coprod\operatorname{Sing}(T_{i})\to
V_{n+1}$

is a dg-Segal epimorphism in $\mathbf{dg-Segal}_{\mathcal{O}}$, or
equivalently, that for all $x,y\in\mathcal{O}$,
$\operatorname{Sing}(T_{n+1})_{(x,y)}=\operatorname{Map}(-,T_{n+1}(x,y))\to
V_{n+1\\\ (x,y)}$ is a split epimorphism.

We have created $F_{*}\to F$ a dg-Segal hypercover with fixed objects such
that for all $n\in\mathbb{N}$, there exists a free dg-category of finite type
$T_{n}$ such that $\operatorname{Sing}(T_{n})=F_{n}$. Now, we have proven in
Theorem 4.5.6 that $\operatorname{Sing}$ is a fully faithful functor. In
consequence, every morphism in the simplicial object $F_{*}$ comes from a
morphism in $\mathbf{dg-cat}_{\mathcal{O}}$, and there exists a simplicial
object $T_{*}\in\mathbf{dg-cat}_{\mathcal{O}}$ such that
$\operatorname{Sing}(T_{*})=F_{*}$. We have finished our proof.
∎

Now that we have constructed our hypercover $F_{*}\to F$, we are on the final
stretch of the proof. Indeed, the last thing we need is to prove that for all
$F\in\mathbf{dg-Segal}$ there exists a dg-category $T$ such that $F$ is DK-
equivalent to $\operatorname{Sing}(T)$, and we have our perfect candidate to
do so.

######  Notation 4.6.9.

Let $F\in\mathbf{dg-Segal}$ be a functor satisfying the dg-Segal conditions.
Let $\operatorname{Sing}(T_{*})=F_{*}\to F$ be a dg-Segal hypercover with
fixed objects as constructed in Theorem 4.6.8. We call $T$ the homotopy
colimit of the simplicial object $T_{*}$. In other words, we define

$T=\operatorname{hocolim}(T_{i}).$

We need now to prove that $\operatorname{Sing}(T)\to F$ is a DK-equivalence.
In order to do that, we will use two DK-equivalences that are easier to prove:
$\operatorname{hocolim}(\operatorname{Sing}(T_{i}))\to\operatorname{Sing}(\operatorname{hocolim}(T_{i}))=\operatorname{Sing}(T)$
and $\operatorname{hocolim}(\operatorname{Sing}(T_{i}))\to F$.

######  Lemma 4.6.10.

Let $F_{*}\to F$ be a dg-Segal hypercover in $\mathbf{dg-Segal}_{\mathcal{O}}$
with $\mathcal{O}=\pi_{0}(F(k))$ constructed as before. Then, for all
$x,y\in\mathcal{O}$, the augmented simplicial complex $T_{*}(x,y)\to F(x,y)$
is a split hypercover in $\mathbf{C(k)}$.

###### Proof.

This follows from the definition of dg-Segal hypercover.
∎

######  Proposition 4.6.11.

Let $F_{*}\to F$ be a dg-Segal hypercover constructed as in Theorem 4.6.8.
Then the homotopy colimit of $F_{*}$ is DK-equivalent to $F$ in $\mathbf{dg-
Segal}_{\mathcal{O}}$.

###### Proof.

The proof of this follows very closely the one in Lemma 4.5.5. Indeed, in
order to prove something is a DK-equivalence, we have two conditions: firstly,
that the morphism

$\left[\operatorname{hocolim}(F_{i})\right]\to\left[F\right]$

is an essentially surjective. But the hypercover has been constructed to have
fixed objects $\pi_{0}(F(k))$, so this condition is verified by construction.

That leaves us with the second condition. We fix $x,y\in\mathcal{O}$. Do we
have that

$(\operatorname{hocolim}F_{i})_{(x,y)}\to F_{(x,y)}$

is a quasi-equivalence? By definition,
$F_{i,(x,y)}=\operatorname{Map}(-,T_{i}(x,y))$ and
$F_{(x,y)}=\operatorname{Map}(-,F(x,y))$, so we can rewrite this condition as
wondering whether

$\operatorname{hocolim}\operatorname{Map}(-,T_{i}(x,y))\to\operatorname{Map}(-,F(x,y))$

is a quasi-equivalence. By Lemma 4.6.10, we know that $T_{*}(x,y)\to F(x,y)$
is a split hypercover of complexes, so for all $n\in\mathbb{N}$,
$T_{n}(x,y)\to T_{*}^{\partial\Delta^{n}}(x,y)$ is a split epimorphism, and in
consequence for all $E\in\mathbf{C(k)}$ the morphism

$\pi_{0}(\operatorname{Map}(E,T_{n}(x,y)))\to\pi_{0}(\operatorname{Map}(E,T_{*}(x,y)^{\partial\Delta^{n}}))\simeq\pi_{0}(\operatorname{Map}(E,T_{*}(x,y))^{\partial\Delta^{n}})$

is an epimorphism. That means that
$\operatorname{Map}(E,T_{*}(x,y))\to\operatorname{Map}(E,F(x,y))$ is a
hypercover of simplicial sets, and by Proposition 4.4.2, we have a quasi-
equivalence

$\operatorname{hocolim}\operatorname{Map}(-,T_{i}(x,y))\simeq\operatorname{Map}(-,F(x,y)).$

So the morphism $\operatorname{hocolim}(F_{i})\to F$ is a DK-equivalence and
we have finished our proof.
∎

Like in the case of the dg-categories, we have proved that the homotopy
colimit is DK-equivalent to $F$, but only with fixed objects. But if we want
to use it, we need to see that it is indeed true in $\mathbf{dg-Segal}$.

######  Proposition 4.6.12.

Let $F$ be a dg-Segal space and $F_{*}\to F$ a dg-Segal hypercover constructed
as in Theorem 4.6.8. Then the homotopy colimit of $F_{*}$ is DK-equivalent to
$F$ in $\mathbf{dg-Segal}$.

###### Proof.

This proof is almost identical to the one in Proposition 4.4.14, and as such
we won’t spend too much time on its details. As was the case there, we have
that the homotopy colimit of the hypercover is DK-equivalent to $F$, but only
with fixed objects. So we’re going to prove that the forgetful functor
$\Phi:\mathbf{dg-Segal}_{\mathcal{O}}\to\mathbf{dg-Segal}$ commutes with those
homotopy colimits. Let us denote the obvious forgetful functor by
$\Xi:\mathbf{dg-Segal}_{\mathcal{O}}\to\coprod\underline{k}/\mathbf{dg-
Segal}$.

For all $G\in\mathbf{dg-Segal}$ we then have a morphism of the form

$\operatorname{Map}(\Phi(F),G)\to\operatorname{Map}(\coprod\underline{k},G)\simeq\prod\operatorname{Map}(\underline{k},G).$

By getting the fiber of this morphism and doing the same with
$\operatorname{hocolim}F_{i}$, we get the following diagram:

${\operatorname{Map}(F,G)}$${\operatorname{Map}(\Xi(F),G)}$${\operatorname{Map}(\Phi(F),G)}$${\prod\operatorname{Map}(\underline{k},G)}$${\operatorname{holim}\operatorname{Map}(F_{i},G)}$${\operatorname{holim}\operatorname{Map}(\Xi(F_{i}),G)}$${\operatorname{holim}\operatorname{Map}(\Phi(F_{i}),G)}$${\prod\operatorname{Map}(\underline{k},G)}$$\scriptstyle{\sim}$$\scriptstyle{f}$$\scriptstyle{=}$

If we can prove that $f$ is a weak equivalence, we have finished. For that,
let us prove that $\Xi$ is fully faithful.

We have an adjunction $\Xi:\operatorname{Ho}(\mathbf{dg-
Segal}_{\mathcal{O}})\rightleftharpoons\operatorname{Ho}(\coprod\underline{k}/\mathbf{dg-
Segal}):\Gamma$ where the right adjoint $\Gamma$ is such that for all
$H\in\coprod\underline{k}/\mathbf{dg-Segal}$ and $L\in
c\mathcal{L}_{\mathbb{S}}$, $\Gamma(H)(L)$ is the following homotopy pullback

${\Gamma(F)(L)}$${F(L)}$${\coprod_{\mathcal{O}_{L}}F(k)}$${\coprod_{\mathcal{O}_{L}}\mathcal{O}}$${\ulcorner^{h}}$

where $\mathcal{O}_{L}=\operatorname{Obj}(L)$ is the set of objects of $L$. It
is trivial that with such a construction,
$\Gamma\Xi=\operatorname{Id}_{\mathbf{dg-Segal}_{\mathcal{O}}}$. We have then
that $\Xi$ is fully faithful and in consequence that
$\operatorname{Map}(\Xi,G)$ is a weak equivalence. By the 2-out-of-3
condition, we have that $f$ is a weak equivalence, and so we have that $\Phi$
commutes with colimits. By Proposition 4.6.11, we know that
$\operatorname{hocolim}(F_{i})$ is DK-equivalent to $F$ in $\mathbf{dg-
Segal}_{\mathcal{O}}$, and that means that we have a DK-equivalence between
$\operatorname{hocolim}(F_{i})$ and $F$ in $\mathbf{dg-Segal}$ too. We have
finished the proof.
∎

Let us see now about $\operatorname{Sing}(T_{i})\to\operatorname{Sing}(T)$.

######  Lemma 4.6.13.

Let $F_{*}\to F$ be a dg-Segal hypercover as constructed before, let $T_{*}$
be a simplicial object in $\mathbf{dg-cat}_{\mathcal{O}}$ with
$\mathcal{O}=\pi_{0}(F(k))$ such that $\operatorname{Sing}(T_{*})=F_{*}$ and
$T=\operatorname{hocolim}T_{i}$. Then the augmented simplicial object
$T_{*}\to T$ is a $c\mathcal{L}_{\mathbb{S}}$-hypercover of dg-categories.

###### Proof.

As we have fixed objects on this, by Lemma 4.4.10 if for all
$x,y\in\mathcal{O}$, the augmented object $T_{*}(x,y)\to T(x,y)$ is a split
hypercover of complexes, then $T_{*}\to T$ is a hypercover of dg-categories
and we have finished.

Let us fix $x,y\in\mathcal{O}$. By Lemma 4.6.10, $T_{*}(x,y)\to F(x,y)$ is a
split hypercover. But as it is a split hypercover, by Lemma 4.4.11 we know
that $T(x,y)=\operatorname{hocolim}(T_{i}(x,y))\simeq F(x,y)$. So the
augmented object $T_{*}(x,y)\to T(x,y)$ is a split hypercover on
$\mathbf{C(k)}$, and in consequence $T_{*}\to T$ is a hypercover on
$\mathbf{dg-cat}$ and we have finished our proof.
∎

Now we just need to put everything together.

######  Theorem 4.6.14.

Let $F$ be a functor that satisfies the dg-Segal conditions. Then there exists
a dg-category $T$ such that the morphism $\operatorname{Sing}(T)\to F$ is a
DK-equivalence.

###### Proof.

Let $\operatorname{Sing}(T_{*})=F_{*}\to F$ be a dg-Segal hypercover
constructed following Theorem 4.6.8. We have then a diagram of this form:

${\operatorname{Sing}(\operatorname{hocolim}(T_{i}))=\operatorname{Sing}(T)}$${F}$${\operatorname{hocolim}(\operatorname{Sing}(T_{i}))=\operatorname{hocolim}(F_{i})}$$\scriptstyle{\phi}$$\scriptstyle{\zeta}$

We have proven that the corresponding augmented object $T_{*}\to T$ is a
hypercover of dg-categories, and by Lemma 4.5.5, the morphism $\phi$ is a DK-
equivalence. We have also proven that $\operatorname{hocolim}(F_{i})\to F$ is
also a DK-equivalence, so by Remark 4.3.20, that means that our morphism
$\operatorname{Sing}(T)\to F$ is also a DK-equivalence and we have finished.
∎

And we are done.

######  Theorem 4.6.15.

Assume Hypothesis 4.3.22 is true. Then, if we take the complete dg-Segal model
structure on
$\operatorname{Fun}^{\mathbb{S}}(c\mathcal{L}_{\mathbb{S}}^{op},\mathbf{sSet})$,
the functor $\operatorname{Sing}:\mathbf{dg-cat}\to\mathbf{dg-Segal_{c}}$ is
essentially surjective. Hence there exists an equivalence of categories of the
form

$\operatorname{Ho}(\mathbf{dg-cat})\to\operatorname{Ho}(\mathbf{dg-
Segal_{c}}).$

##  Chapter 5 Future work

> " SUNDAY
> (The first day of the rest of their lives)"
>
> —Terry Pratchett, Neil Gaiman, Good Omens

Now that we have finished talking about the results that are completely
solved, it is time to tackle results that are in progress and those who we
consider to be interesting prospects in a more general sense.

###  5.1 Complete dg-Segal spaces and DK-equivalences

We saw in the last chapter that the main result of this thesis hinges on one
hypothesis. Let us remind the reader of it here.

######  Hypothesis 5.1.1 (Hypothesis 4.3.22).

Let $f:F\to G$ be a morphism between two functors satisfying the dg-Segal
conditions. Then, $f$ is a DK-equivalence if and only if it is a weak
equivalence in the complete dg-Segal model structure.

Now, this is not a new or surprising condition to have. Indeed, a very similar
situation appears in [Rez01, Theorem 7.7, proof in Section 14]: in the classic
case the weak equivalences for the complete Segal model structure are exactly
the Dwyer-Kan equivalences between complete Segal spaces. In that paper, this
identification between Dwyer-Kan equivalences and complete Segal weak
equivalences is done by the means of a completion functor such that for all
Segal space $W$ there exists a map $i_{W}:W\to\widehat{W}$ with the following
properties:

  1. 1.

The completion $\widehat{W}$ is a complete Segal space.

  2. 2.

The completion map $i_{W}$ is a weak equivalence in the complete Segal model
structure.

  3. 3.

The completion map $i_{W}$ is a Dwyer-Kan equivalence.

Using such a completion map, the author reduces the problem from asking
whether a Dwyer-Kan equivalence $f:W\to W^{\prime}$ between Segal spaces is a
weak equivalence for the complete Segal model structure to whether its
completion $\hat{f}$ is one, and some relatively straightforward arguments
prove the if and only if from the theorem.

The main issue with this proof is how to construct the completion functor
mentioned above. Obviously, a simple fibrant replacement would be enough to
assuage conditions 1 and 2, but not 3. Instead, the author constructs their
complete Segal space by way of defining a certain simplicial space $E(m)$ for
all $m\in\mathbb{N}$ and computing

$\widehat{W}_{n}=diag([m]\to\operatorname{Map}(E(m),W^{F(n)}))=diag([m]\to(W^{E(m)})_{n}).$

We expect the proof of our hypothesis to be something along these lines.
Unfortunately, the construction won’t be as smooth-sailing in our case: Rezk
uses the direct product to define his exponentials, but if there is one thing
that is well known about the monoidal structure of dg-categories, it’s that it
isn’t compatible with its model structure (see [Toë11]: the object
$\Delta_{k}(1,0,1)$ is cofibrant in the model category of dg-categories, but
it is easy to prove that $\Delta_{k}(1,0,1)\otimes\Delta_{k}(1,0,1)$ is not).
We will be forced to define our own version of the monoidal structure.

Let us start at the beginning. We have seen that we have the linearisation
adjunction,
$j_{!}:\operatorname{Fun}(\Delta^{op},\mathbf{sSet})\rightleftharpoons\operatorname{Fun}^{\mathbb{S}}(c\mathcal{L}^{op}_{\mathbb{S}},\mathbf{sSet}):j^{*}$,
and we have defined $E_{k}=j_{!}(E)$. That will be our best candidate in order
to reproduce our $E(m)$ in this new context.

######  Remark 5.1.2.

Attention! In Rezk’s case, $E$ was the nerve of the category with two objects
and an isomorphism between them, $0\tilde{\to}1$. But its equivalent is not
true in our case:

$E_{k}\neq\operatorname{Sing}((0\tilde{\to}1)\otimes k).$

And we should be grateful for that: indeed,
$\operatorname{Sing}((0\tilde{\to}1)\otimes k)=\operatorname{Sing}(k)$, so
$E_{k}$ being equal to it would mean that the complete dg-Segal model
structure is in fact exactly the same as the dg-Segal model structure!

######  Definition 5.1.3.

For all $n\in\mathbb{N}$, we have that

$E_{k}(n)=\mathbb{L}j_{!}(0\tilde{\to}1\tilde{\to}\ldots\tilde{\to}n).$

Following the example of [Rez01], we would want to define our dg-completion
functor as follows:

######  Definition 5.1.4.

Let $X$ be a dg-Segal space. We call the completion of $X$, and we denote by
$\hat{X}$, a complete dg-Segal space of the form

$\hat{X}=\operatorname{hocolim}([i]\to X^{E_{k}(i)}).$

######  Remark 5.1.5.

We attract the reader’s attention to the fact that, if this is defined
correctly, its image by the forgetful functor $j^{*}$ will give us the
completion of $j^{*}(X)$ in classical Segal spaces.

The problem here is that we need to define this exponential, which asks from
us to define a tensor product. We have that, for all $L\in
c\mathcal{L}_{\mathbb{S}}$

$X^{E_{k}(n)}(L)\simeq\operatorname{Map}(\operatorname{Sing}(L)\otimes
E_{k}(n),X).$

But what does $\operatorname{Sing}(L)\otimes E_{k}(n)$ mean? Well, even if
$E_{k}(n)$ itself is not of the form $\operatorname{Sing}(T)$ for a certain
$T\in\mathbf{dg-cat}$, we can write it as
$E_{k}(n)=\operatorname{hocolim}\operatorname{Sing}(L_{i})$ where $L_{i}\in
c\mathcal{L}_{\mathbb{S}}$ for all $i\in\mathbb{N}$. So we technically could
write this tensor product as

$\operatorname{Sing}(L)\otimes
E_{k}(n)=\operatorname{hocolim}\operatorname{Sing}(L)\otimes\operatorname{Sing}(L_{i}).$

Now, our first instinct would be to write this as
$\operatorname{Sing}(L\otimes L_{i})$; but $L\otimes L_{i}$ isn’t a free dg-
category of finite type anymore. Maybe we could reiterate the process by
writing $L\otimes L_{i}$ as a homotopy colimit of another $K_{*}$? We will
see.

######  Remark 5.1.6.

It is important to notice that, should this method succeed (and we see no
reason why it wouldn’t), we not only have finished the proof of our theorem,
but also defined a monoidal structure on $\mathbf{dg-Segal_{c}}$ that is
compatible with the model structure, giving in its wake a possible monoidal
model structure to the category of dg-categories.

###  5.2 The linear simplex category

The free dg-categories of finite type were, sadly, not our first attempt at a
model for dg-categories. Indeed, when we started we expected to have something
a little closer to the simplex category we use in our definition of Segal
spaces. The methods we wanted to use didn’t work with them, though, which is
why we opted for the free dg-categories instead; but as we haven’t lost hope
of being able to bring it back to it, we explain the linear simplex category
here.

####  5.2.1 First constructions

In this text, we have used many different versions of dg-categories that were
some version of "a finite number of objects with morphisms between them being
$k[s]$". For example, when we discussed Tabuada’s cofibrantly generated model
structure for $\mathbf{dg-cat}$ in [Tab10], we constructed a series of objects
of the form $\Delta_{k}(1,s,1)$, for all $s\in\mathbb{Z}$, dg-categories with
two objects and $k[s]$ the cochain complex that is $k$ in degree $s$. We have
also the linearisation functor, that gives us $\Delta_{k}(n,0,1)$, dg-
categories with $n$ objects and $k$ the morphism between $i$ and $i+1$ for all
$0\leq i\leq n-1$.

We are now going to expand on those definitions, and make what will be the
objects of our new linear simplex category.

######  Definition 5.2.1.

Fix $n\in\mathbb{N}$ a natural and let $s\in\mathbb{Z}^{n}$ and
$d\in\mathbb{N}^{n}$. We call $\Delta_{k}(n,s,d)$ a dg-category consisting of
the following data:

  * •

A set of $n+1$ objects, that we will denote by $0,\ldots,n$ for clearness’
sake.

  * •

For all $0\leq i-1<i\leq n$, a complex $\operatorname{Hom}(i-1,i)$
concentrated in degree $s_{i}$, where it is $k^{d_{i}}$ i.e. a complex which
is zero everywhere except for in degree $s_{i}$. In other words, for $0\leq
i<j\leq n$, a complex $\operatorname{Hom}(i,j)$ of the form

$\operatorname{Hom}(i,j)=\otimes_{i<k\leq j}\operatorname{Hom}(k-1,k).$

In other words, a dg-category $\Delta_{k}(n,s,d)$ is as follows:

${0}$${1}$${\ldots}$${n-1}$${n}$$\scriptstyle{k^{d_{1}}\left[s_{1}\right]}$$\scriptstyle{k}$$\scriptstyle{k^{d_{2}}\left[s_{2}\right]}$$\scriptstyle{k}$$\scriptstyle{k^{d_{n}}\left[s_{n}\right]}$$\scriptstyle{k}$$\scriptstyle{k}$

######  Notation 5.2.2.

We denote by $\Delta_{k}(0)$ or $k$ the dg-category $\Delta_{k}(0,s,d)$ with
only one object and $k$ as its complex of endomorphisms. We remind the reader
that this dg-category is the unit for the tensor product.

######  Definition 5.2.3.

We call the linear simplex category, and we denote it by $\Delta_{k}$, the
full subcategory of the category of dg-categories $\mathbf{dg-cat}$ which has
the dg-categories $\Delta_{k}(n,s,d)$ as objects for all $n\in\mathbb{N}$,
$s\in\mathbb{Z}^{n}$ and $d\in\mathbb{N}^{n}$.

One immediate advantage of the objects in $\Delta_{k}$ is that it is way
easier to compute the morphism sets in $\mathbf{dg-cat}$ from
$\Delta_{k}(n,s,d)$ to a general dg-category $T$ than it is to compute them
from a general dg-category to another.

######  Proposition 5.2.4.

Let $T$ be a object in $\mathbf{dg-cat}$ and $\Delta_{k}(n,s,d)$ in
$\Delta_{k}$. Let
$\phi_{*}:\operatorname{Hom}(\Delta_{k}(n,s,d),T)\to\operatorname{Hom}(\coprod\Delta_{k}(0),T)$
the morphism induced by the inclusion
$\phi:\coprod_{n+1}\Delta_{k}(0)\to\Delta_{k}(n,s,d)$ where we send each copy
of $\Delta_{k}(0)$ to a different object in $\Delta_{k}(n,s,d)$. Then, we have
the following isomorphisms:

  * •

The set of morphisms from $\Delta_{k}(0)$ to $T$ is isomorphic to the objects
in $T$, $\operatorname{Hom}(\Delta_{k}(0),T)\simeq\operatorname{Obj}(T)$. In
other words, giving a morphism $\Delta_{k}(0)\to T$ is equivalent to fixing a
point in $T$.

  * •

The fiber of $\phi$ is given by

$\phi_{*}^{-1}(t)\simeq\prod\operatorname{Hom}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],T(t_{i-1},t_{i}))\simeq\prod(Z^{s_{i}}(\operatorname{Hom}_{T}(t_{i-1},t_{i})))^{d_{i}},$

where $Z^{s_{i}}(\operatorname{Hom}_{T}(t_{i-1},t_{i}))$ is the set of
cocycles of $\operatorname{Hom}(t_{i-1},t_{i})$ in degree $s_{i}$, i.e. the
set of objects $a\in\operatorname{Hom}(t_{i-1},t_{i})^{s_{i}}$ such that
$d(a)=0$. In other words, if we fix $n+1$ points in $T$, $t_{0},\ldots,t_{n}$,
then describing the set $\operatorname{Hom}(\Delta_{k}(n,s,d),T)$ comes down
to describing the set of cocycles in degree $s_{i}$ of the cochain complex
$T(t_{i-1},t_{i})$ for all $0<i\leq n$.

###### Proof.

Let us start with
$\operatorname{Hom}(\Delta_{k}(0),T)\simeq\operatorname{Obj}(T)$. Let
$f:\Delta_{k}(0)\to T$ be an object in $\operatorname{Hom}(\Delta_{k}(0),T)$.
Giving such a morphism is equivalent to giving an object in $T$ (the image of
$0$) and a morphism in $T$ (the image of $k=\operatorname{Hom}(0,0)$). We will
define the isomorphism as

$F:\operatorname{Hom}(\Delta_{k}(0),T)\to\operatorname{Obj}(T)$ $\ \ f\mapsto
f(0).$

The surjectivity is obvious. The only thing that we need to check is that $F$
is, in fact, injective: that if we fix a point $t\in T$ in $T$, there is only
one possible morphism $\operatorname{Hom}(0,0)=k\to\operatorname{Hom}(t,t)$.
Let $f:\Delta_{k}(0)\to T$ be a dg-functor such that $F(f)=t$. As it is a
morphism in $\mathbf{dg-cat}$, we know that the diagram

${k}$${\operatorname{Hom}(0,0)=k}$${\operatorname{Hom}(t,t)}$$\scriptstyle{e_{t}}$$\scriptstyle{e_{0}}$$\scriptstyle{f}$

commutes. In particular, that means that the image of the identity of
$\operatorname{Hom}(0,0)=k$ is fixed, and that fixes the entirety of the
morphism $f$. We have that if we fix an object $t\in T$ there is only one
possible morphism $\operatorname{Hom}(0,0)\to\operatorname{Hom}(t,t)$, and the
functor $F$ is injective.

The functor $F$ is an isomorphism and
$\Delta_{k}(0)\simeq\operatorname{Obj}(T)$. In particular, giving a morphism
from $\Delta_{k}(0)$ to $T$ is equivalent to fixing a point in $T$.

Now let us do the fiber, and let us start by the first isomorphism. By
definition of $\phi$, and by the last part of this proof, looking at the fiber
$\phi_{*}^{-1}$ is equivalent to fixing $n+1$ objects $t_{0},\ldots,t_{n}$ in
$T$, and then looking at the morphisms $f:\Delta_{k}(n,s,d)\to T$ such that
$f(i)=t_{i}$ for all $0\leq i\leq n$.

$\phi_{*}^{-1}\simeq\\{f\in\operatorname{Hom}(\Delta_{k}(n,s,d),T)/\
f(i)=t_{i}\ \forall i\in\operatorname{Obj}(\Delta_{k}(n,s,d))\\}.$

We have fixed the images of all the objects of $\Delta_{k}(n,s,d)$. This set
is entirely determined by the morphisms of cochain complexes between the fixed
objects. But we know that the morphisms of objects in $\Delta_{k}(n,s,d)$ are
entirely determined by the ones between $i-1$ and $i$, for all $1\leq i\leq
n$. So we have the following isomorphisms:

$\phi_{*}^{-1}\simeq\\{(f_{i})_{1\leq i\leq n}/\
f_{i}:\Delta_{k}(n,s,d)(i-1,i)=k^{d_{i}}\left[s_{i}\right]\to
T(t_{i-1},t_{i})\\}\simeq\prod\operatorname{Hom}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],T(t_{i-1},t_{i})).$

We have proven the first isomorphism, reducing the computation of the set of
morphisms between these dg-categories to a computation of morphisms of
complexes. But we can go even simpler. Let us attack the second isomorphism.
For legibility’s sake, let us take a cochain complex $A\in\mathbf{C(k)}$, and
we’ll prove that

$\operatorname{Hom}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],A)\simeq
Z^{s_{i}}(A)^{d_{i}}.$

A morphism of complexes from $k^{d_{i}}\left[s_{i}\right]$ to $A$ is of the
form

${\ldots}$${0}$${k^{d_{i}}}$${0}$${\ldots}$${\ldots}$${A^{s_{i}-1}}$${A^{s_{i}}}$${A^{s_{i}+1}}$${\ldots}$$\scriptstyle{f_{s_{i}}}$

We can easily see that this morphism is determined solely by $f_{s_{i}}$. By
definition of a morphism of complexes and of cocycles, we have, then, that

$\operatorname{Hom}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],A)\simeq\\{f\in\operatorname{Hom}_{\mathbf{Mod}}(k^{d_{i}},A^{s_{i}})/\
d\circ f=0\\}\simeq\operatorname{Hom}_{\mathbf{Mod}}(k^{d_{i}},Z^{s_{i}}(A)).$

The only thing we have left to do is prove that
$\operatorname{Hom}_{\mathbf{Mod}}(k^{d_{i}},Z^{s_{i}}(A))\simeq
Z^{s_{i}}(A)^{d_{i}}$. We know that $k^{d_{i}}$ is a free $k$-module with a
basis consisting of $d_{i}$ generators, $e_{1},\ldots,e_{d_{i}}$. We define a
morphism

$F:\operatorname{Hom}_{\mathbf{Mod}}(k^{d_{i}},Z^{s_{i}}(A))\to
Z^{s_{i}}(A)^{d_{i}}$ $f\mapsto(f(e_{1}),\ldots,f(e_{d_{i}})).$

It is a classical result in Algebra that this is a bijection.

We have constructed the isomorphisms

$\phi_{*}^{-1}(x)\simeq\prod\operatorname{Hom}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],T(x_{i-1},x_{i}))\simeq\prod(Z^{s_{i}}(\operatorname{Hom}_{T}(x_{i-1},x_{i})))^{d_{i}},$

and we have finished this proof.
∎

Already we know that every dg-category is fibrant, but the objects in
$\Delta_{k}$ are also cofibrant. Let us get a computational lemma out of the
way first.

######  Lemma 5.2.5.

Let $g:A\to B$ be a trivial fibration in $\mathbf{dg-cat}$, let
$s_{i}\in\mathbb{Z}$ and $x_{i-1},x_{i}\in\operatorname{Obj}(A)$ two objects
of $A$. Then the induced morphism of sets

$Z^{s_{i}}(A(x_{i-1},x_{i}))\to Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))$

is surjective.

###### Proof.

It is just a question of diagram-chasing. Let $\alpha\in
Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))$. As $g$ is a fibration, in particular it is
surjective on the complex of morphisms, and there exists $\beta\in
A(x_{i-1},x_{i})$, but not necessarily a cocycle. At the same time, we know
that $g$ is a weak equivalence, so there is an isomorphism of cohomology
groups $H^{s_{i}}(A(x_{i-1},x_{i}))\simeq H^{s_{i}}(B(g(x_{i-1}),g(x_{i})))$,
which means that there exists a $\beta^{\prime}\in
Z^{s_{i}}(A(x_{i-1},x_{i}))$ such that $g(\beta^{\prime})$ is in the same
equivalence class as $\alpha$,

$g(\beta^{\prime})-\alpha=g(\beta^{\prime})-g(\beta)=g(\beta^{\prime}-\beta)\in{\text{Im}}d_{B,i}.$

There is, then, a $\gamma\in B^{s_{i}-1}(g(x_{i-1}),g(x_{i}))$ such that
$d(\gamma)=g(\beta-\beta^{\prime})$. On the other hand, using again the
surjectivity of $g$, we have a $\gamma^{\prime}\in A^{s_{i}-1}(x_{i-1},x_{i})$
such that $g(\gamma^{\prime})=\gamma$, and in consequence
$d(g(\gamma^{\prime}))=g(\beta-\beta^{\prime})$. By the linearity of $g$, we
have that $g(\beta^{\prime}-d(\gamma^{\prime}))=g(\beta)=\alpha$, and so
$\beta^{\prime}-d(\gamma^{\prime})$ is a preimage of $\alpha$ in
$Z^{s_{i}}(A(x_{i-1},x_{i}))$.

So the morphism is surjective and we have finished.
∎

######  Corollary 5.2.6.

All objects in $\Delta_{k}$ are cofibrant in the model structure on
$\mathbf{dg-cat}$.

###### Proof.

The case of $\Delta_{k}(0)$ is trivial: from Theorem 3.4.9 we know that
$\emptyset\to\Delta_{k}(0)$ is a generating cofibration, so in particular a
cofibration. The dg-category $\Delta_{k}(0)$ is cofibrant.

Let $n\in\mathbb{N}$, and let $s\in\mathbb{Z}^{n}$, $d\in\mathbb{N}^{n}$. We
want to prove that $\Delta_{k}(n,s,d)$ is cofibrant. For that, we will use the
description of a cofibration as a morphism which has the left lifting property
with respect to all trivial fibrations. Let $g:A\to B$ be a trivial fibration
for the model structure for $\mathbf{dg-cat}$, and $f:\Delta_{k}(n,s,d)\to B$
a dg-morphism. We want to find a lift for this commutative square:

${\emptyset}$${A}$${\Delta_{k}(n,s,d)}$${B}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{h}$

As $\emptyset$ is the initial object in $\mathbf{dg-cat}$, the existence of
this lift is equivalent to the following condition: for every trivial
fibration $g:A\to B$ and every $f:\Delta_{k}(n,s,d)\to B$ there exists a
factorization of $f$ through $g$, i.e. there exists a morphism
$h:\Delta_{k}(n,s,d)\to A$ such that $f=g\circ h$. This is, in turn,
equivalent to asking that

$g_{*}:\operatorname{Hom}(\Delta_{k}(n,s,d),A)\to\operatorname{Hom}(\Delta_{k}(n,s,d),B)$

is surjective.

We know that such a morphism is surjective on the objects. As such, we can fix
$n+1$ objects in $A$, $x_{0},\ldots,x_{n}\in A$. Fixing the objects allows us
to use the result in Proposition 5.2.4: we have reduced the problem to proving
that

$\prod_{x_{i-1},x_{i}\in
A}Z^{s_{i}}(A(x_{i-1},x_{i}))^{d_{i}}\to\prod_{x_{i-1},x_{i}\in
A}Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))^{d_{i}}$

is surjective. As we are taking the product and the powers over the same
families of indexes, proving that this morphism is surjective is equivalent to
asking that the morphism

$Z^{s_{i}}(A(x_{i-1},x_{i}))\to Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))$

is. But we have already proven that that is the case in Lemma 5.2.5.

The morphism $g_{*}:Z^{s_{i}}(A(x_{i-1},x_{i}))\to
Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))$ is surjective. The morphism
$\emptyset\to\Delta_{k}(n,s,d)$ is a cofibration, so $\Delta_{k}(n,s,d)$ is
cofibrant and we have finished.
∎

The computations for the morphism sets are not all we are going to need here.
Indeed, we will be using the mapping spaces in the construction of our
simplicial presheaf category, so it would be useful to have a similar result
to the one in Proposition 5.2.4 but for the mapping spaces. We will, luckily,
get one, but before that we will need a couple of preliminary results.

######  Lemma 5.2.7.

Let $n\in\mathbb{N}$, $s\in\mathbb{Z}^{n}$ and $d\in\mathbb{N}^{n}$. The
morphism $\phi:\coprod_{n}\Delta_{k}(0)\to\Delta_{k}(n,s,d)$ is a cofibration
in the model structure of $\mathbf{dg-cat}$.

###### Proof.

Once again, to prove that $\phi$ is a cofibration we use the fact that it is a
cofibration if and only if $\phi$ has the left lifting property with respect
to all trivial fibrations. This time we don’t have the luck to have the
initial object as our top corner, but we can still see this property as some
kind of surjectivity condition. Indeed, $\phi$ has the left lifting property
if and only if for all $g:A\to B$ trivial fibration the morphism

$g_{*}^{\prime}:\operatorname{Hom}(\Delta_{k}(n,s,d),A)\to\operatorname{Hom}(\coprod
k,A)\times_{\operatorname{Hom}(\coprod
k,B)}\operatorname{Hom}(\Delta_{k}(n,s,d),B)$

is surjective.

We want to use Proposition 5.2.4 to compute those $\operatorname{Hom}$, but we
only know how to compute $\operatorname{Hom}(\Delta_{k}(n,s,d),B)$ with fixed
objects. He have, then, to take the coproduct over all possible selections of
$n+1$ points. Like that, we get the following isomorphisms:

$\operatorname{Hom}(\coprod k,A)\times_{\operatorname{Hom}(\coprod
k,B)}\operatorname{Hom}(\Delta_{k}(n,s,d),B)\simeq\operatorname{Obj}(A)^{n+1}\times_{\operatorname{Obj}(B)^{n+1}}\coprod_{\operatorname{Obj}(B)^{n+1}}\prod
Z^{s_{i}}(B(x_{i-1},x_{i}))^{d_{i}}$
$\simeq\coprod_{\operatorname{Obj}(B)^{n+1}}\prod
Z^{s_{i}}(B(x_{i},x_{i+1}))^{d_{i}}.$

We have reduced the problem to proving that

$\coprod_{\operatorname{Obj}(A)^{n+1}}\prod
Z^{s_{i}}(A(x_{i-1},x_{i}))^{d_{i}}\to\coprod_{\operatorname{Obj}(B)^{n+1}}\prod
Z^{s_{i}}(B(g(x_{i-1}),g(x_{i})))^{d_{i}}$

is surjective. Once again, we can ignore the products and the coproducts in
this, and reduce the problem even more by taking those out: the question we
end up with whether for all $g:A\to B$ trivial cofibrations the induced
morphism on cocycles

$Z^{s_{i}}(A(x_{i},x_{i+1}))\to Z^{s_{i}}(B(g(x_{i}),g(x_{i+1})))$

is surjective. But that is exactly the result of Lemma 5.2.5, so we have our
result. The morphism $\phi:\leavevmode\nobreak\
\coprod_{n}\Delta_{k}(0)\to\Delta_{k}(n,s,d)$ is a cofibration and we have
finished.
∎

####  5.2.2 A characterization of Maps

######  Lemma 5.2.8.

Let $T$ be a dg-category. Let $n\in\mathbb{N}$, $s\in\mathbb{Z}^{n}$ and
$d\in\mathbb{N}^{n}$. There exists a homotopy pullback of simplicial sets of
the following form:

${\operatorname{Map}_{\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),T)}$${*}$${\operatorname{Map}_{\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),T)}$${\operatorname{Map}_{\mathbf{dg-
cat}}(\coprod_{n+1}k,T)}$$\scriptstyle{U^{\prime}}$${\ulcorner^{h}}$$\scriptstyle{\phi^{*}}$

where $\phi^{*}$ is the morphism induced by
$\phi:\coprod_{n+1}k\to\Delta_{k}(n,s,d)$ as above, and $U^{\prime}$ is
induced by the forgetful functor $U:\coprod_{n+1}k/\mathbf{dg-
cat}\to\mathbf{dg-cat}$.

In particular, $\operatorname{Map}_{\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),T)$ is the homotopy fiber of $\phi^{*}$.

###### Proof.

Let us take $C_{*}(T)$ a simplicial frame of $T$ in $\coprod k/\mathbf{dg-
cat}$. As $U$ is a right Quillen adjoint, the image $U(C_{*}(T))$ is a
simplicial frame of $U(T)=T$ in $\mathbf{dg-cat}$. By abuse of notation, we
will still denote $U(C_{*}(T))$ by $C_{*}(T)$. We have proven that both
$k=\Delta_{k}(0)$ and $\Delta_{k}(n,s,d)$ are cofibrant, so we don’t need to
take cofibrant replacements. With those choices made, the diagram from the
lemma could be rewritten by saying that there exists a homotopy pullback of
the form

${\operatorname{Hom}_{k\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),C_{*}(T))}$${*}$${\operatorname{Hom}_{\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),C_{*}(T))}$${\operatorname{Hom}_{\mathbf{dg-
cat}}(k\coprod
k,C_{*}(T)).}$$\scriptstyle{U^{\prime}}$$\scriptstyle{\phi^{*}}$

The simplicial set $\operatorname{Hom}_{\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),C_{*}(T))$ is literally the fiber of $\phi^{*}$. But
here we are asking for a homotopy fiber, not just a fiber. For that we will
use the properness of $\mathbf{sSet}$. We know by Corollary 3.2.38 that on a
right proper model category, if we have a diagram $X\to Z\leftarrow Y$ where
at least one of these two arrows is a fibration, then the pullback is
naturally weak equivalent to the homotopy pullback. We have proven in Lemma
5.2.7 that the morphism $\phi:\coprod_{n}\Delta_{k}(0)\to\Delta_{k}(n,s,d)$ is
a cofibration, and according to Proposition 3.2.55, if $\phi$ is a
cofibration, the morphism $\phi^{*}$ is a fibration. So we have a pullback
along a fibration, and by right properness of $\mathbf{sSet}$,
$\operatorname{Hom}_{k\coprod k/\mathbf{dg-cat}}(\Delta_{k}(n,s,d),C_{*}(T))$
is a homotopy pullback of $\phi^{*}$ along a point. As all dg-categories are
fibrant, we don’t need to get a fibrant replacement of this object to get a
homotopy fiber. The object $\operatorname{Hom}_{k\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(n,s,d),C_{*}(T))$ is a homotopy fiber of $\phi^{*}$ over a
point and we have our result. We have finished.
∎

######  Proposition 5.2.9.

Let $T$ be a dg-category and $\Delta_{k}(n,s,d)$ in $\Delta_{k}$. Let
$\phi_{*}:\operatorname{Hom}(\Delta_{k}(n,s,d),T)\to\operatorname{Hom}(\coprod\Delta_{k}(0),T)$
the morphism induced by the inclusion
$\phi:\coprod_{n+1}\Delta_{k}(0)\to\Delta_{k}(n,s,d)$ where we send each copy
of $\Delta_{k}(0)$ to a different object in $\Delta_{k}(n,s,d)$. Then, we have
the following weak equivalences in $\mathbf{sSet}$:

  * •

The mapping space from $\coprod\Delta_{k}(0)$ to $T$ is weak equivalent to the
cartesian product of $n+1$ copies of the nerve of $F(\Delta_{k}(0),T)$, the
category of quasi-representable $T^{op}$-modules with only weak equivalences
as its morphisms. In other words,

$\operatorname{Map}(\coprod\Delta_{k}(0),T)\simeq|T|\times\ldots\times|T|=\times_{n+1}N(F(\Delta_{k}(0),T)).$

  * •

Let $x\in\operatorname{Obj}(T)^{n+1}$ be an $(n+1)$-uple of objects. The
mapping space from $\Delta_{k}(n,s,d)$ to $(T,x)$ in $\coprod k/\mathbf{dg-
cat}$, the category of dg-categories with $n+1$ fixed objects, is weak
equivalent to the product over $1\leq i\leq n$ of the mapping spaces from
$k^{d_{i}}\left[s_{i}\right]$ to $T(x_{i-1},x_{i})$ in the category of cochain
complexes. In other words,

$\operatorname{Map}_{\coprod k/\mathbf{dg-
cat}}((\Delta_{k}(n,s,d)),(T,x))\simeq\prod_{i}\operatorname{Map}_{\mathbf{C(k)}}(k^{d_{i}}\left[s_{i}\right],T(x_{i-1},x_{i})).$

###### Proof.

The first weak equivalence is a direct consequence of Theorem 3.4.38. Indeed,
that result says that if we have two dg-categories, in our case $k$ and $T$,
the simplicial set $\operatorname{Map}(k,T)$ is weak equivalent to the nerve
of the category of quasi-representable $k\otimes T^{op}$-modules with
equivalences as morphisms. But $k$ is the unit for the tensor product, so
$F(k,T)$ is isomorphic to the category of quasi-representable $T^{op}$-modules
with the appropriate morphisms. The Cartesian product is trivial.

For the next part, we will prove it for $n=1$. Indeed, the reasoning stays the
same, and the writing is easier. In that case, we define a Quillen adjunction

${\mathbf{C(k)}}$${k\coprod k/\mathbf{dg-
cat}}$${E}$${\Delta_{k}(1,E)}$${T(x_{1},x_{2})}$${(T,x_{1},x_{2})}$$\scriptstyle{\xi}$$\scriptstyle{\beta}$

where $\Delta_{k}(1,E)$ denotes the dg-category with two objects, $k$ as the
cochain complexes of endomorphisms, and $E$ as the complex of morphisms
between the two objects. In particular,
$\xi(k^{d}\left[s\right])=\Delta_{k}(1,s,d)$.

Let us take $C_{*}(T)$ a simplicial frame in $k\coprod k/\mathbf{dg-cat}$. As
$\beta$ is a right Quillen functor, the cochain complex $\beta(C_{*}(T))$ is
still a simplicial frame. We have already proven in Corollary 5.2.7 that all
$\Delta_{k}$ are cofibrant, and in Corollary 3.4.10 that all
$\beta(\Delta_{k}(1,s,d)=k^{d}\left[s\right]$ are too. By definition of an
adjunction, we have that

$\operatorname{Map}_{\mathbf{C(k)}}(k^{d}\left[s\right],T)=\operatorname{Hom}_{\mathbf{C(k)}}(k^{d}\left[s\right],\beta(C_{*}(T)))\simeq\operatorname{Hom}_{k\coprod
k/\mathbf{dg-cat}}(\Delta_{k}(1,s,d),C_{*}(T))=\operatorname{Map}_{k\coprod
k/\mathbf{dg-cat}}(\Delta_{k}(1,s,d),T)$

We have the weak equivalence and we have finished.
∎

Finally, this last result gives us a way to characterize weak equivalences in
$\mathbf{dg-cat}$ using elements in $\Delta_{k}$. In fact, we don’t even need
the entirety of the objects in $\Delta_{k}$: it suffices with the ones with
two objects and no degree.

######  Theorem 5.2.10.

Let $T$ and $T^{\prime}$ be two dg-categories and $f:T\to T^{\prime}$ a dg-
functor between them. Then $f$ is a weak equivalence in $\mathbf{dg-cat}$ if
and only if the induced morphisms

$\operatorname{Map}(\Delta_{k}(0),f):\operatorname{Map}_{\mathbf{dg-
cat}}(\Delta_{k}(0),T)\to\operatorname{Map}_{\mathbf{dg-
cat}}(\Delta_{k}(0),T^{\prime})$ $\forall s\in\mathbb{Z},\ \
\operatorname{Map}(\Delta_{k}(1,s,1),f):\operatorname{Map}_{\mathbf{dg-
cat}}(\Delta_{k}(1,s,1),T)\to\operatorname{Map}_{\mathbf{dg-
cat}}(\Delta_{k}(1,s,1),T^{\prime})$

are weak equivalences in $\mathbf{sSet}$.

###### Proof.

One implication is trivial. According to Proposition 3.2.56, if $Y$ is a
fibrant object then $\operatorname{Map}(-,Y)$ is a right Quillen functor, and
in particular it preserves weak equivalences between fibrant objects. But in
$\mathbf{dg-cat}$ every object is fibrant, so if $f$ is a weak equivalence
then both $\operatorname{Map}(\Delta_{k}(0),f)$ and
$\operatorname{Map}(\Delta_{k}(1,s,1),f)$ are weak equivalences.

Now let us prove the other implication. Assume that both
$\operatorname{Map}(\Delta_{k}(0),f)$ and
$\operatorname{Map}(\Delta_{k}(1,s,1),f)$ for all $s\in\mathbb{Z}$ are weak
equivalences in $\mathbf{sSet}$, and let us prove that that’s enough to get
that the original dg-functor $f$ is a weak equivalence. A dg-functor in
$\mathbf{dg-cat}$ is a weak equivalence if it is quasi-essentially surjective
and quasi-fully faithful.

A dg-functor $f$ is quasi-essentially surjective if the induced map
$\left[f\right]:\left[T\right]\to\left[T^{\prime}\right]$ is essentially
surjective. That is equivalent to proving that the map
$(\left[T\right]/\cong)\to)\left[T^{\prime}\right]/\cong)$ between the
isomorphism classes of the homotopy categories is surjective. We have assumed
that the map
$\operatorname{Map}(k,f):\operatorname{Map}(k,T)\to\operatorname{Map}(k,T^{\prime})$
is a weak equivalence in $\mathbf{sSet}$. By definition of a weak equivalence
in $\mathbf{sSet}$, that means that for all $i\in\mathbb{N}$ and for all point
$x\in T$ the induced morphism $\pi_{i}(\operatorname{Map}(k,f),x)$ is an
isomorphism. In particular, it is an isomorphism if $i=0$. But we have seen in
Corollary 3.4.39 that the set
$\left[k,T\right]\simeq\pi_{0}(\operatorname{Map}(k,T),x)$ is isomorphic to
the set of classes of isomorphisms in $\left[T\right]$. We have then that the
map

$(\left[T\right]/\cong)\simeq\pi_{0}(\operatorname{Map}(k,T),x)\to\pi_{0}(\operatorname{Map}(k,T^{\prime}),f(x))\simeq(\left[T^{\prime}\right]/\cong)$

is not only surjective, but an isomorphism. The dg-functor $f$ is quasi-
essentially surjective.

A dg-functor $f$ is quasi-fully faithful if for all $x,y\in T$ the induced
morphism $T(x,y)\to T^{\prime}(f(x),f(y))$ is weak equivalence in
$\mathbf{C(k)}$. That is, if we want to prove that $f$ is quasi-fully faithful
we need to prove that for all $i\in\mathbb{Z}$ and for all $x,y\in T$ the
induced morphism $H^{i}(f):H^{i}(T(x,y))\to H^{i}(T^{\prime}(f(x),f(y)))$ is
an isomorphism of groups. We have proven in Proposition 5.2.8 that we have a
fibration $E=\operatorname{Map}(\Delta_{k}(1,s,1),T)\to
B=\operatorname{Map}(k\coprod k,T)$ which has $F=\operatorname{Map}_{k\coprod
k/\mathbf{dg-cat}}(\Delta_{k}(1,s,1),T)$ as its homotopy fiber. That gives us
a long exact sequence of the form

${\ldots}$${\pi_{n}(F,x)}$${\pi_{n}(E,x)}$${\pi_{n}(B,x)}$${\pi_{n-1}(F,x)}$${\ldots}$${\ldots}$${\pi_{1}(E,x)}$${\pi_{1}(B,x)}$${\pi_{0}(F,x)}$

and another one of the same form for
$E^{\prime}=\operatorname{Map}(\Delta_{k}(1,s,1),T^{\prime})\to
B^{\prime}=\operatorname{Map}(k\coprod k,T^{\prime})$ with the fiber
$F^{\prime}=\operatorname{Map}_{k\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(1,s,1),T^{\prime})$. By using the morphisms
$\operatorname{Map}(\Delta_{k}(1,s,1),f)$ and $\operatorname{Map}(k,f)$ we can
induce a morphism of complexes of the form

${\ldots}$${\pi_{n}(F,x)}$${\pi_{n}(E,x)}$${\pi_{n}(B,x)}$${\pi_{n-1}(F,x)}$${\ldots}$${\ldots}$${\pi_{n}(F^{\prime},x)}$${\pi_{n}(E^{\prime},x)}$${\pi_{n}(B^{\prime},x)}$${\pi_{n-1}(F^{\prime},x)}$${\ldots}$

By definition of a weak equivalence in $\mathbf{sSet}$, the morphisms
$\pi_{i}(\operatorname{Map}(\d{(}1,s,1),f),x)$ and
$\pi_{i}(\operatorname{Map}(k,f),x)$ are isomorphisms for all
$i\in\mathbb{Z}$. We have a morphism of long exact sequences which are
isomorphisms in two out of every three. By the Five Lemma, that means that
$\pi_{i}(\operatorname{Map}_{k\coprod k/\mathbf{dg-
cat}}(\Delta_{k}(1,s,1),f),x)$ is also an isomorphism for all $i\geq 1$. But
we have already computed the simplicial sets $\operatorname{Map}_{k\coprod
k/\mathbf{dg-cat}}(\Delta_{k}(n,s,d),T)$ in Proposition 5.2.9; we have that

$\operatorname{Map}_{k\coprod k/\mathbf{dg-
cat}}((\Delta_{k}(1,s,1),(0,1)),(T,(x_{0},x_{1})))\simeq\operatorname{Map}_{\mathbf{C(k)}}(k\left[s\right],T(x_{0},x_{1})).$

We know that for all $i\geq 0$,

$\pi_{i}(\operatorname{Map}_{\mathbf{C(k)}}(k\left[s\right],T(x_{0},x_{1})),x)\simeq
H^{-i-s}(T(x_{0},x_{1})),$

So if we put the isomorphisms next to each other, we have, for all
$s\in\mathbb{Z}$ and all $i\geq 1$ that

${H^{-i-s}(T(x_{0},x_{1}))\simeq\pi_{i}(\operatorname{Map}_{k\coprod
k/\mathbf{dg-
cat}}(\Delta_{k}(1,s,1),T),x)}$${\ldots}$${\longrightarrow\pi_{i}(\operatorname{Map}_{k\coprod
k/\mathbf{dg-cat}}(\Delta_{k}(1,s,1),T^{\prime}),f(x))\simeq
H^{-i-s}(T^{\prime}(f(x_{0}),f(x_{1})))}$$\scriptstyle{\sim}$

and the morphism $f:T\to T^{\prime}$ induces an isomorphism between cohomology
groups of the form $H^{-i-s}$. As we have that isomorphism for every
$s\in\mathbb{Z}$, by making $s$ vary we have the isomorphism on cohomology
groups for every degree. The morphism $f$ is quasi-fully faithful.

The morphism $f:T\to T^{\prime}$ is a weak equivalence in $\mathbf{dg-cat}$,
and we have finished.
∎

####  5.2.3 Expected result

It was at this moment when we realized that we would not be able to use the
results in Section 4.4 for $\Delta_{k}$. Indeed, the conditions on Theorem
4.4.6 are too strong: the linear simplex category is not closed for finite
coproducts. But just because it isn’t possible to use the hypercover results
as we have put them it doesn’t mean we cannot do it some other way. Indeed,
free dg-categories have already very little relationships between the morphism
complexes, so it’s not that much of a stretch to consider that we could
potentially rewrite complete dg-Segal spaces as some kind of simplicial
functors from $\Delta_{k}$ to $\mathbf{sSet}$.

######  Hypothesis 5.2.11.

There exists a full subcategory of
$\operatorname{Fun}^{\mathbb{S}}(\Delta_{k}^{op},\mathbf{sSet})$ that is
equivalent to $\mathbf{dg-Segal_{c}}$.

###  5.3 Other possible prospects

Although the last two sections are the most obvious and direct prospects from
this work, they are not the only ones. We add here some other possible
applications and questions.

####  5.3.1 Automorphisms of dg-cat

At the end of his paper [Toë05], Toën used his result about complete Segal
spaces to compute the group of automorphisms of $\infty$-categories, which
turned out to be just $\mathbb{Z}/2\mathbb{Z}$, containing only the
dualisation as a non-identity automorphism. We could try to do the same thing
for $\mathbf{dg-cat}$.

Indeed, it makes sense to attempt to use our construction for this: if we
think of it as a kind of "presentation by generators and relations", where the
category of free dg-categories $c\mathcal{L}_{\mathbb{S}}$ (or $\Delta_{k}$,
if we have already proven the hypothesis in the last section) would be the
generators and the weak equivalences we described as the conditions on
complete dg-categories, it is common practice to define the morphisms from
$\mathbf{dg-cat}$ to another model category $C$ by looking at the morphisms
from $c\mathcal{L}_{\mathbb{S}}$ to $C$ that turn the relations into weak
equivalences. This method could work with any category, but in particular we
can use it to see the automorphisms of $\mathbf{dg-cat}$.

Sadly, in our case the automorphism group of $\mathbf{dg-cat}$ will not be as
simple as the one for $\infty$-categories. It will contain
$\mathbb{Z}/2\mathbb{Z}$, of course, as the duality is still an invertible
morphism that turns dg-categories into dg-categories, but it won’t be enough.
As we are working with $k$-linear categories, it is not unexpected that the
automorphisms of the ring $k$ will be included in the group of automorphisms,
and maybe some other groups too.

It would also be interesting to explore what the higher homotopy groups of
$Aut(\mathbf{dg-cat})$ would be: we expect the Hochschild homology to appear
at some point.

####  5.3.2 Hypercover definitions

As any invested reader will have noticed, even though the definitions are
different, the construction of $M_{0}$-hypercovers and of dg-Segal hypercovers
follow the exact same structure. In the same vein, the definitions might not
be the same, but they follow the same pattern: we have a definition of some
kind of epimorphism involving the Map functor, and then we ask that for all
$n\in\mathbb{N}$,

$\Phi_{n}:A_{n}\to A^{\partial\Delta^{n}}$

is that kind of epimorphism. This is not a particularity of the hypercovers we
have defined here. We have already seen something similar in the case of
hypercovers in simplicial sets at Definition 4.4.1 and in [Lur09] Lurie
defines a hypercover over an $\infty$-topos to be an augmented simplicial
object such that $\Phi_{n}$ is an effective epimorphism for all $n$. With so
many similar definitions around, there is a natural question that arises: is
it possible to find a general definition that combines all of these?

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