Let $\A$ and $\B$ be two $A_{\infty}$-categories. An $A_{\infty}$-functor $F:\A\rightarrow \B$ is the datum of

* a map $F:\obj (\A)\rightarrow \obj (\B)$,

* for all $n\geq 1$ and all $A_0$, …, $A_n\in\A$ a graded map

\[
F_n: (A_{n-1},A_n)\otimes (A_{n-2}, A_{n-1})\otimes \cdots \otimes (A_0,A_1)\rightarrow \Hom_{\B}(FA_0,FA_n)
\]

of degree $1-n$

such that

\[
\sum (-1)^{r+st}F_u(\boldsymbol{1}^{\otimes r}\otimes m_s\otimes \boldsymbol1^{\otimes t})=\sum (-1)^sm_r(F_{i_1}\otimes F_{i_2}\otimes \cdots \otimes F_{i_r})
\]

where the first sum runs over all decompositions $n=r+s+t$, we put $u=r+1+t$, and the second sum runs over all $1\leq r\leq n$ and all decompositions $n=i_1+\cdots+i_r$; the sign on the right hand side is given by

\[
\]

The $A_{\infty}$-functor $F$ is strict if we have $F_n=0$ for all $n\geq 2$. Clearly the above datum with $F_n=0$ for $n\geq 2$ defines a strict $A_{\infty}$-functor if and only if for each $k\geq 1$, we have

\[
F_1m_k=m_k(F_1\otimes \cdots\otimes F_1).
\]

Let $\A$ and $\B$ be two $A_{\infty}$-categories with strict identities. An $A_{\infty}$-functor $F$ is strictly unital if

\[
\]

\[
F_i(\cdots\otimes e_A\otimes\cdots)=0
\]

for all $i\geq 2$.

The following simple example, due to M. Kontsevich, shows that already between dg categories, $A_{\infty}$-functors are quite interesting.
[[93, 28]]
Let $ I_{2}$ be the $k$-category generated by the category $J_2$ with $2$ objects and precisely one morphism with any given source and target

\[
\begin{tikzcd}
1\ar[r,bend left=4ex,"a"]&2.\ar[l,bend left=4ex,"b"]
\end{tikzcd}
\]

We consider it as a dg category concentrated in degree 0. Let $\A$ be a dg category and $X,Y$ two objects in $\A$.

We will show that $X$ and $Y$ are homotopy equivalent if and only if there exists a strictly unital $A_{\infty}$-functor from $I_2$ to $\A$ such that $1$ is sent to $X$ and $2$ is sent to $Y$.

Let $\mathcal K$ be the Kontsevich dg category defined as the dg $k$-path category given by

\[
\begin{tikzcd}
1\ar[r,bend left=4ex,"a"]\ar[r,bend left=12ex,"r"]\ar[loop left,looseness=10,"h_1"] &2\ar[l,bend left=4ex,"b"]\ar[loop right, looseness=10, "h_2"]
\end{tikzcd}
\]

where $|a|=|b|=0$, $|h_1|=-1=|h_2|$, $|r|=-2$ and $d(a)=d(b)=0$, $d(h_1)=1-ba$, $d(h_2)=1-ab$, $d(r)=-ah_1+h_2a$.
By <cit.>, $\mathcal K$ is a cofibrant resolution of $I_2$.
In particular we have

\begin{align}
\Ext^{n}_{\mathcal K}(i,j)=0\label{Kont}
\end{align}

for $i,j\in\{1,2\}$ and $n\neq 0$.

Suppose $X$ and $Y$ are homotopy equivalent.
Then we have a homotopy equivalence $f: X\rightarrow Y$ and a homotopy inverse $g:Y\rightarrow X$.
By definition there exist $\tilde{h_1}:X\rightarrow X$ and $\tilde{h_2}: Y\rightarrow Y$ of degree $-1$ such that $d(\tilde{h_1})=1-gf$ and $d(\tilde{h_2})=1-fg$.
Set ${h_1}=\tilde{h_1}+g\tilde{h_2}f-gf\tilde{h_1}$ and ${h_2}=\tilde{h_2}$. Let $r=\tilde{h_2}\tilde{h_2}f-\tilde{h_2}f\tilde{h_1}$.
Then $d({h_1})=d(\tilde{h_1})$ and $-f{h_1}+{h_2}f=d(r)$.
Thus $(f,g,h_1,h_2,r)$ defines a dg functor $G: \mathcal K \rightarrow \A$.

Notice that $m_i$ vanishes in $I_2$ for $i\neq 2$ and in $\A$ for $i\geq 3$.
When $i=2$, $m_2$ produces either identities or zeros in $I_2$.
Thus for an $A_{\infty}$-functor $F:I_2\rightarrow \A$, the conditions become

\begin{align}
d(F_1(a))&=0, d(F_1(b))=0\nonumber\\
1=F_1(b)F_1(a)+d(F_2(b,a)), 1&=F_1(a)F_1(b)+d(F_2(a,b))\nonumber\\
\sum (-1)^{r}F_{n-1}(\boldsymbol1^{\otimes r}\otimes m_2\otimes \boldsymbol1^{\otimes t})&=d\circ(F_n)+\sum(-1)^{i_1-1}m_2(F_{i_1}\otimes F_{i_2}) \label{A3}
\end{align}

where $n\geq 3$, $r+t=n-2$ and $i_1+i_2=n$.
When acting on $(a_1,\cdots, a_n)$, the conditions become trivial for $n\geq 3$ if some $a_i$ is an identity.
When each $a_i$ is non identity, $m_2$ in each summand of the left hand side will produce an identity and hence becomes zero by $F_{n-1}$.
Thus we only need to take care of $F_n(a_1,a_2,\cdots, a_n)$ with $a_i\in\{a, b\}$ and $a_i\neq a_{i+1}$.

We define an $A_{\infty}$-functor $F:I_2\rightarrow \A$ as follows: the value of $F_n$ on a sequence $(a_1,a_2,\cdots, a_n)$, where $a_i\in\{a,b\}$ and $a_i\neq a_{i+1}$, is the image under the dg functor $G$ of an element $\tilde{F_n}(a_1,a_2,\cdots,a_n)$, which is constructed by induction, in $\mathcal K(i,j)$ with $(i,j)$ depending on the sequence $(a_1,a_2,\cdots,a_n)$.
More precisely

\begin{align*}
\tilde{F_1}(a)&=a,\tilde{F_1}(b)=b\\
\tilde{F_2}(a,b)&=h_2, \tilde{F_2}(b,a)=h_1.\\
\end{align*}

By induction we have

\begin{align}
\sum(-1)^{i_1}d\circ m_2(\tilde{F_{i_1}}\otimes \tilde{F_{i_2}})(a_1,a_2,\cdots,a_n)=0.\label{Ainf}
\end{align}

where $(i_1,i_2)$ runs through the pair such that $i_1+i_2=n$. Here $(\tilde{F_{i_1}}\otimes \tilde{F_{i_2}})(a_1,a_2,\cdots,a_n)$ is understood as $\tilde{F_{i_1}}(a_1,\cdots,a_{i_1})\otimes \tilde{F_{i_2}}(a_{i_1+1},\cdots,a_n)$.

By <ref>, there exists an element $\tilde{F_n}(a_1,a_2,\cdots,a_n)\in \mathcal K(i,j)^{1-n}$ whose coboundary is the cocycle

\[
\sum(-1)^{i_1}\tilde{F_{i_1}}(a_1,\cdots,a_{i_1})\circ\tilde{F_{i_2}}(a_{i_1+1},\cdots,a_n)
\]

which appears on the left hand side of <ref>.

For example we may put

\[
\tilde{F_3}(b,a,b)=-brb+bh_2^2+h_1^2b-h_1bh_2.
\]

The Kontsevich dg category $\mathcal K$ is quasi-equivalent to the dg $k$-path category $\C$ given by the differential graded quiver

\[
\begin{tikzcd}
1\ar[r,bend left=4ex,"a"]\ar[loop left,looseness=10,"h_1"] &2\ar[l,bend left=4ex,"b"]\ar[loop right, looseness=10, "h_2"]\ar[l,bend right=12ex,"b'"swap]
\end{tikzcd}
\]

where $|a|=|b|=|b'|=0$, $|h_1|=-1=|h_2|$, and $d(a)=d(b)=d(b')=0$, $d(h_1)=1-ba$, $d(h_2)=1-ab'$.
A quasi-equivalence $F:\mathcal K\rightarrow \C$ is given as follows: on the objects we have $F(1)=1$ and $F(2)=2$ and on the arrows we have $F(a)=a$, $F(b)=b'$, $F(h_2)=h_2$,

\begin{align*}
F(h_1) &=h_1+b'(h_2a-ah_1)+(1-b'a)(bh_2-h_1b')a \\
F(r)     &=h_2h_2a-h_2a(h_1+(bh_2-h_1b')a).
\end{align*}

We call this dg category the Cisinski dg category because it is the image of Cisinski's interval $J$ (cf. <cit.>) under the functor $\Lambda$ (cf. <ref>).

The datum of a dg functor $F:\mathcal K\rightarrow \A$ is equivalent to a morphism $\alpha:X_1\rightarrow X_2$ closed of degree $0$ in $\A$ together with a contracting homotopy for $\Cone(\alpha^{\wedge}:X_1^{\wedge}\rightarrow X_2^{\wedge})$, while the datum of a dg functor $G:\C\rightarrow \A$ is equivalent a morphism $\alpha:X_1\rightarrow X_2$ closed of degree $0$ in $\A$ together with null-homotopies of the maps $[0,\;1]^{\intercal}: X_2^{\wedge}\rightarrow \Cone(\alpha^{\wedge})$ and $[1,\;0]:\Cone(\alpha^{\wedge})\rightarrow \Sigma X_1^{\wedge}$.

Let $\B$ be a small $A_{\infty}$-category and $\A$ an $A_{\infty}$-category.
It is known, cf. [93], that there is a natural $A_{\infty}$-category $\Fun_{\infty}(\B,\A)$ of strictly unital $A_{\infty}$-functors between $\B$ and $\A$.
It was constructed in <cit.> using twists of $A_{\infty}$-structures.
Keller further developed an idea in [105] and interpreted it in <cit.> as an internal $\Hom$-object in the tensor category of cocomplete augmented dg cocategories.
Note that when the target $\A$ is a dg category, the $A_\infty$-category $\Fun_{\infty}(\B,\A)$ is again a dg category.
The reason is that we use the $m_j^{\A}$, $j\geq i$, to define the map $m_i$ of the $A_{\infty}$-category
$\Fun_{\infty}(\B,\A)$ and when $\A$ is a dg category, then the $m_{j}^{\A}$ vanish for all $j\geq 3$.

The dg category $\Fun_{\infty}(k\Mor,\A)$ is isomorphic to the dg morphism category $\Mor(\A)$ introduced in <cit.>. More precisely, $\Mor(\A)$ has as objects the closed morphisms $f:A\rightarrow B$ of degree 0 in $\A$. The morphism complex $\Mor(\A)((A,B,f),(A',B',f'))$ has as the degree $m$ component the space of matrices
\end{pmatrix}$
where the morphisms $j:A\rightarrow A'$, $l:B\rightarrow B'$ are of degree $m$ and the morphism $h:A\rightarrow B'$ is of degree $m-1$. The differential $d$ carries a morphism
\end{pmatrix}$
\end{pmatrix}$.
The composition map is the multiplication of matrices.

Now let $\A$ be a dg category and $\B$ be the dg $k$-path category of the following graded quiver with relations

\begin{equation}\label{3term}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

where $|a|=|b|=0$, $d(a)=0$, $d(b)=0$ and $ba=0$.

Our aim is to describe the dg category $\Fun_{\infty}(\B,\A)$ following the construction in [95].

By definition, it has strictly unital $A_{\infty}$-functors $F:\B\rightarrow\A$ as objects.
A strictly unital $A_{\infty}$-functor $F:\A\rightarrow \B$ is given by the datum of

* objects $F(i)=A_i$ in $\A$ with $i=0,1,2$;

* graded maps

\[
F_1:\Hom_{\B}(i,j)\rightarrow \Hom_{\A}(A_i,A_j)
\]

of degree $0$ where $i\leq j$ and a graded map

\[
F_2:\Hom_{\B}(1,2)\otimes\Hom_{\B}(0,1)\rightarrow \Hom_{\A}(A_0,A_2)
\]

of degree $-1$ such that $F_1$ sends identities in $\B$ to identities in $\A$ and

\[
F_1\circ d_{\B}=d_{\A}\circ F_1
\]

\[
F_1m_2-F_2(\boldsymbol1\otimes d_{\B})-F_2(d_{\B}\otimes \boldsymbol1)=m_2(F_1\otimes F_1)+d_{\A}F_2.
\]

Let $f=F_1(a)$, $j=F_1(b)$ and $h=F_2(b,a)$. Then $(f,j,h)$ defines an $A_{\infty}$-functor $\B\rightarrow \A$ if and only if $|f|=|j|=0$, $|h|=-1$ and $d(f)=0$, $d(j)=0$ and $d(h)=-jf$.

Thus we identify a strictly unital $A_{\infty}$-functor $F:\B\rightarrow\A$ with the corresponding diagram

\begin{equation}\label{F}
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[rr,bend right = 8ex,"h"swap]&A_1\ar[r,"j"]&A_2
\end{tikzcd}
\end{equation}

where $|f|=|j|=0$, $|h|=-1$ and $d(f)=0$, $d(j)=0$ and $d(h)=-jf$.

Let $F$ and $F'$ be two strictly unital $A_{\infty}$-functors from $\B$ to $\A$. We identify them with their corresponding diagrams. Let $e_i$ be the unit of the object $i$ in $\B$. An element $H$ of degree $n$ in the graded space $\Hom(F,F')$ of strictly unital morphisms is given by graded morphisms

\[
H_0: {k}\cdot e_i\rightarrow \Hom_{\A}(A_i,A'_i)
\]

of degree $n$ where $i=0,1,2$, graded morphisms

\[
H_1:\Hom_{\B}(i,j)\rightarrow \Hom_{\A}(A_{i},A'_{j})
\]

of degree $n-1$ where $i< j$ and graded morphisms

\[
H_2:\Hom_{\B}(1,2)\otimes\Hom_{\B}(0,1)\rightarrow \Hom_{\A}(A_0,A_2')
\]

of degree $n-2$.

Let $h_i=H_0(e_i)$, $s_1=H_1(a)$, $s_2=H_1(b)$ and $t=H_2(b,a)$. We identify a strictly unital morphism $H$ of degree $n$ with the following diagram

\begin{equation}\label{mordegreen}
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[d,"h_0"swap]\ar[rd,"s_1"red,swap ,red]\ar[rrd,"t"blue,blue]\ar[rr,bend left = 8ex,"h"]&A_1\ar[d,"h_1"swap]\ar[rd,"s_2"red,red]\ar[r,"j"]&A_2\ar[d,"h_2"]\\
&A_0'\ar[r,"f'"swap]\ar[rr,bend right = 8ex,"h'"swap]&A_1'\ar[r,"j'"swap]&A_2'
\end{tikzcd}
\end{equation}

where $|h_i|=n$ for $i=0,1,2$, $|s_1|=|s_2|=n-1$, $|t|=n-2$.

Let $H\in \Hom^n(F,F')$. Denote by $H'{\coloneqq}d(H)\in \Hom^{n+1}(F,F')$. It is given as follows

\begin{align*}
H'_{0} & =d_{\A}\circ H_0, \\
H_1' & =d_{\A}\circ H_1+(-1)^{n+1}m_2^{\A}(F_1'\otimes H_0)+(-1)^{n}m_2^{\A}(H_0\otimes F_1), \\
H_2' & =d_{\A}(H_2)+(-1)^{n}m_2^{\A}(F_1'\otimes H_1) +m_2^{\A}(F_2'\otimes H_0)\\
& \quad\quad +(-1)^nm_{2}^{\A}(H_1\otimes F_1)+(-1)^{n+1}m_2^{\A}(H_0\otimes F_2).
\end{align*}

Via the identification with diagrams, we see that the corresponding diagram for $d(H)$ is

\[
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[d,"h_0'"swap]\ar[rd,"s_1'"red,swap ,red]\ar[rrd,"t'"blue,blue]\ar[rr,bend left = 8ex,"h"]&A_1\ar[d,"h_1'"swap,near end]\ar[rd,"s_2'"red,red]\ar[r,"j"]&A_2\ar[d,"h_2'"]\\
&A_0'\ar[r,"f'"swap]\ar[rr,bend right = 8ex,"h'"swap]&A_1'\ar[r,"j'"swap]&A_2'
\end{tikzcd}
\]

\[
\]

for $i=0,1,2$, and

\[
s_1'=d(s_1)+(-1)^{n+1}(f'\circ h_0)+(-1)^{n}h_1\circ f,
\]

\[
s_2'=d(s_2)+(-1)^{n+1}(j'\circ h_1)+(-1)^{n}h_2\circ j
\]

\[
t'=d(t)+(-1)^{n}(j'\circ s_1)+h'\circ h_0+(-1)^{n}(s_2\circ f)+(-1)^{n+1}(h_2\circ h).\]

Clearly $d^2(H)=0$.

We now describe the compositions.

Let $F$, $F'$ and $F''$ be $A_{\infty}$-functors from $\B$ to $\A$. Let $H'\in \Hom^r(F,F')$ and $H''\in \Hom^s(F',F'')$ be two morphisms in the dg category $\Fun_{\infty}(\B,\A)$.

Assume the corresponding diagram is given by

\[
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[d,"h_0'"swap]\ar[rd,"s_1'"red,swap ,red]\ar[rrd,"t'"blue,blue]\ar[rr,bend left = 8ex,"h"]&A_1\ar[d,"h_1'"swap,near end]\ar[rd,"s_2'"red,red]\ar[r,"j"]&A_2\ar[d,"h_2'"]\\
&A_0'\ar[r,"f'"]\ar[d,"h_0''"swap]\ar[rrd,"t''"blue,blue]\ar[rd,"s_1''"red,swap,red] &A_1'\ar[rd,"s_2''"red, red]\ar[d,"h_1''"swap, near end]\ar[r,"j'"]&A_2'\ar[d,"h_2''"]\\
&A_0''\ar[r,"f''"swap]\ar[rr,bend right = 8ex,"h''"swap]&A_1''\ar[r,"j''"swap]&A_2''
\end{tikzcd}
\]

where, in order to keep the diagram readable, we omit the arrow $h':A_0'\rightarrow A_2'$.

We denote by $\tilde{H}\in \Hom^{r+s}(F,F'')$ the composition of $H$ and $H'$.
Then the corresponding diagram is given as follows:

\[
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[d,"\tilde{h_0}"swap]\ar[rd,"\tilde{s_1}"red,swap ,red]\ar[rrd,"\tilde{t}"blue,blue]\ar[rr,bend left = 8ex,"h"]&A_1\ar[d,"\tilde{h_1}"swap,near end]\ar[rd,"\tilde{s_2}"red,red]\ar[r,"j"]&A_2\ar[d,"\tilde{h_2}"]\\
&A_0''\ar[r,"f''"swap]\ar[rr,bend right = 8ex,"h''"swap]&A_1''\ar[r,"j''"swap]&A_2''
\end{tikzcd}
\]

\[
\tilde{h_i}=h_i''h_i'
\]

for $i=1,2,3$, and

\[
\tilde{s_1}=h_1''s_1'+(-1)^{r}s_1''h_0',
\]

\[
\tilde{s_2}=h_2''s_2'+(-1)^{r}s_2''h_1'
\]

\[
\tilde{t}=h_2''t'+t''h_0'+(-1)^{r+1}s_2''s_1'.
\]

Note that in the expression of $\tilde{t}$, we only have length $2$ paths with suitable signs.
This is due to the vanishing of $m_j^{\A}$ for $j\geq 3$.

Let $\A$ be a dg category.
The homotopy category $\mathcal H_{3t}(\A)$ of 3-term homotopy complexes (= 3-term $h$-complexes, or 3-term complexes up to homotopy) over $\A$ is defined to be $H^0\Fun_{\infty}(\B,\A)$ where $\B$ is the dg category <ref>.
More precisely, its objects are given by diagrams $X$ in $\A$ of the form

\[
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[rr,bend right = 8ex,"h"swap]&A_1\ar[r,"j"]&A_2
\end{tikzcd}
\]

where $|f|=|j|=0$, $|h|=-1$ and $d(f)=0$, $d(j)=0$ and $d(h)=-jf$.

Suppose $X'$ is given by the diagram with objects and morphisms with superscript a prime symbol.

A morphism from $X$ to $X'$ is given by a homotopy equivalence class of 6-tuples $(h_i,s_j,t)$, $i=0,1,2$, $j=1,2$ where

\[
h_i:A_i\rightarrow A_i', |h_i|=0, d(h_i)=0,
\]

\[
s_j:A_{j-1}\rightarrow A_{j}', |s_j|=-1, d(s_1)=f'h_0-h_1f,
\]

\begin{equation}
d(s_2)=j'h_1-h_2j, \label{s2} %\tag{}
\end{equation}

\begin{equation}
t:A_0\rightarrow A_2', |t|=-2, d(t)=h_2\circ h-h'\circ h_0-s_2\circ f-j'\circ s_1.\label{tt}
\end{equation}

Two 6-tuples $(h_i,s_j,t)$ and $(h_i',s_j',t')$ are homotopy equivalent if there exists a 6-tuple $(h''_i,s_j'',t'')$ such that

\[
|h_i''|=-1, h_i-h_i'=d(h_i''),
\]

\[
|s_j''|=-2, s_1-s_1'=d(s_1'')+f'\circ h_0''-h_1''\circ f,
\]

\[
s_2-s_2'=d(s_2'')+j'\circ h_1''-h_2''\circ j,
\]

\[
|t''|=-3, t-t'=d(t'')-j'\circ s_1''+h'\circ h_0''-s_2''\circ f+h_2''\circ h.
\]

The composition of two morphisms given by homotopy equivalence classes of 6-tuples $(h_i',s_j',t'):F\rightarrow F'$ and $(h_i'',s_j'',t''):F'\rightarrow F'$, is the morphism given by the 6-tuple $(\tilde{h}_i,\tilde{s}_j,\tilde{t}):F\rightarrow F''$ where

\begin{align*}
\tilde{h}_i  &=h_i''h_i' ,\\
\tilde{s}_1 &=h_1''s_1'+s_1''h_0', \\
\tilde{s}_2 &=h_2''s_2'+s_2''h_1' \\
\tilde{t}      &=h_2''t'+t''h_0'-s_2''s_1'.
\end{align*}

The identity morphism for $X$ is given by the homotopy equivalence class of the 6-tuple $(h_i,s_j,t)$ where $h_i=\Id$, $s_j=0$ and $t=0$ for $i=0,1,2$ and $j=1,2$.

From the definition, we see that $\mathcal H_{3t}(\A)$ is isomorphic to $\mathcal H_{3t}(\tau_{\leq 0}(\A))$.
Thus we may assume $\A$ is strictly connective when needed.

Clearly if a 6-tuple $(h_i,s_j,t)$ gives rise to an isomorphism in $\mathcal H_{3t}(\A)$, then $h_i$ is a homotopy equivalence for each $i=0,1,2$.
We will prove that the converse is also true.

Let $\A'$ be the dg category obtained from $\A$ by adding a zero object.
The canonical dg functor $\A\rightarrow \A'$ is fully faithful and hence the induced dg functor $\Fun_{\infty}(\B,\A)\rightarrow \Fun_{\infty}(\B,\A')$ is also fully faithful.
Therefore we may assume that the dg category $\A$ contains a zero object.

Let $F$ be an object in $\Fun_{\infty}(\B,\A)$ which corresponds to the diagram <ref>. We have the following diagrams

\[
\begin{tikzcd}
0\ar[r,"0"]\ar[rr, bend left=8ex,"0"]&A_1\ar[d,"1"red,red]\ar[r,"0"]&0\\
A_0\ar[r,"f"]\ar[d,"1"red,red]\ar[rr,bend right=8ex,"0"]&A_1\ar[r,"0"]&0\\
A_0\ar[r,"0"]\ar[rr,bend right=8ex,"0"swap]&0\ar[r,"0"]&0
\end{tikzcd}
\]

where we omit the zero maps.
This gives rise to a graded-split short exact sequence in $\Fun_{\infty}(\B,\A)$:

\[
\begin{tikzcd}
0\ar[r]& A_1^{1}\ar[r] &U\ar[r] &A_0^{0}\ar[r]&0
\end{tikzcd}
\]

where we use the superscript $i$ to indicate the position of the object in the diagram, for example $A_1^1$ stands for the diagram

\[
\begin{tikzcd}
0\ar[r,"0"]\ar[rr,bend right=8ex,"0"swap]& A_1\ar[r,"0"]&0.
\end{tikzcd}
\]

Similarly we have a graded-split short exact sequence in $\Fun_{\infty}(\B,\A)$:

\[
\begin{tikzcd}
\end{tikzcd}
\]

So $F$ can be obtained by graded-split extensions from $A_i^i$, $i=0,1,2$.

Let $\theta:F\rightarrow F'$ be a morphism in $\mathcal H_{3t}(\A)$ given by the homotopy equivalence class of a 6-tuple $(h_i,s_j,t)$ as above.
If $h_i$ is a homotopy equivalence in $\A$ for each $i=0,1,2$, then $\theta$ is an isomorphism.

We use the notations in the above discussion.
For each $F''$ in $\mathcal H_{3t}(\A)$, we have a morphism of graded-split short exact sequences of complexes over $k$

\[
\begin{tikzcd}
\end{tikzcd}
\]

We want to show that the morphisms $\alpha$ and $\gamma$ are quasi-isomorphisms of complexes.
It is enough to show the following claim.

Claim: for each homotopy equivalence $A\rightarrow A'$ in $\A$, and for each $i=0$, $1$, $2$, the induced morphism

\[
(F'',A^i)\rightarrow (F'',A'^i)
\]

is a quasi-isomorphism of complexes.

To see this, we apply $\Hom(-,A^i)$ and $\Hom(-,A'^i)$ to the associated graded-split short exact sequences for $F''$ and get morphisms of graded-split short exact sequences of complexes over $k$

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
\end{tikzcd}
\]

Now we observe that for any pair of objects $B$ and $C$ in $\A$, the complex $\Hom_{\Fun_{\infty}(\B,\A)}(B^i,C^j)$ is isomorphic to a shift of $\Hom_{\A}(B,C)$ if $i\leq j$ and is zero if $j>i$.
Since $A\rightarrow A'$ is a homotopy equivalence in $\A$, the above maps in blue are quasi-isomorphisms of complexes.
This implies that the morphism $(F'',A^i)\rightarrow (F'',A'^i)$ is a quasi-isomorphism.
This proves the claim and hence the proposition.

Denote by $\tilde{\mathrm{Com}}$ the dg path $k$-category of the quiver

\begin{equation}\label{3term'}
\begin{tikzcd}
0\ar[rr,bend left=6ex,"c"]\ar[r,"a"swap]&1\ar[r,"b"swap]&2
\end{tikzcd}
\end{equation}

where ${|}a{|}=0$, ${|}b{|}=0$, ${|}c{|}=-1$, and $d(c)=ba$.

We identify objects in $\D(\A\otimes \tilde{\Com}^{op})$ with sequences

\[
\begin{tikzcd}
X\ar[r,"f"]\ar[rr,"h"swap, bend right=6ex]&Y\ar[r,"j"]&Z
\end{tikzcd}
\]

in $\C_{dg}(\A)$ where $f$ and $j$ are closed morphisms of degree 0 and $h$ is a morphism of degree $-1$ such that $d(h)=jf$.

Each object in $\rep(\tilde{\Com},\A)$ is isomorphic to an object of the form

\[
\begin{tikzcd}
A^{\wedge}\ar[r,"f^{\wedge}"]\ar[rr, bend right =6ex,"h^{\wedge}"swap]&B^{\wedge}\ar[r,"j^{\wedge}"]&C^{\wedge}
\end{tikzcd}
\]

where $A,B,C$ are objects in $\A$, the maps $f$ and $j$ are of degree $0$, the map $h$ is of degree $-1$ and $d(h)=jf$.

We observe that $\tilde{\Com}$ is a cofibrant dg category.
Recall that for dg categories $\A$ and $\B$ where $\B$ is cofibrant, we have a canonical bijection <cit.>

\[
\Hom_{\Hqe}(\B,\A)=\Hom_{\dgcat}(\B,\A)/\mathrm{htp} \iso \Iso(\rep(\B,\A))
\]

where the homotopy class of a dg functor $F:\B\rightarrow \A$ is sent to the isomorphism class of the dg bimodule $\Hom_{\A}(-,F(-))$.
Thus each object in $\rep(\tilde{\Com},\A)$ is isomorphic to an object given by a dg functor $\tilde{\Com}\rightarrow \A$.

The following theorem is claimed by Kontsevich, modulo some results by Toën.
We give a direct proof in the special case where $\B$ is the dg category <ref>.

We have a canonical isomorphism

\[
\Fun_{\infty}(-,-)\rightarrow \rep_{dg}(-,-)
\]

of bifunctors $\Hqe_{k-\mathrm{cf}}^{op}\times \Hqe\rightarrow \Hqe$, where $\Hqe_{k-\mathrm{cf}}$ is the
full subcategory of $\Hqe$ whose objects are the dg categories whose morphism
complexes are cofibrant dg $k$-modules.

Let $\A'$ be a full dg subcategory of $\A$ and $\B$ be a small dg category.
Let $F:\rep(\B,\A')\rightarrow \rep(\B,\A)$ be the functor induced by the inclusion $\A'\rightarrow \A$.
Then an object $X\in \rep(\B,\A)$ lies in the essential image of $F$ if and only if for each object $B$ in $\B$, $X(-,B)$ is quasi-isomorphic to a dg $\A$-module represented by an object in $\A'$.

Let $\A''$ be the closure of $\A'$ under homotopy equivalences in $\A$. The inclusion of
$\A'$ into $\A''$ is a quasi-equivalence and therefore induces a quasi-equivalence in
$\rep_{dg}(\B,?)$. By the above theorem, it also induces a quasi-equivalence in
$\Fun_\infty(\B,?)$. Thus, we may assume that $\A'$ is stable under homotopy equivalences
in $\A$. Now the inclusion $\Fun_\infty(\B,\A') \to \Fun_\infty(\B,\A)$ is clearly an isomorphism
onto the full subcategory of $A_\infty$-functors whose values on objects lie in $\A'$.
Thanks to the theorem, this implies the statement.

Let $\A$ be a small dg category and $\C$ be a dg category given by a differential graded quiver $Q$ without relations and whose differential $d$ is such that $Q$ admits a finite filtration

\[
F_0\subset F_1\subset F_2\subset\ldots\subset F_N=Q
\]

such that all $F_p$ have the same objects as $Q$, and the bimodule of arrows of $F_0$ vanishes and $d(F_p)$ is contained in $kF_{p-1}$.
Then for each dg functor $F:\C\rightarrow \A$, the associated dg $\C$-$\A$-bimodule $_{F}\A_{\A}=\Hom_{\A}(-,F(-))$ admits a cofibrant resolution

\[
\begin{tikzcd}
0\ar[r]& \ker(p) \ar[r,"u"] &\bigoplus_{i\in Q}\Hom_{\A}(-,F(i))\otimes \Hom_{\C}(i,-)\ar[r,"p"]                          & \Hom_{\A}(-,F(-))\ar[r]& 0.
\end{tikzcd}
\]

Let $F:\C\rightarrow \A$ be a dg functor.
Then we have a short exact sequence of dg $\A\otimes \C^{op}$-modules

\[
\begin{tikzcd}
0\ar[r]& \ker(p) \ar[r,"u"] &\bigoplus_{i\in Q}\Hom_{\A}(-,F(i))\otimes \Hom_{\C}(i,-)\ar[r,"p"]                          & \Hom_{\A}(-,F(-))\ar[r]& 0.
\end{tikzcd}
\]

where the dg module $\ker(p)$ is isomorphic to

\[
\bigoplus_{\alpha:i\rightarrow j\text{ in $Q$}}\Hom_{\A}(-,F(i))\otimes\Hom_{\C}(j,-)\otimes k\alpha
\]

as a graded module.
It has a finite increasing filtration

\[
0\subset M_0\subset M_{1}\subset \ldots\subset M_{N}=\ker(p)
\]

\[
M_{k}=\bigoplus_{\alpha:i\rightarrow j\text{ in $F_k$}}\Hom_{\A}(-,F(i))\otimes\Hom_{\C}(j,-)\otimes k\alpha.
\]

The subquotient $M_{k+1}/M_{k}$ is isomorphic to the shift of the representable dg bimodule

\[
\bigoplus_{\alpha:i\rightarrow j\text{ in $F_{k+1}\backslash F_{k}$}}\Sigma^{|\alpha|}\Hom_{\A}(-,F(i))\otimes\Hom_{\C}(j,-).
\]

Thus $\Cone(u)$ is a cofibrant resolution of the dg bimodule $_{F}\A_{\A}$ associated with the dg functor $F:\C\rightarrow \A$.

Let $\B$ be the dg category defined by <ref>.
We have a canonical isomorphism $\mu:\Fun_{\infty}(\B,\A)\rightarrow \rep_{dg}(\B,\A)$ in $\mathrm{Hqe}$.

Let $\B'$ be the dg category $\tilde{\mathrm{Com}}$ defined by <ref>.
We have a quasi equivalence $\B'\rightarrow \B$.

We define a dg functor $\tilde{\mu}: \Fun_{\infty}(\B,\A)\rightarrow \rep_{dg}(\B',\A)$ as follows:

Let $F$ be an object in $\Fun_{\infty}(\B,\A)$ with the corresponding diagram <ref>.
Let $M$ be the dg $\B'^{op}\otimes \A$ module which corresponds to the diagram

\[
\begin{tikzcd}
A_0^{\wedge}\ar[r,"f^{\wedge}"]\ar[rr,bend right=8ex,"-h^{\wedge}"swap]&A_1^{\wedge}\ar[r,"j^{\wedge}"]&A_2^{\wedge}.
\end{tikzcd}
\]

Denote by $X=2^{\wedge}\otimes A_0^{\wedge}$, $Y=1^{\wedge}\otimes A_0^{\wedge}\oplus 2^{\wedge}\otimes A_1^{\wedge}\oplus 2^{\wedge}\otimes \Cone(\Id_{A_0^{\wedge}})$ and $Z=0^{\wedge} \otimes A_0^{\wedge}\oplus 1^{\wedge}\otimes A_1^{\wedge}\oplus 2^{\wedge}\otimes A_2^{\wedge}$ the cofibrant dg $\B'^{op}\otimes \A$-modules.
By Lemma <ref>, we have an exact sequence of dg $\B'^{op}\otimes \A$-modules:

\[
\begin{tikzcd}
0\ar[r]& X\ar[r,"\alpha"]&Y\ar[r,"\beta"]&Z\ar[r,"\gamma"]&M\ar[r]&0
\end{tikzcd}
\]

\[
\alpha^{\intercal}=
\begin{bmatrix}
b^{\wedge}\otimes \Id& \Id\otimes f^{\wedge}&1\otimes {[}0,1{]}^{\intercal}
\end{bmatrix},
\]

\[
\beta=
\begin{bmatrix}
a^{\wedge}\otimes \Id&0&-(ba)^{\wedge} \otimes {[}0,1{]}-c^{\wedge}\otimes {[}s^{-1},0{]} \\
-\Id\otimes f^{\wedge}&b^{\wedge}\otimes \Id&0\\
0&-\Id\otimes j^{\wedge}&\Id\otimes {[}-h^{\wedge}{,}(jf)^{\wedge}{]}
\end{bmatrix}
\]

and $\gamma$ is the obvious morphism.

The object $F$ is sent by $\tilde{\mu}$ to the totalization of the sequence

\[
\begin{tikzcd}
\end{tikzcd}
\]

So $\tilde{\mu}(F)$ is quasi-isomorphic to $M$.

Let $H:F\rightarrow F'$ be a morphism of degree $n$ in $\Fun_{\infty}(\B,\A)$ given by the diagram <ref>.

Recall that for a dg $\A$-module $M$, $s:M\rightarrow \Sigma M$ is the graded map of degree $-1$ which is $\Id: {M^n}\rightarrow (\Sigma M)^{n-1}=M^{n}$ on each degree $n$.
Clearly $d(s)=0$.

We have the following diagram in $\C_{dg}(\B'^{op}\otimes\A)$

\[
\begin{tikzcd}
0\ar[r]& X\ar[r,"\alpha"]\ar[d,"u"]&Y\ar[r,"\beta"]\ar[d,"v"]\ar[rd,dashed,"r"]&Z\ar[r]\ar[d,"w"]&0\\
0\ar[r]& X'\ar[r,"\alpha'"swap]&Y'\ar[r,"\beta'"swap]&Z'\ar[r]&0
\end{tikzcd}
\]

\[
u=\Id\otimes h_0^{\wedge},
\]

\[
\begin{bmatrix}
\Id\otimes h_0^{\wedge}&0&0\\
0&\Id\otimes h_1^{\wedge}&\Id\otimes{[}s_1^{\wedge}\circ s^{-1},(f'h_0-h_1f)^{\wedge}{]}\\
0&0&\Id\otimes \begin{bmatrix}(-1)^{n}s\circ h_0^{\wedge}\circ s^{-1}\ \ \ 0\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h_0^{\wedge}\end{bmatrix}
\end{bmatrix},
\]

\[
\begin{bmatrix}
\Id\otimes h_0^{\wedge}& 0&0\\
0&\Id\otimes h_1^{\wedge}&0\\
0&0&\Id\otimes h_2^{\wedge}
\end{bmatrix},
\]

\[
\begin{bmatrix}
\Id\otimes s_1^{\wedge}&0&-b^{\wedge}\otimes {[}0,s_1^{\wedge}{]}\\
0&\Id\otimes s_2^{\wedge}& -\Id\otimes {[}t^{\wedge},(s_2\circ f)^{\wedge}{]}
\end{bmatrix}.
\]

The morphism $H:F\rightarrow F'$ is sent by the dg functor $\tilde{\mu}$ to the morphism

\[
\begin{bmatrix}
\end{bmatrix}.
\]

We have that

\[
\alpha'u=(-1)^n v\alpha,
\]

\[
d(u)=\Id\otimes (d(h_0))^{\wedge},
\]

\[
\begin{bmatrix}
\Id\otimes d(h_0)^{\wedge}&0&0\\
0&\Id\otimes d(h_1)^{\wedge}&\Id\otimes {[}s_1'^{\wedge} \circ s^{-1},(f'd(h_0)-d(h_1)f)^{\wedge}{]}\\
0&0&\Id\otimes\begin{bmatrix}(-1)^{n+1}s\circ (d(h_0))^{\wedge}\circ s^{-1}\ \ 0\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d(h_0)\end{bmatrix}
\end{bmatrix},
\]

\[
\begin{bmatrix}
\Id\otimes d(h_0)^{\wedge}&0&0\\
0&0&\Id\otimes d(h_1)^{\wedge}
\end{bmatrix},
\]

\[
d(r)+\beta'v-(-1)^n w\beta=
\begin{bmatrix}
\Id\otimes s_1'^{\wedge}&0&-b^{\wedge}\otimes{[}0,s_1'^{\wedge}{]}\\
0&\Id\otimes s_2'^{\wedge}&-\Id\otimes {[}t'^{\wedge},(s_2'\circ f)^{\wedge}{]}
\end{bmatrix}
\]

\[
\]

\[
\]

\[
\]

So the functor $\tilde{\mu}:\Fun_{\infty}(\B,\A)\rightarrow \rep_{dg}(\B',\A)$ is indeed a dg functor.

By Lemma <ref>, the functor $\tilde{\mu}$ is quasi-dense.

To show the quasi-full faithfulness, we may assume that the dg category $\A$ contains a zero object.
We adopt the notations used in the discussions before Proposition <ref>.
In particular, the object $U\in \Fun_{\infty}(\B,\A)$ corresponds to the following diagram

\[
\begin{tikzcd}
A_0\ar[r,"f"]\ar[rr,bend right=8ex,"0"swap]&A_1\ar[r,"0"]&0.
\end{tikzcd}
\]

And we have graded-split short exact sequences in $\Fun_{\infty}(\B,\A)$:

\[
\begin{tikzcd}
\end{tikzcd}
\]

\[
\begin{tikzcd}
0\ar[r]&A_2^2\ar[r] &F\ar[r]&U\ar[r]&0.
\end{tikzcd}
\]

Thus it suffices to show that $\tilde{\mu}$ induces quasi-isomorphisms between objects of the form $A^i$ where $A\in \A$ and $i=0,1,2$.

This can be checked using the fact that $\tilde{\mu}(F)$ is a cofibrant resolution of $M$.

Denote by $\tilde{\mathrm{Sq}}$ the dg quotient of $k\Sq$ by its full dg subcategory consisting of the object 10, i.e. the dg path $k$-category of the quiver

\begin{equation}
\begin{tikzcd}
10\ar[loop left,"h"]\ar[r,"l"swap]&11
\end{tikzcd}
\end{equation}

where ${|}f{|}=0={|}j{|}={|}g{|}={|}l{|}$, ${|}h{|}=-1$, $d(h)=\Id_{10}$, and $jf=lg$.

We identify objects in $\D(\A\otimes\tilde{\Sq}^{op})$ with diagrams

\[
\begin{tikzcd}
N\ar[loop left,"h"]\ar[r,"l"swap]&Z
\end{tikzcd}
\]

in $\C_{dg}(\A)$ where $f,g,j,l$ are closed morphisms of degree 0 such that $jf=lg$ and $h$ is a morphism of degree $-1$ such that $d(h)=\Id_{N}$.

Let $F:\tilde{\mathrm{Com}}\rightarrow \tilde{\mathrm{Sq}}$ be the dg functor such that $F(0)=00$, $F(1)=01$, $F(2)=11$ and $F(a)=f$, $F(b)=j$ and $F(c)=lhg$.
Let $G:k\mathrm{Sq}\rightarrow \tilde{\mathrm{Sq}}$ be the obvious dg quotient functor.
We have the following cospan of dg functors

\[
\begin{tikzcd}
\end{tikzcd}\;.
\]

By <cit.>, the dg functor $\Id\otimes G^{op}:\A\otimes k\Sq^{op}\rightarrow \A\otimes\tilde{\Sq}^{op}$ is a dg quotient of $\A\otimes k\Sq^{op}$ modulo $\A\otimes \{01\}$ and hence a localization functor, i.e. the restriction functor along $\Id\otimes G^{op}$ is fully faithful.

Let $\mathcal S$ be the right orthogonal of the objects $(A, 01)^{\wedge}$ in $\D(\Sq)$, where $A$ runs through objects in $\A$.
More precisely, $\mathcal S$ is the full triangulated subcategory of $\D(\mathrm{Sq})$ consisting of squares

\[
\begin{tikzcd}
\end{tikzcd}
\]

where $N$ is an acyclic dg $\A$-module.

$F$ is a quasi-equivalence and $G$ induces an equivalence $\mathcal S\xrightarrow{\sim}\D(\A\otimes\tilde{\Sq}^{op})$.

It is enough to show that $F$ induces a quasi-isomorphism of complexes

\[
\Hom_{\tilde{\mathrm{Com}}}(0,2)\rightarrow \Hom_{\tilde{\mathrm{Sq}}}(00,11).
\]

On the left hand side, the complex is given by

\[
\cdots\rightarrow 0\rightarrow k\cdot c\xrightarrow{\sim} k\cdot(ba)\rightarrow 0\rightarrow \cdots
\]

which is contractible. On the right hand side, the complex is given by

\[
\cdots\rightarrow k\cdot(lh^3g)\xrightarrow{\sim} k\cdot(lh^2g)\xrightarrow{0} k\cdot(lhg)\xrightarrow{\sim} k\cdot(lg)\rightarrow 0\rightarrow\cdots
\]

which is also contractible. Thus $F$ is a quasi-equivalence.

As a consequence of Lemma <ref>, we have a fully faithful functor

\[
\begin{tikzcd}
\D(\A\otimes \tilde{\mathrm{Com}}^{op})\ar[r] &\D(\mathrm{Sq})
\end{tikzcd}
\]

which sends an object

\begin{equation}
\label{sequence}\tag{$\bigstar\bigstar$}
\begin{tikzcd}
&X\ar[r,"f"]\ar[rr,bend right = 6ex,"h"swap]&Y\ar[r,"j"]&Z
\end{tikzcd}
\end{equation}

\begin{equation}\label{square}\tag{$\bigstar\bigstar\bigstar$}
\begin{tikzcd}
\end{tikzcd}\;.
\end{equation}

The following is a direct consequence of Lemma <ref> and Lemma <ref>.

Assume the dg category $\A$ contains a contractible object, e.g. $Z^0(\A)$ is an additive category.
Each object

\[
\begin{tikzcd}
\end{tikzcd}
\]

in $\rep(\mathrm{Sq})$, where $N$ is acyclic, is isomorphic in $\D(\mathrm{Sq})$ to an object of the form

\[
\begin{tikzcd}
IA^{\wedge}\ar[r] &C^{\wedge}
\end{tikzcd}
\]

If $X$ and $Z$ are already representable dg modules,
then the isomorphism can be chosen such that when restricted to both ends of the squares, the morphisms are identities in $\D(\A)$.

Recall that the homotopy category $\mathcal H_{3t}(\A)$ of 3-term h-complexes over $\A$ is defined to be $H^0(\Fun_{\infty}(\B,\A))$ where $\B$ is the dg category defined by <ref>.
Then we have a fully faithful functor

\[
F: \mathcal H_{3t}(\A)\iso \rep(\tilde{\Com},\A)\rightarrow \rep(\Sq)
\]

sending a 3-term h-complex over $\A$

\begin{equation}\label{3t}
\begin{tikzcd}
A_0\ar[r,"f"]\ar[rr,bend right = 8ex,"h"swap]&A_1\ar[r,"j"]&A_2
\end{tikzcd}
\end{equation}

where $|f|=|j|=0$, $|h|=-1$ and $d(f)=0$, $d(j)=0$ and $d(h)=-jf$, to the following object in $\rep(\Sq)$

\begin{equation}\label{3tsquare}
\begin{tikzcd}
A_0^{\wedge}\ar[r,"f^{\wedge}"]\ar[d,"i"swap]                                              &A_1^{\wedge}\ar[d,"j^{\wedge}"]\\
IA_0^{\wedge}\;\;\ar[r,"{[}-h^{\wedge}{,}-d(h^{\wedge}){]}"swap]                  &\;\;A_2^{\wedge}\mathrlap{.}
\end{tikzcd}
\end{equation}

A 3-term h-complex over $\A$ given by <ref> is a homotopy (resp. left, resp. right) short exact sequence if the corresponding object <ref> in $\rep(\Sq)$ is homotopy bicartesian (resp. cartesian, resp. cocartesian).

Let $\C_{dg}^{\leq 0}(k)$ be the full dg subcategory of $\C_{dg}(k)$ consisting of complexes concentrated in non-positive degrees.
Consider a sequence in $\C_{dg}^{\leq 0}(k)$

\[
\begin{tikzcd}
U\ar[r,"f"]\ar[rr,bend right = 8ex,"h"swap]&V\ar[r,"j"]&W.
\end{tikzcd}
\]

where $|f|=|j|=0$, $|h|=-1$ and $d(f)=0$, $d(j)=0$ and $d(h)=-jf$.
It is a homotopy left exact sequence and we say that it is a homotopy kernel of $j:V\rightarrow W$, provided that the canonical morphism

\[
U\xrightarrow{[f\;,h]^{\intercal}} \tau_{\leq 0}\Sigma^{-1}\Cone (j)
\]

is a quasi-isomorphism of complexes, or equivalently, for each $n\leq 0$ and each pair of elements $(v,w)\in Z^{n}V\times W^{n-1}$ such that $d(w)=-j(v)$, there exists an element $u\in Z^n U$, unique up to a coboundary, such that there exists a pair of elements $(v',w')\in V^{n-1}\times W^{n-2}$ satisfying $v-f(u)=d(v')$, $w-h(u)=-d(w')-j(v')$.
Then we have

A 3-term h-complex given by <ref> is a homotopy left (resp. right) exact sequence if and only if for each object $A$ in $\A$, the corresponding 3-term h-complex over $\C_{dg}^{\leq 0}(k)$

\[
\begin{tikzcd}
\tau_{\leq 0}(A,A_0)\ar[r,"\tau_{\leq 0}{(}A{,}f{)}"]\ar[rr,bend right = 8ex,"\tau_{\leq 0}{(}A{,}h{)}"swap]&\tau_{\leq 0}(A,A_1)\ar[r,"\tau_{\leq 0}{(}A{,}j{)}"]&\tau_{\leq 0}(A,A_2),
\end{tikzcd}
\]

\[
\begin{tikzcd}
\tau_{\leq 0}(A_2,A)\ar[r,"\tau_{\leq 0}{(}j{,}A{)}"]\ar[rr,bend right = 8ex,"\tau_{\leq 0}{(}h{,}A{)}"swap]&\tau_{\leq 0}(A_1,A)\ar[r,"\tau_{\leq 0}{(}f{,}A{)}"]&\tau_{\leq 0}(A_0,A)
\end{tikzcd}
\]

is a homotopy left exact sequence.
It is a homotopy short exact sequence if and only if both of the above sequences are homotopy left exact.

Recall that for a dg category $\A$, we denote by $\pretr(\A)$ its pretriangulated hull.
We observe that the dg category $\Fun_{\infty}(\B,\A)$ can be seen a non-full dg subcategory of $\pretr(\A)$ via the dg functor $\mathrm{Tot}_{\A}:\Fun_{\infty}(\B,\A)\rightarrow \pretr(\A)$. On objects $\mathrm{Tot}_{\A}$ sends an $A_{\infty}$-functor $F:\mathcal B\rightarrow \A$ which corresponds to the diagram <ref>, to the formal sum $(\Sigma^2A_0\oplus \Sigma A_1\oplus A_0, q)$ where $q$ is the matrix

\[
\begin{bmatrix}
\end{bmatrix}.
\]

Let $d(q)$ be the matrix with $d(q)_{ij}=d(q_{ij})$.
Clearly $d(q)+q^2=0$.
The dg functor $\mathrm{Tot}_{\A}$ sends a morphism which corresponds to the diagram <ref>, to the matrix

\[
\begin{bmatrix}
(-1)^{n-1}s_1&(-1)^n h_1&0\\
\end{bmatrix}.
\]

Since the dg functor $\mathrm{Tot}_{\A}$ is not full,
the induced functor $H^0(\mathrm{Tot}_{\A}):\mathcal H_{3t}(\A)\rightarrow \tr(\A)$ is neither full nor faithful in general.

For two objects $F$ and $F'$ in $\Fun_{\infty}(\B,\A)$, we use $(F,F')$ to indicate the morphism complex in $\Fun_{\infty}(\B,\A)$ and use $\A(F,F')$ to indicate the morphism complex in $\pretr(\A)$ of the objects $\mathrm{Tot}_{\A}(F)$ and $\mathrm{Tot}_{\A}(F')$.
In the following, we will describe the functor $\mathrm{Tot}_{\A}:\Fun_{\infty}(\B,\A)\rightarrow \pretr(\A)$ in more detail.
We assume the dg category $\A$ contains a zero object.
Note that $\mathrm{Tot}_{\A}(A_0^0)=\Sigma^2 A_0^{\wedge}$ and $\mathrm{Tot}_{\A}(A_1^1)=\Sigma A_1^{\wedge}$.

Let $V'$ and $W$ be the following objects in $\Fun_{\infty}(\B,\A)$

\[
\begin{tikzcd}
A_1'\ar[r,"j'"swap]\ar[rr,bend left=8ex,"0"]&A_2'\ar[r,"0"swap]&0,
\end{tikzcd}
\]

\[
\begin{tikzcd}
0\ar[r,"0"swap]\ar[rr,bend left=8ex,"0"]&A_0\ar[r,"f"swap]&A_1.
\end{tikzcd}
\]

Let $M$ be the complex

\[
(\Sigma^{2}\A(A_2',A_0)\oplus \Sigma \A(A_1',A_0)\oplus \Sigma \A(A_2',A_1),d)
\]

where $d(h_1,s_1,s_2)=(d(h_1), d(s_1)+(-1)^{n}(h_1\circ j), d(s_2)+(-1)^{n+1}f\circ h_1)$ where $h_1:A_2'\rightarrow A_0$ is of degree $n+2$ and $s_1$ and $s_2$ are of degree $n+1$.

We have the following commutative diagrams in $\C(k)$, which relates the complex $(F,F')$ with $\A(F',F)$

\[
\begin{tikzcd}
0\ar[r]&\A(A_1', \Sigma^{-2}A_1)\ar[r,"\alpha"]& \Sigma^2(V', W)\ar[r,"\beta"] & M\ar[r]&0
\end{tikzcd}
\]

\[
\begin{tikzcd}
0\ar[r]& (F',A_2^{2}) \ar[r,tail]\ar[d, equal]   & (F',F)\ar[r,two heads] \ar[d, tail]                    & (F', U)\ar[r, two heads]\ar[d,tail]                              &0\\
0\ar[r]& \A(F', A_2^{2}) \ar[r,tail]                  & \A(F',F)\ar[r,two heads]\ar[d,two heads]       & \A(F', U)\ar[r]\ar[d, two heads,"\gamma"]               &0\\
&                                                        &M\ar[r,equal]                                                   &M                                                                            &
\end{tikzcd}
\]

where $\alpha(t)=(0,0,t)$, $\beta(h_i,s_j,t)=(h_1,s_1,s_2)$ and the action of $\gamma$ on the degree $n$ component is given as follows

\[
\gamma(\begin{bmatrix}h_0'&s_1&h_1\\(-1)^{n+1}s_1'&(-1)^n h_1'&s_2\\t&s_2&h_2'\end{bmatrix})=\begin{bmatrix}h_1&s_1&(-1)^{n+1}s_2\end{bmatrix}.
\]

Let $\mathrm{Filt}_3(\pretr(\A))$ be the dg category with objects the twisted objects $(Y=\oplus_{i\in I}\Sigma^{n_i}Y_i^{\wedge},d)$ with a 3-step filtration $I_0\subset I_1\subset I_2=I$ such that $\oplus_{i\in I_0}\Sigma^{n_i}Y_i^{\wedge}\subseteq \oplus_{i\in I_1}\Sigma^{n_i}Y_i^{\wedge}\subseteq \oplus_{i\in I_2}\Sigma^{n_i}Y_i^{\wedge}$ are dg submodules.
Then we have a localization dg functor $\mathrm{Filt}_3(\pretr\A)\rightarrow \pretr(\A)$
which sends an object $(Y,I_0\subseteq I_1\subseteq I_2)$ to $Y$.
It admits a fully faithful left adjoint $\pretr(\A)\rightarrow \mathrm{Filt}_3(\pretr(\A))$ which sends a dg module $Y=(\oplus_{i\in I}\Sigma^{n_i}Y_i^{\wedge},d)$ to $(Y,\emptyset\subseteq \emptyset\subseteq I)$.
Then the totalization dg functor $\mathrm{Tot}_{\A}:\Fun_{\infty}(\B,\A)\rightarrow \pretr(\A)$ is the composition of a fully faithful funtor $i:\Fun_{\infty}(\B,\A)\rightarrow \mathrm{Filt}_3(\A)$ and the localization dg functor $\mathrm{Filt}_3(\pretr(\A))\rightarrow \pretr(\A)$.
Indeed, for each object $F\in \Fun_{\infty}(\B,\A)$, the dg module $\mathrm{Tot}_{\A}(F)$ has a natural filtration given by

\[
i(F)=(\mathrm{Tot}_3(\A), A_2^{\wedge}\subseteq \Cone(j^{\wedge}:A_1^{\wedge}\rightarrow A_2^{\wedge})\subseteq \mathrm{Tot}_{\A}(F)).
\]

§ BASIC DIAGRAM LEMMAS

§.§ Homotopy kernels and cokernels

Recall the morphism category $\Mor$ defined in subsection <ref>

\[
\begin{tikzcd}
\end{tikzcd}
\]

and the dg category $\B$ defined in <ref>

\[
\begin{tikzcd}
0\ar[r,"f"] &1\ar[r,"j"]&2.
\end{tikzcd}
\]

where $|f|=|j|=0$, and $d(f)=d(j)=0$, $jf=0$.
Let $i:k\Mor\rightarrow \B$ be the inclusion dg functor which sends object $0$ to $0$ and object $1$ to $1$.
Let $i':k\Mor\rightarrow \B$ be the inclusion dg functor which sends object $0$ to $1$ and object $1$ to $2$.
Let $\Res:\Fun_{\infty}(\B,\A)\rightarrow\Fun_{\infty}(k\Mor,\A)\iso\Mor(\A)$ (resp. $\Res'$) be the restriction functor along $i$ (resp. $i'$).

By homotopy kernel of an object $j:B\rightarrow C$ in $\Mor(\A)$ (cf. Example <ref>), we mean a homotopy left exact squence $X$ (cf. Definition <ref>)

\[
\begin{tikzcd}
A\ar[rr,bend right=8ex,"h"swap]\ar[r,"f"]&B\ar[r,"j"]&C.
\end{tikzcd}
\]

Sometimes we say $f:A\rightarrow B$ or $A$ is the homotopy kernel of $j$.

Dually we define the homotopy cokernel of an object in $\Mor(\A)$.

Assume the dg category $\A$ contains a zero object.
An object $f:A\rightarrow B$ in $\Mor(\A)$ is a homotopy monomorphism if its homotopy kernel is zero, or equivalently, for each $A'\in\A$, the morphism $\Hom_{\A}(A',A)\rightarrow \Hom_{\A}(A',B)$ induces an isomorphism in each negative degree cohomology and an injection in degree zero cohomology. Dually we have the notion of homotopy epimorphism.

A homotopy kernel need not be the homotopy kernel of its homotopy cokernel (if they exist).
Also, a homotopy kernel need not be a homotopy monomorphism.

An object $f:A\rightarrow B$ in $\Mor(\A)$ is a homotopy monomorphism if and only if the object in $\rep(\Sq)$

\[
\begin{tikzcd}
\end{tikzcd}
\]

is a homotopy cartesian square.

In the above definition, we have that $\Res'(X)=j$.
Compare Definition <ref> with Definition <ref>.
The two definitions are compatible with each other, due to the following observation:

Let $j:B\rightarrow C$ be an object in $\Mor(\A)$.
Consider a morphism $\theta$ from $j':B'\rightarrow C'$ to $j$ in $H^0(\Mor(\A))$ as follows

\[
\begin{tikzcd}
\end{tikzcd}
\]

Suppose we are given an object

\[
\begin{tikzcd}
A'\ar[r,"f'"]\ar[rr,bend right=8ex,"h'"swap]&B'\ar[r,"j'"]&C'
\end{tikzcd}
\]

in $\mathcal H_{3t}(\A)$.
Then we have the following morphism $\mu$ in $\mathcal H_{3t}(\A)$ which restricts to the morphism $\theta$

\[
\begin{tikzcd}
A'\ar[r,"f'"]\ar[rd,"0"{red,swap},red]\ar[rrd,"0"blue,blue]\ar[d,equal]\ar[rr,"h'",bend left=8ex]&B'\ar[r,"j'"]\ar[d,"h_1"swap]\ar[rd,"s_2"red,red]&C'\ar[d,"h_2"]\\
A'\ar[r,"h_1f'"swap]\ar[rr,"h_2h'-s_2f'"swap,bend right=8ex]&B\ar[r,"j"swap]&C\mathrlap{.}
\end{tikzcd}
\]

The morphism $\mu$ is an isomorphism if and only if the morphism $\theta$ is an isomorphism, cf. Proposition <ref>.

Recall that $\mathcal H_{3t}$ is defined as $H^0(\Fun_{\infty}(\B,\A))$ where $\B$ is the dg category defined by <ref>.
Let us recall the universal property of homotopy left exact sequences.
Suppose we have 3-term h-complexes $X_1$ and $X_2$ with $X_2$ being homotopy left exact.
Then each morphism $\theta:\Res'(X_1)\rightarrow \Res'(X_2)$ in $H^0(\Mor(\A))$ extends uniquely to a morphism $\mu:X_1\rightarrow X_2$ in $\mathcal H_{3t}(\A)$ such that $\theta=\Res'(\mu)$.
Indeed, we have the following

Suppose we have 3-term h-complexes $X_i$ in $\A$, $i=1$, $2$, of the form

\[
\begin{tikzcd}
A_i\ar[rr,bend right=8ex,"h_i"swap]\ar[r,"f_i"]&B_i\ar[r,"j_i"]&C_i
\end{tikzcd}
\]

where $X_2$ is homotopy left exact. Then any morphism in $Z^0(\Mor(\A))$

\[
\begin{tikzcd}
\end{tikzcd}
\]

extends to a morphism in $Z^0(\Fun_{\infty}(\B,\A))$

\[
\begin{tikzcd}
A_1\ar[rd,"s_1"{red,swap},red]\ar[rrd,"t"{blue},blue]\ar[d,"a"swap]\ar[rr,bend left=8ex,"h_1"]\ar[r,"f_1"]&B_1\ar[r,"j_1"]\ar[d,"b"swap]\ar[rd,"s_2"{red},red]&C_1\ar[d,"h_3"]\\
A_2\ar[rr,bend right=8ex,"h_2"swap]\ar[r,"f_2"swap]&B_2\ar[r,"j_2"swap]&C_2\mathrlap{.}
\end{tikzcd}
\]

Put $u=h_2f_1$ and $v=h_3h_1-s_2f_1$. Then the pair $(u,v)$ satisfies $d(v)=-h_3j_1f_1-(j_2h_2-h_3j_1)f_1=-j_2h_2f_1=-j_2u$.

Since $X_2$ is homotopy left exact, we have a closed morphism $a:A_1\rightarrow A_2$ of degree $0$ such that there exists a pair $(-s_1,-t)$ where $s_1$ is a morphism from $A_1$ to $B_2$ of degree $-1$, $t$ is a morphism from $A_1$ to $C_2$ of degree $-2$ satisfying $u-f_2a=d(-s_1)$ and $v-h_2a=-d(-t)-j(-s_1)$. So the 6-tuple $(a,b,h_3,s_1,s_2,t)$ is a morphism from $X_1$ to $X_2$ in $Z^0(\Fun_{\infty}(\B,\A))$.

Let $f:X\rightarrow Y$ be an object in $\Mor(\A)$ and

\[
\begin{tikzcd}
\end{tikzcd}
\]

the associated object $S$ in the category $\rep(\Cosp)$.
Recall that we have a fully faithful functor $\mathcal H_{3t}(\A)\hookrightarrow \rep(\Sq,\A)$.
Then a homotopy kernel of the object $f$ in $\mathcal H_{3t}(\A)$ can be identified with a homotopy pullback of the cospan $S$.
Hence by Proposition <ref>, the homotopy kernel of an object in $\Mor(\A)$ is unique up to a unique isomorphism if it exists in $\mathcal H_{3t}(\A)$.

§.§ Restriction of morphisms in $\mathcal H_{3t}(\A)$

Let $\overline{\Sq}$ be the dg $k$-path category of the following graded quiver

\[
\begin{tikzcd}
\end{tikzcd}
\]

where $f$, $g$, $j$ and $k$ are closed morphisms of degree $0$ and $h$ is a morphism
of degree $-1$ such that $d(h)=kg-jf$.
The obvious dg functor $\overline{\Sq}\rightarrow \Sq$ is a quasi-equivalance and induces an equivalence of categories $\rep(\Sq,\A)\rightarrow\rep(\overline{\Sq},\A)$.
We will transport the definitions and results in the previous sections to objects in $\rep(\overline{\Sq},\A)$.
One reason for working with this category is that objects in $\rep(\overline{\Sq},\A)$ are isomorphic to objects given by dg functors $F:\overline{\Sq}\rightarrow \A$, because the dg category $\overline{\Sq}$ is cofibrant.

If we take a span (resp. cospan) of representable dg modules, if its homotopy pushout (resp. pullback) exists, we can assume the homotopy pushout (resp. pullback) restricts to the original span (resp. cospan), and is
not merely isomorphic to it.

Let $f:A\rightarrow B$ and $f':A'\rightarrow B'$ be two objects in $\Mor(\A)$.
Then a morphism $\eta:f\rightarrow f'$ in $H^0(\Mor(\A))$ gives rise to a class of objects in $\rep(\overline{\Sq},\A)$

\[
\begin{tikzcd}
\end{tikzcd}
\]

which are isomorphic to each other: two homotopic morphisms from $f$ to $f'$ give rise to homotopic dg functors $\overline{\Sq}\rightarrow \A$ in the sense of <cit.>.

We have a quasi-equivalence

\[
\rep_{dg}(\overline{\Sq},\A)\iso\Mor(\Mor(\A)).
\]

In particular, we have an epivalence

\[
\rep(\overline{\Sq},\A)\rightarrow\Fun(\Mor,H^0(\Mor(\A))).
\]

Recall that we have quasi-equivalences $k\overline{\Sq}\iso k\Mor\otimes k\Mor$ and $\Mor(\A)\iso \rep_{dg}(k\Mor,\A)$.
Then for any small dg category $\B$, we have canonical bijections

\begin{align*}
\Hqe(\B,\rep_{dg}(\overline{\Sq},\A))&\iso\Hqe(\overline{\Sq}\otimes \B,\A)\\
&\iso\Hqe(k\Mor\otimes k\Mor\otimes \B,\A)\\
\end{align*}

Therefore we have a quasi-equivalence

\[
\rep_{dg}(\overline{\Sq},\A)\iso \rep_{dg}(k\Mor,\Mor(\A)).
\]

Thus we have an epivalence

\[
\rep(\overline{\Sq},\A)\iso \rep(k\Mor,\Mor(\A))\xrightarrow{\Dia} \Fun(\Mor,H^0(\Mor(\A))).
\]

Recall the restriction functors $\Res,\Res':\Fun_{\infty}(\B,\A)\rightarrow \Mor(\A)$ defined before Definition <ref>.
Suppose we have a morphism $\alpha:X\rightarrow X'$ in $\mathcal H_{3t}(\A)$ of the form

\begin{equation}\label{morphism3term}
\begin{tikzcd}
&A_0\ar[r,"f"]\ar[d,"h_0"swap]\ar[rd,"s_1"red,swap ,red]\ar[rrd,"t"blue,blue]\ar[rr,bend left = 8ex,"h"]&A_1\ar[d,"h_1"swap]\ar[rd,"s_2"red,red]\ar[r,"j"]&A_2\ar[d,"h_2"]\\
&A_0'\ar[r,"f'"swap]\ar[rr,bend right = 8ex,"h'"swap]&A_1'\ar[r,"j'"swap]&A_2'
\end{tikzcd}
\end{equation}

Then by the above discussion, the two morphisms $\Res(\alpha)$, $\Res'(\alpha)$ give rise to two objects in $\rep(\overline{\Sq},\A)$.

As above, let $\alpha:X\rightarrow X'$ be a morphism in $\mathcal H_{3t}(\A)$.
We call $\Res(\alpha)\in \rep(\overline {\Sq},\A)$ the restriction of $\alpha:X\rightarrow X'$ to $f:A\rightarrow B$.
Similarly, we call $\Res'(\alpha)$ the restriction of $\alpha:X\rightarrow X'$ to $j:B\rightarrow C$.

Suppose we have objects $X$ and $X'$ in $\mathcal H_{3t}(\A)$ of the following form

\[
\begin{tikzcd}
A_0\ar[rr,bend right=8ex,"h"swap]\ar[r,"f"]&A_1\ar[r,"j"]&A_2.
\end{tikzcd}
\]

where for $X'$ we add to each term a prime symbol superscript.
Let $\alpha:X\rightarrow X'$ be a morphism in $\mathcal H_{3t}(\A)$ as in <ref>.

1) Suppose $h_2:A_2\rightarrow A_2'$ is a homotopy equivalence.
Let $X''\in \rep(\overline{\Sq},\A)$ be the restriction of $\alpha$ to $f:A_0\rightarrow A_1$.
Then the following statements hold:

(a) If $X'$ is homotopy left exact, then $X$ is a homotopy left exact if and only if $X''$ is homotopy cartesian.

(b) If $X''$ is homotopy cocartesian, then $X$ is homotopy right exact if and only if so is $X'$.

2)Suppose $h_0:A_0\rightarrow A_0'$ is a homotopy equivalence.
Let $X'''$ be the restriction of $\alpha$ to $j:A_1\rightarrow A_2$. Then the following statements hold:

(a) If $X$ is homotopy right exact, then $X'$ is homotopy right exact if and only if $X'''$ is homotopy cocartesian.

(b) If $X'''$ is homotopy cartesian, then $X$ is homotopy left exact if and only if so is $X'$.

We prove 1) and then 2) follows by duality.
For simplicity, we omit the symbol $^{\wedge}$ of the representable dg modules $A^{\wedge}$.
Since $h_2$ is a homotopy equivalence, we may assume $h_2$ is the identity of ${A_2}$.
The morphism $\alpha$ is given by the following diagram in $\C(\A)$:

\[
\begin{tikzcd}
&V=\Sigma^{-1}\Cone(j)\ar[r,"{[}-1{,}0{]}"]\ar[dd,"\begin{bmatrix}h_1\;\;0\\-s_2\;\; 1\end{bmatrix}"]&A_1\ar[r,"j"]     \ar[dd,"h_1"] \ar[rdd,"s_2"red,red] &A_2\ar[dd,equal]\\
\end{tikzcd}
\]

where diagonal maps are the homotopies making the diagram commutative in $\mathcal H(\A)$.
Then $X''$ is given by

\[\begin{tikzcd}
\end{tikzcd}\;.
\]

Put $U=\Cone(f)$, $U'=\Cone(f')$ and $V_3=\Sigma^{-1}\Cone(h_1,f')$.
We have a canonical morphism $r=(h,j):U\rightarrow A_2$ and similarly a morphism $r':U'\rightarrow A_2'$.
From the object $X$, we have the following diagram in $\D(\A)$

\[
\begin{tikzcd}
A_0\ar[r,"f"]\ar[d,"u"swap]&A_1\ar[r,""]\ar[d,equal]&U\ar[r]\ar[d,"r"]&\Sigma A_0\ar[d]\\
V\ar[r]&A_1\ar[r,"j"]&A_2\ar[r]\ar[d]&\Sigma V\ar[d]\\
& &C(r)\ar[r,equal]&C(r)
\end{tikzcd}.
\]

Put $Y=\Sigma^{-1}\Cone(h_1)$ and $W=\Cone(u')$.
Similarly we have the following diagrams in $\D(\A)$

\[
\begin{tikzcd}
\Sigma Y \ar[r,equal]\ar[d,"-\begin{bmatrix}1\;\;0\\s_2\;\;j'\end{bmatrix}"swap]& \Sigma Y\ar[d,"{[}1{,}0{]}"]&\\
\Sigma V\ar[r,"{[}-1{,}0{]}"swap]&\Sigma A_1&
\end{tikzcd},
\begin{tikzcd}
V_3\ar[r,"\begin{bmatrix}-1\;\;0\;\;0\\0\;\;-1\;\;0\end{bmatrix}"]\ar[d,"{[}0{,}-1{,}0{]}"swap]&A_1\oplus A_0'\ar[d,"{[}0{,}1{]}"]\ar[r,"{[}h_1{,}f'{]}"]&A_1'\\
A_0' \ar[d,"\begin{bmatrix}0\\-f'\end{bmatrix}"swap]\ar[r,equal]& A_0'\ar[d,"0"]&\\
\Sigma Y\ar[r,"{[}-1{,}0{]}"swap]&\Sigma A_1&
\end{tikzcd},
\]

\[
\begin{tikzcd}
& W\ar[rr,equal]&&W
\end{tikzcd}.
\]

We have a morphism from the sequence in blue to the mapping triangle of $v$ as follows

\[
\begin{tikzcd}
V_3\ar[r,"v"]\ar[d,equal]&V\ar[r,"\begin{bmatrix}0\;\;0\\h_1\;\;0\\-s_2\;\;1\end{bmatrix}"]\ar[d,equal]&W\ar[r,"\begin{bmatrix}0\;\;0\;\;0\\1\;\;0\;\;0\\0\;\;1\;\;0\end{bmatrix}"]\ar[d,"\theta"]&\Sigma V_3\ar[d,equal]\\
V_3\ar[r,"v"swap]&V\ar[r,"{[}0{,}1{]^{\intercal}}"swap]&\Cone(v)\ar[r,"{[}1{,}0{]}"swap]&\Sigma V_3\mathrlap{.}
\end{tikzcd}
\]

By direct inspection, we see that $\Cone(v)$ is isomorphic to the mapping cone of

\[
\Cone(\Id_{\Sigma^{-1}A_1})\rightarrow W.
\]

The map $\theta$ is the canonical inclusion from $W$ to this cone.
One checks directly that the rightmost square commutes and the middle square commutes up to homotopy.

Now we see that the composition of the canonical maps $A_0\xrightarrow{[f,h_0,s_1]^{\intercal}} V_3$ and $v:V_3\rightarrow V$ is homotopic to the canonical map $A_0\rightarrow V$.

(a) Since $X'$ is homotopy left exact, the map $u':A_0'\rightarrow V'$ induces a quasi-isomorphism

\[
\tau_{\leq 0}\RHom(A,\Sigma^{-i}A_0')\rightarrow\tau_{\leq 0 }\RHom(A,\Sigma^{-i}V')
\]

for each $A'\in \A$.
Thus the induced map

\[
\tau_{\leq 0}\Hom(A',V_3)\rightarrow \Hom(A', V)
\]

is a quasi-isomorphism for each $A\in \A$.

So $X$ is homotopy left exact if and only if $X''$ is homotopy cartesian.

(b) Put $U_3=\Cone((f,h_0)^{\intercal})$.
Consider the following diagrams

\[
\begin{tikzcd}
A_0\ar[r]\ar[d,equal]&A_0'\oplus A_1\ar[r]\ar[d]&U_3\ar[d]\\
\end{tikzcd},
\begin{tikzcd}
& Z\ar[r,equal]& Z
\end{tikzcd},
\begin{tikzcd}
\]

Since $X''$ is homotopy cocartesian, the canonical map $U_3\rightarrow A_1'$ induces a quasi-isomorphism

\[
\tau_{\leq 0}\RHom(A_1',A')\rightarrow \tau_{\leq 0}\RHom(U_3, A')
\]

for each $A'\in\A$. This implies that the canonical map $U\rightarrow U'$ induces a quasi-isomorphism

\[
\tau_{\leq 0}\RHom(U', A')\rightarrow \tau_{\leq 0}\RHom(U, A')
\]

for each $A'\in\A$. From the rightmost square, we infer that $X$ is homotopy right exact if and only if so is $X'$.

Suppose we have the following diagram in $\C_{dg}(\A)$ with lower square being homotopy cartesian and the upper square an object in $\rep(\overline{\Sq},\A)$.
Then the outer square (i.e. the square on the right hand side) is homotopy cartesian if and only if so is the upper square.

\[
\begin{tikzcd}
X_1\arrow[r,"f"]\arrow[d,"g"swap,""{name=1} ]\ar[rd,"h"blue,blue]&X_2\arrow[d,"j",""{name=2}]\\
\arrow[phantom,from=1,to=2,""]\arrow[phantom,from=3,to=4,""]
\end{tikzcd}\;\;,\;\;
\begin{tikzcd}
X_1\ar[r,"f"]\ar[d,"g'g"swap]\ar[rd,"h'g+j'h"{near start,blue},blue]&X_2\ar[d,"j'j"]\\
\end{tikzcd}
\]

We may assume that the dg category $\A$ admits a zero object and direct sums.
Note that direct sums of homotopy cartesian squares remain homotopy cartesian.
Note also that squares of the form

\[
\begin{tikzcd}
\end{tikzcd}
\]

are trivially homotopy cartesian.

The statement follows by applying Proposition <ref> to one of the following two diagrams

\[
\begin{tikzcd}X_1\ar[rd,blue,"s"]\ar[rr,bend left=8ex,blue,"h'g+j'h"blue]\ar[r,"u"]\ar[d,"g"swap]&X_2\oplus X_5\ar[r,"v"]\ar[d,"r"]&X_6\ar[d,equal] &&X_1\ar[r]\ar[d,equal]&X_2\oplus X_3\ar[d]\ar[r] &X_4\ar[d]\\
X_3\ar[rr,bend right=8ex,"h'"{blue,swap},blue]\ar[r,"w"swap]&X_4\oplus X_5 \ar[r,"z"swap] &X_6&&X_1\ar[r]&X_2\oplus X_5\ar[r]&X_6
\end{tikzcd}
\]

where $r=\begin{bmatrix}j\;\;0\\0\;\;\Id\end{bmatrix}$, $s=\begin{bmatrix}h\\0\end{bmatrix}$, $u=\begin{bmatrix}f\\g'g\end{bmatrix}$, $v={[}j'j{,}-k'{]}$, $w=\begin{bmatrix}f'\\g'\end{bmatrix}$ and $z={[}j'{,}-k'{]}$.

Suppose we have a homotopy pullback square $X_2$

\[
\begin{tikzcd}
\end{tikzcd}
\]

and an object $X_3$ in $\rep(\overline{\Sq},\A)$

\[
\begin{tikzcd}
\end{tikzcd}
\]

where $g$ is a morphism from $D$ to $E$.
Then the triple $(c,gd,s_3)$ satisfies $d(s_3)=hgd-fc=-[-h,\;f]\begin{bmatrix}gd\\c\end{bmatrix}$ and hence we have a morphism $a:A\rightarrow B$ such that there exists a triple $(s,s_1,t)$ such that

\[
\begin{bmatrix}gd\\c\end{bmatrix}-\begin{bmatrix}e\\b\end{bmatrix}a=\begin{bmatrix}d(s_1)\\d(s)\end{bmatrix}\]

\[
\]

So we have the following diagram

\[
\begin{tikzcd}
A\ar[d,"d"swap]\ar[rr,bend left=12ex,"s",dashed]\ar[r,dashed,"a"]\ar[rr,bend left=6ex,"c"]\ar[rd,"s_1"{red,swap},red,dashed]\ar[rrd,"t"{blue},blue,dashed,bend left=2ex]&B\ar[r,"b"]\ar[d,"e"swap]\ar[rd,"s_2"red,red,]&C\ar[d,"f"]\\
\end{tikzcd}
\]

where the left hand square $X_1$ is an object in $\rep(\overline{\Sq},\A)$ and $(-s,0,-t)$ is a homotopy between $X_3$ and the composition of $X_1$ and $X_2$, all viewed as morphisms in $Z^0(\Mor(\A))$.
So we have

Keep the notations as above.
If we have homotopy pullback squares $X_2$ and $X_3$, then there exists a pair $(a,s_1)$ such that $X_1$ is also homotopy pullback and that the composition of $X_1$ and $X_2$ is homotopic to $X_3$ viewed as morphisms in $Z^0(\Mor(\A))$.

§.§ Homotopy pushouts/pullbacks of homotopy exact sequences

Let $X$ be a homotopy right exact sequence over $\A$

\[
\begin{tikzcd}
A\ar[rr,bend right=8ex,"h"swap]\ar[r,"f"]&B\ar[r,"j"]&C.
\end{tikzcd}
\]

Let $\overline{\alpha}:A\rightarrow A'$ be a morphism in $H^0(\A)$.
Let $S\in \D(\mathrm{Sp})$ be the following span

\[
\begin{tikzcd}
\end{tikzcd}.
\]

By Lemma <ref>, it is determined by $\overline{\alpha}$ up to an isomorphism and this isomorphism restricts to the identity on each component in $H^0(\A)$.
Assume that the span $S$ admits a homotopy pushout.

We have a homotopy right exact sequence $X'$ and a morphism $\mu:X\rightarrow X'$

\begin{equation}\label{const}
\begin{tikzcd}
&A\ar[r,"f"]\ar[d,"h_0=\alpha"swap]\ar[rd,"s_1"red,swap ,red]\ar[rrd,"t"blue,blue]\ar[rr,bend left = 8ex,"h"]&B\ar[d,"h_1"swap]\ar[rd,"s_2"red,red]\ar[r,"j"]&C\ar[d,equal]\\
&A'\ar[r,"f'"swap]\ar[rr,bend right = 8ex,"h'"swap]&B'\ar[r,"j'"swap]&C
\end{tikzcd}
\end{equation}

such that the restriction of $\mu$ to $f:A\rightarrow B$ is a homotopy pushout of $S$ (cf. Subsection <ref>).
Furthermore, if both $X$ and $X'$ are homotopy short exact, then the homotopy pushout of $S$ is homotopy bicartesian.

Let $X''$ be the pushout of $S$

\[
\begin{tikzcd}
&A\ar[r,"f"]\ar[d,"h_0=\alpha"swap]\ar[rd,"s_1"red,swap ,red]&B\ar[d,"h_1"]\\
\end{tikzcd}\;.
\]

Consider the morphism

\[
(h,-j,0):M=\Cone(A\xrightarrow{[f,\alpha]^{\intercal}} B\oplus A')\rightarrow C.
\]

Since $X''$ is the homotopy pushout of $S$, we have a morphism $j':B'\rightarrow C$ such that there exists a graded morphism $(-t, s_2, h'):M\rightarrow C$ of degree $-1$ such that

\[
d(-t,s_2,h')=(-j's_1+h, -j+j'h_1, -j'f').
\]

So we have

\[
(-d(t)-h'\alpha-s_2f, d(s_2), d(h'))=(-j's_1+h, -j+j'h_1, -j'f').
\]

Thus we have a morphism $\mu:X\rightarrow X'$ of the form <ref> where $X'$ is the 3-term h-complex on the second row.

By Proposition <ref>, we deduce that $X'$ is homotopy right exact and that if both $X$ and $X'$ are homotopy short exact sequences, then $X''$ is homotopy bicartesian.

The following is a direct consequence of Lemma <ref> and Proposition <ref>.

Let $\A$ be an additive dg category.
For a homotopy cocartesian square $X$ as follows

\[
\begin{tikzcd}
\end{tikzcd}\;,
\]

the morphism $f$ has a homotopy cokernel if and only if $k$ has a homotopy cokernel.
If this is the case, there is a morphism from the homotopy cokernel of $f$ to that of $k$ such that when restricted to the third term, it restricts to the identity, and that when restricted to $f:A\rightarrow B$, it restricts to $X$.
Here we still denote by $X$ the corresponding morphism from $f$ to $k$ in $H^0(\Mor(\A))$ given by the homotopy cocartesian square $X$.

Suppose we are given a homotopy cartesian square $X\in \rep(\overline{\Sq},\A)$ of the form

\[
\begin{tikzcd}
\end{tikzcd}
\]

where all terms are representable, together with a morphism $\alpha$ from $f':A'\rightarrow B$ to $k:C\rightarrow D$ in $H^0(\Mor(\A))$

\[
\begin{tikzcd}
\end{tikzcd}\;.
\]

We still denote by $X$ the associated morphism from $f:A\rightarrow B$ to $k:C\rightarrow D$ given by the homotopy cartesian square $X$. Then there exists a morphism $\beta: f'\rightarrow f$ in $H^0(\Mor(\A))$ such that $X\circ\beta=\alpha$ and that $\beta$ restricts to $\Id_{B}$.

Consider the morphism

\[
A'\xrightarrow{[f',-g',s']^{\intercal}} M= \Sigma^{-1}\Cone(B\oplus C\xrightarrow{[j,k]} D).
\]

Since $X$ is homotopy cartesian, we have a morphism $r:A'\rightarrow A$ such that there exists a graded morphism $[u,v,w]^{\intercal}:A'\rightarrow M$ of degree $-1$ such that

\[
d([u,v,w]^{\intercal})=[f,-g,s]^{\intercal}\circ r-[f',-g',s']^{\intercal}.
\]

So we have $d(u)=fr-f'$, $d(v)=g'-gr$ and $d(w)=-ju-kv+s'-sr$.

We have the following diagram

\[
\begin{tikzcd}
A'\ar[rrd,bend left=8ex,"f'"]\ar[rdd,bend right=8ex,"g'"swap]\ar[rrdd,dashed,bend left=5ex,red,"s'"]\ar[rd,dashed,"r"swap]&&\\
\end{tikzcd}
\]

So we have a morphism $\beta:f'\rightarrow f$ in $H^0(\Mor(\A))$ as follows

\[
\begin{tikzcd}
\end{tikzcd}\;
\]

such that the composition of $\beta$ with $X$ is homotopic to $\alpha$.

We also have the following direct consequence of Lemma <ref> and Proposition <ref>.

Let $j:B\rightarrow C$ and $j':C\rightarrow D$ be two objects in $H^0(\Mor(\A))$.
Assume that they both admit homotopy kernels which are homotopy short exact sequences

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap, bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}\;\;,\;\;
\begin{tikzcd}
A'\ar[r,"f'"]\ar[rr,"h'"swap, bend right=8ex]&C\ar[r,"j'"]&D
\end{tikzcd}
\]

If the cospan $L$

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pullback, then we have the following diagram

\[
\begin{tikzcd}
A\ar[dd,bend right=12ex,"h''"{swap,blue},blue]\ar[r,equal]\ar[d,"u"swap]\ar[rd,"s'"{red},red]\ar[rdd,"t"{red},red]&A\ar[d,"f"swap]\ar[dd,bend left=8ex,"h"blue,blue, near end]&\\
E\ar[r,"w"swap,near start]\ar[d,"v"swap]\ar[rr,bend left=6ex,"h'''"{blue,near end},blue]\ar[rd,"s"{red,swap},red]&B\ar[r,"j'j"swap]\ar[d,"j"swap]&D\ar[d,equal]\\
A'\ar[r,"f'"swap]\ar[rr,"h'"{swap,blue},bend right=8ex,blue]&C\ar[r,"j'"swap]&D
\end{tikzcd}
\]

a) The left bottom square is a homotopy pullback square.

b) The sequence on the leftmost column is a homotopy left exact sequence.

c) The sequence in the middle row is a homotopy left exact sequence which is homotopy short exact if the homotopy pullback of the cospan $L$ is homotopy short exact.

d) The 6-tuple $(\Id_{A},w,f',s',s,t)$ is a morphism from the 3-term h-complex in the first column to that in the second column in $\mathcal H_{3t}(\A)$.

e) We have $h'''=j's+h'v$ and the 6-tuple $(v,j,\Id_{D},s,0,0)$ is a morphism from the 3-term h-complex in the second row to that in the bottom row in $\mathcal H_{3t}(\A)$.

Since the diagram functor $\Dia: H^0(\Mor(\A)) \to \Mor(H^0(\A))$ is an epivalence,
an object $f: A \to B$ of $\Mor(\A)$ is a homotopy equivalence if and only if
this holds for any object $f': A' \to B'$ such that $\Dia(f)$ and $\Dia(f')$ are
isomorphic in $\Mor(H^0(\A))$.
It is direct to verify the following lemma.
Consider an object

\[
\begin{tikzcd}
\end{tikzcd}\;\;.
\]

of $\rep(\overline{\Sq},\A)$ which is homotopy cartesian.
If the morphism $j$ is a homotopy equivalence, then so is the morphism $g$.

§ EXACT DG CATEGORIES

Let $\A$ be a dg category over $k$ which is additive, i.e. $H^0{\A}$ is additive.
For convenience, we will assume that $Z^0(\A)$ is an additive category.

For an object $f:A\rightarrow B$ in $\rep(\Mor)$, denote by $[f]$ its isomorphism class and by $\overline{f}$ the corresponding morphism in $H^0(\A)$, i.e.  the corresponding object in $\Fun(\Mor,\D(\A))$.

Recall that by Lemma <ref>, the diagram functor $\Dia:\D(\Mor)\rightarrow \Mor(\D(\A))$ is an epivalence.
It restricts to an epivalence $H^0(\Mor(\A))\rightarrow \Mor(H^0(\A))$.
Therefore if a property of objects in $H^0(\Mor(\A))$ is stable under isomorphisms, we will also say that the corresponding objects in $\Mor(H^0(\A))$ have the same property.
We emphasize that if the object $f:A\rightarrow B$ is isomorphic to $f':A'\rightarrow B'$ and $g:B\rightarrow C$ is isomorphic to $g':B'\rightarrow C'$ in $H^0(\Mor(\A))$, it is not necessarily true that $gf$
is isomorphic to $g'f'$.
But this will not cause confusion because in what follows, we consider objects in $H^0(\Mor(\A))$ up to isomorphism.
For example, when we say “compositions of deflations are deflations",
it means that $gf$ is a deflation whenever $g$ and $f$ are deflations and then $g'f'$ is also a deflation,
although $g'f'$ and $gf$ may not be isomorphic.

An exact structure on $\A$ is a class $\mathcal{S}\subseteq \mathcal H_{3t}(\A)$ stable under isomorphisms, consisting of homotopy short exact sequences (called conflations)

\[
\begin{tikzcd}
A\ar[r, tail,"i"]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,two heads, "p"]&C\\
\end{tikzcd}
\]

where $i$ is called an inflation and $p$ is called a deflation, such that the following axioms are satisfied

Ex0$\Id_{0}$ is a deflation.

Ex1Compositions of deflations are deflations.

Ex2Given a deflation $p:B\rightarrow C$ and any map $c: C'\rightarrow C$ in $Z^0(\A)$, the object

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pullback

\[
\begin{tikzcd}
\end{tikzcd}
\]

and ${p'}$ is also a deflation.

$\Ex2^{op}$Given an inflation $i: A\rightarrow B$ and any map $a:A\rightarrow A'$ in $Z^0(\A)$, the object

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pushout

\[
\begin{tikzcd}
\end{tikzcd}
\]

and ${i'}$ is also an inflation.

We call $(\A,\mathcal {S})$ or simply $\A$ an exact dg category.

Let $(\A,\mathcal S)$ and $(\A',\mathcal S')$ be exact dg categories.
A morphism $F:\A\rightarrow \A'$ in $\Hqe$ is exact if the induced functor $\mathcal H_{3t}(\A)\rightarrow \mathcal H_{3t}(\A')$ sends objects in $\mathcal S$ to objects in $\mathcal S'$.
We denote by $\Hqe_{ex}(\A,\A')$ the subset of $\Hqe(\A,\A')$ consisiting of exact morphisms.

Note that the class $\mathcal S\subseteq \mathcal H_{3t}(\A)$ is stable under isomorphisms.
Let $\overline{\imath}$ be a morphism in $H^0(\A)$.
If a representative $i$ of $\overline{\imath}$ is an inflation, then any representative in
$\overline{\imath}$ is also an inflation.
Thus, being an inflation is a property of the morphism $\overline{\imath}$.
We have an even stronger property: If we view $\ol{\imath}:A_0\rightarrow A_1$ as an
object in $\Mor(H^0(\A))$, then being an inflation is stable under isomorphisms in $\Mor(H^0(\A))$, cf. Lemma <ref>.

In the setting of Axiom ${\Ex}2$, since the homotopy pullback is unique up to a unique isomorphism
in $\mathcal H_{3t}(\A)$, by Lemma <ref>, the object ${j'}$ is unique up to a canonical isomorphism
in $H^0(\Mor(\A))$.

Let $\A$ and $\A'$ be additive dg categories.

a) By Lemma <ref> and its dual, the exact dg structures on $\A$ are in bijection with the exact structures on $\tau_{\leq 0}\A$.

b) Let $F:\A\rightarrow \A'$ be a quasi-equivalence of dg categories.
Then $F$ induces an equivalence of categories $\mathcal H_{3t}(\A)\iso\mathcal H_{3t}(\A')$ which
preserves and reflects the property of being homotopy short exact.
Thus, the quasi-equivalence $F$ induces a bijection between the class of exact dg structures
on $\A$ and that on $\A'$.

For two objects $C$ and $A$ in $\A$, consider the set of `equivalence classes of extensions'
$\mathbb E(C,A)$ (resp. $\mathbb E'(FC,FA)$), to be defined in Definition <ref>, which is associated with the exact structure on $\A$ (resp. $\A'$).
The quasi-equivalence $F$ induces a bijection between $\mathbb E(C,A)$ and $\mathbb E'(C,A)$, cf. Lemma <ref>.

c) Let $\A$ be an exact dg category and $\A\rightarrow \pretr(\A)$ be the inclusion of $\A$ into its pretriangulated hull.
Let $\A'$ be the additive closure of $\A$ in $\pretr(\A)$.
Since $H^0(\A)$ is an additive category, the inclusion of $\A\rightarrow \A'$ is a quasi-equivalence.
By b), we may replace $\A$ by $\A'$ and assume that the dg category $\A$ is such that $Z^0(\A)$ is an additive category.

Note that an object $f:A\rightarrow B$ in $H^0(\Mor(\A))$ is a deflation (resp. an inflation) if and only if it admits a homotopy kernel (resp. homotopy cokernel) (cf. Definition <ref>) which is a conflation. Observe that homotopy equivalences are both inflations and deflations by axioms
Ex0, Ex2 and Ex$2^{op}$.

Let $\A$ be an additive category which we consider as a dg category concentrated in degree 0.

Let $A$, $B$ and $C$ be objects in $\A$ and consider a 3-term h-complex $X$ in $\A$

\[
\begin{tikzcd}
A\ar[rr,"h"swap,bend right=8ex]\ar[r,"f"]&B\ar[r,"j"]&C.
\end{tikzcd}
\]

Since $\A$ is concentrated in degree zero, the homotopy $h$ is zero.
By Example <ref>, the 3-term h-complex $X$ is a homotopy short exact sequence if and only if the corresponding sequence $0\rightarrow A\xrightarrow {f}B\xrightarrow{j}C\rightarrow 0$ is a kernel-cokernel pair in the additive category $\A$.
Also, two kernel-cokernel pairs are isomorphic (resp. equivalent) if and only if the corresponding homotopy short exact sequences are isomorphic (resp. equivalent).

By <cit.>, the axioms of a Quillen exact structure on $\A$ correspond to the axioms of an exact dg structure on $\A$.
Therefore, to endow an additive category $\A$ with a Quillen exact structure is the same as to endow $\A$ with an exact dg structure.

Let $\A$ be a pretriangulated dg category.
By Remark <ref>, we may assume that the dg category $\A$ is strictly pretriangulated.
Then each morphism $f:A\rightarrow B$ admits a homotopy cokernel

\[
\begin{tikzcd}
A\ar[r,"f"] \ar[rr, bend right=8ex,"h"swap]&B\ar[r,"j"]&\Cone(f).
\end{tikzcd}
\]

Dually, each morphism admits a homotopy kernel.
It is not hard to check that the class of all homotopy short exact sequences in $\A$ defines an exact structure on $\A$.
If not stated otherwise, we always consider this maximal exact structure on a pretriangulated dg category.

A full dg subcategory $\A'$ of an exact dg category $\A$ is extension-closed provided that for each conflation in $\A$

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,"j"]&C,
\end{tikzcd}
\]

if $A$ and $C$ belong to $\A'$, then so does $B$.

Let $\A'$ be an extension-closed subcategory of an exact dg category $(\A,\mathcal S)$.
Let $X$ be a conflation in $\A$ with all terms in $\A'$.
Then $X$ can be seen as a 3-term h-complex in $\A'$.
By Remark <ref>, the object $X$ is a homotopy short exact sequence in $\A'$.

We have an inclusion functor $\mathcal H_{3t}(\A')\hookrightarrow \mathcal H_{3t}(\A)$.
Let $\mathcal S'$ be the class of objects in $\mathcal H_{3t}(\A')$ whose image in $\mathcal H_{3t}(\A)$ belongs to $\mathcal S$.
We show that the class $\mathcal S'$ defines an exact dg structure on $\A'$.
Axiom ${\Ex}0$ is direct to check.
Axiom ${\Ex}1$ follows from Lemma <ref> and the fact that $\A'$ is extension-closed in $\A$.
Consider the span $S$ of a morphism in $\A'$ along an inflation in $\A'$.
It admits a homotopy pushout in $\A$, and by Proposition <ref>,
the homotopy pushout $X$ is an extension of objects in $\A'$ and hence remains an object in $\A'$.
Again by Proposition <ref>, the pushout of the inflation in $\A'$ is an inflation in $\A$ and
the conflation that this inflation belongs to is in $\mathcal S'$.
Thus Axiom $\Ex2^{op}$ is satisfied.
Dually, one checks Axiom $\Ex2$.

We consider a special case of <cit.> where the torsion pair $\mathcal S=\X\perp\Y$ is a t-structure.
Let $\T$ be an algebraic triangulated category and $\mathcal S$ a thick subcategory with a t-structure $\mathcal S=\X\perp \Y$.
Assume that $\T$ has torsion pairs $\T=\X\perp \X^{\perp}={^{\perp}\Y}\perp \Y$ where $(-)^{\perp}$ and $^{\perp}(-)$ both denote the Hom orthogonal.
Put $\Z=\X^{\perp}\cap {^{\perp}\Sigma \Y}$ and $\U=\T/\mathcal S$.
So we have a sequence of functors $\Z\hookrightarrow \T\xrightarrow{\pi}\T/\mathcal S=\U$.
By <cit.>, we have that the above composition is an equivalence of $k$-categories $F:\Z\iso \U$.
Indeed, the triangulated structure on $\Z$ is canonical and does not depend on the categories $\mathcal X$ and $\mathcal Y$, once we fix a dg enhancement for $\T$.
We denote the dg enhancements of the above categories by adding a subscript $dg$ to the notations of the corresponding categories.
In particular, we have that $\U_{dg}$ is the dg quotient $\T_{dg}/\mathcal S_{dg}$.
Claim: the composition of dg functors $\Z_{dg}\hookrightarrow \T_{dg}\rightarrow \U_{dg}$ induces a quasi-equivalence  $\tau_{\leq 0}\Z_{dg}\iso \tau_{\leq 0}\U_{dg}$.
Indeed, let $Z$ be an object in $\Z$. Then $\Sigma^{-1}Z$ is in the subcategory $\Sigma^{-1}\X^{\perp}\cap {^{\perp}\Y}\subseteq \Sigma^{-1}\X^{\perp}\cap {^{\perp}\Sigma \Y}$.
We take the triangle of $\Sigma^{-1}Z$ associated with the t-structure $\T=\X\perp\X^{\perp}$

\[
\begin{tikzcd}
V\rightarrow \Sigma^{-1}Z\rightarrow Z'\rightarrow \Sigma V.
\end{tikzcd}
\]

We see that $Z'$ is in $\Z$ and $F(Z')\iso \Sigma^{-1}F(Z)$.
For $i\geq 0$ and $W\in \Z$, we have

\[
\Hom_{\T}(Z',\Sigma^{-i}W)\iso \Hom_{\T}(Z,\Sigma^{-i+1}W).
\]

Therefore by induction we have

\[
\Hom_{\T}(Z,\Sigma^{-i+1}W)\iso \Hom_{\U}(F(Z),\Sigma^{-i+1}F(W))
\]

for $\mbox{$i\geq 0$}$.
Since $\U_{dg}$ is pretriangulated, we have that $\Z_{dg}$ is an exact dg category with conflations given by all homotopy short exact sequences in $\Z_{dg}$.
Note that in general $\Z_{dg}$ is not a pretriangulated dg category.

The following diagram lemma is useful.
It is a direct consequence of Lemma <ref> and Axiom $\Ex2$ (or $\Ex2^{op}$).

Let $(\A,\mathcal S)$ be an exact dg category.
Suppose we are given conflations $X$

\[
\begin{tikzcd}
A\ar[r,"f"]          \ar[rr, bend right=8ex,"h"swap]                  &B\ar[r,"j"]&C
\end{tikzcd}
\]

and $X'$

\[
\begin{tikzcd}
A'\ar[r,"f'"]          \ar[rr, bend right=8ex,"h'"swap]                  &B'\ar[r,"j'"]&C'
\end{tikzcd}.
\]

Let $\alpha:X\rightarrow X'$ be a morphism in $\mathcal H_{3t}(\A)$.
Denote by $\overline a:A\rightarrow A'$ (resp. $\overline c:C\rightarrow C'$) the restriction of $\alpha$ to $A$ (resp. $C$).
Then $\alpha$ can be written as a composition $X\xrightarrow{\beta} \tilde{X}\xrightarrow{\gamma} X'$ where $\tilde{X}$ is a conflation with ends $A'$ and $C$, and the morphism $\beta$ restricts to $\overline a:A\rightarrow A'$ and $\Id_{C}$, and the morphism $\gamma$ restricts to $\Id_{A'}$ and $\overline{c}:C\rightarrow C'$.
If we use the biadditive bifunctor $\mathbb E$ which is to be defined in <ref>, then we have $[X']\in \mathbb E(C,A')$ and the morphism $\beta$ (resp. $\gamma$) shows that $[\tilde{X}]=\overline a_*[X]$ (resp. $[\tilde{X}]=\overline c^{*}[X']$).

Combining Lemma <ref> and Lemma <ref>, we have

Let $\A$ be an exact dg category.
If a morphism $\alpha:X\rightarrow X'$ of conflations restricts to homotopy equivalences on both ends, it restricts to a homotopy equivalence between the middle terms.

Let $(\A,\mathcal S)$ be an exact dg category and $A,C$ two objects in $H^0\A$.
Let $X_i$, $i=1,2$, be two conflations of the form

\begin{equation}\label{twoconflations}
\begin{tikzcd}
A\ar[r,"f_i"]\ar[rr,"h_i"swap,bend right=8ex]&B_i\ar[r,"j_i"]&C
\end{tikzcd}.
\end{equation}

A morphism $\theta:X_1\rightarrow X_2$ in $\mathcal H_{3t}(\A)$ is called an equivalence if it restricts to $\Id_{A}$ and $\Id_{C}$.
Note that by Corollary <ref>, an equivalence of conflations is necessarily an isomorphism in $\mathcal H_{3t}(\A)$.
For a conflation, we denote by $[X]$ the equivalence class to which $X$ belongs.

We define $\mathbb E_{\mathcal S}(C,A)$ to be the set of equivalence classes of conflations

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}
\]

with fixed ends $A$ and $C$.
When there is no risk of confusion, we will simply denote it by $\mathbb E(C,A)$.

Let $F:\A\rightarrow \A'$ be a quasi-equivalence of dg categories.
Suppose $\A$ is an exact dg category.
Equip $\A'$ with the induced exact structure from the quasi-equivalence $F$.
For two objects $C$, $A$ in $\A$, consider the sets $\mathbb E(C,A)$ (resp. $\mathbb E'(FC,FA)$), which is associated with the exact structure on $\A$ (resp. $\A'$).
The quasi-equivalence $F$ induces a bijection

\[
\mathbb E(C,A)\rightarrow \mathbb E'(FC,FA).
\]

It is direct to check that the above map is an injection.
We show that it is a surjection.
For an object $X\in\mathcal H_{3t}(\A)$, we denote by $FX$ the object in $\mathcal H_{3t}(\A')$ induced by $F$.

Let $[X']$ be an element in $\mathbb E'(FC,FA)$.
Suppose $X'$ is of the form

\[
\begin{tikzcd}
FA\ar[r,"f'"]\ar[rr,"h'"swap,bend right=8ex]&B'\ar[r,"j'"]&FC\mathrlap{.}
\end{tikzcd}
\]

Since $F:\A\rightarrow\A'$ is a quasi-equivalence, we have a homotopy equivalence $FB\iso B'$ for some $B\in\A$.
Hence we have an isomorphism in $H^0(\Mor(\A'))$ of the form

\[
\begin{tikzcd}
\end{tikzcd}
\]

By Lemma <ref>, we may assume $B'=FB$ and $j'=F(j)$ for some $j:B\rightarrow C$ in $Z^0(\A)$.
The morphism $j:B\rightarrow C$ is a deflation.
Suppose its homotopy kernel $Y$ is of the form

\[
\begin{tikzcd}
A'\ar[r,"\tilde{f}"]\ar[rr,"\tilde{h}"swap,bend right=8ex]&B\ar[r,"j"]&C\mathrlap{.}
\end{tikzcd}
\]

Then we have an isomorphism $\theta: X'\rightarrow FY$ in $\mathcal H_{3t}(\A)$ which restricts to the identity of $j':FB\rightarrow FC$ in $H^0(\Mor(\A'))$.
Let $\overline{a'}:FA'\rightarrow FA$ be the restriction of $\theta$ to $FA$.
Then we have a morphism $a:A\rightarrow A'$ such that $F(a)$ is a homotopy inverse of $a'$.
We have an isomorphism in $H^0(\Mor(\A'))$ of the form

\[
\begin{tikzcd}
\end{tikzcd}
\]

By Lemma <ref>, we infer that $X'$ is equivalent to $FX$ where $X$ is of the form

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C\mathrlap{.}
\end{tikzcd}
\]

Suppose $(\A, \mathcal S)$ is an exact dg category.
The following statements hold:

a) The diagram

\[
\begin{tikzcd}
A\ar[r,"{[}1\ 0{]}^{\intercal}"]\ar[rr,"0"swap,bend right =8ex]&A\oplus B\ar[r,"{[}0\ 1{]}"]\ar[r]&B\\
\end{tikzcd}
\]

is a conflation for $A,B\in \A$.

b)If a morphism $g:B\rightarrow C$ in $Z^0(\A)$ admits a homotopy kernel and for some $h:B'\rightarrow B$ the composition $gh:B'\rightarrow C$ is a deflation, then $g$ is a deflation.

b)$^{\text{op}}$If a morphism $f:A\rightarrow B$ admits a homotopy cokernel and for some $e:B\rightarrow B'$ the composition $ef:A\rightarrow B'$ is an inflation, then $f$ is an inflation.

c)The direct sum of two conflations is a conflation.

d)Axiom ${\Ex1}^{op}$ holds.

Suppose we are in the situation of Axiom Ex2.

Claim 1: $B\oplus C'\xrightarrow{[j,c]} C$ is a deflation.

We have a conflation $X\in\mathcal H_{3t}(\A)$ of the form

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,"j"]&C\mathrlap{.}\\
\end{tikzcd}
\]

By Axiom Ex2, the cospan

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pullback $Y$:

\[
\begin{tikzcd}
{B'}\ar[r," {j'}"]\ar[d," {b}"swap]\ar[rd,"s_2"red,red]& {C'}\ar[d," {c}"]\\
{B}\ar[r," {j}"swap]& {C}
\end{tikzcd}\;\;.
\]

By the dual of Proposition <ref>, we have a morphism $\mu:X'\rightarrow X$ in $\mathcal H_{3t}(\A)$

\[
\begin{tikzcd}
&A\ar[r,"f'"]\ar[d,equal]\ar[rd,"s_1"red,swap ,red]\ar[rrd,"t"blue,blue]\ar[rr,bend left = 8ex,"h'"]&B'\ar[d,"b"swap]\ar[rd,"s_2"red,red]\ar[r,"j'"]&C'\ar[d,"c"]\\
&A\ar[r,"f"swap]\ar[rr,bend right = 8ex,"h"swap]&B\ar[r,"j"swap]&C
\end{tikzcd}
\]

where $X'$ is the 3-term h-complex in the first row.

By Axiom Ex2, the morphism ${j'}$ is also a deflation.
So $X'$ is a conflation and in particular a homotopy short exact sequence.
Hence, by the dual of Proposition <ref>, $Y$ is homotopy bicartesian.

Let $X''$ be the 3-term h-complex

\[
\begin{tikzcd}
{B'}\ar[r,"\begin{bmatrix} {j'}\\ {b}\end{bmatrix}"]\ar[rr,"s_2"swap,bend right=8ex]& {C'}\oplus  {B}\ar[r,"{[} {c}{,}-{j}{]}"]&C\\
\end{tikzcd}\;\;.
\]

Then $X''$ is a homotopy short exact sequence.

Claim 2: $X''$ is a conflation.

Consider the conflation $X$ and the object $f':A\rightarrow B'$ in $H^0(\Mor(\A))$.
The span $S$

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pushout $Z$

\[
\begin{tikzcd}
A\ar[rd,"s_3"{red},red]\ar[r,"f"]\ar[d,"f'"swap]& B\ar[d,"h_1=\begin{bmatrix}0\\\Id\end{bmatrix}"]\\
B'\ar[r,"f''=\begin{bmatrix}j'\\b\end{bmatrix}"swap]&C'\oplus B
\end{tikzcd}.
\]

where $s_3=\begin{bmatrix}-h'\\s_1\end{bmatrix}$.
By Axiom Ex$2^{op}$, the morphism $B'\xrightarrow{[b,j']^{\intercal}} B\oplus C'$ is an inflation.
So by Proposition <ref>, there exists a morphism $X\rightarrow \tilde{X''}$ in $\mathcal H_{3t}(\A)$

\[
\begin{tikzcd}
&A\ar[r,"f"]\ar[d,"f'"swap]\ar[rd,"s_3"red,swap ,red]\ar[rrd,"t"blue,blue]\ar[rr,bend left = 8ex,"h"]&B\ar[d,"h_1"swap]\ar[rd,"s_2"red,red]\ar[r,"j"]&C\ar[d,equal]\\
&B'\ar[r,"f''"swap]\ar[rr,bend right = 8ex,"h''"swap]&C'\oplus B\ar[r,"j''"swap]&C
\end{tikzcd}
\]

where $\tilde{X''}$ is the 3-term h-complex in the second row.
By the uniqueness of homotopy cokernels,
the homotopy right exact sequence $\tilde{X''}$ is isomorphic to $X''$.
By Axiom Ex$2^{op}$, $\tilde{X''}$ is a conflation.
This proves Claim 2.
Then the morphism $B\oplus C'\xrightarrow{[j,c]} C$ is a deflation.
This proves Claim 1.
Note that during the proof we only use Axioms Ex2 and Ex$2^{op}$,
so the dual of the claims also holds.

Now we are ready to prove $\mathrm{b})$.
Suppose we have a homotopy left exact sequence

\[
\begin{tikzcd}
K\ar[r,"f"] \ar[rr,"h"swap,bend right=8ex]&B\ar[r,"g"]&C
\end{tikzcd}
\]

From the cospan

\[
\begin{tikzcd}
B'\ar[r,"gh"swap, two heads]&C
\end{tikzcd},
\]

we know that the morphism $B\oplus B'\xrightarrow{[g,gh]} C$ is a deflation by claim 1.
Then the morphism

\[
{[}g\ 0{]}:B\oplus B'\xrightarrow{\begin{bmatrix}1&-h\\0&1\end{bmatrix}} B\oplus B'\xrightarrow{\begin{bmatrix}g,gh\end{bmatrix}} C
\]

is also a deflation by Axiom Ex1.

The 3-term h-complex

\[
\begin{tikzcd}
K\oplus B'\ar[r,"u"]\ar[rr,"{[}h{,}\;0{]}"swap,bend right=8ex]&B\oplus B'\ar[r,"v"]&C\mathrlap{,}\\
\end{tikzcd}
\]

where $u=\begin{bmatrix}f&0\\0&\Id_{B'}\end{bmatrix}$ and $v={[}g{,}\;0{]}$, is homotopy left exact, as the direct sum of a homotopy left exact sequence and a trivial 3-term h-complex.
Therefore it is a conflation.

We have the following morphism of homotopy left exact sequences

\[
\begin{tikzcd}
K\oplus B'\ar[rrd,"0"{blue,near end},blue,bend right=2ex]\ar[rd,"0"{red,swap},red,bend right=2ex]\ar[r,"u"]\ar[d,"{[}1{,} 0{]}"swap]\ar[rr,bend left=8ex,"{[}h{,}\;0{]}"] &B\oplus B'\ar[rd,"0"red,red]\ar[d,"{[}1{,}0{]}"swap]\ar[r,"v"]&C\ar[d,equal]\\
K\ar[r,"f"swap] \ar[rr,"h"swap,bend right=8ex]&B\ar[r,"g"swap]&C
\end{tikzcd}
\]

where the left square is homotopy bicartesian, we infer that the morphism $g:B\rightarrow C$ is a deflation by Ex$2^{op}$. This proves Statement b).

Since the composition

\[
\Id_B:B\xrightarrow{\begin{bmatrix}0\\1\end{bmatrix}} A\oplus B\xrightarrow{[0,\;1]} B
\]

is a deflation and the morphism $A\oplus B\xrightarrow{[0,\;1]} B$ admits a homotopy kernel,
it is a deflation by Statement b).
So Statement a) follows.

Let $X_i$, $i=1$, $2$, be a conflation of the form

\[
\begin{tikzcd}
A_i\ar[r,"f_i"]\ar[rr,bend right=8ex,"h_i"swap]&B_i\ar[r,"j_i"]&C_i\mathrlap{\;.}
\end{tikzcd}
\]

Clearly the direct sum $X_1\oplus X_2$ is homotopy short exact and the morphism

\[
\begin{tikzcd}
B_1\oplus B_2\ar[r,"\begin{bmatrix}j_1\ 0\\0\ j_2\end{bmatrix}"]&C_1\oplus C_2
\end{tikzcd}
\]

can be written as a composition

\[\begin{tikzcd}
B_1\oplus B_2\ar[r,"r=\begin{bmatrix}1\ 0\\0\ j_2\end{bmatrix}"]&B_1\oplus C_2\ar[r,"s=\begin{bmatrix}j_1\ 0\\0\ 1\end{bmatrix}"]&C_1\oplus C_2
\end{tikzcd}.
\]

Observe that the morphism $B_1\oplus B_2\xrightarrow{r} B_1\oplus C_2$ is a homotopy pullback of the deflation $j_2:B_2\rightarrow C_2$ along $B_1\oplus C_2\rightarrow  C_2 $ and hence is a deflation.
Similarly, the morphism $B_1\oplus C_2\xrightarrow{s} C_1\oplus C_2$ is also a deflation.
So the composition $B_1\oplus B_2\rightarrow C_1\oplus C_2$ is a deflation.
This proves Statement c).

It remains to prove Axiom ${\Ex1}^{op}$ and then one proves b)$^{op}$ dually.

Let $f:A\rightarrow B$ be an inflation with the homotopy cokernel given by

\[
\begin{tikzcd}
{A}\ar[r," {f}"]\ar[rr,bend right=8ex,"h"swap]& {B}\ar[r,"j"]&C
\end{tikzcd}
\]

Let $f':B\rightarrow B'$ be another inflation.
Our aim is to show that the morphism $f'f$ is also an inflation.

By Axiom ${\Ex2}^{op}$, the span

\[
\begin{tikzcd}
\end{tikzcd}
\]

admits a homotopy pushout

\[
\begin{tikzcd}
{B}\ar[r," {f'}"]\ar[d," {j}"swap]\ar[rd,"s"red,red]& {B'}\ar[d," {j'}"]\\
{C}\ar[r," {f''}"swap]& {C'}
\end{tikzcd}
\]

which is homotopy bicartesian.
Thus by the dual of Claim 1,
the morphism $[ {j'}, {f''}]: {B'}\oplus  {C}\rightarrow  {C'}$ is a deflation.
By the dual of Corollary <ref>, the morphism ${j'}$ admits a homotopy kernel which is isomorphic to $f'f:A\rightarrow B'$.

We have the following commutative diagram in $\A$

\[
\begin{tikzcd}
{B'}\oplus  {B}\ar[rd,"{[}0{,}-s{]}"red,red]\ar[r,"\begin{bmatrix}1\ 0\\0\  {j}\end{bmatrix}"]\ar[d,"{[}1{,} {f'}{]}"swap]& {B'}\oplus  {C}\ar[d,"{[} {j'}{,} {f''}{]}"]\\
{B'}\ar[r," {j'}"swap]& {C'}
\end{tikzcd}.
\]

By Axiom ${\Ex1}$, the morphism $[ {j'}, {f''}]\begin{bmatrix}1&0\\0&{j}\end{bmatrix}$ is a deflation.
By the above diagram, the morphism ${j'}[1, {f'}]$ is homotopic to it and hence is also a deflation.
So by Statement b), the morphism ${j'}$ is a deflation.
Hence the morphism $f'f$ is an inflation.
This proves Statement d).

§.§ The biadditive bifunctor $\mathbb E$

Suppose we are given an element $[X]\in \mathbb E(C,A)$, where $X$ is as follows

\begin{equation}\label{X}
\begin{tikzcd}
A\ar[r,"{f}"]       \ar[rr,bend right=8ex,"{h}"swap]      &B\ar[r,"{j}"]         &C
\end{tikzcd}\;\;,
\end{equation}

and a map $\overline{a}:A\rightarrow A'$ in $H^0(\A)$.
Below we will prove that there exists a unique equivalence class of conflations $[X']\in \mathbb E(C,A')$,
such that for each representative $X$ of $[X]$,
there exists a representative $X'$ of $[X']$ with a morphism $\theta:X\rightarrow X'$
which restricts to the identity on $C$ and the morphism $\overline{a}:A\rightarrow A'$.

By Axiom ${\Ex2}^{op}$, the existence of $[X']$ is clear from Proposition <ref>.

Let us clarify its uniqueness.
Suppose we have morphisms $\theta_1:X\rightarrow X_1$ and $\theta_2:X\rightarrow X_2$ with the required property.
So the objects $X_1  $ and $X_2  $ are both conflations.
By Proposition <ref>, we may assume that the restriction of $\theta_1$ to $f:A\rightarrow B$ is a homotopy cocartesian square.
Let $S_i\in\rep(\overline{\Sq},\A)$, $i=1$, $2$, be the restriction of $\theta_i$ to $f:A\rightarrow B$.
Assume that $X_i  $, $i=1,2$, is of the form

\[
\begin{tikzcd}
A'  \ar[r,"f_i"]\ar[rr,bend right=8ex,"h_i"swap]&B_i\ar[r,"j_i"]&C
\end{tikzcd}\;,
\]

and that $S_1$ is a homotopy pushout of the cospan $L$

\[
\begin{tikzcd}
\\A'
\end{tikzcd}.
\]

By the universal property of homotopy pushouts, cf. Remark <ref>, there is a unique morphism $\mu$ in $\rep(\overline{\Sq},\A)$ from $S_1$ to $S_2$ which restricts to $\Id_{L}$.
We still denote by $S_i$ the morphism in $H^0(\Mor(\A))$ from $f$ to $f_i$ given by the square $S_i$ for $i=1$, $2$.
By Lemma <ref>, the morphism $\mu:S_1\rightarrow S_2$ yields morphisms $\Id_{f}$, $S_1$, $S_2$ and a morphism $\alpha$ from $f_1:A'\rightarrow B_1$ to $f_2:A'\rightarrow B_2$ which restricts to $\Id_{A'}$ and such that we have $\alpha\circ S_1=S_2\circ \Id_{f}$ in $H^0(\Mor(\A))$.
By the universal property of homotopy pushouts, the morphism $\alpha$ induces a morphism $\theta_3:X_1\rightarrow X_2$ in $\mathcal H_{3t}(\A)$ such that we have $\theta_3\circ \theta_1=\theta_2$.
Then the morphism $\theta_3$ is an equivalence between $X_1$ and $X_2$.
Since $\theta_3$ is necessarily an isomorphism by Corollary <ref>, we have that the morphism $\alpha:S_1\rightarrow S_2$ is an isomorphism.
So we have

Let $\A$ be an exact dg category.
Suppose we have a morphism $\alpha:X\rightarrow X'$ of conflations in $\mathcal H_{3t}(\A)$ given by the diagram <ref>.
If the morphism $h_2:A_2\rightarrow A_2'$ is a homotopy equivalence, then the left square is homotopy bicartesian.

The operation $\overline{a}:A\rightarrow A'\mapsto \overline{a}_*:\mathbb E(C,A)\rightarrow \mathbb E(C,A')$,
where an element $[X]$ is sent to $\overline{a}_*[X]{\coloneqq}[X']$ which is specified by the above mentioned property,
together with its dual operation $\overline{c}:C'\rightarrow C\mapsto \overline{c}^{*}:\mathbb E(C,A)\rightarrow \mathbb E(C',A)$,
makes $\mathbb E:H^{0}(\A)^{op}\times H^{0}(\A)\rightarrow  \Set$ into a bifunctor.

In section <ref>, we will prove that $H^0(\A)$ carries a canonical extriangulated structure in the
sense of Nakaoka–Palu [111] whose extension bifunctor is $\mathbb{E}$.

Suppose we are given $[X]\in \mathbb E(C,A)$, $\overline{a}:A\rightarrow A'$ and $\overline{c}:C'\rightarrow C$, where $X$ is given by the diagram <ref>.

Put $\overline a_*[X]=[X_1]$, $\overline c^{*}[X_1]=[X_2]$ and $\overline c^*[X]=[X_3]$.
Then the conflations $X_1$, $X_2$ and $X_3$ are of the following form respectively

\[
\begin{tikzcd}
A'\ar[r,"f_1"]\ar[rr,bend right=8ex,"h_1"swap]&E_1\ar[r,"j_1"]&C &A'\ar[r,"f_2"]\ar[rr,bend right=8ex,"h_2"swap]&E_2\ar[r,"j_2"]&C'&A\ar[r,"f_3"]\ar[rr,bend right=8ex,"h_3"swap]&E_3\ar[r,"j_3"]&C'
\end{tikzcd}.
\]

By the definition of $[X_2]$, we have a homotopy cartesian square which gives rise to a morphism $\alpha$ in $H^0(\Mor(\A))$ from $j_2:E_2\rightarrow C'$ to $j_1:E_1\rightarrow C$.
We also have a morphism $\beta$ from $j_3:E_3\rightarrow C'$ to $j_1:E_1\rightarrow C'$.
By Lemma <ref>, we have a morphism $\gamma$ from $j_3:E_3\rightarrow C'$ to $j_2:E_2\rightarrow C'$ which restricts to $\Id_{C'}$ and such that $\alpha\circ \gamma=\beta$.
The associated diagram in $\D(\A)$ is as follows:

\[
\begin{tikzcd}[cramped,sep=small]
&A\ar[rr,"\overline{f_3}"]\ar[dd,dotted,bend left=4ex]\ar[ld,equal]&
&E_3\ar[rr,"\overline{\jmath_3}"]\ar[dd, dotted,bend left=4ex]\ar[ld]&
\end{tikzcd}
\]

By universal property of the homotopy pullbacks (cf. Remark <ref>), the morphism $\gamma$ induces a morphism from $X_3$ to $X_2$.
It clearly restricts to $\overline a:A\rightarrow A'$.
Therefore $\overline a_*[X_3]=[X_2]$.

Our next aim is to show that $\mathbb E(C,A)$ is an abelian group for each pair $(C,A)$ of objects in $H^0(\A)$ such that $\mathbb E:H^0(\A)^{op}\times H^0(\A)\rightarrow \Ab$ is a biadditive bifunctor.
We start by describing the addition operation on $\mathbb E(C,A)$.

Let $[X]$ and $[X']$ be two elements in $\mathbb E(C,A)$ given by the diagram <ref>.
Put $\overline a=[1\ 1]:A\oplus A\rightarrow A$ and $\overline c=[1\ 1]^{\intercal}:C\rightarrow C\oplus C$. Recall from Proposition <ref> that the direct sum $X\oplus X'$ is also a conflation.
Then the sum $[X]+[X']$ is defined to be $\overline c^{*}\overline a_*[X\oplus X']$.
The above addition operation is well-defined, associative and commutative and that the equivalence class of

\[
\begin{tikzcd}
A\ar[r,"\begin{bmatrix}1\\0\end{bmatrix}"]\ar[rr,"0"swap,bend right=8ex]&A\oplus C\ar[r,"{[}0{,}1{]}"]&C
\end{tikzcd}
\]

is a zero element.

It is direct to check the well-definedness and commutativity.
To show that the above conflation is a zero element, observe that we have the following morphism in $\mathcal H_{3t}(\A)$

\[
\begin{tikzcd}
A\ar[rrd,"0"{blue,near start},blue]\ar[d,"\begin{bmatrix}0\\1\end{bmatrix}"swap]\ar[rr,bend left=8ex,"h"]\ar[r,"f"]\ar[rd,"s"{red,swap},red,bend right=2ex]&B\ar[rd,"0"red,red,bend left=2ex]\ar[d,"{[}0{,}\;j{,}\;1{]^{\intercal}}"]\ar[r,"j"]&C\ar[d,"{[}1{,}\;1{]^{\intercal}}"]\\
A\oplus A\ar[rr,bend right=8ex,"h'"swap]\ar[r,"f'"swap]&A\oplus C\oplus B\ar[r,"j'"swap]&C\oplus C
\end{tikzcd}
\]

where $f'=\begin{bmatrix}1&0\\0&0\\0&f\end{bmatrix}$, $j'=\begin{bmatrix}0&1&0\\0&0&j\end{bmatrix}$, $h'=\begin{bmatrix}0&0\\0&h\end{bmatrix}$, $s=[0,\;h,\;0]^{\intercal}$.

To show the associativity, observe that we have the equality

\[
\begin{bmatrix}1&0\\0&1\\0&1\end{bmatrix}\circ\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}\circ\begin{bmatrix}1\\1\end{bmatrix}:A\rightarrow A\oplus A\oplus A.
\]

Conflations in the equivalence class of zero elements are called splitting.

Let $X$ be a conflation as follows

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,"j"]&C
\end{tikzcd}.
\]

The following statements are equivalent:

1)$X$ is a splitting conflation.

2)$\overline f$ is a split monomorphism in $H^{0}(\A)$.

2)$^{\text{op}}$$\overline{\jmath}$ is a split epimorphism in $H^{0}(\A)$.

The implication $1)\Rightarrow 2)$ follows from Corollary <ref>.

We show $2)\Rightarrow 1)$.

Suppose $\overline f$ is a split monomorphism in $H^{0}(\A)$.
Then we have an isomorphism from $A\rightarrow A\oplus C'$ to $f:A\rightarrow B$ in $H^0(\Mor(\A))$, which restricts to $\Id_{A}$.
So their homotopy cokernels are isomorphic and hence $C'$ is homotopy equivalent to $C$.
So we have an isomorphism in $H^0(\Mor(\A))$ from $f'=(1\ 0)^{\intercal}: A\rightarrow A\oplus C$ to $f:A\rightarrow B$ which restricts to $\Id_{A}$.
Then it induces an isomorphism $\alpha$ in $\mathcal H_{3t}(\A)$

\[
\begin{tikzcd}
A\ar[d,"h_0"swap]\ar[r,"{[}1{,}0{]^{\intercal}}"]\ar[rr,bend left=8ex,"0"]&A\oplus C\ar[r,"{[}0\ 1{]}"]\ar[d,"h_1"]&C\ar[d,"h_2"]\\
A\ar[r,"f"swap]\ar[rr,bend right=8ex,"h"swap]&B\ar[r,"j"swap]&C
\end{tikzcd}
\]

where we have omitted the diagonal morphisms and $h_0$ is homotopic to $\Id_{A}$.
Let $c$ be a homotopy inverse of $h_2$.
Then the morphism $c$ gives rise to a morphism $\beta$

\[
\begin{tikzcd}
A\ar[d,equal]\ar[r,"{[}1{,}0{]^{\intercal}}"]\ar[rr,bend left=8ex,"0"]&A\oplus C\ar[r,"{[}0\ 1{]}"]\ar[d,"\begin{bmatrix}1\;\;0\\0\;\;c
\end{bmatrix}"]&C\ar[d,"c"]\\
A\ar[r,"{[}1{,}0{]^{\intercal}}"swap]\ar[rr,bend right=8ex,"0"]&A\oplus C\ar[r,"{[}0\ 1{]}"swap]&C
\end{tikzcd}\;\;.
\]

So the composition $\alpha\circ \beta $ is an equivalence between $X$ and a splitting conflation and hence $X$ is a splitting conflation.

Let $X$ be a conflation of the form

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}.
\]

1) The element $\overline{\jmath}^*[X]$ is zero in $\mathbb E(B,A)$.

2)For any zero morphism $0:W\rightarrow C$ in $H^0(\A)$, the corresponding element $0^*[X]$ is zero in $\mathbb E(W,A)$.

Let $[X]$ be an element in $\mathbb E(C,A)$ where $X$ is of the form

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}.
\]

Then $[-X]$, where $-X$ is of the form

\[
\begin{tikzcd}
A\ar[r,"-f"]\ar[rr,"-h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}
\]

is an inverse of $[X]$ under the addition defined above. So this addition makes $\mathbb E(C,A) $ into an abelian group.

It is direct to check that the element $[-X]$ is well-defined.
The following diagram in $\A$

\[
\begin{tikzcd}
A\ar[rr,bend left=8ex,"h"]\ar[d,swap,"\begin{bmatrix}-1\\1\end{bmatrix}"]\ar[r,"f"]&B\ar[d,"\begin{bmatrix}1\\1\end{bmatrix}"]\ar[r,"j"]&C\ar[d,"\begin{bmatrix}1\\1\end{bmatrix}"]\\
A\oplus A\ar[rr,"h'"swap,bend right=8ex]\ar[r,"f'"swap]&B\oplus B\ar[r,"j'"swap]&C\oplus C
\end{tikzcd}
\]

where the diagonal morphisms are zero and where $f'=\begin{bmatrix}-f&0\\0&f\end{bmatrix}$, $j'=\begin{bmatrix}j&0\\0&j\end{bmatrix}$ and $h'=\begin{bmatrix}-h&0\\0&h\end{bmatrix}$, gives a morphism $\alpha:X\rightarrow -X\oplus X$ in $\mathcal H_{3t}(\A)$.
We apply Lemma <ref> to the morphism $\alpha$.

Put $a=(1\ 1):A\oplus A\rightarrow A$, $a'=(-1\ 1)^{\intercal}:A\rightarrow A\oplus A$ and $c=(1\ 1)^{\intercal}:C\rightarrow C\oplus C$.

Then we have $\alpha=\gamma\circ\beta$ where $\beta:X\rightarrow\tilde{X}$, $\gamma:\tilde{X}\rightarrow -X\oplus X$ and $[\tilde{X}]=\overline{a'}_{*}[X]=\overline c^*[-X\oplus X]$.

So we have

\[
[X]+[-X]=\overline a_{*}\overline c^*[-X\oplus X]=\overline a_{*}[\tilde{X}]=\overline a_{*}\overline {a'}_{*}[X]=0_*[X]=0.
\]

This shows that $[-X]$ is an additive inverse of $[X]$.

The bifunctor $\mathbb E$ is a biadditive bifunctor from $H^0(\A)^{op}\times H^0(\A)$ to the category $\Ab$ of abelian groups.

Let $X_i$, $i=1$, $2$, be a conflation of the form

\[
\begin{tikzcd}
A\ar[r,"f_i"]\ar[rr,"h_i"swap,bend right=8ex]&B_i\ar[r,"j_i"]&C
\end{tikzcd}
\]

and $\overline b:A\rightarrow A'$ a morphism in $H^0(\A)$.

Put $a=(1\ 1):A\oplus A\rightarrow A$, $a'=(1\ 1):A'\oplus A'\rightarrow A'$, $c=(1\ 1)^{\intercal}:C\rightarrow C\oplus C$ and $d=\begin{bmatrix}b&0\\0&b\end{bmatrix}:A\oplus A\rightarrow A'\oplus A'$.

Put $[\tilde{X_i}]=\overline b_*[X_i]$, $i=1,2$.
Then we have $d_*[X_1\oplus X_2]=[\tilde{X_1}\oplus \tilde{X_2}]$.

So we have

\[
\begin{aligned}
\overline b_*([X_1]+[X_2])&=\overline b_*\overline a_*\overline c^*[X_1\oplus X_2]=\overline a'_*\overline d_*\overline c^{*}[X_1\oplus X_2]\\
&=\overline a'_*\overline c^{*}\overline d_*[X_1\oplus X_2]=\overline a'_*\overline c^{*}[\tilde{X_1}\oplus \tilde{X_2}]\\
&=\tilde{X_1}+\tilde{X_2}=\overline b_*[X_1]+\overline b_*[X_2].
\end{aligned}
\]

So the map $\overline{b}_*$ is a morphism of abelian groups.

Let $X$ be a conflation of the form

\[
\begin{tikzcd}
A\ar[r,"f"]\ar[rr,"h"swap,bend right=8ex]&B\ar[r,"j"]&C
\end{tikzcd}.
\]

Suppose we have a morphism $\overline b_i:A\rightarrow A'$ in $H^0(\A)$ for each $i=1$, $2$.

Put $d'=\begin{bmatrix}b_1&0\\0&b_2
\end{bmatrix}$ and $a''=[1,1]^{\intercal}:A\rightarrow A\oplus A$.
Put $[Y_i]=(\overline b_i)_*[X]$ for $i=1$, $2$.

We have a canonical morphism $\alpha:X \rightarrow X\oplus X$ which is componentwise given by the morphism $(1\ 1)^{\intercal}$.

By Lemma <ref>, the morphism $\alpha$ factorizes as $\alpha=\gamma\circ \beta$ where we have $\beta: X\rightarrow \tilde{X}$, $\gamma:\tilde{X}\rightarrow X\oplus X$ and $[\tilde{X}]=\overline a''_*[X]=\overline c^*[X\oplus X]$.

Then we have

\[
\begin{aligned}
(\overline b_1+\overline b_2)_*[X]&=\overline a'_*\overline d'_*\overline a''_*[X]=\overline a'_*\overline d'_*\overline c^*[X\oplus X]\\
&=\overline a'_*\overline c^*\overline d'_{*}[X\oplus X]=\overline a'_*\overline c^*[Y_1\oplus Y_2]\\
&=Y_1+Y_2=(\overline b_1)_*[X]+(\overline b_2)_*[X].
\end{aligned}
\]

Thus we have
$(\overline b_1+\overline b_2)_*=(\overline b_1)_*+(\overline b_2)_*$.
This together with its dual, shows that the bifunctor $\mathbb E$ is biadditive.

For two elements $\delta_i=[X_i]\in \mathbb E(C_i,A_i)$, $i=1$, $2$, let $\delta_1\oplus \delta_2$ be the element in

\[
\mathbb E(C_1\oplus C_2, A_1\oplus A_2)\simeq \mathbb E(C_1,A_1)\oplus \mathbb E(C_2, A_2)\oplus \mathbb E(C_1, A_2)\oplus \mathbb E(C_2, A_1)
\]

corresponding to the element $(\delta_1,\delta_2,0,0)$.
$\delta_1\oplus \delta_2$ is the equivalence class of $X_1\oplus X_2$.

Denote by $a_i:A_1\oplus A_2\rightarrow A_i$ the canonical projection for $i=1$, $2$.
Denote by $c_i:C_i\rightarrow C_1\oplus C_2$ the canonical inclusion for $i=1$, $2$.

It is enough to show that we have $c_j^*(a_i)_*[X_1\oplus X_2]=0$ for $i\neq j$ and $c_i^*(a_i)_*[X_1\oplus X_2]=[X_i]$.

We show the case when $i=1$.
We have $(a_1)_*[X_1\oplus X_2]=[X_1\oplus Y]$ where $Y$ is the conflation

\[
\begin{tikzcd}
0\ar[r]\ar[rr,"0"swap,bend right=8ex]&C_2\ar[r,equal]&C_2
\end{tikzcd}.
\]

So we have $c_1^*(a_1)_*[X_1\oplus X_2]=[X_1]$.

By Corollary <ref>, we have $c_2^*(a_1)_*[X_1\oplus X_2]=0$.

§.§ DG nerve of an exact dg category

In this subsection, we link our notion of exact dg category to Barwick's notion of exact
$\infty$-category [8]. Standard references for $\infty$-categories are [104, 101, 23, 94, 102].

One of the aims of this section is to prove the following theorem. The construction of
the dg nerve will be recalled in <ref> and the notion of an exact structure on an $\infty$-category
in Definition <ref>. The notion of exact structure on a dg category is given in Definition <ref>.

Let $\A$ be a dg category such that $H^0(\A)$ is an additive category.
Then there is a bijection between the class of exact structures on the dg category $\A$ and the class of
exact structures on the $\infty$-category $N_{dg}(\A)$.
Moreover, if $\A$ is a pretriangulated dg category endowed with the maximal exact structure
(given by all homotopy short exact sequences), then its dg nerve is a stable $\infty$-category
endowed with the maximal exact structure (where each morphism is both an inflation and a deflation).

This extends in particular a theorem by Faonte [39], who showed that
if a dg category is pretriangulated, then its dg nerve is a stable $\infty$-category.

We begin by recalling some basic definitions concerning simplicial sets. We follow the treatment in [66].
The simplicial category $\Delta$ is the full subcategory of the category $\cat$ of small categories consisting of the totally ordered sets

\[
\]

for $n\geq 0$.
The simplicial category $\Delta$ is generated by the injective morphisms $d^i:[n-1]\rightarrow [n]$ for $n\geq 1$ and $0\leq i\leq n$, where the image of $d^i$ does not include $i$, and the surjective morphisms
$s^i:[n]\rightarrow [n-1]$ for $0\leq i\leq n-1$, where $s^i$ identifies $i$ and $i+1$.
It is not hard to check that these satisfy the following relations called cosimplicial identities

* $d^jd^i=d^id^{j-1}$ for $i<j$,

\begin{equation*}
\begin{cases}
d^is^{j-1}, & (i<j) \\
\Id, & (i=j\  \text{or}\  j+1)\\
d^{i-1}s^j, & (i>j+1)
\end{cases}
\end{equation*}

* $s^js^i=s^{i-1}s^j$ for $i>j$.

Moreover, one can show that the category $\Delta$ has a presentation given by these generators
and relations, cf. <cit.>.
For a category $\C$, the category of simplicial objects in $\C$ is the
category $\Fun(\Delta^{op},\C)$ of $\C$-valued presheaves on $\Delta$.
The most important examples are the category $\sSet$ of simplicial sets and the
category $\sMod k$ of simplicial $k$-modules.

For $n\geq 0$, we have the standard $n$-simplex $\Delta^n$ defined as the functor
represented by the object $[n]$. For a simplicial set $X$ and an integer $n\geq 0$, we write
$X_n$ for the set $X([n])$ of $n$-simplices of $X$. Notice that by the Yoneda lemma,
this set identifies with

\[
\Hom_{\sSet}(\Delta^n,X).
\]

Thus, an $n$-simplex $x$ of $X$ can be viewed as a morphism $x:\Delta^n\rightarrow X$.
For a morphism $d^i:[n-1]\rightarrow[n]$ in $\Delta$, by abuse of notation, we put $d_i=X(d^i):X_{n}\rightarrow X_{n-1}$, which is a face map.
Similarly for a morphism $s^j:[n]\rightarrow [n-1]$, we put $s_j=X(s^j):X_{n-1}\rightarrow X_n$, which is a degeneracy map.
The set $\pi_0(X)$ of connected components of $X$ is defined as the coequalizer of the double arrow

\[
\begin{tikzcd}(X_1\ar[r,"d_1",shift left=0.8ex]\ar[r,"d_0"swap,shift right =0.8ex]&X_0).\end{tikzcd}
\]

This extends to a functor

\[
\pi_0:\sSet\rightarrow \Set,\;\; X\mapsto \pi_0(X).
\]

It admits a right adjoint

\[
\const:\Set\rightarrow \sSet,\;\; A\mapsto \const(A)
\]

where $\const(A)$ is the constant functor $\Delta^{op}\rightarrow \Set$ with value $A$.
So we have the following adjunction

\[
\begin{tikzcd}
\sSet\ar[r,shift left=0.8ex,"\pi_0"]&\Set\ar[l,shift left=0.8ex,"\const"]
\end{tikzcd}
\]

where $\pi_0$ is the left adjoint.

The inclusion functor $\Delta\rightarrow \cat$ extends to a unique colimit-preserving functor

\[
h(-):\sSet\rightarrow \cat
\]

which associates to each simplicial set $X$ its homotopy category $hX$. It can be described
as the path category of the quiver $(X_0,X_1,d_1,d_0)$ modulo the relations given

\[
d_1(h)\sim d_0(h)\circ d_2(h)
\]

for all elements $h\in X_2$.
The nerve functor $N(-):\cat\rightarrow \sSet$ is right adjoint to $h(-)$.
More explicitly, for a small category $\C$ we have

\[
\]

for each $n\geq 0$. For any $n\geq 0$, the boundary of the standard $n$-simplex is

\[
\begin{tikzcd}
\partial \Delta^n=\bigcup_{E} \Delta^{E}\subset\Delta^n
\end{tikzcd}
\]

where $E$ runs through the proper full subcategories of $[n]$ and $\Delta^{E}$ is the nerve $N(E)$.
Similarly, for integers $n\geq 1$ and $0\leq k\leq n$, the $k$-th horn of $\Delta^n$ is

\[
\Lambda^n_k=\bigcup_{k\in E} \Delta^{E}\subset \Delta^n
\]

where $E$ runs through proper full subcategories of $[n]$ which contain the object $k$.
Obviously, we have $\Lambda^n_k\subsetneq \partial \Delta^n\subsetneq \Delta^n$.

A simplicial set $X$ is an $\infty$-category provided each map $\Lambda^n_k\rightarrow X$ can be lifted to a map $\Delta^n\rightarrow X$ along $\Lambda^n_k\rightarrow \Delta^n$ whenever $n\geq 2$ and $0<k<n$.
It is a Kan complex if the same is true whenever $n\geq 2$ and $0\leq k\leq n$.

For two simplicial sets $X$ and $Y$, their product $X\times Y$ is defined by $(X\times Y)_n=X_n\times Y_n$.
The mapping simplicial set $\Map(X,Y)$ is defined by putting

\[
\Hom_{\sSet}(\Delta^n,\Map(X,Y))\iso\Hom_{\sSet}(\Delta^n\times X,Y)
\]

for each $n\geq 0$.
The simplicial set $\Map(X,Y)$ is an $\infty$-category (resp. Kan complex) whenever $Y$ is.

Let $\E$ be an $\infty$-category.
An element $x\in \E_0$ is called an object or a vertex of $\E$.
An element $f\in \E_1$ is called a morphism or an arrow of $\E$.
Put $x=d_1(f)$ and $y=d_0(f)$.
We write $f$ as $f:x\rightarrow y$ and say that $f$ is a morphism from $x$ to $y$.
A morphism $f:x\rightarrow y$ is an isomorphism if it becomes an isomorphism in the homotopy category $h(\E)$ and in this case, we say that $x$ is isomorphic to $y$ as objects of the $\infty$-category $\E$.
Let $\sigma$ be a $2$-simplex in $\mathcal E$. Put $f=d_2(\sigma)$, $g=d_0(\sigma)$ and $h=d_1(\sigma)$.
Then $\sigma$ corresponds to a diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

which identifies $h$ with the composition $g\circ f$ and we shall write $g\circ f\sim h$.
The homotopy category $h\mathcal E$ can be described alternatively as follows (cf. [14], also [23]): let us fix two objects $x$ and $y$ in $\E$. We define the following four relations on the set of morphisms of $\E$ from $x$ to $y$:

$f\sim_1g$ if $f1_x\sim g$;

$f\sim_2g$ if $1_yf\sim g$;

$f\sim_3 g$ if $g1_x\sim f$;

$f\sim_4g$ if $1_yg\sim f$;

It is a standard result that the above four relations are equal and that they are equivalence relations on the set of morphisms from $x$ to $y$ in $\E$.

For two objects $x$ and $y$ in $\E$, put

\[
\Hom_{h'\E}(x,y)=\{\text{morphisms from }x\text{ to }y\}/\sim_1.
\]

Given a morphism $f:x\rightarrow y$ in $\E$, we write $[f]$ for its equivalence class in $\Hom_{h'\E}(x,y)$.
For any triple of objects $(x,y,z)$ in $\E$, the composition map is given by

\[
\begin{tikzcd}
\Hom_{h'\E}(y,z)\times\Hom_{h'\E}(x,y)\ar[r]& \Hom_{h'\E}(x,z)\\
({[}g{]},{[}f{]})\ar[r,mapsto]&{[}g{]}\circ {[}f{]}
\end{tikzcd}
\]

where $[g]\circ [f]=[h]$ whenever $gf\sim h$.
It is standard that this map is well-defined and that the above produces a category $h'\E$.
By the universal property of $h\E$, there exists a functor $h\E\rightarrow h'\E$ which is the
identity on the set of objects.
It is standard that this functor is fully faithful and thus we have a canonical isomorphism of categories $h\E\iso h'\E$.

Let $f:X\rightarrow Y$ be a morphism of simplicial sets.
It is a weak homotopy equivalence provided the induced map $\pi_0\Map(Y,K)\iso\pi_0\Map(X,K)$ is a bijection for each Kan complex $K$.
The category $\sSet$ carries the Kan-Quillen model structure whose weak equivalences are the weak homotopy equivalences and whose cofibrations are the injections of simplicial sets.
Fibrations in the Kan-Quillen model structure are called Kan fibrations.
The fibrant objects in the Kan-Quillen model structure are exactly the Kan complexes.

The triple $(\sSet,\times,\Delta^0)$ is a closed symmetric monoidal model category with the Kan-Quillen model structure, where the internal Hom is given by $\Map$.
A simplicial category is a category enriched over $\sSet$.
A simplicial category $\C$ is fibrant if for each pair $(X,Y)$ of objects in $\C$, the Hom simplicial set $\C(X,Y)$ is fibrant, i.e. a Kan complex.

Let $\C$ and $\D$ be two $\infty$-categories. A morphism $f:\C\rightarrow \D$ of $\infty$-categories is an equivalence of $\infty$-categories if there exists a morphism $g:\D\rightarrow \C$ such that $gf$ is isomorphic to $\mathrm{id_{\C}}$ in the $\infty$-category $\Map(\C,\C)$ and $fg$ is isomorphic to $\mathrm{id_{\D}}$ in $\Map(\D,\D)$.
The induced functor $h(f):h(\C)\rightarrow h(\D)$ of an equivalence of $\infty$-categories $f:\C\rightarrow \D$ is in particular an equivalence of categories.

A morphism $f:X\rightarrow Y$ of simplicial sets is a categorical equivalence provided the induced morphism

\[
\Map(Y,\C)\rightarrow\Map(X,\C)
\]

is an equivalence of $\infty$-categories for each $\infty$-category $\C$.

$\sSet$ carries the Joyal model structure whose weak equivalences are the
categorical equivalences and whose cofibrations are the injections of simplicial sets.
The fibrant objects in the Joyal model structure are exactly the $\infty$-categories and
the categorical equivalences between $\infty$-categories are exactly the equivalences of $\infty$-categories.
We will write $\sSet_{Quillen}$ or $\sSet_{Joyal}$ when we want to stress the model structure we are using.
Let $\C$ and $\D$ be two $\infty$-categories and $f:\C\rightarrow \D$ a morphism in $\mathrm{Ho}(\sSet_{Joyal})$.
The morphism $f$ yields an isomorphism class of functors $h(\C)\rightarrow h(\D)$.
In the special case of $\C=\Delta^0$, we obtain a canonical bijection

\[
\Hom_{\mathrm{Ho}(\sSet_{Joyal})}(\Delta^0, \D)\iso \Iso(h(\D)).
\]

The triple $(\sSet,\times,\Delta^0)$ is a closed symmetric monoidal model category with the Joyal model structure, where the internal Hom is given by $\Map$, cf. <cit.>.

Let $K$ be a Kan complex and $x, y\in K_0$ two 0-simplices.
Define $x$ to be homotopic to $y$, written $x\sim y$, if and only if there is a 1-simplex $z\in K_1$ such that $d_1z=x$ and $d_0z=y$.
One checks that homotopy of vertices is an equivalence relation and that $\pi_0K=K_0/\sim$.
If $v$ is a vertex of $K$, then one defines $\pi_0(K,v)$ to be the pointed set $\pi_0X$ with basepoint the equivalence class $[v]$ of $v$.
Let $n> 0$ and $v\in K_0$.
For a simplicial set $X$, we will denote the map $X\rightarrow \Delta^0\xrightarrow{v} K$ by $v$ and refer to it as the constant map with value $v$.
For example when $X=\Delta^n$, we will still denote by $v$ the corresponding $n$-simplex in $K$.
Let $F$ be the pullback of the following diagram

\[
\begin{tikzcd}
\Delta^0\ar[r,"v"]&\Map(\partial\Delta^n,K).
\end{tikzcd}
\]

It is clear that $F$ is a Kan complex. The pointed set $\pi_0(F,v)$ carries a canonical
group structure, which is abelian  for $n\geq 2$. It is called the
nth homotopy group $\pi_n(K,v)$ of $K$ at $v$.

By Whitehead's theorem, a morphism $f:K\rightarrow K'$ of Kan complexes is a weak homotopy equivalence if and only if it satisfies the following conditions:

$(a)$ The map of sets $\pi_0(f):\pi_0(K)\rightarrow \pi_0(K')$ is a bijection;

$(b)$ For each vertex $x\in K$ with image $y=f(x)\in K'$ and each $n\geq 1$, the map of homotopy groups $\pi_n(f):\pi_n(K,x)\rightarrow\pi_n(K',y)$ is an isomorphism.

For each simplicial set $K$, we have the free simplicial $k$-module $FK$ generated by $K$ so that $(FK)_n$ is the free $k$-module generated by the set $K_n$ for each $n\geq 0$ and the structure maps are extended from those of $K$ by $k$-linearity.
This determines the free functor $F:\sSet \rightarrow \sMod k$.
Let $U$ be the forgetful functor $\sMod k\rightarrow \sSet$.
For a simplicial $k$-module $X$, the underlying simplicial set of $X$ is $U(X)$.
Then we have the adjunction

\[
\begin{tikzcd}
\sSet\ar[r, shift left=1ex,"F"]&\sMod k\ar[l,shift left=1ex,"U"]
\end{tikzcd}
\]

where $F$ is the left adjoint.

In order to relate dg categories with $\infty$-categories, we need the Dold-Kan correspondence ([26]) as a key ingredient. Note that we are working with cochain complexes.
We denote by $\C(k)^{\leq 0}$ the category of nonpositively graded cochain complexes.

Let $X$ be a simplicial $k$-module.

\[
\]

\[
d^{-n}=\sum_{0\leq i\leq n}(-1)^id_i:C^{-n}(X)\rightarrow C^{-n+1}(X)
\]

for $n\geq 0$.
The unnormalized cochain complex $C^{*}(X)$ is given by

\[
\cdots\rightarrow C^{-n}(X)\xrightarrow{d^{-n}} C^{-n+1}(X)\rightarrow \cdots C^{-2}(X)\xrightarrow{d^{-2}} C^{-1}(X)\xrightarrow{d^{-1}} C^0(X)\rightarrow 0\rightarrow \cdots.
\]

For each $n\geq 0$, put

\[
N^{-n}(X)=\bigcap_{1\leq i\leq n}\ker(d_i)\subseteq C^{-n}(X)=X_n.
\]

The map $d^{-n}$ carries $N^{-n}(X)$ into $N^{-n+1}(X)$ and therefore we obtain the normalized cochain complex $N^*(X)$

\[
\cdots\rightarrow N^{-n}(X)\xrightarrow{d^{-n}} N^{-n+1}(X)\rightarrow \cdots \rightarrow N^{-2}(X)\xrightarrow{d^{-2}} N^{-1}(X)\xrightarrow{d^{-1}} N^0(X)\rightarrow 0\rightarrow \cdots\ .
\]

The assignment $X\mapsto N^{*}(X)$ determines the normalized cochain complex functor $N:\sMod k\rightarrow \C(k)^{\leq 0}$.

The normalized cochain complex functor $N:\sMod k\rightarrow \C(k)^{\leq 0}$ is an equivalence of categories.

Its quasi-inverse is the  Dold-Kan construction $DK:\C(k)^{\leq 0}\rightarrow \sMod k$:
A concrete description of $DK$ can be found in <cit.>.
The construction of the functor $DK$ remains valid if we replace $\Mod k$ by any additive category and it is an equivalence of categories if the additive category has split retractions (i.e. it is weakly idempotent complete).
In our case, for a complex $V\in\C(k)^{\leq 0}$, we identify the $k$-module $DK(V)_n$ with

\[
\Hom_{\sMod k}(N(F(\Delta^n)),V)
\]

where $F(\Delta^n)$ is the free simplicial $k$-module generated by $\Delta^n$.

The category $\C(k)^{\leq 0}$ carries the projective model structure (cf. [123] and <cit.>) such that

* the weak equivalences are the quasi-isomorphisms of complexes;

* the cofibrations are the monomorphisms with componentwise projective cokernel;

* the fibrations are the morphisms which are epimorphisms in strictly negative degrees.

By the Dold-Kan correspondence, this induces a model structure on $\sMod k$ which is called the projective model structure.

Let $f:X\rightarrow Y$ be a morphism in $\C(k)^{\leq 0}$.
We will need the following explicit factorization of $f$ into a trivial cofibration followed by a fibration.
For a $k$-module $M$ and any $n\leq -1$, denote by $D^n(M)$ the following complex

\[
\dots\rightarrow 0\rightarrow M^{n}\xrightarrow{\Id} M^{n+1}\rightarrow 0\rightarrow \dots
\]

where $M^n=M$ is in degree $n$ and $M^{n+1}=M$ is in degree $n+1$.
For each $n\leq -1$, let $u^n: Q^n \to Y^n$ be a surjective morphism with projective $Q^n$.
Let $P(Y)$ be the following contractible complex

\[
\bigoplus_{n\leq -1} D^n(Q^{n}).
\]

For each $n\leq -1$, the surjective map $u^n:Q^n\rightarrow Y^n$ induces a map $s^n: D^n(Q^{n})\rightarrow Y$ which is surjective in degree $n$.
Let $v:P(Y)\rightarrow Y$ be the morphism given by the $s^n$, $n\leq -1$.
Clearly $v$ is surjective in each strictly negative degree.
Then we have the following factorization of $f$

\begin{equation}\label{factor}
X\xrightarrow{g=\begin{bmatrix}1\\0\end{bmatrix}} X\oplus P(Y)\xrightarrow{h=\begin{bmatrix}f\;\; v\end{bmatrix}} Y.
\end{equation}

Here the morphism $g$ is a trivial cofibration and $h$ is a fibration.
Note that for a complex $Q\in \C(k)^{\leq 0}$ which is componentwise projective, each null-homotopic map $Q\rightarrow Y$ factors through the map $v:P(Y)\rightarrow Y$.

Let $f:X\rightarrow Y$ be a morphism of simplicial $k$-modules.
By <cit.>, the map $U(f)$ is a Kan fibration if and only if the map $N(f)$ is a fibration in the projective model structure of $\C(k)^{\leq 0}$.
In particular, the underlying simplicial set of a simplicial $k$-module is a Kan complex.

For a simplicial $k$-module $X$, the additive operation $X\times X\rightarrow X$, $(x,y)\in X_n\times X_n \mapsto x+y\in X_n$ induces a map $\pi_n(X,0)\times \pi_n(X,0)\rightarrow \pi_n(X,0)$ for each $n\geq 0$.
This gives an abelian group structure on $\pi_n(X,0)$.
The Eckmann-Hilton argument shows that it agrees with the usual group structure on $\pi_n(X,0)$ for $n>0$.
An element $[\alpha]\in\pi_n(X,0)$ is an equivalence class of maps $\alpha:\Delta^n\rightarrow X$ such that $d_i(\alpha)=0$ for $i=0,\dots,n$.
Such maps $\alpha:\Delta^n\rightarrow X$ are in bijection with elements in $Z^{-n}N^*(X)$.
Unwinding the definition of the equivalence relation, we see that two such maps $\alpha,\beta:\Delta^n\rightarrow X$ are equivalent if and only if the corresponding elements in $Z^{-n}N^*(X)$ are cohomologous.
Therefore $\pi_n(X,0)$ can be identified with $H^{-n}N^*(X)$.
Fix a vertex $x\in X_0$.
The morphism $\theta_x: X\rightarrow X$ of simplicial sets, defined by sending $y\in X_n$ to $x+y\in X_n$, induces an isomorphism of pointed sets $\pi_n(X,0)\iso\pi_n(X,x)$ for $n>0$.

Let $f:X\rightarrow Y$ be a morphism of simplicial $k$-modules.
By the above discussions, the map $U(f)$ between the underlying simplicial sets
is a weak homotopy equivalence if and only if $N(f):N^*(X)\rightarrow N^{*}(Y)$ is a weak equivalence in the projective model structure.
Thus, the projective model structure on $\sMod k$ is the right transferred model structure
(cf. <cit.>) from $\sSet_{Quillen}$ along $U$.
In particular, $\sMod k$ is cofibrantly generated and the adjunction

\[
\begin{tikzcd}
\sSet\ar[r, shift left=1ex,"F"]&\sMod k\ar[l,shift left=1ex,"U"]
\end{tikzcd}
\]

is a Quillen adjunction, cf.  <cit.>.

Let $\I$ be a small category and $F:\I\rightarrow \C(k)^{\leq 0}$ a functor.
Since $\C(k)^{\leq 0}\iso \sMod k$ is a simplicial model category (<cit.>), one has the local definition of homotopy limits of $F$ (<cit.>).

In general, one also has a nice global definition of homotopy limits of $F$ since $\sMod k$ is a combinatorial model category in the sense of Jeff Smith (cf. also <cit.>),
and the global and local definitions of homotopy limits coincide (<cit.>).

We recall the global definition of homotopy limits and will use it for computation.
We follow Lurie's treatment as in <cit.>.
Let $\I$ be a small category and $\const: \sMod k\rightarrow \Fun(\I,\sMod k)$ the functor sending an object $X\in \sMod k$ to the constant functor $\const(X):\I\rightarrow \sMod k$ with value $X$.
It has a right adjoint $\lim:\Fun(\I,\sMod k)\rightarrow \sMod k$ sending a functor
$F:\I\rightarrow \sMod k$ to $\lim F$. Thus we have the following adjunction

\[
\begin{tikzcd}
\sMod k\ar[r,"\const",shift left=0.8ex]&\Fun(\I,\sMod k).\ar[l,"\lim",shift left=0.8ex]
\end{tikzcd}
\]

Since $\sMod k$ is combinatorial, $\Fun(\I,\sMod k)$ carries the injective model structure, where

* A morphism $\alpha:F\rightarrow G$ is a weak equivalence if and only if for each $i\in\I$, the map $\alpha_i:F(i)\rightarrow G(i)$ is a weak equivalence in $\sMod k$.

* A morphism $\alpha:F\rightarrow G$ is a cofibration if and only if for each $i\in\I$, the map $\alpha_i:F(i)\rightarrow G(i)$ is a cofibration in $\sMod k$ (with the projective model structure).

It follows that $\const: \sMod k\rightarrow \Fun(\I,\sMod k)$ is a left Quillen functor with respect to the injective model structure on $\Fun(\I, \sMod k)$.
Let $F:\I\rightarrow \sMod k$ be an object in $\Fun(\I,\sMod k)$ and $A$ an object in $\sMod k$.
Let $\alpha: \const(A)\rightarrow F$ be a natural transformation and $\beta:A\rightarrow \lim F$ the corresponding morphism in $\sMod k$.
Then $\alpha$ exhibits $A$ as a homotopy limit of $F$ if for some weak equivalence $F\rightarrow F'$ where $F'$ is fibrant in $\Fun(\I,\sMod k)$, the composite map $A\rightarrow \lim F\rightarrow \lim F'$ is a weak equivalence in $\sMod k$.

Note that $\sSet_{Quillen}$ is also combinatorial.
It is immediate that the right Quillen functor $U:\sMod k\rightarrow \sSet$ is compatible with homotopy limits.

Let $\I$ be the cospan category $\Cosp$ defined in subsection <ref>.
Then an object $F$ in $\Fun(\I,\C(k)^{\leq 0})$

\[
\begin{tikzcd}
X\ar[r,"k"swap]& Z
\end{tikzcd}
\]

is fibrant in the injective model structure if and only if both $j$ and $k$ are fibrations in $\C(k)^{\leq 0}$.

Thus using the factorization in <ref>, we obtain a fibrant replacement

\[
\begin{tikzcd}
&Y\oplus P(Z)\ar[d,"\begin{bmatrix}j\;\; v\end{bmatrix}"]\\
X\oplus P(Z)\ar[r,"\begin{bmatrix}k\;\; v\end{bmatrix}"swap]& Z.
\end{tikzcd}
\]

By diagram chase, we see that the pullback of this diagram is quasi-isomorphic to

\[
\tau_{\leq 0}\Sigma^{-1}\Cone(X\oplus Y\xrightarrow{[k,\;j]} Z).
\]

Take a cofibrant replacement $r:Q\rightarrow \tau_{\leq 0}\Sigma^{-1}\Cone(X\oplus Y\xrightarrow{[k,j]} Z)$.
Then any null-homotopic map $Q\rightarrow Z$ factors through $v:P(Z)\rightarrow Z$.
Thus we obtain a commutative diagram in $\C(k)^{\leq 0}$

\[
\begin{tikzcd}
X\oplus P(Z)\ar[r]&Z\mathrlap{.}
\end{tikzcd}
\]

Let $F'$ be the cospan obtained from the above diagram by restriction.
Then we have a weak equivalence $F\rightarrow F'$ in $\Fun(\Cosp,\C(k)^{\leq 0})$ and the above diagram exhibits $Q$ as a homotopy pullback of $F'$.

Therefore, a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

is homotopy pullback if and only if the canonical map

\[
U\rightarrow \tau_{\leq 0}\Sigma^{-1}\Cone(X\oplus Y\xrightarrow{[k,\;j]} Z)
\]

is a quasi-isomorphism of connective complexes.

Let $\Cat_{\Delta}$ be the category of simplicial categories, i.e. categories which are enriched over $\sSet$.
There is a Quillen equivalence

\[
\begin{tikzcd}
\sSet\ar[r,"\C(-)", shift left=0.8ex]&\Cat_{\Delta}\ar[l, shift left=0.8ex,"N"]
\end{tikzcd}
\]

relating Bergner's model structure on $\Cat_{\Delta}$ with $\sSet_{Joyal}$, cf. <cit.> and [29] for an explicit description of the mapping spaces in $\C(S)$ for a general simplicial set $S$.
The functor $N:\Cat_{\Delta}\rightarrow \sSet$ is called the simplicial nerve functor or homotopy coherent nerve functor.
An ordinary category $\B$ can be considered as a simplicial category via the functor $\const:\Set\rightarrow \sSet$.
Then the simplicial nerve of $\B$ agrees with the ordinary nerve of $\B$.

The simplicial category $\C(\Delta^2)$ has the elements of $\{0,1,2\}$ as objects.
The mapping spaces $\Hom_{\C(\Delta^2)}(0,1)$ and $\Hom_{\C(\Delta^2)}(1,2)$ are isomorphic to $\Delta^0$.
The mapping space $\Hom_{\C(\Delta^2)}(0,2)$ is isomorphic to $\Delta^1$.
The composition map

\[
\Hom_{\C(\Delta^2)}(1,2)\times \Hom_{\C(\Delta^2)}(0,1)\rightarrow \Hom_{\C(\Delta^2)}(0,2)
\]

can be identified with the map $1:\Delta^0\rightarrow \Delta^1$.

In analogy with the nerve functor $N$ taking a small category to an $\infty$-category, there
is a dg nerve functor $N_{dg}$ which turns a small dg category into an $\infty$-category.
A construction is given by Lurie in <cit.> using an explicit description of
the $n$-simplices due to Hinich and Schechtman <cit.>.
The dg nerve is proved in <cit.> to be a right Quillen functor
in a Quillen adjunction given by a cosimplicial object in $\dgcat$.
The cosimplicial object is made explicit in <cit.>: it takes
the poset $[n]$ to the cobar-bar construction of the associated $k$-category $k[n]$.
To make the Quillen adjunction more precise,
let us endow $\dgcat$ with the Dwyer-Kan model structure and
$\sSet$ with the Joyal model structure. Then the Quillen adjunction is (cf. [101])

\begin{equation}\label{Lambdadgnerve}
\begin{tikzcd}
\sSet\ar[r,shift left=0.8ex,"\Lambda"]&\dgcat\ar[l,shift left=0.8ex,"N_{dg}"]
\end{tikzcd}
\end{equation}

where $N_{dg}$ is the right Quillen functor.
In particular, as each small dg category is fibrant in the Dwyer-Kan model structure, the
dg nerve sends quasi-equivalences to equivalences of $\infty$-categories.

Let us describe the functor $\Lambda$ in more detail.
For a standard $n$-simplex $\Delta^n$, the dg category
$\Lambda(\Delta^n)$ is defined as follows: let $Q^{(n)}$ be the
graded quiver with vertex set $[n]$ and for each chain $I=\{i<p_1<\cdots<p_l<j\}$ with $l\geq 0$ an arrow

\[
I:i\rightarrow j
\]

of degree $l$. Then $\Lambda(\Delta^n)$ is the dg $k$-category whose underlying graded category is the free graded $k$-category generated by $Q^{(n)}$ with differential given by

\[
d(I)=\sum_{1\leq m\leq l}(-1)^{l-m}(I \backslash \{p_m\}-\{p_m< \cdots<p_l<j\}\circ\{i<p_1<\cdots<p_m\})
\]

on each chain $I=\{i<p_1<\cdots<p_l<j\}$.

For a morphism $\sigma:[m]\rightarrow [n]$ in $\Delta$, the dg functor $\Lambda(\Delta^{\sigma}):\Lambda(\Delta^m)\rightarrow \Lambda(\Delta^n)$ is given by

\begin{equation*}
\begin{cases}
\sigma(I) & \text{if }\sigma|_{I}\text{ is injective,} \\
\Id_p & \text{if }I=\{i<j\}\text{ and }\sigma(i)=\sigma(j)=p,\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}

We now construct a natural dg functor (cf. [95])

\[
\theta_{m,n}: \Lambda(\Delta^m\times \Delta^n)\rightarrow \Lambda(\Delta^m)\otimes \Lambda (\Delta^n).
\]

A chain of a finite ordered set is a totally ordered finite subset.
Recall that we have the following presentation of $\Delta^m\times \Delta^n$ (<cit.>)

\[
\begin{tikzcd}
\coprod_{1\leq i<j\leq \left(\begin{smallmatrix}m+n\\m\end{smallmatrix}\right)} \Delta^{n_{c{(i)}\cap c{(j)}}}\ar[r,shift left=0.8ex,"u"]\ar[r,shift right=0.8ex,"v"swap]&\coprod_{1\leq i\leq \left(\begin{smallmatrix}m+n\\n\end{smallmatrix}\right)}\Delta^{n_{c{(i)}}}\ar[r]&\Delta^m\times \Delta^n
\end{tikzcd}
\]

where each $c(i)$ is a maximal chain of the ordered set $[m]\times [n]$ and $n_{c(i)}=m+n$ is the length of the chain $c(i)$ for each $i$.
The maximal chain $c(i)$ corresponds to an $(m,n)$-shuffle, i.e. a strictly increasing map of ordered sets $\sigma^{(i)}:[m+n]\rightarrow [m]\times [n]$.
The chain $c(i)\cap c(j)$ corresponds to the equalizer $\mathrm{Eq}_{i,j}$ of the maps $\sigma^{(i)}$ and $\sigma^{(j)}$.
The maps $u$ and $v$ then correspond to the inclusion of
$\mathrm{Eq}_{i,j}$ into $[m+n]$.

For an $(m,n)$-shuffle $\sigma:[m+n]\rightarrow [m]\times [n]$, put $\sigma_{-}:[m+n]\rightarrow [m]$ and $\sigma_{+}:[m+n]\rightarrow [n]$ the components of $\sigma$, i.e. $\sigma(j)=(\sigma_{-}(j),\sigma_{+}(j))$ for $0\leq j\leq m+n$.

Now for each $1\leq i\leq \left(\begin{smallmatrix}m+n\\n\end{smallmatrix}\right)$ define

\[
F_{c(i)}:\Lambda(\Delta^{n_{c(i)}})\rightarrow \Lambda (\Delta^m)\otimes \Lambda(\Delta^n)
\]

to be the dg functor given by $\Lambda(\sigma^{(i)}_{-})$ and $\Lambda(\sigma^{(i)}_{+})$.
Then by definition $(F_{c(i)})_{1\leq i\leq \left(\begin{smallmatrix}m+n\\n\end{smallmatrix}\right)}$ induces a dg functor

\[
\theta_{m,n}:\Lambda(\Delta^m\times\Delta^n)\rightarrow \Lambda(\Delta^m)\otimes\Lambda(\Delta^n).
\]

It extends by colimit to

\[
\theta_{X,Y}:\Lambda(X\times Y)\rightarrow \Lambda(X)\otimes\Lambda(Y)
\]

for simplicial sets $X$ and $Y$.

For any pair of simplicial sets $(X,Y)$, the above dg functor $\theta_{X,Y}$ is a quasi-equivalence.

This can be proved using the fact that the coherent nerve commutes with products (up to categorical
weak equivalence), which is proved in <cit.>.
We give a direct proof after Lemma <ref>.
The key ingredient is to express each $\infty$-category as the localization of the nerve of an ordinary category by a subcategory, cf. <cit.>.

Since right adjoints of monoidal functors preserve internal Homs, we obtain the following corollary.

Let $\A$ be a small dg category and $X$ a simplicial set.
We have an equivalence of $\infty$-categories

\[
\phi_{X,\A}: N_{dg}(\rep_{dg}(\Lambda(X),\A))\rightarrow \Map(X,N_{dg}(\A)).
\]

In particular, we have an equivalence of categories

\[
\delta_{X,\A}: \rep(\Lambda(X),\A)\rightarrow h(\Map(X,N_{dg}(\A))).
\]

Let $Y$ be a simplicial set.
Put $\mathcal H=\mathrm{Ho}(\sSet_{Joyal})$.
Then by adjunction we have

\[
\begin{aligned}
\Hom_{\mathcal H}(Y,\Map(X,N_{dg}(\A)))&\iso \Hom_{\mathcal H}(Y\times X,N_{dg}(\A))\\
&\iso \Hom_{\Hqe}(\Lambda(Y\times X),\A) \\
&\xleftarrow{\Hom(\theta_{X,Y},\A)} \Hom_{\Hqe}(\Lambda(X)\otimes \Lambda(Y),\A)\\
&\iso \Hom_{\Hqe}(\Lambda(Y),\rep_{dg}(\Lambda(X),\A))\\
&\iso \Hom_{\mathcal H}(Y,N_{dg}\rep_{dg}(\Lambda(X),\A)).
\end{aligned}
\]

Thus we have a natural isomorphism in $\mathrm{Ho}(\sSet_{Joyal})$

\[
N_{dg}(\rep_{dg}(\Lambda(X),\A))\iso \Map(X,N_{dg}(\A)).
\]

We have the following diagram of functors

\[
\begin{tikzcd}
k\mbox{-}\cat\ar[d,shift left=0.8ex, "F"]\ar[r,hook]&\dgcat_k\ar[d,shift left=0.8ex,"N_{dg}"]\\
\cat\ar[r,"N"swap]\ar[u,"k(-)",shift left=0.8ex]&\infty\mbox{-}\cat\ar[u,shift left=0.8ex,"\Lambda"]
\end{tikzcd}
\]

where $k(-)$ is the $k$-linearization functor and $F$ is the forgetful functor.
For a $k$-category $\C$, we have an isomorphism of $\infty$-categories $N(F(\C))\iso N_{dg}(\C)$.
For a small category $\I$, we have a natural morphism $N(\I)\rightarrow N_{dg}(k(\I))$ which induces a canonical dg functor $\Lambda(N(\I))\rightarrow k(\I)$.

The canonical dg functor $\Lambda(N(\I))\rightarrow k\I$ is a quasi-equivalence for each small category $\I$.

We first consider the special case when $\I$ is the path category $P(Q)$ of a quiver $Q$.
Recall that $N(\I)$ is the colimit of the functor $\Delta(N(\I))\rightarrow \sSet$ sending $(\Delta^n,x:\Delta^n\rightarrow N(I))$ to $\Delta^n$ (cf. <cit.>).
Notice that the full subcategory $\J$ of $\Delta(N(\I))$ consisting of $n$-simplices of $N(\I)$

\[
(i_0\xrightarrow{\alpha_0} i_1\xrightarrow{\alpha_1}\ldots\xrightarrow{\alpha_{n-1}} i_n)
\]

where $\alpha_j$ is an arrow in $Q$ for $0\leq j\leq n-1$, is a cofinal subcategory.
Notice also that the morphisms $(\Delta^n,p)\rightarrow (\Delta^m,q)$ in $\J$ are induced by inclusions of paths, i.e. $q$ is a part of $p$ as a path. Therefore $\Lambda(N(\I))$ is the colimit

\[
\colim_{p:\Delta^n\rightarrow N(\I)} \Lambda(\Delta^n)
\]

where $p$ runs through the paths in $Q$ and $n$ is the length of $p$.

For any pair of objects $(i,j)$, let $\mathcal S_{i,j}$ be the set of paths from $i$ to $j$. Let $\Lambda'$ be the dg category whose set of objects is $Q_0$ and the Hom complex $\Hom_{\Lambda'}(i,j)$ is given by the complex

\[
\bigoplus_{p\in\mathcal S_{i,j}}\Hom_{\Lambda(\Delta^{m})}(0,m)
\]

where $m$ is the length of the path $p$.
The composition map

\begin{align*}
\Hom_{\Lambda(\Delta^m)}(0,m)\otimes\Hom_{\Lambda(\Delta^n)}(0,n)\rightarrow \Hom_{\Lambda(\Delta^{m+n})}(0,m+n)
\end{align*}