We define extension quasi-categories for exact quasi-categories in an analogous way to the special case of ordinary exact categories. We show that these form an Omega-spectrum, generalizing a theorem of Retakh. In so doing, we give an explicit Kan fibrant resolution of these extension quasi-categories, which relies on a factorization property. Finally, we show that the homotopy groups of extension quasi-categories are naturally isomorphic to the higher extension groups of the extriangulated category given by the homotopy category.

§ INTRODUCTION

Roughly speaking, an exact category $\c$ is an additive category together with a collection of short exact sequences subject to certain constraints~[11], see~[15] for a modern approach. This allows for homological algebra to be performed in a more general context than abelian categories. Non-abelian examples include the category of vector bundles of a scheme and the category of Banach spaces. In addition to homological algebra, exact categories provide a natural framework for~$K$--theory. Quillen first introduced the higher algebraic~$K$--groups in this context~[23].

The definition has recently been extended by Barwick to quasi-categories in order to generalize definitions and results in~$K$--theory~[1, 2, 4]. This broadens the scope considerably; it captures both (nerves of) exact categories and stable quasi-categories. Moreover, every extension closed subcategory of a stable quasi-category has the structure of an exact quasi-category. In fact, all exact quasi-categories occur in this manner~[16]. One would thus expect more constructions and results about exact categories to generalize, providing potentially useful applications. In particular, one could study homological algebra from a quasi-categorical point of view.

In order to better understand the homological algebra of exact quasi-categories~$$, we define quasi-categorical analogues of extension categories, denoted by~$^n_(B,A)$. These are well understood if~$$ is the nerve of an ordinary exact category.  For a start, it is well-known that the Yoneda Ext-groups~$^n_(B,A)$ can be recovered as the zeroth homotopy groups of~$^n_(B,A)$ \cite[Proposition~XII.4.4]{Mac67}.

For an exact quasi-category~$$, we declare that~$^0_(B,A)_(B,A)$, the mapping space from~$B$ to~$A$. We will define maps~$R_n^n_(B,A)→Ω^n+1_(B,A)$, resulting in a spectrum~$_(B,A)$. The main result of ours is the following.
\begin{reptheorem}{theorem:IsOmegaSpectrum}
Let~$\C$ be an exact quasi-category and let~$A$ and~$B$ be objects in~$\C$.
Then the spectrum~$\Extcat_\C(B,A)$ is an~$\Omega$--spectrum. In other words, all of the maps~$R_n$ are weak equivalences.
\end{reptheorem}

This is a generalization of a theorem of Retakh, who proved the same result for abelian categories \cite[Theorem~2(b)]{Ret86}. It follows that we have isomorphisms
\begin{equation}\label{eq:hermanniso}
\begin{tikzcd}
\pi_{-n}\Extcat_{\C}(B,A)\simeq \pi_{0}\Extcat^n_{\C}(B,A) \arrow[r]& \pi_{1}\Extcat^{n+1}_{\C}(B,A)
\end{tikzcd}
\end{equation}
of homotopy groups. This means, in particular, that the higher structure of~$_(B,A)$ descends to the abelian group structure on classical~$$--groups.

To prove Theorem~\ref{theorem:IsOmegaSpectrum} it is required of us to find a manageable model for~$Ω^n_(B,A)$. Similarly to how you construct injective resolutions in homological algebra, this is achieved by finding Kan fibrant replacements. We define a new variant~$$ of the~$$--functor [14], enjoying the property that~$^n_(B,A)$ is a Kan complex. Having applied $$ only once, this is more computationally feasible compared to the usual transfinite composite~$^n_(B,A)→^∞^n_(B,A)$:

\begin{reptheorem}{proposition:fExIsKan}
The simplicial set~$\fEx\Extcat^n_{\C}(B,A)$ is a Kan complex (or~$\infty$--groupoid).
\end{reptheorem}

The proof of \Cref{proposition:fExIsKan} relies on a factorization property of~$^n_(B,A)$. We refer to \Cref{prop:VW3.1} for the details. In the case of an ordinary exact category~$$̧, this factorization property was first addressed by Hermann <cit.> in order to explicitly describe the isomorphisms (<ref>). See <cit.> for details. Initially, it was not clear whether factorization was possible for all exact categories, but this has now been shown by Volkov–Witherspoon <cit.>. Our <Ref> then generalizes their result to the quasi-categorical framework.

There is a notion extriangulated categories, defined by Nakaoka–Palu [21], which simultaneously generalizes exact and triangulated categories. Recently, Nakaoka–Palu have shown that the homotopy category of an exact quasi-category has a natural extriangulated structure <cit.>. This extends the well-known result that the homotopy category of a stable quasi-category is canonically triangulated <cit.>, and moreover the homotopy category of the nerve of an exact category is obviously exact. Note that the homotopy category of a stable quasi-category is hardly ever exact. Building on Nakaoka–Palu's work, we show that the Retakh spectra $\Extcat_{\C}(-,-)$ determine the extriangulation on $\homcat\C$.

For a non-negative integer $n$, Gorsky–Nakaoka–Palu <cit.> define the $n$–extension groups for extriangulated categories, ultimately inspired by Yoneda's classic monograph on extension categories [28]. We show that the higher structure contained in the Retakh spectra $\Extcat_{\C}(-,-)$ descends to the $n$–extension groups in the homotopy category.

Let $\C$ be an exact quasi-category, and consider $(\homcat\C,\pi_0\Extcat^1_{\C}(-,-))$ as an extriangulated category.
The bifunctor $\pi_0\Extcat^n_{\C}(-,-)$ is naturally isomorphic to the $n$th extension functor of $\homcat \C$.

Outline. Section <ref> contains little to no original ideas. It is concerned with basics of exact quasi-categories and diagram lemmas. We define the extension categories for quasi-categories in Section <ref>. Moreover, the factorization property of exact quasi-categories is proven. We follow up with an overview of the Retakh spectrum in Section <ref>. Although Theorem <ref> is stated here, technical details are postponed. The reader only interested in the application to extriangulated categories, can safely skip straight to Section <ref> from here. Section <ref> is all about the factorized $\fEx$–functor and proofs of technical lemmas. Only the basic intuition on $\fEx$ is required to read and follow Section <ref>, where we prove Theorems <ref> and <ref>. In the final Section <ref> we discuss how Retakh spectra determine the extriangulation on homotopy categories. In particular, Theorem <ref> is proven.

Notation and conventions. Throughout, we fix a Grothendieck universe $U$. Simplicial sets should thus be understood as simplicial $U$–sets unless otherwise is stated. We refer to Shulman <cit.> for a detailed treatment of Grothendieck universes.

A category $J$ will be identified with its nerve $\mathrm{N}(J)$, so that it can be regarded as a quasi-category. An $(\infty,1)$-functor between quasi-categories $\C$ and $\D$ (i.e. a morphism of the underlying simplicial sets) will simply be referred to as a functor.

A homotopy coherent diagram of $J$ in a quasi-category $\C$ is a functor $J\to \C$. A homotopy commutative diagram is a functor from $J$ into the homotopy category $\homcat\C$.

Recall that the limit (resp. colimit) of a diagram $D\colon J\to \C$ arises as a left (resp. right) adjoint of the diagonal functor $\C\to \C^{J}$.
Suppose that $\C$ is a quasi-category in which all natural maps of the form

\begin{equation*}
\coprod_{i=1}^n X_i \to \prod_{i=1}^n X_i
\end{equation*}

are homotopy equivalences. In this case, we make the assumption that finite coproducts and finite product coincide. More precisely, the functor defining the limit $\prod_{i=1}^n X_i$ coincides strictly with the one defining the colimit $\coprod_{i=1}^n X_i$. The term biproduct will be used to emphasise that it is both a product and a coproduct, and it will be denoted by $\bigoplus_{i=1}^n X_i$.

Let $\bigoplus\limits_{i=1}^n X_i$ and $\bigoplus\limits_{j=1}^m Y_j$ be biproducts in $\C$. Then a map $f\colon \bigoplus\limits_{i=1}^n X_i\to \bigoplus\limits_{j=1}^m Y_j$ is uniquely determined, up to homotopy, by its components $f_{j,i}\colon X_i\to Y_j$. We will thus write $f$ as a matrix $(f_{j,i})_{i,j}$.

The $0$–simplices of a quasi-category $\C$ are often referred to as objects. Similarly, $1$–simplices are referred to as maps or morphisms.
We write $f\sim g$ when the  $1$–simplicies $f$ and $g$ are homotopic. A degenerate 1-simplex $s_0X$ is denoted by X[r,equal]    X.

For two simplicial sets $X$ and $Y$, we denote by $\hom (X,Y)$ the function complex (or internal hom) whose set of $n$–simplices is $\hom(X,Y)_n=\Hom_{\sSet}(X\times [n], Y)$. In the case of $\hom ([1],\C)$, where $\C$ is a quasi-category, there is the subcomplex $\hom_{\C}(B,A)$ whose $0$–simplices are the $1$–simplices $B\to A$ in $\C$. Note that this space models the homotopy function complex, as defined in [7]. We refer to <cit.> and <cit.> for details. For an ordinary category $\c$ we denote by~$_(X,Y)$ the set of maps~$X→Y$.

\section{Exact quasi-categories}\label{sec:exqcat}

We first review exact quasi-categories, a generalization of exact categories.
Since the underlying category of an exact category is additive, it is only sensible to introduce the quasi-categorical generalization.
\begin{definition}\label{addcat}
A quasi-category~$\C$ is \emph{additive} if the following hold.
\begin{enumerate}[label=(Add\arabic*), leftmargin=1.3cm]
\item\label{addcat1} There is a zero object 0 in~$\C_0$, which is to say that~$\hom_\C (0,X)$ and~$\hom_\C (X,0)$ are contractible for all~$X\in\C_0$.
\item\label{addcat2} Finite products and coproducts exist in~$\C$.
\item\label{addcat3} For any finite family of objects~$\{X_1,\dots, X_n\}$, the natural map
\begin{equation*}
\coprod_{i=1}^n X_i \to \prod_{i=1}^n X_i,
\end{equation*}
induced by the identity maps~$X_i\to^{1}X_i$, is a homotopy equivalence.
\item For all~$X,Y\in\C_0$, the Hom-set~$\Hom_{h\C}(X,Y)\coloneqq \pi_0\hom_{\C}(X,Y)$ admits an abelian group structure:~$f+g$ is the composite
\begin{center}
\begin{tikzcd}[ampersand replacement=\&]
X\arrow[r,"{\begin{pmatrix} 1 \\ 1 \end{pmatrix}}"] \& X\coprod X\arrow[r,"{\begin{pmatrix} f & 0 \\ 0 & g \end{pmatrix}}"] \& Y\prod Y \arrow[r,"{\begin{pmatrix} 1 & 1 \end{pmatrix}}"] \& Y
\end{tikzcd}
\end{center}
in~$\homcat \C$.
\end{enumerate}
\end{definition}
A bicomplete quasi-category~$$ is additive precisely when the homotopy category~$$ is additive as an ordinary category. Analogously to how  biproducts arise in additive categories, the axioms \ref{addcat2} and \ref{addcat3} permit the assumption that~$$ has a finite biproducts, which we denote by~$⊕_i=1^n$. More precisely, we may choose the product and coproduct functors
\begin{equation*}
\prod,\coprod\colon\C^{n}\to \C
\end{equation*}
in such a way that they coincide strictly -- for (co)limits in quasi-categories are homotopy invariant.

Barwick defines exact quasi-categories \cite[Definition~3.1]{Bar15} by adapting Keller's minimal set of axioms [15] to the quasi-categorical setting.
\begin{definition}
Let~$\C$ be an additive quasi-category and let~$\C\cof$ and~$\C\fib$ be subcategories that contain all objects in~$\C$, as well as all homotopy equivalences. The maps in~$\C\cof$ will be referred to as \emph{cofibrations}, whereas morphisms in~$\C\fib$ are \emph{fibrations}. The triple~$(\C,\C\cof,\C\fib)$ is called an \textit{exact quasi-category} if the following axioms hold.
\begin{enumerate}[label=(Ex\arabic*), leftmargin=1.1cm]
\item\label{Ex1} For any zero object~$0$ in~$\C_0$, all morphisms of the form~$0\to X$ are cofibrations and those of the form~$X\to 0$ are fibrations.
\item\label{Ex2} Pushouts of cofibrations exist and are cofibrations, and dually for pullbacks of fibrations.
\item\label{Ex3} The following are equivalent for a homotopy coherent square
\begin{center}
\begin{tikzcd}
A\arrow[r,"i"]\arrow[d,"f"] & E\arrow[d,"p"] \\
C\arrow[r,"g"] & B
\end{tikzcd}
\end{center}
\begin{enumerate}[label=(Ex3.\arabic*), leftmargin=1.4cm]
\item\label{Ex3a} The square is a pullback, the map~$g$ is a cofibration and~$p$ is a fibration.
\item\label{Ex3b} The square is a pushout, the map~$i$ is a cofibration and~$f$ is a fibration.
\end{enumerate}
\end{enumerate}
\end{definition}
If the subcategories~$$ and~$$ are implicitly specified, we simply say that~$$ is exact. A triple~$(,,)$ satisfying the axioms above is called an \textit{exact structure} on~$$.

Whenever a cofibration appears in a diagram, it will be drawn as follows:~$$ (like a monomorphism in an ordinary category). A fibration will be drawn as a two-headed arrow~$$.

Nerves of exact categories are exact quasi-categories, where the the class of cofibrations consists of the admissible monomorphisms, and the fibrations are the admissible epimorphisms. Moreover, any stable quasi-category \cite[Definition~1.1.1.9]{Lur17} can be seen as an exact quasi-category where~$==$.

The axioms above provide a framework for exact sequences, in more or less the usual fashion.
\begin{definition}
Let~$\C$ be an exact quasi-category. An \emph{exact sequence} in~$\C$ is a square
\begin{center}
\begin{tikzcd}
A\arrow[r]\arrow[d, two heads] & E\arrow[d] \\
0\arrow[r, tail] & B
\end{tikzcd}
\end{center}
where~$0$ is a zero object and the equivalent criteria in \ref{Ex3} are met.
\end{definition}
Equivalently, the exact sequences are the bicartesian squares
\begin{center}
\begin{tikzcd}
A\arrow[r,"i",tail]\arrow[d,"",two heads]\arrow[dr, phantom, "\square"] & E\arrow[d,"p",two heads] \\
0\arrow[r,"",tail] & B
\end{tikzcd}
\end{center}
As for ordinary exact categories, exact sequences can be drawn horizontally
\begin{equation*}
\begin{tikzcd}
A\arrow[r,"i",tail]&E\arrow[r,"p",two heads]&B,
\end{tikzcd}
\end{equation*}
omitting the zero object. There is not really any loss of information in such notation, for the choice of morphisms to/from zero is irrelevant up to homotopy.

\begin{definition}\label{def:mapofseq}
A \textit{map of exact sequences} is simply a map of bicartesian squares.
Such maps will mostly be depicted as homotopy commutative diagrams of the form
\begin{equation*}
\begin{tikzcd}
A\arrow[r,"i",tail]\arrow[d]&E\arrow[r,"p",two heads]\arrow[d]&B\arrow[d] \\
A'\arrow[r,"i'",tail]&E'\arrow[r,"p'",two heads]&B'
\end{tikzcd}
\end{equation*}
\end{definition}

Exact functors also have a completely analogous definition.
\begin{definition}\label{def:exfunc}
Let~$\C$ and~$\mathscr{D}$ be exact quasi-categories. A functor~$F\colon \C\to \mathscr{D}$ is \textit{exact} if it preserves zero objects, cofibrations and pushouts of cofibrations.
\end{definition}
When checking that a functor is exact, one may equivalently consider the dual statement for fibrations \cite[Proposition~4.8]{Bar15}. Exact sequences are obviously preserved by exact functors, and by considering split exact sequences it is seen that they preserve the additive structure as well.

Full subcategories~$𝒟$ of a given exact quasi-category~$$ inherit the exact structure if they are \textit{closed under extensions}, i.e. for all exact sequences
\begin{center}
\begin{tikzcd}
A\arrow[r,tail] & E\arrow[r,two heads] & B
\end{tikzcd}
\end{center}
with~$A,B∈𝒟_0$, we have that~$E∈𝒟_0$. The cofibrations~$𝒟$ are the maps in~$∩𝒟$ whose cofiber are in~$𝒟$, and dually for fibrations. The inclusion functor
is then exact.

It turns out that all exact quasi-categories occur as extension closed subcategories of stable quasi-categories.
\begin{theorem}[{\cite[Theorem~1]{Kle20}}]\label{thm:Kle20.1}
Given a small exact quasi-category~$\C$, there exists a stable hull~$\mathcal{H}^{\rm st}(\C)$ into which~$\C$ embeds exactly and universally.
\end{theorem}
This can be seen as a quasi-categorical version of the Gabriel--Quillen embedding theorem for exact categories, which states that they embed exactly into abelian categories.

A number of results concerning exact quasi-categories generalize to results about exact quasi-categories. We will make frequent use of our two next lemmas, first generalized by Barwick.
\begin{lemma}[{\cite[Lemma~4.5]{Bar15}}]\label{Bar15.4.5}
A homotopy coherent square
\begin{center}
\begin{tikzcd}
A\arrow[r,"i"]\arrow[d,"f"] & E\arrow[d,"p"] \\
C\arrow[r,"g"] & B
\end{tikzcd}
\end{center}
in an exact quasi-category is bicartesian if either of the following conditions hold.
\begin{enumerate}
\item\label{Bar15.4.5.1} The square is a pushout and~$i$ is a cofibration.
\item\label{Bar15.4.5.2} The square is a pullback and~$p$ is a fibration.
\end{enumerate}
\end{lemma}
\begin{lemma}[{\cite[Lemma~4.7]{Bar15}}]\label{Bar15.4.7}
\begin{center}
\begin{tikzcd}
A\arrow[r,"i"]\arrow[d,"f"] & E\arrow[d,"p"] \\
C\arrow[r,"g"] & B
\end{tikzcd}
\end{center}
be a homotopy coherent square satisfying one of the properties \eqref{Bar15.4.5.1} or \eqref{Bar15.4.5.2} in \Cref{Bar15.4.5}. Then we have an exact sequence
\begin{center}
\begin{tikzcd}[ampersand replacement=\&]
A \arrow[r,"{\begin{pmatrix}-i \\ f \end{pmatrix}}",tail]\& E\oplus C \arrow[r,"{\begin{pmatrix}p & g \end{pmatrix}}",two heads] \& B.
\end{tikzcd}
\end{center}
In particular, the maps~${\begin{pmatrix}-i \\ f \end{pmatrix}}$ and~${\begin{pmatrix}i \\ -f \end{pmatrix}}$ are cofibrations and~${\begin{pmatrix}p & g \end{pmatrix}}$ is a fibration.
\end{lemma}

Generalizations of celebrated diagram lemmas will also be helpful in later sections. The first relates pushouts to maps of exact sequences.

\begin{lemma}[{\cite[Proposition~A.1]{Kle20}}]\label{Buh2.12}
A pushout
\begin{center}
\begin{tikzcd}
A \arrow[r, "i",tail] \arrow[d] \arrow[dr, phantom, "\ulcorner",very near end]  & E \arrow[d] \\

C \arrow[r,tail]               & F
\end{tikzcd}
\end{center}
where~$i$ is cofibration, can be extended to a map of exact sequences
\begin{center}
\begin{tikzcd}
A \arrow[r, "i",tail] \arrow[d] \arrow[dr, phantom, "\ulcorner",very near end]  & E \arrow[d]\arrow[r,two heads,"p"] & B\arrow[d,equal] \\

C \arrow[r,tail]               & F \arrow[r,two heads] & B
\end{tikzcd}
\end{center}
Conversely, the existence of such a map of exact sequences implies that the square is a pushout. A dual statement holds for pullbacks along fibrations.
\end{lemma}

Secondly, a map of exact sequences has a canonical factorization.
\begin{lemma}[{\cite[Proposition~A.2]{Kle20}}]\label{Buh3.1}
Any map of exact sequences
\begin{center}
\begin{tikzcd}
A \arrow[r, "i_A",tail] \arrow[d,"f'"]   & E \arrow[d,"f"]\arrow[r,two heads,"p_B"] & B\arrow[d,"f''"] \\

C \arrow[r,tail,"i_C"]               & F \arrow[r,two heads,"p_D"] & D
\end{tikzcd}
\end{center}
can be factored
\begin{center}
\begin{tikzcd}
A \arrow[r, "i_A",tail] \arrow[d,"f'"]\arrow[dr, phantom, "\square"]   & E \arrow[d,"g"]\arrow[r,two heads,"p_B"] & B\arrow[d,equal] \\
C \arrow[r, "j",tail] \arrow[d,equal]   & Z \arrow[d,"h"]\arrow[r,two heads,"q"]\arrow[dr, phantom, "\square"] & B\arrow[d,"f''"] \\
C \arrow[r,tail,"i_C"]               & F \arrow[r,two heads,"p_D"] & D
\end{tikzcd}
\end{center}
where the squares marked by~$\square$ are bicartesian.
\end{lemma}

The renowned Five Lemma is a consequence of \Cref{Buh3.1}. It will be proved in \Cref{app:diaproofs}, since there is no significant difference from the special case of ordinary exact categories.
\begin{lemma}[Five lemma]\label{5lem}
Consider a map of exact sequences
\begin{center}
\begin{tikzcd}
A \arrow[r, "i_A",tail] \arrow[d,"f'"]   & E \arrow[d,"f"]\arrow[r,two heads,"p_B"] & B\arrow[d,"f''"] \\

C \arrow[r,tail,"i_C"]               & F \arrow[r,two heads,"p_D"] & D
\end{tikzcd}
\end{center}
If~$f'$ and~$f''$ are homotopy equivalences (resp. cofibrations, resp. fibrations), so is~$f$.
\end{lemma}

Lastly, the~$3×3$--lemma will come in handy. It is also proved in \Cref{app:diaproofs}.
\begin{lemma}[$3\times 3$--lemma]\label{lemma:3x3}
Consider the homotopy coherent diagram with exact columns
\begin{center}
\begin{tikzcd}
A' \arrow[d,tail,"i_A"]\arrow[r,"f'"] & E' \arrow[r,"g'"]\arrow[d,tail,"i_E"] & B'\arrow[d,tail,"i_B"] \\
A \arrow[r,"f"]\arrow[d,two heads,"p_A"] & E \arrow[r,"g"]\arrow[d,two heads,"p_E"] & B\arrow[d,two heads,"p_B"] \\
A''\arrow[r,"f''"]& E''\arrow[r,"g''"] & B''
\end{tikzcd}
\end{center}
If the middle row and one of the other rows are exact, then the remaining row is exact.
\end{lemma}

\section{Quasi-categories of extensions}
\label{section:extensioncategories}
Fix an exact quasi-category~$$ and let $$ and $$ be arbitrary (not necessarily full) subcategories of $$. We define the quasi-categories of~$n$--extensions~$^n_(,)$.

Recall that~$[1]×[1]$ consists of two~$2$--simplices glued along their~$1$--faces to obtain a square. A short exact sequence is thus a homotopy coherent diagram~$𝔼[1]×[1]→$ which defines a bicartesian square
\begin{center}
\begin{tikzcd}
A\arrow[r,"i",tail]\arrow[d,"",two heads]\arrow[dr, phantom, "\square"] & E\arrow[d,"p",two heads] \\
0\arrow[r,"",tail] & B
\end{tikzcd}
\end{center}
The corners in~$[1]×[1]$ are commonly labeled~$(k,l)$ for~$k,l=0,1$. Every inclusion~$i_l,k[0]→[1]×[1]$ induces a map~$eval_k,l([1]×[1],)→$ which restricts to~$eval_k,l^1_(,)→$. In particular a short exact sequence
\begin{center}
\begin{tikzcd}
\mathbb{E}\colon &  F' \arrow[r, tail] & E \arrow[r, two heads] & F
\end{tikzcd}
\end{center}
satisfies~$eval_0,0𝔼=F'$,~$eval_0,1𝔼=0$,~$eval_1,0𝔼=E$ and~$eval_1,1𝔼=F$.

\begin{definition}
Formally, we define the~\emph{$1$--extension} category
\[
\Extcat^1_{\C}(\B,\A)\coloneqq e^{-1}(\B\times \A),
\]
where $e$ is the functor
\begin{equation*}
\begin{tikzcd}[column sep=6em]
e\colon \Extcat^1_{\C}(\C,\C)\arrow[r,"{\mathrm{eval}_{0,0}\times\mathrm{eval}_{1,1}}"] & \C\times \C.
\end{tikzcd}
\end{equation*}
In other words,~$\Extcat^1_{\C}(\B,\A)$ is the subcategory of~$\hom([1],\C)$ generated exact sequences starting in~$\A$ and ending in~$\B$
\end{definition}

%While we are at it, let $\hom_{\C}(\B,\A)$ be $f^{-1}(\B\times\A)$, where $f$ is the functor
%\begin{tikzcd}[column sep=4em]
%f\colon \map([1],\C)\arrow[r,"{\mathrm{eval}_0\times\mathrm{eval}_1}"%] & \C\times \C.

Note that the~$1$--simplices in~$^1_(,)$ are maps of exact sequences, as defined in \Cref{def:mapofseq}. If $$ and $$ both consist of a single object, say $A$ and $B$, respectively, and all of its morphisms are homotopic to the identity, we write $_^1(B,A)$. The Five Lemma has a very nice interpretation in this special case.
\begin{proposition}
\label{proposition:Ext1IsKan}
All maps in~$\Extcat_{\C}^1(B,A)$ are homotopy equivalences. Consequently, the quasi-category~$\Extcat_{\C}^1(B,A)$ is a Kan complex.
\end{proposition}
\begin{proof}
The first assertion is a direct consequence of \Cref{5lem}. Thus, the quasi-category~$\Extcat_{\C}^1(B,A)$ is a Kan complex \cite[Corollary~1.4]{Joy02}.
\end{proof}

Lemma~\ref{Buh2.12} implies that~$^1_(B,A)$ does in fact define a bifunctor
\[
\Extcat^1_\C\colon \C^{\mathrm{op}}\times \C \to \kan
\]
into the quasi-category of Kan complexes.
Indeed, given a morphism~$A→A'$ and an exact sequence
\begin{center}
\begin{tikzcd}
A \arrow[r, tail] & E \arrow[r, two heads] & B
\end{tikzcd}
\end{center}
computing the pushout produces a map of exact sequences:
\begin{center}
\begin{tikzcd}
A \arrow[r, tail] \arrow[d]\arrow[dr, phantom, "\square"]& E \arrow[r, two heads] \arrow[d]& B\arrow[d, equal] \\
A'\arrow[r, tail] & E' \arrow[r, two heads] & B
\end{tikzcd}
\end{center}
Dually, we compute pullbacks to deduce functoriality in the contravariant component.

Before defining the higher extension categories, let us first explain how one may interpret the~$n$--extensions when~$$ is the nerve of an exact category.  In this case, the~\textit{$n$--extensions} of~$B$ by~$A$ can be represented as diagrams
\begin{center}
\begin{tikzcd}[scale cd= 0.8, column sep=0.75em]
& E_1 \arrow[rr, "d_1"] \arrow[rd, "p_1", two heads] &                                                       & E_2 \arrow[rr, "d_2"] \arrow[rd, "p_2", two heads] &                                                       & \cdots \arrow[rr, "d_{n-2}"] \arrow[rd, "p_{n-2}", two heads] &                                                               & E_{n-1} \arrow[rr, "d_{n-1}"] \arrow[rd, "p_{n-1}", two heads] &                                                           & E_n \arrow[rd, "d_n", two heads] &   \\
A\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "d_0", tail] \arrow[rd, two heads] &                                                    & E_{1.5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_1", tail] \arrow[rd, two heads] &                                                    & E_{2.5}\arrow[rr, phantom, "\cdots"] \arrow[ru, "i_2", tail] \arrow[rd, two heads] &                                                               & E_{(n-2).5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_{n-2}", tail] \arrow[rd, two heads] &                                                                & E_{(n-1).5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_{n-1}", tail] \arrow[rd, two heads] &                                  & B \\
& 0 \arrow[ru, tail]                                 &                                                       & 0 \arrow[ru, tail]                                 &                                                       & \cdots \arrow[ru, tail]                                       &                                                               & 0 \arrow[ru, tail]                                             &                                                           & 0 \arrow[ru, tail]               &
\end{tikzcd}
\end{center}
Of course, it is common to simply think of
\begin{center}
\begin{tikzcd}
A \arrow[r, tail , "d_0"] & E_1 \arrow[r, "d_1"] & E_2 \arrow[r, "d_2"] & \cdots \arrow[r, "d_{n-1}"] & E_n \arrow[r, two heads, "d_n"] & B
\end{tikzcd}
\end{center}
as the exact sequence. But such a view-point does not encode exactness, at least not for our purposes. An alternative approach is to rather look at the complementary diagram
\begin{equation}
\label{eq:compdia}
\begin{tikzcd}[scale cd= 0.8, column sep=0.75em]
& E_1 \arrow[rd, "p_1", two heads] &                                                       & E_2  \arrow[rd, "p_2", two heads] &                                                       & \cdots  \arrow[rd, "p_{n-2}", two heads] &                                                               & E_{n-1} \arrow[rd, "p_{n-1}", two heads] &                                                           & E_n \arrow[rd, "d_n", two heads] &   \\
A\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "d_0", tail] \arrow[rd, two heads] &                                                    & E_{1.5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_1", tail] \arrow[rd, two heads] &                                                    & E_{2.5}\arrow[rr, phantom, "\cdots"] \arrow[ru, "i_2", tail] \arrow[rd, two heads] &                                                               & E_{(n-2).5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_{n-2}", tail] \arrow[rd, two heads] &                                                                & E_{(n-1).5}\arrow[rr, phantom, "\diamondsuit"] \arrow[ru, "i_{n-1}", tail] \arrow[rd, two heads] &                                  & B \\
& 0 \arrow[ru, tail]                                 &                                                       & 0 \arrow[ru, tail]                                 &                                                       & \cdots \arrow[ru, tail]                                       &                                                               & 0 \arrow[ru, tail]                                             &                                                           & 0 \arrow[ru, tail]               &
\end{tikzcd}
\end{equation}
which does contain the information of exactness. Moreover, the boundary maps~$d_j$ are uniquely determined as the composition of~$p_j$ and~$i_j$ -- so there is no loss of information. Do note that these diagrams are advantageous when working with quasi-categories due to the fact that the composites~$d_j$ of~$i_j$ and~$p_j$ have not been chosen.

The higher extension quasi-categories are defined recursively:
\begin{definition}
For~$n\geq 2$ we define the~\emph{$n$--extension} quasi-category~$\Extcat_{\C}^n(\B,\A)$ as the pullback of
\begin{center}
\begin{tikzcd}
& \Extcat^{n-1}(\B,\C) \arrow[d, "\mathrm{eval}_{0,0}"]  \\

\Extcat^{1}(\C,\A) \arrow[r, "\mathrm{eval}_{1,1}"]             & \C
\end{tikzcd}
\end{center}
\end{definition}

If $$ and $$ consist of a single object, say $A$ and $B$, respectively, and all of its morphisms are homotopic to the identity, we write $_^n(B,A)$. Note that~$^n_(B,A)$ defines a functor into the quasi-category of quasi-categories. Much like in the case of~$^1_(B,A)$, this follows from Lemma~\ref{Buh2.12}.

The objects in~$^n_(B,A)$ are to be regarded as~$n$ conjoined exact sequences, as depicted in \eqref{eq:compdia}. As a matter of notation, we will omit the zeros from the elements and write a typical object~$𝔼$ of~$^n_(B,A)$ as
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
&                           & E_1 \arrow[rd, two heads,"p_1"]  &                                 & E_2 \arrow[rd, two heads, "p_2"] &                                    & E_n\arrow[rd, two heads, "d_n"] &    \\

& A \arrow[ru, tail, "d_0"] &                                  & E_{1.5} \arrow[ru, tail, "i_1"] &                                  & \cdots \arrow[ru, tail, "i_{n-1}"] &                                 & B
\end{tikzcd}
\end{center}

A map~$f𝔼→𝔽$ of such~$n$--extensions, or~$1$--simplex in~$^n_(B,A)$, is then given by a homotopy coherent diagram
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
& E_1 \arrow[rd, "p^E_1", two heads] \arrow[dd, "f_1"] &                                                   & E_2 \arrow[rd, "p^E_2", two heads] \arrow[dd, "f_2"] &                                & E_n \arrow[rd, "d^E_n", two heads] \arrow[dd, "f_n"] &              \\
A \arrow[ru, "d^E_0"] \arrow[dd,equal] &                                                      & E_{1.5} \arrow[ru, "i^E_1"] \arrow[dd, "f_{1.5}", near start] &                                                      & \cdots \arrow[ru, "i^E_{n-1}"] &                                                      & B \arrow[dd,equal] \\
& F_1 \arrow[rd, "p^F_1", two heads]                   &                                                   & F_2 \arrow[rd, "p^F_2", two heads]                   &                                & F_n \arrow[rd, "d^F_n", two heads]                   &              \\
A \arrow[ru, "d^F_0"]            &                                                      & F_{1.5} \arrow[ru, "i^F_1"]                       &                                                      & \cdots \arrow[ru, "i^F_{n-1}"] &                                                      & B
\end{tikzcd}
\end{center}

At a first glance, our notion of maps between exact sequences may seem stronger than what is conventional. If~$$ is the nerve of an exact category, a map of~$n$--extensions is usually given as a commutative diagram
\begin{center}
\begin{tikzcd}
A\arrow[r,tail]\arrow[d,equal]& E_1\arrow[r]\arrow[d,"f_1"]&E_2\arrow[r]\arrow[d,"f_2"]&\cdots\arrow[r]&E_{n-1}\arrow[d,"f_{n-1}"]\arrow[r]&E_n\arrow[r,two heads]\arrow[d,"f_n"]&B\arrow[d,equal]  \\
A'\arrow[r,tail]& F_1\arrow[r]&F_2\arrow[r]&\cdots\arrow[r]&F_{n-1}\arrow[r]&F_n\arrow[r,two heads]&B'
\end{tikzcd}
\end{center}
without mention of maps~$f_i.5E_i.5→F_i.5$. In our setting we must require the~$f_i.5$s to exist, but they can be induced if~$$ is an ordinary exact category. For example, we can induce~$f_1.5E_1.5→F_1.5$ since the left-most square in
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
& E_1 \arrow[rr] \arrow[rd, "p_1", two heads] \arrow[dd, "f_1"] &                                          & E_2 \arrow[dd, "f_2"] \arrow[rr] &                                          & \cdots \\
A \arrow[ru,tail] \arrow[dd,equal] &                                                               & E_{1.5} \arrow[ru,tail] \arrow[dd, "f_{1.5}" near start,dashed] &                                                        &  &        \\
& F_1 \arrow[rr,crossing over] \arrow[rd, two heads]                          &                                          & F_2 \arrow[rr]                   &                                          & \cdots \\
A \arrow[ru,tail]            &                                                               & F_{1.5} \arrow[ru,tail]                       &                                                        &                        &
\end{tikzcd}
\end{center}
commutes. Then using the fact that fibrations in nerves of exact categories are epimorphisms forces
\begin{center}
\begin{tikzcd}
E_{1.5} \arrow[r] \arrow[d, "f_{1.5}"] & E_2 \arrow[d, "f_2"] \\
E_{1.5} \arrow[r]                      & F_2
\end{tikzcd}
\end{center}
to commute. Proceeding in such a manner yields ~$f_i.5$ for~$1≤i ≤n-1$. But this process does not translate to exact quasi-categories; fibrations are not necessarily epimorphisms. The discussion thus far is summarized in
\begin{proposition}
\label{proposition:is ClassicalExt}
If~$\C$ is the nerve of an exact category, then~$\Extcat^n_\C(B,A)$ is the nerve of the classical category of~$n$--extensions.
\end{proposition}

Extension categories enjoy a factorization property, which will be crucial later on.

\begin{proposition}\label{prop:VW3.1}
Let~$n\geq 1$.
A map~$f\colon\mathbb{E}\to\mathbb{F}$ in~$\Extcat^n_{\C}(B,A)$
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
& E_1 \arrow[rd, "p^E_1"', two heads] \arrow[dd, "f_1" description]\arrow[rr,"d^E_1"] &                                                   & E_2 \arrow[rd, "p^E_2"', two heads] \arrow[dd, "f_2" description]\arrow[r,"d^E_2"] &         \cdots\arrow[r,"d^E_{n-1}"]                       & E_n \arrow[rd, "d^E_n"', two heads] \arrow[dd, "f_n" description] &              \\
A \arrow[ru, "d^E_0"] \arrow[dd,equal] &                                                      & E_{1.5} \arrow[ru, "i^E_1"'] \arrow[dd, "f_{1.5}" description, near start] &                                                      & \cdots \arrow[ru, "i^E_{n-1}"'] &                                                      & B \arrow[dd,equal] \\
& F_1 \arrow[rd, "p^F_1"', two heads] \arrow[rr,crossing over]                  &                                                   & F_2 \arrow[rd, "p^F_2"', two heads]\arrow[r]                     &                    \cdots\arrow[r]              & F_n \arrow[rd, "d^F_n", two heads]                   &              \\
A \arrow[ru, "d^F_0"]            &                                                      & F_{1.5} \arrow[ru, "i^F_1"']                       &                                                      & \cdots \arrow[ru, "i^F_{n-1}"'] &                                                      & B
\end{tikzcd}
\end{center}
can be factored
\begin{center}
\begin{tikzcd}
\mathbb{E}\arrow[r,"f'"] & {\rm cyl}(f)\arrow[r,"p"]& \mathbb{F}
\end{tikzcd}
\end{center}
where~$f'$ is a term-wise cofibration, and there exists a factorization~$1_{\mathbb{F}}\sim pm$ where~$m$ is a term-wise cofibration.
\end{proposition}
\begin{proof}
The case~$n=1$ is easily deduced from \Cref{proposition:Ext1IsKan}; choose~$f'$ as the identity on~$\mathbb{E}$ and~$p=f$.

For~$n\geq 2$, the exact sequence~${\rm cyl}(f)$ will be chosen as follows
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=0.5em, column sep=0.75em,ampersand replacement=\&,row sep=5em,column sep=1em]
\& F_1\oplus E_2 \arrow[rd, two heads,"{\begin{pmatrix}p^F_1 & 0 \\ 0 & 1 \end{pmatrix}}" description]  \&                                          \& F_2\oplus E_3\oplus E_2 \arrow[rd, two heads,"{\begin{pmatrix}p^F_2 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}}" description]  \&                                      \& \cdots  \\
A \arrow[ru,tail,"{\begin{pmatrix}d^F_0 \\ 0 \end{pmatrix}}" description]  \&                                                        \& F_{1.5} \oplus E_2 \arrow[ru,tail,"{\begin{pmatrix}i^F_1 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix}}" description]  \&                                                        \&  F_{2.5} \oplus E_3 \arrow[ru,tail,"{\begin{pmatrix}i^F_2  & 0 \\ 0  & 0 \\ 0 & 1 \end{pmatrix}}" description]  \&                 \\
%    \begin{tikzcd}[column sep=0.75em,ampersand replacement=\&,row sep=7em,column sep=1em]
\cdots \arrow[rd, two heads,"{\begin{pmatrix}p^F_{n-2} & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}}" description] \&                                                  \&  F_{n-1}\oplus E_n\oplus E_{n-1} \arrow[rd, two heads,"{\begin{pmatrix}p^F_{n-1} & 0 & 0  \\ 0 & 1 & 0 \end{pmatrix}}" description]  \&                                                  \& F_n \oplus  E_n \arrow[rd,two heads,"{\begin{pmatrix}d^F_{n} & 0 \end{pmatrix}}" description]  \&              \\
\&  F_{(n-2).5} \oplus E_{n-1} \arrow[ru,tail,"{\begin{pmatrix}i^F_{n-2}  & 0 \\ 0  & 0 \\ 0  & 1 \end{pmatrix}}" description]  \&                                                                \& F_{(n-1).5} \oplus E_{n} \arrow[ru,tail,"{\begin{pmatrix}i^F_{n-1}  & 0 \\ 0 & 1 \end{pmatrix}}" description]  \&                                             \& B
\end{tikzcd}
\end{center}
We deduce from \Cref{Bar15.4.7} that~$\mathrm{cyl}(f)$ is an~$n$--extension.
Since~$\mathbb{F}$ is a direct summand of~$\mathrm{cyl}(f)$, the map~$p\colon \mathrm{cyl}(f)\to \mathbb{F}$ is chosen as the projection, and~$m\colon \mathbb{F} \to \mathrm{cyl}(f)$ as the inclusion. For~$f'\colon \mathbb{E}\to \mathrm{cyl}(f)$ we choose the map in ~$\Extcat^n_{\C}(B,A)$ with components
\begin{align*}
f'_1 &= \begin{pmatrix}f_1 \\ d^E_1 \end{pmatrix}, \\
f'_{j.5} &= \begin{pmatrix}f_{j.5} \\ i^E_j \end{pmatrix} \text{ for } 1\leq j \leq n-1 \\
f'_j &= \begin{pmatrix}f_i \\ d^E_j \\ 1_{E_j} \end{pmatrix} \text{ for } 2\leq j \leq n-1 \\
f'_n &= \begin{pmatrix}f_n \\ 1_{E_n} \end{pmatrix}.
\end{align*}
Clearly, we have that~$f=pf'$, so it remains to show that~$f'$ has cofibrations in all components. For~$j\neq 1$, we can use \Cref{Bar15.4.7} (with the squares transposed). Only~$f'_1$ has not been addressed. We have a map of exact sequences
\begin{center}
\begin{tikzcd}[ampersand replacement=\&]
A \arrow[r, "d^E_0"] \arrow[d,equal] \& E_1 \arrow[d, "f_1'"] \arrow[r,"p^E_1", two heads] \arrow[dr,phantom,"\square"] \& E_{1.5} \arrow[d, "f_{1.5}'"] \\
A \arrow[r,"{\begin{pmatrix}d^F_0 \\ 0 \end{pmatrix}}"']                  \& F_1\oplus E_2 \arrow[r,"{\begin{pmatrix}p^F_1 & 0 \\ 0 & 1 \end{pmatrix}}"',two heads]         \& F_{1.5}\oplus E_2
\end{tikzcd}
\end{center}
in which the right square, by \Cref{Buh2.12}, is bicartesian. As \ref{Ex3} applies, the map~$f_1'$ is a cofibration since~$f_{1.5}'$ is.
\end{proof}

In recent work of Volkow--Witherspoon~{\cite[Lemma~3.1]{VW20}}, this factorization property was shown to hold for ordinary exact categories. Our proof for exact quasi-categories can be specialized to this case. We will use it in \Cref{lemma:LoopsOnExt^n}, via~\cref{lem:lemma1}, to prove that all loops in~$^n_(B,A)$ are homotopic to a loop of length 1, as Hermann does for ordinary exact categories [12].

\section{The Retakh spectrum}
\label{section:overview}
Inspired by Retakh~[24], we organize all of the extension categories in an~$Ω$--spectrum object~$_(B,A)$, whose~$n$th entry is~$^n_(B,A)$. Such a construction comes with weak equivalences~$^n_(B,A)→Ω^n+1_(B,A)$. We thus start with a recap of loop spaces in simplicial sets.

Let~$X$ be any simplicial set. The~\emph{loop space} on~$X$, based at~$x$, is defined to be the homotopy pullback of
\begin{center}
\begin{tikzcd}
& \left [ 0 \right ] \arrow[d, "x"]\\
\left [ 0 \right ] \arrow[r, "x"] & X
\end{tikzcd}
\end{center}
It can be computed by first replacing one~$x[0]→X$ with a Kan fibration~$pPX X$, where~$PX$ contractible, and then computing the ordinary pullback [3]. The problem of computing loop spaces can thus be solved by finding such a Kan fibration. This is analogous to how derived functors in homological algebra are calculated by first finding projective or injective resolutions.

There are two evaluation maps~$eval_0$ and~$eval_1$ on~$([1],X)$ induced by applying~$(-,X)$ to the inclusions~$0[0]→[1]$ and~$1[0]→[1]$ of~$0$ and~$1$ into~$[1]$. These maps send a 1-simplex~$e$ in~$X$, which is an object in~$([1],X)$, to its source~$d_1e$ and target~$d_0 e$ in~$X$. The reduced mapping space~$(([1],0),(X,x))$, of~$1$--simplices starting at~$x$, may then be defined as the pre-image~$eval_0^-1 x$. Whenever~$X$ is a Kan complex, one may mimic the construction of loop spaces in topological spaces. For in this case, the evaluation~$eval_1$ is a Kan fibration~\cite[Corollary~5.3 on p.~21]{GJ09}. Hence the loop space of~$X$ based at~$x$, denoted by~$Ω(X,x)$, fits into the pullback diagram
\begin{center}
\begin{tikzcd}
\Omega(X,x)\arrow[r] \arrow[d] & \hom(([1],0),(X,x)) \arrow[d, "\mathrm{eval}_1"]\\
\left [ 0 \right ] \arrow[r, "x"] & X
\end{tikzcd}
\end{center}
For future reference, we state this as a proposition:
\begin{proposition}
\label{proposition:loopsOnKan}
If~$X$ is a Kan complex, then the loop space~$\Omega (X,x)$ on~$X$ based at~$x$ is the pre-image of~$x$ under the first evaluation~$\mathrm{eval}_1\colon \hom(([1],0),(X,x))\to X$.
\end{proposition}

There is a rather natural choice of basepoint in~$^n_(B,A)$. It is the split exact sequence~$σ_1(B,A)$:
\begin{center}
\begin{tikzcd}
A\arrow[r, tail] & A\oplus B \arrow[r, two heads] & B
\end{tikzcd}
\end{center}
for~$n=1$ and~$σ_n(B,A)$
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
&   	                  & A \arrow[rd]  &              & \cdots \arrow[rd] & &  B \arrow[rd, equal] &    \\

& A \arrow[ru, equal]  &               & 0 \arrow[ru] &              & 0\arrow[ru] &                     & B
\end{tikzcd}
\end{center}
otherwise. To define the advertised~$Ω$--spectrum object, we must first find a suitable model for~$Ω(_^n(B,A),σ_n (B,A))$, the loop space of~$^n_(B,A)$ based at~$σ_n(B,A)$, with the correct homotopy type.

Let us first consider loops on~$_^1(B,A)$, as the corresponding homotopy type is easily calculated:
\begin{lemma}
\label{lemma:loopsOnExt1}
The pre-image~$\mathrm{eval}_1^{-1}(\sigma_1(B,A))$ is the loop space on~$\Extcat_\C^1 (B,A)$, based at~$\sigma_1(B,A)$.
\end{lemma}
\begin{proof}
Propositions~\ref{proposition:Ext1IsKan} and~\ref{proposition:loopsOnKan} give the result.
\end{proof}
More explicitly, the objects of~$Ω(_^1(B,A),σ_1(B,A))$ are the maps
\begin{equation*}
\sigma_1(B,A)\to \sigma_1(B,A)
\end{equation*}
i.e. diagrams
\begin{center}
\begin{tikzcd}
A \arrow[r, tail] \arrow[d,equal]   & A\oplus B \arrow[d,"\phi"]\arrow[r,two heads] & B\arrow[d,equal] \\

A \arrow[r,tail]               & A\oplus B \arrow[r,two heads] & B
\end{tikzcd}
\end{center}
whereas the maps are homotopy coherent cubes of such diagrams and so on for higher simplices.

The loop spaces of higher extension categories are not as easily determined: Although~$_^n(B,A)$ is a quasi-category, it is not a Kan complex in general. And so it is required of us to do some work in order to model its loop space. Technicalities are postponed till section~\ref{section:adaptationEx} and~\ref{section:loopsandmainproof}. But for the purpose of presenting the~$Ω$--spectrum of extension categories we shall give a brief description of the objects in our model for~$Ω(_^n(B,A),σ_1(B,A))$ where~$n≥2$. As we shall prove in \Cref{lemma:LoopsOnExt^n}, they are simple zigzags
\begin{center}
\begin{tikzcd}
\sigma_n(B,A) \arrow[r]  & \mathbb{E} & \sigma_n(B,A) \arrow[l]
\end{tikzcd}
\end{center}
of length one in~$_(B,A)$, starting and terminating in~$σ_n(B,A)$.
All of the data contained in such a zigzag can be depicted
\begin{center}
\begin{tikzcd}[scale cd= 0.8, row sep=1em]
& A \arrow[dd,tail]  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &   & B \arrow[dd] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[dd,equal] &                                        & 0\arrow[ru,tail]\arrow[dd] & & 0\arrow[ru,tail]\arrow[dd] &                           & B \arrow[dd,equal] \\
& E_1 \arrow[rd,two heads]         & &   \cdots\arrow[rd,two heads]   &  & E_n \arrow[rd, two heads] &              \\
A \arrow[ru,tail]            &                                        & E_{1.5}\arrow[ru,tail] && E_{(n-1).5} \arrow[ru,tail] &                           & B            \\
& A \arrow[uu,tail]  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &  & B \arrow[uu] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[uu,equal] &                                        &0\arrow[ru,tail]\arrow[uu] & & 0 \arrow[ru,tail]\arrow[uu] &                           & B \arrow[uu,equal]
\end{tikzcd}
\end{center}
in our exact category~$$.

We now define the \emph{zeroth Retakh functor}
\begin{equation*}
R_0\colon\internalhom_\C (B,A)\to \Omega(\Extcat_\C^1 (B,A),\sigma_1(B,A)),
\end{equation*}
which will serve as the first structure map in the Retakh spectrum.
Every map~$fB→A$ induces a map
\begin{center}
\begin{tikzcd}[row sep=6em,ampersand replacement=\&]
A\oplus B\arrow[r,"{\begin{pmatrix} 1 & f \\ 0 & 1 \end{pmatrix}}"] \& A\oplus B
\end{tikzcd}
\end{center}
which in turn can be seen as a endomap of~$σ_1(B,A)$ in~$_^1(B,A)$ (by \Cref{lemma:loopsOnExt1})
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
A\arrow[r,tail]\arrow[d,equal] \& A\oplus B\arrow[d,"{\begin{pmatrix}[0.75] 1 & f \\ 0 & 1 \end{pmatrix}}"] \arrow[r] \& B\arrow[d,equal] \\
A\arrow[r,tail] \& A\oplus B \arrow[r] \& B
\end{tikzcd}
\end{equation*}
Functoriality follows since the construction hinges upon the universal property of the biproduct.

Recall that the homotopy category of an additive quasi-category is additive. In particular, the path components of~$$--spaces are abelian groups. Thus we can pick an additive inverse~$-1_B$ of~$1_B$, up to homotopy. For~$n≥1$, we now describe the~$n$\emph{th Retakh functor} on objects
\begin{equation*}
R_n\colon \Extcat_\C^n(B,A)\to \Omega(\Extcat_\C^{n+1}(B,A),\sigma_{n+1}(B,C)),
\end{equation*}
generalizing Hermann's constructive interpretation of Retakh's work [12]. It sends an~$n$--extension~$𝔼$
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=1em, ampersand replacement=\&]
\& E_1\arrow[rd,two heads,"p_1"]  \& \& \cdots\arrow[rd,two heads,"p_{n-1}"] \& \& E_n\arrow[rd,two heads,"d_n"] \& \\
A\arrow[ru,tail,"d_0"]  \& \& E_{1.5}\arrow[ru,tail,"i_1"] \& \& E_{(n-1).5} \arrow[ru,tail,"i_{n-1}"] \& \& B
\end{tikzcd}
\end{center}
to the loop
\begin{equation*}
\begin{tikzcd}[scale cd= 0.8, row sep=1em, ampersand replacement=\&,column sep=2.25em]
\& A \arrow[dd,tail,"d_0"]  \arrow[rd,two heads] \& \& \cdots\arrow[rd,two heads] \&   \& B \arrow[dd,tail,"{\begin{pmatrix}1 \\ 0 \end{pmatrix}}"'] \arrow[rd,equal]   \&              \\
A \arrow[ru,equal] \arrow[dd,equal] \&                                        \& 0\arrow[ru,tail]\arrow[dd] \& \& 0\arrow[ru,tail]\arrow[dd] \&                           \& B \arrow[dd,equal] \\
\& E_1 \arrow[rd,two heads,"p_1"]         \& \&   \cdots\arrow[rd,two heads,"d_n"]   \&  \& B\oplus B \arrow[rd, two heads,"{\begin{pmatrix}1 & 1 \end{pmatrix}}" near start] \&              \\
A \arrow[ru,tail,"d_0"]            \&                                        \& E_{1.5}\arrow[ru,tail,"i_1"] \& \& B \arrow[ru,tail,"{\begin{pmatrix}-1 \\ 1 \end{pmatrix}}" near end] \&                           \& B            \\
\& A \arrow[uu,tail,"d_0"]  \arrow[rd,two heads] \& \& \cdots\arrow[rd,two heads] \&  \& B \arrow[uu,tail,"{\begin{pmatrix}0 \\ 1 \end{pmatrix}}"] \arrow[rd,equal]   \&              \\
A \arrow[ru,equal] \arrow[uu,equal] \&                                        \& 0\arrow[ru,tail]\arrow[uu] \& \& 0 \arrow[ru,tail]\arrow[uu] \&                           \& B \arrow[uu,equal]
\end{tikzcd}
\end{equation*}
based at~$σ_n+1(B,A)$. The middle row is the~$(n+1)$--extension~$(𝔼,ρ_B)$, the concatenation of~$𝔼$ with the short exact sequence~$ρ_B$:
\begin{center}
\begin{tikzcd}[ampersand replacement=\&,column sep=2.25em]
B \arrow[r, tail, "{\begin{pmatrix}
\end{pmatrix}}"] \& B\oplus B \arrow[r, two heads, "{\begin{pmatrix}
1 & 1
\end{pmatrix}}"] \& B
\end{tikzcd}
\end{center}

Altogether we have defined for every pair~$(B,A)∈_0×_0$ a spectrum~$_(B,A)$ with structure maps~$R_n_^n(B,A)→Ω(_^n+1(B,A),σ_n+1(B,C)) $. And, moreover, it is in fact an~$Ω$--spectrum, to be proven in Propositons~\ref{proposition:R1} and~\ref{proposition:Rn}.

\begin{theorem}
\label{theorem:IsOmegaSpectrum}
The spectrum~$\Extcat_\C(B,A)$ is an~$\Omega$--spectrum. In other words, all of the Retakh maps~$R_n$ are weak equivalences.
\end{theorem}

\begin{corollary}
The~$(-n)$th homotopy group of~$\Extcat_\C(B,A)$ is~$\pi_0 \Extcat_\C^n (B,A)$.
\end{corollary}

Note that~$_(B,A)$ only adds to the already existing abelian group structure on the mapping space~$_(B,A)$. Indeed, the group structure on~$Ω(^1_(B,A),σ_1(B,A))$ is given by concatenating loops of the form
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
A\arrow[r,tail]\arrow[d,equal] \& A\oplus B\arrow[d,"{\begin{pmatrix}[0.75] 1 & f \\ 0 & 1 \end{pmatrix}}"] \arrow[r] \& B\arrow[d,equal] \\
A\arrow[r,tail] \& A\oplus B \arrow[r] \& B
\end{tikzcd}
\end{equation*}
But a concatenation of such reduces to computing a matrix product
\[
\begin{pmatrix}
1 & f\\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & g\\
0 & 1
\end{pmatrix}
\sim
\begin{pmatrix}
1 & f+g\\
0 & 1
\end{pmatrix}
\]
which incorporates the same algebraic structure as~$_(B,A)$.

If~$$ is the nerve of a small abelian category~$𝒜$, \Cref{theorem:IsOmegaSpectrum} specializes to Retakh's result that the extension categories~$_𝒜^n(B,A)$ form an~$Ω$--spectrum [24]. Moreover,~$π_0^n_𝒜(B,A)$ is easily seen to be the abelian group of Yoneda~$n$--extensions. In particular,~$π_0 ^n(B,A)$ generalizes the standard Ext-groups from abelian categories to exact quasi-categories. On the other hand, if we rather consider~$D^b(𝒜)$, the bounded derived category of~$𝒜$, we see that the spectrum~$_D^b(𝒜)(B,Σ^n A)$ coincides with the Retakh spectrum~$_𝒜(B,A)$. Indeed, it is well-known that Yoneda Ext-groups are isomorphic to Hom-groups in the bounded derived category~\cite[Prop~3.2.2]{Ver96}.

The construction of the Retakh spectrum determines a functor
\begin{equation*}
\Extcat_{\C}\colon\C\op\times\C\to \Sp(\Cat_{\infty})
\end{equation*}
where~$(_∞)$ is an appropriate category of~$Ω$--spectra. Specifically, the quasi-category $(_∞)$ is that of spectrum objects \cite[Section~8]{Lur06} in~$_∞$, the quasi-category of small quasi-categories. In this framework, a map~$X^∙→Y^∙$ of~$Ω$--spectra simply becomes a family of maps~$X^n→Y^n$ between the corresponding components.

\begin{proposition}\label{prop:nattransExt}
An exact functor
\begin{equation*}
F\colon \mathscr{C} \to \mathscr{D}
\end{equation*}
induces a canonical natural transformation
\begin{equation*}
\eta_F\colon \Extcat_{\C}(-,-)\Longrightarrow \Extcat_{\mathscr{D}}(F-,F-)
\end{equation*}
of functors~$\C\op\times\C\to\Sp(\Cat_{\infty})$.
\end{proposition}
\begin{proof}
Let~$(B,A)$ be an object of~$\C\op\times\C$. Then~$F$ induces functors
\begin{equation*}
\Extcat^n_{\C}(B,A) \to \Extcat^n_{\D}(FA,FB)
\end{equation*}
sending an exact sequence of length~$n$ in~$\C$ to its image in~$\D$. Assembling these functors yields a map of spectra
\begin{equation*}\label{eq:nattransExt}
\Extcat_{\C}(B,A) \to \Extcat_{\D}(FB,FA).
\end{equation*}
More generally, given an~$n$--simplex
\begin{equation*}
(B_1,A_1) \to (B_2,A_2) \to \cdots \to (B_n,A_n)
\end{equation*}
in~$\C\op\times\C$, one checks that the~$[1]\times [n]$--shaped diagram
\begin{equation}\label{eq:nattransExth}
\begin{tikzcd}[scale cd= 0.8]
\Extcat_{\C}(B_1,A_1) \arrow[r]\arrow[d] &\Extcat_{\D}(FB_1,FA_1)\arrow[d] \\
\Extcat_{\C}(B_2,A_2) \arrow[r]\arrow[d] &\Extcat_{\D}(FB_2,FA_2)\arrow[d] \\
\vdots \arrow[d] &\vdots\arrow[d] \\
\Extcat_{\C}(B_n,A_n) \arrow[r]&\Extcat_{\D}(FB_n,FA_n)
\end{tikzcd}
\end{equation}
is homotopy coherent.

A natural transformation of quasi-categories can be given by a simplicial map
\begin{equation*}
\eta_F \colon \C\op\times \C \to \hom([1],\Sp(\Cat_{\infty})).
\end{equation*}
We choose this map as that sending an~$n$--simplex to the diagram \eqref{eq:nattransExth}. It can be checked that the boundary and degeneracy maps behave as expected, yielding the desired map of simplicial sets.
\end{proof}

\section{An adaption of the subdivision and Ex-functors}
\label{section:adaptationEx}
This section is all about introducing a version of Kan's~$$-functor which is tailored for the extension categories~$^n_(B,A)$. We call it the \textit{factorized Ex-functor} and denote it by~$$. Only the basic intuition behind~$$ is needed to follow Section~\ref{section:awesomestuff}, and in particular following the proofs of how~$^n_(B,A)$ is a Kan complex and that the~$n$th Retakh functor is a weak equivalence.

We spend the next few paragraphs defining the~\emph{factorized subdivision functor}
\begin{equation*}
\fsd\colon \Delta\to {\rm Cat}.
\end{equation*}
The objects in~$[m]$ will be generated recursively.
Let~$[0]=[0]$ be the discrete category with one object and one identity morphism.
Suppose that~$[m-1]$ has been defined, where~$m≥1$, and that its objects are certain~$m-1$ tuples
\begin{equation*}
\end{equation*}
of pairwise disjoint subsets of~${0,1,…, m-1}.$
We may also define~$X$, for any totally ordered set~$X$ with~$m$ elements, simply by setting up a monotone bijection~$X→[m-1]$.
The objects of ~$[m]$ will then be certain~$m$--tuples
\begin{equation*}
(M_1, \dots , M_{m-1}, M_m)
\end{equation*}
of pairwise disjoint subsets of~$$\{0,1,\dots, m\}.$$
We generate them from the objects of~$[m-1]$ according to the procedure given by \ref{ob-fsd1}--\ref{ob-fsd3} below.
\begin{enumerate}[label=(O\arabic*)]
\item\label{ob-fsd1} For each~$0\leq j\leq m$ and each object~$(M_1, \dots , M_{m-1})$ in~$\fsd[0,\dots, \widehat{j},\dots, m]$, we include~$(M_1, \dots , M_{m-1},\emptyset)$. Although duplicates may occur, no element is counted twice.
\item\label{ob-fsd2} For each object~$(M_1, \dots , M_{m-1},\emptyset)$ introduced in \ref{ob-fsd1}, with the exception of those of the form~$(j,\emptyset,\dots,\emptyset)$, we add~$(M_1, \dots , M_{m-1},M_m)$, where~$M_m$ is the set~$\{0,\dots,m\}\setminus\left(\bigcup\limits_{i=1}^{m-1} M_i\right)$ of remaining integers~$\leq m$.
\item\label{ob-fsd3} We add the object~$(01\cdots m,\emptyset,\dots,\emptyset)$.
\end{enumerate}
The category~$[m]$ will then be the preorder (i.e. category with at most one morphism connecting any pair of objects) generated by the following morphisms.
\begin{enumerate}[label=(M\arabic*)]
\item\label{mor-fsd1} For each~$0\leq j\leq m$ and each morphism
\begin{equation*}
f\colon (M_1, \dots , M_{m-1})\to (N_1, \dots , N_{m-1})
\end{equation*} in~$\fsd[0,\dots, \widehat{j},\dots, m]$, we add a morphism
\begin{equation*}
f^j\colon (M_1, \dots , M_{m-1},\emptyset) \to (N_1, \dots , N_{m-1},\emptyset).
\end{equation*}
\item\label{mor-fsd2} For each object~$(M_1, \dots , M_{m-1},M_m)$ introduced in \ref{ob-fsd2}, we add morphisms
\begin{equation*}
h_{(M_1, \dots , M_{m-1})}\colon(M_1, \dots , M_{m-1},M_m)\to (M_1, \dots , M_{m-1},\emptyset).
\end{equation*}
\item\label{mor-fsd3}
$f^j\colon (M_1, \dots , M_{m-1},\emptyset) \to (N_1, \dots , N_{m-1},\emptyset)$
be a morphism introduced in \ref{mor-fsd1}. If~$(M_1, \dots , M_{m-1},\emptyset)$ is not of the form~$(i,\emptyset,\dots,\emptyset)$, we include a morphism
\begin{equation*}
\widehat{f^j}\colon (M_1, \dots , M_{m-1},M_m) \to (N_1, \dots , N_{m-1},N_m).
\end{equation*}
If~$(M_1, \dots , M_{m-1},\emptyset) = (i,\emptyset,\dots,\emptyset)$, we add
\begin{equation*}
\widehat{f^j}\colon (M_1, \dots , M_{m-1},\emptyset) = (i,\emptyset,\dots,\emptyset) \to (N_1, \dots , N_{m-1},N_m).
\end{equation*}
Since~$\fsd[m]$ is required to be a preorder, we add relations so that all resulting subdiagrams commute.
\item\label{mor-fsd4} For each object~$(M_1,M_2,\cdots, M_m)$ introduced in \ref{ob-fsd2}, we have a morphism
\begin{equation*}
i_{(M_1,M_2,\cdots, M_m)} \colon (M_1,M_2,\cdots, M_m) \to (01\cdots m,\emptyset,\dots,\emptyset)
\end{equation*}
We introduce the notation~$i_j$ for the composites~$$(j,\emptyset,\dots,\emptyset)\to (01\cdots m,\emptyset,\dots,\emptyset).$$
\end{enumerate}

We note that all morphisms in~$[m]$ fall into one of the cases \ref{mor-fsd1}--\ref{mor-fsd4}.

From this point on, if an object~$(M_1…,M_m)$ has~$M_i=∅$ for all~$k<i≤m$, then we exclude the empty tail and simply write~$(M_1,…,M_k)$.

The recursive procedure above produces the following category for~$[1]$:
\begin{equation*}
\begin{tikzcd}
0 \arrow[r,"i_0"] & 01 & \arrow[l,"i_1"'] 1,
\end{tikzcd}
\end{equation*}
whereas the category~$[2]$ is graphically displayed as follows:
\begin{center}
\begin{tikzcd}[scale cd= 0.8]
&  &       &                         & 1 \arrow[lldddd,"i_1^2"'] \arrow[lddddd,"\widehat{i_1^2}"'] \arrow[rddddd,"\widehat{i_1^0}"] \arrow[rrdddd,"i_1^0"] &               &       &  &                                                          \\
&  &       &                         &                                                               &               &       &  &                                                          \\
&  &       &                         &                                                               &               &       &  &                                                          \\
&  &       &                         &                                                               &               &       &  &                                                          \\
&  & 01 &                         &                                                               &               & 12 &  &                                                          \\
&  &       & (01,2) \arrow[lu,"h_{01}"'] \arrow[r,"i_{(01,2)}"] & 012                                                 & (12,0) \arrow[l,"i_{(12,0)}"']\arrow[ru,"h_{12}"] &       &  &                                                          \\
&  &       &                         & (02,1) \arrow[d,"h_{02}"] \arrow[u,"i_{(02,1)}"]                                        &               &       &  &                                                          \\
0 \arrow[rruuu,"i_0^2"] \arrow[rrrr,"i_0^1"] \arrow[rrruu,"\widehat{i_{0}^2}"] \arrow[rrrru,"\widehat{i_{0}^1}"] &  &       &                         & 02                                                         &               &       &  & 2 \arrow[llll,"i_2^1"'] \arrow[llllu,"\widehat{i_2^1}"'] \arrow[llluu,"\widehat{i_2^0}"'] \arrow[lluuu,"{i_2^0}"']
\end{tikzcd}
\end{center}
For this particular example, the usual barycentric subdivision~$[2]$ is clearly seen in the interior, although three of its~$0$--simplices have been given unusual indices. From the geometrical point of view we introduce the following terminology:
\begin{itemize}
\item Objects of the form~$j$ are the \emph{singletons}.
\item The full subcategory generated by objects~\ref{ob-fsd1} is the~\emph{boundary} of~$\fsd[m]$.
\item The full subcategory generated by singletons and objects~\ref{ob-fsd2} is the~\emph{thickening} of the boundary in~$\fsd[m]$.
\item The object~$01\cdots m$ is the~\emph{central point}.
\end{itemize}
With this in mind, the morphisms~\ref{mor-fsd2} are the irreducible morphisms from the thickening to the boundary, and morphisms~\ref{mor-fsd4} are the irreducible morphisms from the thickening to the central point.

To complete the definition of the functor~$Δ→Cat$, we need to define its action on morphisms. Recall how the simplex category~$Δ$ is generated by the coface maps~$δ^m_i[m-1]→[m]$ and the codegeneracy maps~$σ^m_i[m+1]→[m]$, subject to the cosimplicial relations~\cite[p.~4]{GJ09}. We proceed to define~$$ on the coface and codegeneracy maps.

As a map of objects, we set~$δ^m_i$ to be
\begin{equation}\label{eq:fsddeltaob}
\begin{tikzcd}[row sep=1em]
{{\rm ob}\fsd[m-1]}\arrow[r,"\fsd\delta^m_i"]  & {{\rm ob}\fsd[m],} \\
(M_1,\dots, M_{m-1})\arrow[r,mapsto] & (\delta^m_i M_1,\dots, \delta^m_i M_{m-1},\emptyset).
\end{tikzcd}
\end{equation}
A morphism~$f$ in~$[m-1]$ will be sent to~$f^i$, introduced in \ref{mor-fsd1}.
By considering the way in which \ref{mor-fsd1} generates morphisms in~$[m]$ from those in~$[m-1]$, it is easy to see that~$δ^m_i$ is a fully faithful functor for each~$i≤m$.

The action on the degeneracy maps will be chosen as follows on the level of objects:
\begin{equation}\label{eq:fsdsigmaob}
\begin{tikzcd}[row sep=1em]
{{\rm ob}\fsd[m+1]}\arrow[r,"\fsd\sigma^m_j"]  & {{\rm ob}\fsd[m],} \\
(M_1,\dots,M_m,M_{m+1})\arrow[r,mapsto] & (\sigma^m_j M_1,\dots, \sigma^m_j M_{m}).
\end{tikzcd}
\end{equation}
Let us proceed with the tedious task of checking that these are indeed functors:
\begin{lemma}
The formulas~\eqref{eq:fsddeltaob} and~\eqref{eq:fsdsigmaob} define functors~$\fsd\delta^m_i$ and~$\fsd\sigma^m_j$, respectively.
\end{lemma}
\begin{proof}
For case of~$\fsd\delta^m_i$, we refer to the above discussion. We can claim that~$\fsd\sigma^m_j$ are functors by showing the following: if there exists a morphism~${\bf M}\to {\bf N}$ in~$\fsd[m+1]$, then there exists a morphism~$\fsd\sigma^m_j({\bf M})\to \fsd\sigma^m_j({\bf N})$ in~$\fsd[m]$. After all, we are mapping a preorder into a preorder, so there is at most one morphism connecting any pair of objects.

We proceed by induction on~$m$. The base case~$m=0$ concerns functors into the category~$[0]$, where the claim trivially holds. Let~$m>0$ and suppose that the claim holds for~$m-1$. Given a morphism~$g\colon{\bf M}\to {\bf N}$ in~$\fsd[m+1]$, we check that there exists a morphism~$\fsd\sigma^m_j g\colon \fsd\sigma^m_j({\bf M})\to \fsd\sigma^m_j({\bf N})$ in~$\fsd[m]$. This is achieved by considering the cases \ref{mor-fsd1}--\ref{mor-fsd4}.

If~$g$ is in \ref{mor-fsd1}, it is of the form~$f^i$ for some~$f$ in~$\fsd[0,\dots,\hat{i},\dots,m+1]$. Since the functors of the form~$\fsd\sigma^{m-1}_k$ have been assumed to exist, they can be exploited to prove the existence of a morphism~$\fsd\sigma^m_j g\colon\fsd\sigma^m_j({\bf M})\to \fsd\sigma^m_j({\bf N})$. Indeed, consider the cosimplicial relation
\begin{equation*}
\sigma^m_j\circ \delta^{m+1}_i =
\begin{cases}
\delta^{m}_i\circ \sigma^{m-1}_{j-1} & \text{if } 0\leq i<j\leq m, \\
{\rm id} & \text{if } i=j,j+1, \\
\delta^m_{i-1}\circ \sigma^{m-1}_{j} & \text{if } 0\leq j+1<i\leq m.
\end{cases}
\end{equation*}
Thus, if~${\bf O}$ is any object in~$\fsd[m]$, we have that
\begin{equation*}
\fsd \sigma^m_j(\delta^{m+1}_i{\bf O}) \coloneqq {\begin{cases}
(\fsd\delta^{m}_i\circ \fsd\sigma^{m-1}_{j-1})({\bf O}) & \text{if } 0\leq i<j\leq m \\
{\bf O} & \text{if } i=j,j+1 \\
(\fsd\delta^m_{i-1}\circ \fsd\sigma^{m-1}_{j})({\bf O}) & \text{if } 0\leq j+1<i\leq m. \\
\end{cases}}
\end{equation*}
Since the induction hypothesis guarantees that both~$\fsd\sigma^{m-1}_{j-1}$ and~$\fsd\sigma^{m-1}_{j}$ induce morphisms from~$g$, it follows that~$\fsd\sigma^m_j$ does. This is because~$g$ is contained in the image of~$\delta^{m+1}_i$.

Should~$g$ be in~\ref{mor-fsd2}, the domain and codomain are sent to the same object. The identity then works as the image of~$g$.

The case of \ref{mor-fsd3} can be reduced to~\ref{mor-fsd1}. Since the last component is removed in \eqref{eq:fsdsigmaob}, the domain and codomain of the morphism~$\widehat{f^j}$ in \ref{mor-fsd3} are sent to the same as the domain and codomain, respectively, of~$f^j$.

The last case \ref{mor-fsd4} can be addressed by the explicit formula~$\fsd \sigma^m_j(i_{\bf M})=i_{\sigma^m_j\bf M}$, whenever~$\sigma^m_j\bf M$ does not agree with the central point in~$\fsd[m]$. Otherwise we define~$\fsd \sigma^m_j$ as the identity at~$0\cdots m$.
\end{proof}

Since the cosimplicial identities hold in~$Δ$, it immediately follows that the functors of the form~$δ^m_j$ and~$σ^m_k$ satisfy the same relations when considered as maps of objects. Since the categories~$[m]$ are preorders, these relations hold when applied to morphisms as well. We are therefore in a position to conclude:

\begin{lemma}
We have defined a functor~$\fsd\colon \Delta\to \Cat$. In other words, it is a cosimplicial category.
\end{lemma}

We will compare the factorizing subdivision~$[m]$ with the standard simplex~$[m]$ via the subdivision~$[m]$ as introduced by Kan in~[14]. To do so it will be beneficial to define~$[m]$ in a recursive way that resembles the description of~$[m]$. This is achieved by removing the thickening of faces that occur in~\ref{ob-fsd2},~\ref{mor-fsd2} and~\ref{mor-fsd3}. So we simply omit these constraints and alter~\ref{mor-fsd4} to rather require maps from the boundary to the central point~$01⋯m$. With such a description there is an evident map~$r[n] →[n]$ obtained by collapsing the information in~\ref{ob-fsd2},~\ref{mor-fsd2} and~\ref{mor-fsd3} onto~\ref{ob-fsd1} and~\ref{mor-fsd1}. The result is in fact a map~$ℓ→$ of cosimplicial categories.

\begin{lemma}
\label{lemma:fsdcontractible}
For every~$m\geq 0$ the factorizing subdivision~$\fsd[m]$ is contractible. In particular, the map~$\ell_m\colon \fsd[m]\to \sd[m]$ is a weak equivalence.
\end{lemma}

\begin{proof}
Consider the full subcategory~$\mathrm{c}[m]$ of~$\fsd[m]$ generated by singletons, objects in the thickening and the central point. This category is contractible since the object~$01\cdots m$ is terminal. So it suffices to prove that~$\fsd[m]$ is weakly equivalent to~$\mathrm{c}[m]$ in the Quillen model structure on simplicial sets. It is well-known that taking the opposite of a category preserves homotopy types. Hence we rather consider the opposites of~$\mathrm{c}[m]$ and~$\fsd[m]$. This allows us to deform along opposites of~\ref{mor-fsd2} from the boundary to the thickening.

We define a retract of the inclusion~$i\colon \mathrm{c}[m]^{\mathrm{op}}\hookrightarrow \fsd[m]^{\mathrm{op}}$. Recall that the boundary of~$\fsd[m]$, the union of its faces, is the full subcategory generated by the objects~\ref{ob-fsd1}. When we consider the opposite of~$\fsd[m]$ every object~\ref{ob-fsd1}, excluding the singletons, admit a unique irreducible morphism~$h_{M_1,\dots,M_{m-1}}^{\mathrm{op}}$ to the thickening, defined by~\ref{mor-fsd2}. To define a retract~$r\colon \fsd[m]^{\mathrm{op}} \to \mathrm{c}[m]^{\mathrm{op}}$ we simply send every~$h_{M_1,\dots,M_{m-1}}^{\mathrm{op}}$ to the identity at~$(M_1,\dots,M_{m-1},M_m)$. The morphisms in the boundary, which are all of the form~$(M_1,\dots,M_{m-1},\emptyset)\to (N_1,\dots,N_{m-1},\emptyset)$, are then forced to be sent to their respective morphisms in the thickening~\ref{mor-fsd3}. Geometrically, the functor~$r$ collapses the boundary of~$\fsd[m]^{\mathrm{op}}$ onto~$c[m]\op$. Since~$r$ fixes singletons as well as objects in~\ref{ob-fsd2} and~\ref{ob-fsd3}, it is clear that~$r\circ i$ is the identity on~$\mathrm{c}[m]^{\mathrm{op}}$.

There is a rather evident natural transformation~$\eta$ from the identity on~$\fsd[m]^{\mathrm{op}}$ to~$i\circ r$ defined by utilizing the irreducible morphisms~$h_{M_1,\dots,M_{m-1}}^{\mathrm{op}}$ from the boundary to the thickening: an object~$\mathbf M =(M_1,\dots,M_{m-1},\emptyset)$ in~\ref{ob-fsd1}, except the singleton, associates uniquely to the morphism
\[
\eta_{\mathbf M}=(M_1,\dots,M_{m-1},\emptyset)\to (M_1,\cdots,M_{m-1},12\dots m\setminus \cup M_i).
\]
Such a natural transformation is equivalent to a homotopy~$\fsd[m]^{\mathrm{op}}\times [1]\to \fsd[m]^{\mathrm{op}}$ from the identity to~$i \circ r$.
\end{proof}

Any simplicial set~$X$ is canonically presentable as a colimit of~$m$--simplices \cite[p.~7]{GJ09}. More precisely, the simplex category~$Δ$, defines, for a fixed simplicial set~$X$, a slice category~$Δ↓X$ and~$X=_[m]→X [m] $. Hence we can lift the factorizing subdivision from the simplex category to all simplicial sets by declaring~$X= _[m]→X [m] $.

\begin{lemma}
\label{lemma:preservefibration}
Every horn inclusion~$\fsd \Lambda^k[m]\hookrightarrow \fsd[m]$ is an acyclic cofibration.
\end{lemma}
\begin{proof}
The inclusion is clearly a cofibration, so we need only verify that it is in fact a weak equivalence. The~$k$--horn~$\Lambda^k [m]$ is the union of all faces in~$[m]$ except the~$k$th one. It can for example be presented as the coequalizer of
\[
\coprod_{0\leq i<j\leq m} [m-2] \rightrightarrows \coprod_{i \neq k} [m-1]
\]
where both of the arrows are coproducts of face inclusions, whence they are cofibrations. Since both~$\sd$ and~$\fsd$ preserve colimits, we can apply them to the coequalizer and obtain~$\sd\Lambda^k[m]$ and~$\fsd\Lambda^k[m]$ as coequalizers. By Lemma~\ref{lemma:fsdcontractible}, the natural map~$\ell_m\colon \fsd[m]\to \sd[m]$ then provides a level-wise weak equivalence of cofibrant coequlizer diagrams. Hence we conclude that the induced map~$\fsd\Lambda^k[m]\to \sd \Lambda^k[m]$ is a weak equivalence, and that~$\fsd\Lambda^k[m]$ is contractible. This means that the inclusion~$\fsd \Lambda^k[m]\hookrightarrow \fsd[m]$ is a map of contractible spaces; a weak equivalence.
\end{proof}

Analogous to how Kan defines the well-known functor~$$ in terms of the subdivision~$$, we define a variant which is tailored for extension categories.

\begin{definition}
\label{definition:fExisRightAdjoint}
The factorizing~$\Ex$--functor is defined by~$\fEx X_m=\Hom_{\sSet}(\fsd[m],X)$ with face and degeneracy maps induced from the cosimplicial structure of~$\fsd$.
\end{definition}
The next observation is immediate from the definition.
\begin{proposition}
\label{proposition:isadjoint}
The factorizing~$\Ex$--functor~$\fEx$ is right adjoint to the factorizing subdivision~$\fsd$.
\end{proposition}

We turn our attention to proving that the factorizing~$$--functor preserves homotopy types. This is done in a very similar fashion as to how you would prove the analogous result for Kan's~$$--functor. First, we derive a map~$ηX→X$ from the natural map~$ℓ_m[m]→[m]$, as described in the discussion preceding Lemma~\ref{lemma:fsdcontractible}, and the~\emph{last vertex map}~$v_m[m] →[m]$ \cite[p.~183]{GJ09}. On the level of simplices, this is achieved by pre-composing an~$m$--simplex~$[m]→X$ with~$ℓ_m$ and~$v_m$.  Taking Lemmas~\ref{lemma:fsdcontractible} and~\ref{lemma:preservefibration} into account we more or less follow the acyclic models argument outlined by Goerss--Jardine~\cite[Chapter~3.4]{GJ09} directly.
\begin{theorem}
\label{theorem:fexpreserveshomotopytypes}
The map~$\eta\colon X\to \fEx X$ is a weak equivalence for any simplicial set~$X$.
\end{theorem}

We end this section with three important diagram lemmas. The two first are vital when proving the last.

\begin{lemma}\label{lem:lemma1}
Let~$J$ be a finite partially ordered set (considered as a category), and let~$D\colon J \to \Extcat^n_{\C}(B,A)$ be a be homotopy coherent diagram. Then~$D$ extends to a diagram
\begin{equation*}
H\colon J\times [1] \to \Extcat^n_{\C}(B,A)
\end{equation*}
with the following properties:
\begin{enumerate}
\item\label{item:Lemma11}$H_1\coloneqq H|_{J\times 1}$ is~$D$,
\item\label{item:Lemma12} all arrows in~$H_0\coloneqq H|_{J\times 0}$ are term-wise cofibrations and
\item\label{item:Lemma13} if~$i$ is an object in~$J$ which does not occur as the target of any morphism, then~$H$ is the identity when restricted to~$i\times [1]$.
\end{enumerate}
\end{lemma}
\begin{proof}
Pick an arbitrary morphism~$f\colon i\to j$ in~$J$. We shall first describe a procedure to define~$H\colon J \times [1] \to \Extcat_{\C}^n (B,A)$ satisfying that~$H(f,0)$ is a term-wise cofibration. Moreover, if~$D(g)\colon D(k) \to D(j)$ is a term-wise cofibration, then also~$H(g,0)$ should be a term-wise cofibration. The same goes for morphisms~$D(g)\colon D(k) \to D(i)$. Applying the factorization in Proposition~\ref{prop:VW3.1} to~$D(f)\colon D(i)\to D(j)$ provides a homotopy coherent diagram
\begin{center}
\begin{tikzcd}
D(i) \arrow[r, "D(f)' ", tail] \arrow[d, equal] & \mathrm{cyl} D(f) \arrow[d, "p"]\\
D(i) \arrow[r, "D(f)"] & D(j)
\end{tikzcd}
\end{center}
which is equivalent to a functor~$G\colon [1]\times [1]\to \Extcat^n_{\C}(B,A)$ satisfying properties \eqref{item:Lemma11}--\eqref{item:Lemma13} above. We extend~$G$ to a functor~$H\colon J\times [1]\to \Extcat^n_{\C}(B,A)$. An object~$k\neq i,j$ in~$J$ is mapped to~$H(k,0)=H(k,1)=D(k)$. On the level of morphisms~$g\colon k\to  \ell$ we can thus automatically define~$H(g,0)=H(g,1)=D(g)$ whenever~$k$ and~$\ell$ both differ from~$j$. If~$g$ starts at~$k=j$, then we choose a pre-composition~$H(g,0)=D(g)\circ p$. And lastly, if~$g$ terminates at~$\ell=j$, then we post-compose with the right inverse~$m$ of~$p$ resulting in~$H(g,0)=m\circ D(g)$. The procedure is functorial: the only non-trivial case to check is~$g\colon k \to j$ and~$h\colon j\to  \ell$, and here we have that~$H(g,0)=D(g) \circ p$ and~$H(h,0)=m \circ D(h)$, whence~$H(h,0)\circ H(g,0) \sim D(g)\circ D(h)$. Observe that if a morphism of the form~$D(g)\colon D(k) \to D(j)$ is a term-wise cofibration, then also~$H(g,0)$ is a term-wise cofibration, since the class of cofibrations is closed under composition.

To finish the proof we shall iterate the above construction to turn every morphism into a term-wise cofibration. Since~$J$ is a finite partially ordered set, there is a finite set of irreducible morphisms~$f_I$ which uniquely factorizes all morphisms in~$J$. Since~$J$ is a finite partially ordered set, any object~$k$ admits a~\emph{height}~$h$. It is defined as the length of the longest chain of irreducible morphisms terminating at~$k$. We define the height of an irreducible morphism~$f_I$ as the height of its target.

Iterate the above process on the set~$f_I$, of irreducible morphisms, by first picking those whose height is equal to~$1$, then proceeding to~$2$ and so on. This is to guarantee that whenever we apply the procedure to an irreducible morphism~$f\colon i\to j$, no morphism out of~$j$ has already been replaced by a term-wise cofibration. Moreover, irreducible morphisms with smaller height will still be replaced by term-wise cofibrations. Indeed, this follows by the last sentence in the first paragraph. In the end, every irreducible morphism is replaced by a term-wise cofibration. If we denote by~$[1]\amalg_{[0]}[1]$ the concatenation of~$[1]$ by~$[1]$, head to tail.
What we have produced is a simplicial map
\[
\hat{H}\colon J\times \left([1]\amalg_{[0]}\cdots \amalg_{[0]} [1]\right) \to \Extcat^n_{\C}(B,A)
\]
where finitely many copies of~$[1]$ are concatenated. But~$\Extcat^n_{\C}(B,A)$ is a quasi-category, so choosing finitely many compositions determine a functor~$H\colon J\times [1] \to \Extcat^n_{\C}(B,A)$. The functor is constructed to satisfy the three asserted properties \eqref{item:Lemma11}--\eqref{item:Lemma13}.
\end{proof}

\begin{lemma}\label{lem:lemma2}
Let~$J$ be a finite connected partially ordered set (considered as a category) with at least three objects. Let~$D\colon J \to \Extcat^n_{\C}(B,A)$ be a homotopy coherent diagram. Assume that all morphisms of the form~$D(f)$, where~$f$ is a morphism in~$J$, are term-wise cofibrations. Then the diagram~$D$ has a cocone.
\end{lemma}
\begin{proof}
Recall that~$D$ has a cocone if it can be extended to a homotopy coherent diagram~$D^{\vartriangleright}\colon J^{\vartriangleright}\to \Extcat^n_{\C}(B,A)$, where the category~$J^{\vartriangleright}$ is obtained from~$J$ by adding a terminal object~$*$. We refer to~$D^{\vartriangleright}(*)$ as the cocone of~$D$ when its incident morphisms are obvious from context.

We prove the claim by induction on the number~$\ell$ of minimal objects in~$J$. The base case~$\ell=1$ amounts to the existence of an initial object~$i$ in~$J$. We prove the base case by induction on the number of objects in~$J$, denoted by~$m$.

The base case~$m=3$ only pertains to the diagrams depicted as the solid part of the diagrams below.
\begin{equation*}
\begin{tikzcd}[row sep=1em, column sep=1.25em]
& D(i) \arrow[ld] \arrow[rd] &                      &  &                                                 & D(i) \arrow[ld] \arrow[rd] &                                         &     \\
D(a) \arrow[rd, dashed] &                         & D(b) \arrow[ld, dashed] &  & D(a) \arrow[rr, "D(f)"] \arrow[rd, "\alpha"', dashed] &                         & D(b) \arrow[ld, dashed] \arrow[rd, dotted] &     \\
& \mathbb{P}                       &                      &  &                                                 & \mathbb{P} \arrow[rr, dotted]    &                                         & \mathbb{Q}
\end{tikzcd}
\end{equation*}
In both cases, we construct an object~$\mathbb{P}$ by taking the pushout of the arrows out of~$i$, adding new dashed arrows above. The category on the left then obtains a cocone. One more pushout is needed to identify a cocone in the right-hand diagram, which is that of~$D(f)$ along~$\alpha$.

Suppose that the claim has been proven for~$m-1$. If~$J$ has~$m$ objects, let~$a$ be a maximal one. The induction hypothesis then gives a cocone~$\mathbb{P}$ of the restricted homotopy coherent diagram
\begin{equation*}
D\vert_{J\setminus a}\colon J\setminus a\to \Extcat^n_{\C}(B,A),
\end{equation*}
obtained by removing~$a$ and all its incident arrows from~$J$. Let $\mathbb{P}$ be a cocone of $D\vert_{J\setminus a}$. A cocone of~$D$ can be found by constructing an object~$\mathbb{Q}$ as well as morphisms~$\mathbb{P}\to \mathbb{Q}$ and~$D(a)\to \mathbb{Q}$, such that for all immediate predecessors~$a_1$ of~$a$, the square
\begin{equation*}
\begin{tikzcd}
D(a_1) \arrow[r] \arrow[d] & D(a) \arrow[d] \\
\mathbb{P} \arrow[r]             & \mathbb{Q}
\end{tikzcd}
\end{equation*}
commutes. Suppose that~$a$ has~$k$ immediate predecessors~$a_1$,~$a_2$,~$\dots$,~$a_k$. The solid part of the following diagram depicts this situation.
\begin{center}
\begin{tikzcd}[scale cd=0.8]
&         &          & D(a)\arrow[r, dashed]   & \mathbb{Q}_1\arrow[r, dashed]  & \mathbb{Q}_2\arrow[r, dashed]  & \cdots\arrow[r, dashed]  & \mathbb{Q}_k \\
D(a_1)\arrow[urrr, "D(f_1)"]\arrow[drrrr, "{\alpha_1}", swap] &  D(a_2)\arrow[urr]\arrow[drrr] &  \cdots  & D(a_k)\arrow[u,"D(f_k)"]\arrow[dr,"{\alpha_k}"] &              &              &        &              \\
&         &          &        & \mathbb{P} \arrow[uu, dashed]\arrow[uur, dashed] \arrow[uurrr, dashed]   &              &        &
\end{tikzcd}
\end{center}
The pushout of~$\alpha_1$ along~$D(f_1)$ gives an object~$\mathbb{Q}_1$ and the dashed morphisms into it. Next, we take the pushout of~$\alpha_2$ along the composite~$D(a_2)\to D(a) \to \mathbb{Q}_1$, producing~$\mathbb{Q}_2$. This procedure terminates after~$k$ steps, producing a cocone~$\mathbb{Q}=\mathbb{Q}_k$ of~$D$.

Having just proved the claim in the case where~$J$ has a unique minimal object, we now deal with the inductive step. Suppose that the claim holds when~$J$ has fewer than~$\ell$ minimal objects, for some~$\ell\geq 2$. If~$J$ contains exactly~$\ell$ minimal objects~$a_1,\dots,a_{\ell}$, we let~$J_{1}$ be the subposet of~$J$ containing the objects~$x$ such that~$x\geq a_1$. The base case (shown above) provides a cocone~$\mathbb{P}_1$ of the subdiagram given by~$J_{1}$.

Now, let~$J'$ be the full subcategory of~$J$ spanned by the objects of~$\bigcup_{i\geq 2}J_{\geq a_i}$.
We extend~$J'$ to~$J''$ by adding an object~$p_1$ and morphisms~$a\to p_1$ for all~$a$ in~$ J_1\cap J'$, such that all resulting triangles commute. The construction of~$\mathbb{P}_1$ in the last paragraph paves the way for a homotopy coherent diagram~$D''\colon J''\to \Extcat^n_{\C}(B,A)$ such that~$D''(p_1)=\mathbb{P}_1$. Since~$J''$ is a partially ordered set with~$\ell-1$ minimal objects, it follows from the induction hypothesis that this diagram has a cocone~$\mathbb{R}$. We claim that~$\mathbb{R}$ serves as a cocone for~$J$. We have suitable morphisms from objects in~$D(J')$ to~$\mathbb{R}$ by definition, and we may choose composites via~$\mathbb{P}_1$ to get suitable morphisms out of objects in~$D(J_1)$.
\end{proof}

In a Kan complex, all horns can be filled. Two subcategories of~$[m]$ are therefore of particular interest.
Consider this statement in \ref{ob-fsd1}, and add the extra assumption that~$j>0$. In full:
\begin{enumerate}[label=($\Lambda$\arabic*)]
\item\label{ob-fsd01} For each~$0< j\leq m$ and each object~$(M_1, \dots , M_{m-1})$ in~$\fsd[0,\dots, \widehat{j},\dots, m]$, we include~$(M_1, \dots , M_{m-1},\emptyset)$. Although duplicates may arise, no element is counted twice.
\end{enumerate}
The objects given by this refining of \ref{ob-fsd1} describe~$Λ^0[m]$, the factorizing subdivision applied to the~$0$--horn. We depict~$Λ^0[m]$ as the solid part of:

\begin{equation*}
\begin{tikzcd}[scale cd=0.8]
&  &    &                                                                          & 1 \arrow[lldddd, "i_1^2"'] \arrow[lddddd, "\widehat{i_1^2}"', dashed] &  &  &  &                                                                    \\
&  &    &                                                                          &                                                                       &  &  &  &                                                                    \\
&  &    &                                                                          &                                                                       &  &  &  &                                                                    \\
&  &    &                                                                          &                                                                       &  &  &  &                                                                    \\
&  & 01 &                                                                          &                                                                       &  &  &  &                                                                    \\
&  &    & {(01,2)} \arrow[lu, "h_{01}"', dashed] \arrow[r, "{i_{(01,2)}}", dashed] & 012                                                                   &  &  &  &                                                                    \\
&  &    &                                                                          & {(02,1)} \arrow[d, "h_{02}",dashed] \arrow[u, "{i_{(02,1)}}", dashed]        &  &  &  &                                                                    \\
0 \arrow[rruuu, "i_0^2"] \arrow[rrrr, "i_0^1"] \arrow[rrruu, "\widehat{i_{0}^2}", dashed] \arrow[rrrru, "\widehat{i_{0}^1}", dashed] &  &    &                                                                          & 02                                                                    &  &  &  & 2 \arrow[llll, "i_2^1"'] \arrow[llllu, "\widehat{i_2^1}"', dashed]
\end{tikzcd}
\end{equation*}
The procedures put forward in \ref{ob-fsd2} and \ref{ob-fsd3} can then be applied to the objects in~$Λ^0[m]$ (i.e. those described in \ref{ob-fsd01}), to generate a larger subcategory~$_0^+[m]$ of~$[m]$. This subcategory is a "fat"~$0$--horn because it also contains the thickening of~$j$--faces,~$j>0$, as well as the central point. The dashed arrows above illustrate~$n=2$.

\begin{lemma}
\label{lemma:MotherLemma}
Any homotopy coherent diagram of the form
\begin{equation*}
\mathbb{E}\colon \fsd \Lambda^0[m]\to \Extcat^n_{\C}(B,A)
\end{equation*}
fills out to
\begin{equation*}
\mathbb{E}^+\colon\fsd_0^+[m]\to \Extcat^n_{\C}(B,A).
\end{equation*}
\end{lemma}
\begin{proof}
We are given the objects generated by \ref{ob-fsd01}. The first step is to fill out to the thickening. Finally, we add the central point of \ref{ob-fsd3}, and make sure that all morphisms in \ref{mor-fsd4} are included. These two steps are achieved by invoking Lemmas~\ref{lem:lemma1} and~\ref{lem:lemma2}, respectively.

To address \ref{ob-fsd2}, \ref{mor-fsd2} and \ref{mor-fsd3} we ought to provide a filler to the thickening. This amounts to a homotopy

\begin{equation*}
H\colon \fsd \Lambda^0[m]\times {[1]} \to \Extcat^n_{\C}(B,A)
\end{equation*}
such that~$H$ is the identity when restricted to~$j\times{[1]}$, for all singletons~$j$. Lemma~\ref{lem:lemma1} thus provides a filler. The restriction
\begin{equation*}
\begin{tikzcd}
\fsd \Lambda^0[m] \arrow[r,"{({\rm id},0)}"] & {\fsd \Lambda^0[m]\times [1]} \arrow[r] & \Extcat^n_{\C}(B,A)
\end{tikzcd}
\end{equation*}
of~$H$ to the thickening satisfy that all entries are term-wise cofibrations. Hence it admits a cocone~$\mathbb{P}$ by \Cref{lem:lemma2}.  Altogether we have defined a filler~$\mathbb{E}^+$ which is equal to~$H(-,0)$ on the thickening and maps the central point to~$\mathbb{P}$.
\end{proof}

\section{Loop spaces of extension categories}\label{section:loopsandmainproof}
\label{section:awesomestuff}
We have constructed a modified~$$--functor to Kan's~$$--functor. This was achieved by modifying the subdivision functor~$$, as we did in the beginning of Section~\ref{section:adaptationEx}. The technical~\Cref{lemma:MotherLemma} will now be used to prove that~$^n_(B,A)$ is a Kan complex and that the~$n$th Retakh functor is a weak equivalence.

\begin{theorem}
\label{proposition:fExIsKan}
Let~$\C$ be an exact quasi-category in which~$A$ and~$B$ are objects.
Then the simplicial set~$\fEx\Extcat_{\C}^n(B,A)$ is a Kan complex.
\end{theorem}
\begin{proof}
Let~$\mathbb{E}\colon\Lambda^k[m]\to\fEx\Extcat^n_{\C}(B,A)$ be a~$k$--horn in~$\fEx\Extcat^n_{\C}(B,A)$. We devise a horn-filling algorithm extending~$\mathbb{E}$ to an~$m$--simplex~$\mathbb{F}\colon [m]\to\fEx\Extcat^n_{\C}(B,A)$. Our algorithm does not depend on the choice of~$k$, up to a re-indexing of~$0$--simplices. So we will only consider~$k=0$. Moreover, utilizing the left adjoint~$\fsd$ we may rather extend a diagram~$\mathbb{E}$ defined on~$\fsd \Lambda^0[m]$ to an~$m$--simplex~$\mathbb{F}\colon \fsd[m]\to \Extcat^n_{\C}$ (Definition~\ref{definition:fExisRightAdjoint}).

We treat the special cases~$m=1$ and~$m=2$ to give the reader a geometric understanding of the horn-filling procedure. If~$m=1$, the~$0$--horn is simply the discrete category with a single object~$1$. Imposing that
\begin{center}
\begin{tikzcd}
\mathbb{F}(0)\arrow[r,equal]& \mathbb{F}(01)\arrow[r,equal]& \mathbb{F}(1)
\end{tikzcd}
\end{center}
gives a suitable horn filler. For~$m=2$, we consider a diagram indexed by the~$0$--horn~$\fsd\Lambda^0[2]$, which lifts to the solid part of
\begin{center}
\begin{tikzcd}[scale cd=0.8]
&  &       &                                         & 1 \arrow[lldddd] \arrow[lddddd] \arrow[rddddd, dashed] \arrow[rrdddd, dashed] &                                    &       &  &                                                                                \\
&  &       &                                         &                                                                       &                                    &       &  &                                                                                \\ &  &       &                                         &                                                                       &                                    &       &  &                                                                                \\&  &       &                                         &                                                                       &                                    &       &  &                                                                                \\
&  & 01 &                                         &                                                                       &                                    & 12 &  &                                                                                \\

&  &       & (01,2) \arrow[lu] \arrow[r] & 012 \arrow[r, equal]                                    & (12,0) \arrow[ru, equal] &       &  &                                                                                \\
&  &       &                                         & (02,1) \arrow[d] \arrow[u]                                &                                    &       &  &                                                                                \\
0 \arrow[rruuu] \arrow[rrrr] \arrow[rrruu] \arrow[rrrru]  &  &       &                                         & 02                                                                 &                                    &       &  & 2 \arrow[llll] \arrow[llllu] \arrow[llluu, dashed] \arrow[lluuu, dashed]
\end{tikzcd}
\end{center}
utilizing Lemma~\ref{lemma:MotherLemma}. Note that the solid part is equivalent to a homotopy coherent diagram in~$\fEx\Extcat^n_\C(B,A)$ indexed by~$\fsd_0^+[2]$, the thick~$0$--horn. Proceed as indicated by the dashed arrows to complete a diagram~$\mathbb{F}\colon \fsd[2]\to \Extcat^n_{\C}(B,A)$.

For a general~$m\geq 2$ we do the same procedure, but some extra care is required when adding identities in the end. Lemma~\ref{lemma:MotherLemma} gives a diagram~$\mathbb{E}^+$ defined on the thick~$0$--horn~$\fsd_0^+[m]$, which restricts to~$\mathbb{E}$ on~$\fsd\Lambda^0[m]$. Comparing~$\fsd_0^+[m]$ and~$\fsd[m]$, we see that the objects in the interior of~$\fsd[m]$'s zeroth face~$\fsd[1,\dots,m]$ and their duplicates in the thickening are missing. The boundary of~$\fsd[1,\dots,m]$, i.e. the union over~$\fsd[1,\dots,\hat{k},\dots,m]$, its thickening in~$\fsd[m]$ and the central point~$0\cdots m$ defines a subcategory~$I[m-1]$ equivalent to~$\fsd[m-1]$. Similarly, the thickening~$T[m-1]$ of~$\fsd[1,\dots, m]$ in~$\fsd[m]$, is of course equivalent to~$\fsd[m-1]$. So we extend~$\mathbb{E}^+$ to include~$T[m-1]$ by mapping irreducible morphisms from~$T[m-1]$ onto~$I[m-1]$ to identities. Lastly, the~$m$--simplex~$\mathbb{F}$ is obtained by also sending irreducible morphisms from the thickening~$T[m-1]$ onto~$\fsd[1,\dots,m]$ to identities. These two steps of adding identities specializes to the above picture for~$m=2$.
\end{proof}

Recall that~$eval_1(([1],0),(^n_(B,A),σ_n(B,A)))→^n_(B,A) $ is the evaltuation at~$1$, which is induced by the inclusion of the terminal object~$1$ into~$[1]$.
\begin{lemma}
\label{lemma:LoopsOnExt^n}
The pre-image~$\mathrm{eval}_1^{-1}(\sigma(B,A))$ models the loop space on~$\fEx\Extcat^n_\C(B,A)$ based at~$\sigma(B,A)$.
\end{lemma}
\begin{proof}
This is a direct consequence of \Cref{proposition:loopsOnKan} and \Cref{proposition:fExIsKan}.
\end{proof}
This is a fairly explicit description of the loop space. In particular, the objects are simple zigzags~$$\sigma_n(B,A)\to \mathbb{E} \from \sigma_n(B,A),$$ i.e. diagrams
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=1em]
& A \arrow[dd,tail]  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &   & B \arrow[dd] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[dd,equal] &                                        & 0\arrow[ru,tail]\arrow[dd] & & 0\arrow[ru,tail]\arrow[dd] &                           & B \arrow[dd,equal] \\
& E_1 \arrow[rd,two heads]         & &   \cdots\arrow[rd,two heads]   &  & E_n \arrow[rd, two heads] &              \\
A \arrow[ru,tail]            &                                        & E_{1.5}\arrow[ru,tail] && E_{(n-1).5} \arrow[ru,tail] &                           & B            \\
& A \arrow[uu,tail]  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &  & B \arrow[uu] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[uu,equal] &                                        &0\arrow[ru,tail]\arrow[uu] & & 0 \arrow[ru,tail]\arrow[uu] &                           & B \arrow[uu,equal]
\end{tikzcd}
\end{center}
Notice how~$eval_1^-1(σ_n(B,A))$ is the space of automaps on~$σ_n(B,A)$ inside the Kan complex~$^n_(B,A)$.
One model for this space has a particularly neat description in terms of simplices in~$^n_(B,A)$. See~\cite[Remark~1.2.2.5]{Lur09} for generalities. An~$m$--simplex in~$eval_1^-1(σ_n(B,A))$ is an~$(m+1)$--simplex~$S$ in~$^n_(B,A)$ with the following properties. The zeroth~$0$--simplex in~$S$ is equal to~$σ_n(B,A)$, also the zeroth face~$d_0S$ is equal to~$s_0^m σ_n(B,A)$, the~$m$--dimensional copy of~$σ_n(B,A)$. The~$i$th face of~$S$, considered as an~$m$--simplex in~$eval_1^-1(σ_n(B,A))$, is the~$m$--simplex~$d_i+1S$ in~$^n_(B,A)$. Here is a simple illustration of a~$1$--simplex between two loops~$γ$ and~$γ'$ together with the zeroth face map:
\begin{center}
\begin{tikzpicture}[scale=0.9, every node/.style={scale=0.9}]

\node[above] at (0,3) {$s_0\sigma_n(B,A)$};
\node[left] at (-1,1.5) {$\gamma$};
\node[right] at (1,1.5) {$\gamma'$};
\node[below] at (0,0) {$\sigma_n(B,A)$};

\begin{scope}[on background layer]
\draw[fill=gray!20] (0,0) -- (2,3) -- (-2,3) -- cycle;
\end{scope}

\draw[->] (2,1.5) -- (3,1.5);
\draw (3,0) -- (5,3);
\node[right] at (4, 1.5) {$\gamma'$};
\node[above] at (2.5,1.5) {$d_0$};

\end{tikzpicture}
\end{center}

Let us proceed with a proof of~\Cref{theorem:IsOmegaSpectrum}. The technique used to show that the Retakh functor~$R_n$ is a weak equivalence depends on~$n$. The case~$n=1$ is significantly simpler than~$n≥2$: the quasi-category~$^1_(B,A)$ is a Kan complex, whereas we need to consider~$^n_(B,A)$ for higher~$n$. Nonetheless, all Retakh maps are proven to be weak equivalences via their homotopy fiber. If~$FX→Y$ is a simplicial map between Kan complexes, then for all~$y$ in~$Y$ the comma category~$F↓y$ models the homotopy fiber at~$y$. We refer to either~\cite[p.~298]{BK72} or \cite[Theorem~4.1.3.1]{Lur09} for details. Its description is very similar to the above mapping space. An~$m$--simplex in~$F↓y$ is an~$(m+1)$--simplex~$S$ in~$Y$ so that the~$(m+1)$st~$0$--simplex is equal to~$y$, whereas~$d_m+1S$ is in the image of~$F$. A~$1$--simplex and its first face is typically depicted
\begin{center}
\begin{tikzcd}
F(x)\arrow[d] & F(x)\arrow[rr, "F(e)"]\arrow[dr] & & F(x')\arrow[dl]\\
y & & y &
\end{tikzcd}
\end{center}

Recall the zeroth Retakh functor
\begin{equation*}
R_0\colon\internalhom_\C (B,A)\to \Omega(\Extcat_\C^1 (B,A),\sigma_1(B,A))
\end{equation*}
mapping~$fB→A$ to the endomap
\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
A\arrow[r,tail]\arrow[d,equal] \& A\oplus B\arrow[d,"{\begin{pmatrix}[0.75] 1 & f \\ 0 & 1 \end{pmatrix}}"] \arrow[r] \& B\arrow[d,equal] \\
A\arrow[r,tail] \& A\oplus B \arrow[r] \& B
\end{tikzcd}
\end{equation*}

\begin{proposition}
\label{proposition:R1}
The zeroth Retakh functor
\begin{equation*}
R_0\colon\internalhom_\C (B,A)\to \Omega(\Extcat_\C^1 (B,A),\sigma_1(B,A))
\end{equation*}
is a weak homotopy equivalence.
\end{proposition}
\begin{proof}
It suffices to show that the homotopy fiber, modeled as the comma category~$R_0\downarrow \gamma$, is contractible for all loops~$\gamma\in \Omega(\Extcat_\C^1 (B,A),\sigma_1(B,A))$.  Any loop is a endomap of~$\sigma_1(B,A)$, and all endomaps of~$\sigma_1(B,A)$ are homotopic to a loop in the image of~$R_0$. Hence the homotopy fiber~$R_0\downarrow \gamma$ is easily seen to deformation retract onto the identity at~$\gamma$. This shows that the homotopy fiber is contractible at any loop~$\gamma$.
\end{proof}

In Section~\ref{section:overview} we defined the~$n$th Retakh functor
\begin{equation*}
R_n \colon \Extcat^n_{\C}(B,A) \to \Omega(\Extcat^{n+1}_{\C}(B,A),\sigma_{n+1}(B,A)).
\end{equation*}
In short, the loop~$R_n𝔼$ is given by
\begin{center}
\begin{tikzcd}
\sigma_{n+1}(B,A) \arrow[r] & (\mathbb{E},\rho_B) &\arrow[l]\sigma_{n+1}(B,A).
\end{tikzcd}
\end{center}
where~$(𝔼,ρ_B)$ is the concatenation of~$𝔼$ and~$ρ_B$:
\begin{center}
\begin{tikzcd}[ampersand replacement=\&,column sep=2.25em]
B \arrow[r, tail, "{\begin{pmatrix}
\end{pmatrix}}"] \& B\oplus B \arrow[r, two heads, "{\begin{pmatrix}
1 & 1
\end{pmatrix}}"] \& B.
\end{tikzcd}
\end{center}

Fix a loop~$γ$ in~$Ω(^n+1_(B,A),σ_n+1(B,A))$, which is a diagram in~$^n+1_(B,A)$ of the form
\begin{center}
\begin{tikzcd}
\sigma_{n+1}(B,A) \arrow[r] & \mathbb{F} &\arrow[l]\sigma_{n+1}(B,A).
\end{tikzcd}
\end{center}
Now we aim to construct a universal map of loops~$U_γR_n 𝔼→γ$ from the image of the Retakh functor. To better see how such a map can be constructed, we depict~$γ$ in full:
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=1em]
& A \arrow[dd,tail,"i_1"]  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &   & B \arrow[dd,"\gamma_1"] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[dd,equal] &                                        & 0\arrow[ru,tail]\arrow[dd] & & 0\arrow[ru,tail]\arrow[dd] &                           & B \arrow[dd,equal] \\
& F_1 \arrow[rd,two heads]         & &   \cdots\arrow[rd,two heads]   &  & F_{n+1} \arrow[rd, two heads,"p_{n+1}", near start] &              \\
A \arrow[ru,tail,"i_1"]            &                                        & F_{1.5}\arrow[ru,tail] && F_{n.5} \arrow[ru,tail,"i_{n+1}"] &                           & B            \\
& A \arrow[uu,tail,"i_1"']  \arrow[rd,two heads] & & \cdots\arrow[rd,two heads] &  & B \arrow[uu,"\gamma_2"'] \arrow[rd,equal]   &              \\
A \arrow[ru,equal] \arrow[uu,equal] &                                        &0\arrow[ru,tail]\arrow[uu] & & 0 \arrow[ru,tail]\arrow[uu] &                           & B \arrow[uu,equal]
\end{tikzcd}
\end{center}
It is determined by~$𝔽$ together with two right inverses~$γ_1$ and~$γ_2$ of~$p_n+1$, i.e.~$p_n+1γ_i∼1_B$. Consequently, a composite
\begin{equation*}
-\gamma_1+\gamma_2 \coloneqq \begin{pmatrix} -1 & 1 \end{pmatrix} \begin{pmatrix}\gamma_1 \\ \gamma_2 \end{pmatrix}\colon B\to F_{n-1}
\end{equation*}
factors through~$i_n+1$, which is to say that we have 2-simplex
\begin{center}
\begin{tikzcd}
B \arrow[rr,"-\gamma_1+\gamma_2"]\arrow[rd,"b"'] && F_{n-1} \\
& F_{n.5}\arrow[ru,"i_{n+1}"'] &
\end{tikzcd}
\end{center}
in~$$. Note that, up to homotopy, the map~$b$ is the unique map such that~$-γ_1+γ_2∼i_n+1∘b$. This is because of how it is defined from the universal mapping property of the short exact sequence
\begin{center}
\begin{tikzcd}
F_{n.5}\arrow[r, tail, "i_{n+1}"] & F_{n+1}\arrow[r, two heads, "p_{n+1}"] & B.
\end{tikzcd}
\end{center}
Lemma~\ref{Buh2.12} provides a map of~$n$--extensions
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=1em]
& F_1 \arrow[rd, two heads,] \arrow[dd, equal] &                                                       & \cdots \arrow[rd, two heads] &                                    & E_n\arrow[rd, two heads] \arrow[dd, "u_n"]\arrow[rddd,phantom,"\square"] &\\

A \arrow[ru, tail]\arrow[dd, equal] &                                                   & F_{1.5} \arrow[ru, tail] \arrow[dd, equal] &                                                    & F_{(n-1).5} \arrow[ru, tail]\arrow[dd,equal] &                        &B\arrow[dd,"b"]\\

& F_1 \arrow[rd, two heads]                   &                                                       & \cdots \arrow[rd, two heads]                   &                                    & F_n\arrow[rd, two heads]      &    \\

A \arrow[ru, tail]                  &                                                   & F_{1.5} \arrow[ru, tail]                       &                                                     & F_{(n-1).5} \arrow[ru, tail] &                                     & F_{n.5}
\end{tikzcd}
\end{center}
in which~$E_n$ is the pullback. Let~$𝔼$ be the top row, then the above diagram extends to a map of loops~$U_γR_n𝔼→γ$ given by
\begin{equation*}
\begin{tikzcd}
\sigma_{n+1}(B,A) \arrow[r,equal]\arrow[d] & \sigma_{n+1}(B,A)\arrow[d] \\
(\mathbb{E},\rho_B) \arrow[r,"u_{\gamma}"] & \mathbb{F} \\
\sigma_{n+1}(B,A)\arrow[u] \arrow[r,equal] & \sigma_{n+1}(B,A)\arrow[u]
\end{tikzcd}
\end{equation*}
where~$u_γ$ is the map
\begin{center}
\begin{tikzcd}[scale cd=0.8, row sep=1em, ampersand replacement=\&]
\& \cdots \arrow[rd, two heads] \&                                    \& E_n\arrow[rd, two heads] \arrow[dd,"u_n"]\arrow[rddd,phantom,"\square"] \& \& B\oplus B\arrow[rd,"{\begin{pmatrix} 1 & 1 \end{pmatrix}}"]\arrow[dd,"{\begin{pmatrix} \gamma_1 & \gamma_2 \end{pmatrix}}" near end] \& \\
A \arrow[ru, tail] \arrow[dd, equal] \&                                                    \& F_{(n-1).5} \arrow[ru, tail]\arrow[dd,equal] \&                        \& B\arrow[dd,"b"]\arrow[ru,"{\begin{pmatrix} -1 \\ 1 \end{pmatrix}}"]\arrow[rd,phantom,"\square"] \& \& B\arrow[dd,equal]\\
\& \cdots \arrow[rd, two heads]                   \&                                    \& F_n\arrow[rd, two heads]      \&   \& F_{n+1}\arrow[rd,"p_{n+1}"] \\
A \arrow[ru, tail]                       \&                                                     \& F_{(n-1).5} \arrow[ru, tail] \&                                     \& F_{n.5} \arrow[ru,"i_{n+1}"] \& \& B
\end{tikzcd}
\end{center}
If~$γ=R_n𝔼$, then both~$-γ_1+γ_2$ and~$i_n+1$ agree with~$

so that~$b∼1_B$. The Five Lemma (Lemma~\ref{5lem}) thus implies that~$u_n$ is a homotopy equivalence so that~$U_γ$ is the identity on~$R_n𝔼$.

Let~$J$ be an indexing category, and consider~$DJ →Ω^n+1_(B,A)$ a homotopy coherent diagram. We can apply the above steps to all of~$D$, resulting in a diagram contained in the image of~$R_n$. Indeed, no matter how complicated~$D$ is when translated to a diagram in~$$, the above process only hinges upon the universal mapping property into short exact sequences as well as functoriality of pullbacks. We summarize the discussion thus far.
\begin{lemma}
\label{lemma:universal}
A homotopy coherent diagram~$D\colon J \to \Omega\Extcat^{n+1}_{\C}(B,A)$, indexed by some category~$J$, extends to a diagram
\[
H\colon J \times [1] \to \Omega\Extcat^{n+1}_{\C}(B,A)
\]
by declaring~$H|_{\gamma\times [1]}=U_\gamma$ for all~$\gamma=D(j)$. Moreover, the extended diagram~$H$ enjoys the following properties:
\begin{enumerate}
\item~$H_1\coloneqq H|_{J\times 1}$ is~$D$,
\item~$H_0\coloneqq H|_{J\times 0}$ is in the image of~$R_n$ and
\item if~$i$ is an object in~$J$ such that~$D(i)=R_n\mathbb{E}$, then~$H$ is the identity when restricted to~$i\times [1]$.
\end{enumerate}
\end{lemma}

The zeroth Retakh functor was proven a weak equivalence in Proposition~\ref{proposition:R1}, so in order to finish a proof of Theorem~\ref{theorem:IsOmegaSpectrum} we need only verify that the~$n$th Retakh functor is a weak equivalence.

\begin{proposition}
\label{proposition:Rn}
The~$n$th Retakh functor
\[
R_n\colon \Extcat^n_{\C}(B,A)\to \Omega(\Extcat^{n+1}_{\C}(B,A),\sigma_n(B,A))
\]
is a weak equivalence for all~$n\geq 1$.
\end{proposition}
\begin{proof}
The objects~$A$ and~$B$ are fixed within~$\C$, so we simply write~$\sigma=\sigma_{n+1}(B,A)$ and~$\Extcat^{n+1}=\Extcat^{n+1}_{\C}(B,A)$ throughout the proof. As in the proof of Proposition~\ref{proposition:R1}, we show that the homotopy fiber~$R_n\downarrow \gamma$ is contractible for a fixed loop~$\gamma$:
\begin{center}
\begin{tikzcd}
{\sigma}\arrow[r,"\gamma_1"] & \mathbb{F} & {\sigma} \arrow[l,"\gamma_2"'].
\end{tikzcd}
\end{center}
To do so, we prove that the universal map of loops~$U_\gamma\colon R_n \mathbb{E}_1\to \gamma$ is initial in~$R_n\downarrow \gamma$. The definition of an initial object~\cite[Definition~4.1]{Joy02} requires that every simplicial sphere
\[
\partial[m]\to R_n\downarrow \gamma
\]
whose restriction to~$0$ is~$U_\gamma$ can be filled. We first treat the case~$m=1$ to get a better geometric grasp of the algorithm for general~$m$.

Step 1: The~$0$--dimensional simplicial sphere~$\partial[1]\to R_n\downarrow \gamma$ consists of two loops~$R_n\mathbb{E}_1$ and~$R_n\mathbb{E}_2$ which constitute a~$2$--horn
\begin{center}
\begin{tikzcd}
{R_n\mathbb{E}_1}\arrow[rd, "U_\gamma"] & & {R_n\mathbb{E}_2}\arrow[ld]\\
& {\gamma}&
\end{tikzcd}
\end{center}
whose~$0$--simplices are ~$R_n\mathbb{E}_1,R_n\mathbb{E}_2, \gamma$. Notice how the left-most map is~$U_\gamma$.

Step 2: A~$1$--simplex in~$\Omega \Extcat^{n+1}$ is a~$2$--simplex in~$\fEx \Extcat^{n+1}$. The data in Step 1 thus defines a diagram~$\mathbb{F}'$~in~$\fEx \Extcat^{n+1}$ depicted

\begin{center}
\begin{tikzpicture}[scale=0.9, every node/.style={scale=0.9}]

\node[above] at (0,3) {$\mathbb{F}'_3$};
\node[left] at (-2,2) {$\mathbb{F}'_1$};
\node[right] at (2,2) {$\mathbb{F}'_2$};
\node[below] at (0,0) {$\mathbb{F}'_0$};

\node[left] at (-1,1) {$R_n\mathbb{E}_1$};
\node[right] at (0,1.5) {$\gamma$};
\node[right] at (1,1) {$R_n\mathbb{E}_2$};
\node[above] at (-1.5,2.45) {$s_0\sigma$};
\node[above] at (1.5,2.45) {$s_0\sigma$};

\begin{scope}[on background layer]
\draw[fill=gray!20] (0,0) -- (0,3) -- (-2,2) -- cycle;
\draw[fill=gray!20] (0,0) -- (0,3) -- (2,2) -- cycle;
\end{scope}

\end{tikzpicture}
\end{center}
Every~$0$--simplex~$\mathbb{F}'_i$ is equal to~$\sigma$. Define~$\mathbb{F}(123)=s_0^2 \sigma$, where~$s_0^2\sigma$ is the degenerate~$2$--dimensional copy of~$\sigma$, to obtain a~$3$--horn~$\mathbb{F}$ from~$\mathbb{F}'$.

Step 3: We fill out~$\mathbb{F}$ to~$\mathbb{F}^+$ according to Lemma~\ref{lemma:MotherLemma}. Define a diagram~$D$ by restricting~$\mathbb{F}^+$ to the boundary spanned by~$\mathbb{F}'_0,\mathbb{F}'_1,\mathbb{F}'_2$, its thickening as well as the central point. This translates to a diagram~$D\colon [1]\to \Omega\Extcat^{n+1}$ from~$R_n\mathbb{E}_1$ to~$R_n\mathbb{E}_2$. Although~$D(01)$ need not be in the image of~$R_n$, this is easily fixed by applying Lemma~\ref{lemma:universal}:~we have that~$D$ extends to a homotopy~$H$ from a diagram in the image of~$R_n$ to~$D$. In particular,~$H(-,0)$ defines a filling of~$\mathbb{F}^+$ to the thickening of the third face. We further extend~$\mathbb{F}^+$ to all of~$[3]$ by adding equalities from the zeroth face to its thickening.

This completes the case~$m=1$. Indeed, the constructed~$3$--simplex~$[3]\to \fEx\Extcat^{n+1}$ has a third face of the form

\begin{center}
\begin{tikzpicture}[scale=0.9, every node/.style={scale=0.9}]

\node[left] at (-2,2) {$\mathbb{F}'_1$};
\node[right] at (2,2) {$\mathbb{F}'_2$};
\node[below] at (0,0) {$\mathbb{F}'_0$};

\node[left] at (-1,1) {$R_n\mathbb{E}_1$};
\node[right] at (1,1) {$R_n\mathbb{E}_2$};
\node[above] at (0,2) {$s_0\sigma$};
\node[] at (0,1) {$R_n f$};

\begin{scope}[on background layer]
\draw[fill=gray!20] (0,0) -- (2,2) -- (-2,2) -- cycle;
\end{scope}

\end{tikzpicture}
\end{center}
which encodes an arrow~$R_nf\colon R_n\mathbb{E}_1\to R_n\mathbb{E}_2$ and fills the horn in step 1:
\begin{center}
\begin{tikzcd}
{R_n\mathbb{E}_1}\arrow[rd, "U_\gamma"] \arrow[rr, "R_n f"] & & {R_n\mathbb{E}_2}\arrow[ld]\\
& {\gamma}&
\end{tikzcd}
\end{center}

There is nothing special about~$m=1$, except for the possibility of drawing pictures. Here is the general algorithm outlined above:

Step 1: The simplicial sphere~$\partial[m]\to R_n\downarrow \gamma$ has~$0$--simplices~$R_n\mathbb{E}_i\to \gamma~$, which we index by~$i=1,\dots,m+1$, and the first map agrees with~$U_\gamma$. This data defines an~$(m+1)$--horn in~$\Omega \Extcat^{n+1}$ since all arrows terminate in~$\gamma$. We order the~$0$--simplices~$R_n\mathbb{E}_1,\dots,R_n\mathbb{E}_{m+1}, \gamma$.

Step 2: For every~$i=1, \dots,m+1$ we now have an~$m$--simplex~$\Gamma_i$ in~$\Omega \Extcat^{n+1}$ spanned by
\[
\]
Let~$S_1,\dots,S_{m+1}$ be the corresponding~$(m+1)$--simplices in~$\fEx\Extcat^{n+1}$. We also introduce~$S_0=s_0^{m+1}\sigma$. These simplices assemble into an~$(m+2)$--dimensional horn
\begin{equation*}
\mathbb{F}\colon \Lambda^{m+2}[m+2]\to \fEx\Extcat^{n+1}.
\end{equation*}
Indeed, it suffices to check that~$d_iS_j=d_{j-1}S_i$ for~$i<j$ and~$j\neq m+1$ \cite[p.~10]{GJ09}. If ~$j>0$, then applying~$d_0$ to~$S_j$ always produces~$s_0^{m}\sigma=d_{j-1}s_0^{m+1}$. So we need only consider~$i>0$. The~$m$--simplex~$d_iS_j$ corresponds to~$d_{i-1}\Gamma_i$ spanned by
\[
R_n\mathbb{E}_1,\dots,\hat{R_n\mathbb{E}}_i,\dots,\hat{R_n\mathbb{E}}_j,\dots R_n\mathbb{E}_{m+1},\gamma
\]
where the~$j$th term was removed first. If we apply~$d_{j-1}$ to~$\Gamma_i$, we get the same output: we first remove~$R_n\mathbb{E}_i$ in the construction of~$\Gamma_i$ so that reindexing to~$j-1$ removes~$R_n\mathbb{E}_j$ thereafter.

Step 3: Fill out~$\mathbb{F}$ to~$\mathbb{F}^+$ according to Lemma~\ref{lemma:MotherLemma}. To further extend it to an~$(m+2)$--simplex we proceed as in the proof of Theorem~\ref{proposition:fExIsKan}. But instead of adding equalities to include the thickening of the~$(m+2)$nd face, we rather invoke Lemma~\ref{lemma:universal}. Thereafter, we include the~$(m+2)$nd face by sending irreducible morphisms from its thickening to identities. If we interpret the~$(m+2)$nd face as an~$(m+1)$--simplex of loops, then this simplex is necessarily in the image of~$R_n$. The simplicial sphere~~$\partial[m]\to R_n\downarrow \gamma$ is thus filled out.
\end{proof}

\section{Extriangulated homotopy categories and higher extensions}\label{section:extriangulated}

It has recently been shown that the homotopy category~$$ of an exact quasi-category~$$ has a natural extriangulated structure \cite[Theorem~4.22]{NP20}, when equipped with the bifunctor~$π_0^1_(B,A)$ as well as a readily accessible additive realization. In this section, we use of the Retakh Ext-$Ω$--spectrum to shed new light on Nakaoka--Palu's result. More specifically, our claim is that the Retakh spectrum determines the extriangulation of the homotopy category as well as extriangulated functors between them. We will also prove in \Cref{thm:higherext} that higher extension categories induce the higher extension groups when passing to the homotopy category.

We need a few preliminary definitions to review the relatively new theory of extriangulated categories.

\begin{definition}
Let~$\c$ be an additive category and let~$\mathbf{E}\colon \c\op \times \c \to \Ab$ be an additive bifunctor into the category of abelian groups. We refer to group elements~$\xi\in \mathbf{E}(B,A)$ as \emph{$\mathbf{E}$--extensions}. A \emph{morphism of~$\mathbf{E}$--extensions}~$\xi\in\mathbf{E}(B_1,A_1)$ and~$\xi'\in\mathbf{E}(B_2,A_2)$ is a pair of morphisms~$(a\colon A_1\to A_2,b\colon B_1\to B_2)$ such that~$\mathbf{E}(b,A_2)(\xi')=\mathbf{E}(B_1,a)(\xi)$.
\end{definition}

Having defined a notion of a morphism of~$\mathbf{E}$--extensions, it is quickly checked that there is a category of such. We denote this by~$\mathbf{E}\mhyphen\!\Ext_{\c}^1$.

The role played by~$\mathbf{E}$--extensions is that of equivalence classes of exact sequences. For example, if~$\C$ is an exact quasi-category we have an additive bifunctor
\begin{equation*}
\Extcat_{\C}^1(-,-)\colon \C\op\times\C\to \Sp
\end{equation*}
where $\Sp$ is the quasi-category of spectrum objects of Kan complexes. This is the first component in the Retakh spectrum functor~$\Extcat_{\C}(-,-)$. It induces a biadditive functor
\begin{equation*}
\mathbf{E}\coloneqq\pi_0\Extcat_{\C}^1(-,-)\colon (\homcat\C)\op\times\homcat\C\to \Ab.
\end{equation*}
which sends a tuple~$(B,A)$ to the set of equivalence classes of exact sequences in~$\C$ of the form~$ A\mono E\epi B$ (i.e. to a connected component of~$\Extcat^1_{\C}(B,A)$). An~$\mathbf{E}$--extension is simply an equivalence class of exact sequences. Since the homomorphism~$\mathbf{E}(B,a)$ sends an exact sequence~$A\mono E\epi B$ to the bottom of the following commutative diagram
\begin{center}
\begin{tikzcd}
A\arrow[r]\arrow[d,"a"]\arrow[dr, phantom, "\square"] & E\arrow[r]\arrow[d,] & B\arrow[d,equal] \\
A'\arrow[r] & Z\arrow[r] & B
\end{tikzcd}
\end{center}
and dually for~$\mathbf{E}(b,A)$, it follows by \Cref{Buh3.1} that a morphism of~$\mathbf{E}$--extensions is an equivalence class of morphisms of complexes
\begin{center}
\begin{tikzcd}
A\arrow[r]\arrow[d,"a"] & E\arrow[r]\arrow[d,"e"] & B\arrow[d,"b"] \\
A'\arrow[r] & E'\arrow[r] & B' \\
\end{tikzcd}
\end{center}
which is uniquely determined by the pair~$(a,b)$.

\begin{lemma}\label{exfunclem}
Let~$\mathbf{E}\colon\c\op\times\c\to\Ab$ and~$\mathbf{E}'\colon\c{'}\op\times\c'\to\Ab$ be additive bifunctors. If~$F\colon\c\to\c'$ is an additive functor, then a natural transformation~$$\eta\colon\mathbf{E}(-,-)\to\mathbf{E}'(F\op-,F-)$$ induces a functor
$F_{\eta}\colon \mathbf{E}\mhyphen\!\Ext_{\c}\to \mathbf{E}'\mhyphen\!\Ext_{\c'}$. If~$F$ is an equivalence and~$\eta$ is a natural isomorphism, then~$F_{\eta}$ is an equivalence.
\end{lemma}
\begin{proof}
The maps~$\eta_{B,A}\colon\mathbf{E}(B,A)\to\mathbf{E}'(FB,FA)$ establish a suitable map of objects. If~$(a,b)\colon \xi_1\to\xi_2$ is a morphism of~$\mathbf{E}$--extensions~$\xi_1\in\mathbf{E}(B_1,A_1)$ and~$\xi_2\in\mathbf{E}(B_2,A_2)$, the commutativity of the diagram
\begin{center}
\begin{tikzcd}
\mathbf{E}(B_1,A_1)\arrow[r,"\eta_{B_1,A_1}"]\arrow[d,"{\mathbf{E}(B_1,a)}"'] & \mathbf{E}'(FB_1,FA_1)\arrow[d,"{\mathbf{E}'(FB_1,a)}"]\\
\mathbf{E}(B_1,A_2)\arrow[r,"\eta_{B_1,A_2}"]&\mathbf{E}'(FB_1,FA_2) \\
\mathbf{E}(B_2,A_2)\arrow[r,"\eta_{B_2,A_2}"]\arrow[u,"{\mathbf{E}(b,A_2)}"]&\mathbf{E}'(FB_2,FA_2)\arrow[u,"{\mathbf{E}'(b,FA_2)}"']
\end{tikzcd}
\end{center}
shows that~$(Fa,Fb)$ is a morphism~$\eta_{B_1,A_1}\xi_1\to\eta_{B_2,A_2}\xi_2$ of~$\mathbf{E}'$--extensions.

If~$F$ is an equivalence and~$\eta$ is a natural isomorphism, it is clear that~$F_{\eta}$ is essentially surjective. Full fidelity follows from the full fidelity of~$F$.
\end{proof}

Since we have a natural isomorphism~$\pi_{-1}\Extcat_{\C}\to \pi_{0}\Extcat^1_{\C}$, between two bifunctors~$(\homcat\C)\op\times\homcat \C\to\Ab$, \Cref{exfunclem} provides an equivalence
\begin{equation}\label{eq:natiso}
\mathbf{E}'\mhyphen\!\Ext_{\homcat\C} \to \mathbf{E}\mhyphen\!\Ext_{\homcat\C}
\end{equation}
where~$\mathbf{E}=\pi_{0}\Extcat^1_{\C}$ and~$\mathbf{E}'=\pi_{-1}\Extcat_{\C}$.

Unlike~$\pi_0\Extcat_{\C}^1$, an arbitrary additive bifunctor~$\mathbf{E}$ need not be directly tied to exact sequences. In the general case, it necessary to manually link the elements of the abstract group~$\mathbf{E}(B,A)$ to concrete diagrams of the form~$A\to E\to B$, or rather equivalence classes of such.

\begin{definition}
Let~$\c$ be an additive category. Two diagrams of the form~$A\to E\to B$ and~$A\to E'\to B$ are said to be \emph{equivalent} if there exists an isomorphism~$e\colon E\to E'$ rendering the following diagram commutative.
\begin{center}
\begin{tikzcd}
&E\arrow[dr]\arrow[dd,"e",]& \\
A\arrow[ur]\arrow[dr] && B\\
\end{tikzcd}
\end{center}
The equivalence class containing~$A\to E\to B$ is denoted~$[A\to E\to B]$.
\end{definition}

It will be convenient to regard the equivalence classes of sequences as objects in a category~$\seq_{\c}$.
A morphism in this category from~$[A\to E\to B]$ to~$[A'\to E'\to B']$ is pair of morphisms~$(a\colon A\to A',b\colon B\to B')$
for which there exists a commutative diagram as follows:
\begin{center}
\begin{tikzcd}
A\arrow[r]\arrow[d,"a"] & E\arrow[r]\arrow[d,"e"] & B\arrow[d,"b"] \\
A'\arrow[r] & E'\arrow[r] & B' \\
\end{tikzcd}
\end{center}
Composition in this category is clearly well defined and associative.

The attentive reader might have had a \textit{dejà vu} experience when reading the last paragraph. Indeed, the category of~$\pi_0\Extcat_{\C}^1$--extensions is remarkably similar to~$\seq_{\homcat\C}$. The difference is that~$\seq_{\homcat\C}$ contains more objects, giving rise to an embedding~$\iota_{\C}\colon\mathbf{E}\mhyphen\!\Ext^1_{\homcat\C}\embed \seq_{\homcat\C}$, where~$\mathbf{E}=\pi_0\Extcat_{\C}^1$. This is an example of the general notion of realization.

\begin{definition}
Let~$\mathbf{E}\colon \c\op \times \c \to \Ab$ be an additive bifunctor.
A \emph{realization} is a functor
\begin{equation*}
\mathfrak{s}\colon\mathbf{E}\mhyphen\!\Ext_{\c}^1\to \seq_{\c}
\end{equation*}
such that an~$\mathbf{E}$--extension~$\xi\in \mathbf{E}(B,A)$ is sent to an equivalence class~$[A\to E\to B]$ with appropriate endpoints.
Equivalently, a \emph{realization} is a correspondence~$\mathfrak{s}$ that sends an~$\mathbf{E}$--extension~$\xi\in \mathbf{E}(B,A)$ to an equivalence class~$[A\to E\to B]$, satisfying the following property: If~$(a,b)$ is a morphism of~$\mathbf{E}$--extensions that are realized by~$[A\to E\to B]$ and~$[A'\to E'\to B']$, there exists a morphism~$e\colon E\to
E'$ such that the diagram
\begin{center}
\begin{tikzcd}
A\arrow[r]\arrow[d,"a"] & E\arrow[r]\arrow[d,"e"] & B\arrow[d,"b"] \\
A'\arrow[r] & E'\arrow[r] & B' \\
\end{tikzcd}
\end{center}
is commutative.

A realization~$\mathfrak{s}$ is \emph{additive} if~$\mathfrak{s}(_{B}0_{A})=[\sigma_1(B,A)]$, where~$_{B}0_{A}$ is the zero element in~$\mathbf{E}(B,A)$ and~$\sigma_1(B,A)$ is the split exact sequence, and~$\mathfrak{s}(\xi_1\oplus\xi_2)=\mathfrak{s}(\xi_1)\oplus\mathfrak{s}(\xi_2)$.
\end{definition}

A realization restricts to a map from~$\mathbf{E}(B,A)$ to a subset of the set of equivalence classes of the form~$[A\to E\to B]$. Since~$\mathbf{E}(B,A)$ is an abelian group, this map gives a group structure to this subset. Moreover, an additive realization gives rise to a group structure that behaves similarly to Yoneda Ext-groups in exact categories.

The embedding~$\iota_{\C}\colon\mathbf{E}\mhyphen\!\Ext^1_{\homcat\C}\embed \seq_{\homcat\C}$, where~$\mathbf{E}=\pi_{0}\Extcat^1_{\C}$, was our motivating example of a realization. It is easy to see that this is additive. Composing with the above equivalence \eqref{eq:natiso} yields an additive realization
\begin{equation*}
\mathbf{E}'\mhyphen\!\Ext^1_{\homcat\C}\to\mathbf{E}\mhyphen\!\Ext^1_{\homcat\C}\embed \seq_{\homcat\C}
\end{equation*}

We now have all ingredients to define what is meant by an extriangulated category and extriangulated functors.

\begin{definition}[{\cite[Definition~2.12]{NP19}}]\label{extricat}
Let~$\c$ be an additive category, and consider a bifunctor~$\mathbf{E}\colon\c\op\times\c\to\Ab$ as well as an additive realization~$\mathfrak{s}$ of~$\mathbf{E}$. The triple~$(\c,\mathbf{E},\mathfrak{s})$ is an \emph{extriangulated category} if the follows axioms hold.
\begin{enumerate}[label=(ET\arabic*), leftmargin=1.42cm]
\setcounter{enumi}{2}
\item[(ET3)$\;\;\;$]\label{ET3} Let~$\xi_1\in \mathbf{E}(B_1,A_1)$ and~$\xi_2\in \mathbf{E}(B_2,A_2)$ be~$\mathbf{E}$--extensions with realizations
\begin{equation*}
\mathfrak{s}(\xi_1) = [A_1\to E_1\to B_1] \qquad \mathfrak{s}(\xi_2) = [A_2\to E_2\to B_2]
\end{equation*}
If the solid part of the following diagram commutes
\begin{center}
\begin{tikzcd}
A_1\arrow[r]\arrow[d] & E_1\arrow[r]\arrow[d] & B_1\arrow[d,dashed] \\
A_2\arrow[r] & E_2\arrow[r] & B_2
\end{tikzcd}
\end{center}
the dashed morphism exists and the resulting diagram is commutative.
\item[(ET3$\op$)]\label{ET3op} Dual of \ref{ET3} \cite[(ET3$\op$) in Definition~2.12]{NP19}.
\item[(ET4)$\;\;\;$]\label{ET4} Let~$\xi_1\in \mathbf{E}(B_1,A_1)$ and~$\xi_2\in \mathbf{E}(C,B)$ be~$\mathbf{E}$--extensions with realizations
\begin{equation*}
\mathfrak{s}(\xi_1) = [A\to E\to B], \qquad \mathfrak{s}(\xi_2) = [E\to F\to C].
\end{equation*}
There exists a commutative diagram
\begin{center}
\begin{tikzcd}
A\arrow[r,"i"]\arrow[d,equal] & E\arrow[r,"p"]\arrow[d] & B\arrow[d,"j"] \\
A\arrow[r] & F\arrow[r]\arrow[d] & G\arrow[d,"q"] \\
& C\arrow[r,equal] & C
\end{tikzcd}
\end{center}
and an~$\mathbf{E}$--extension~$\xi_3\in\mathbf{E}(G,A)$ which is realized by the middle row. Moreover, we require that
\begin{enumerate}[label=(ET4.\arabic*), leftmargin=1.38cm]
\item~$\mathfrak{s}(\mathbf{E}(C,p)(\xi_2))=[B\to G\to C]$,
\item~$\mathbf{E}(j,A)(\xi_3)=\xi_1$,
\item~$\mathbf{E}(G,i)(\xi_3)=\mathbf{E}(q,E)(\xi_2)$.
\end{enumerate}
\item[(ET4$\op$)]\label{ET4op} Dual of \ref{ET4} \cite[Remark~2.22]{NP19}.
\end{enumerate}
\end{definition}

\begin{definition}[{\cite[Definition~2.32]{B-TS20}}]\label{extrifunc}
Let~$(\c,\mathbf{E},\mathfrak{s})$ and~$(\c',\mathbf{E}',\mathfrak{s}')$ be extriangulated categories. An \emph{extriangulated functor} consists of an additive functor~$F\colon \c\to \c'$ for which there exists a natural transformation~$\eta \colon \mathbf{E}(-,-)\to \mathbf{E}'(F\op-,F-)$ such that diagram of functors
\begin{center}
\begin{tikzcd}
\mathbf{E}\mhyphen\!\Ext_{\c} \arrow[r,"F_{\eta}"]\arrow[d,"\mathfrak{s}"] & \mathbf{E}'\mhyphen\!\Ext_{\c'}\arrow[d,"\mathfrak{s}'"]\\
\seq_{\c} \arrow[r,"\seq_{F}"] & \seq_{\c'}
\end{tikzcd}
\end{center}
commutes, where~$F_{\eta}$ is as in \Cref{exfunclem} and~$\seq_{F}$ sends the class~$[A\to E\to B]$ to the class~$[FA\to FE\to FB]$.
An extriangulated functor is an \emph{equivalence} if~$F$ is an equivalence of categories.
\end{definition}

As stated above, the homotopy category of an exact quasi-category admits a natural extriangulation. More precisely:

\begin{theorem}[{\cite[Theorem~4.22 and Proposition~4.28]{NP20}}]\label{NP20.4.22}
Let~$\C$ be an exact quasi-category, and consider the additive bifunctor
\begin{equation*}
\mathbf{E}=\pi_0\Extcat_{\C}^1(-,-)\colon (\homcat\C)\op\times \homcat\C \to \Ab,
\end{equation*}
as well as the additive realization
\begin{equation*}
\mathbf{E}=\pi_{0}\Extcat^1_{\C}\embed^{\iota_{\C}} \seq_{\homcat\C}.
\end{equation*}
Then the triple~$(\homcat\C,\mathbf{E},\iota_{\C})$ is an extriagulated category. Furthermore, if~$F\colon \C\to \D$ is an exact functor, then the induced functor~$\homcat F\colon \homcat\C\to \homcat\D$ between homotopy categories is an extriangulated functor.
\end{theorem}

An extriangulated category is \textit{topological} if it is equivalent to the homotopy category of some exact quasi-category.

We noted above that we have a natural isomorphism of bifunctors
\begin{equation*}
\varphi\colon \pi_{-1}\Extcat_{\C}(-,-) \Longrightarrow \pi_0\Extcat_{\C}^1(-,-).
\end{equation*}
A realization of~$\pi_{-1}\Extcat_{\C}(-,-)$ is achieved by precomposing the realization~$\iota_{\C}$ with the equivalence \eqref{eq:natiso}. It now follows from \Cref{exfunclem} that the identity functor~$1_{\homcat\C}$ on~$ \homcat\C$ yields an extriangulated equivalence
\begin{equation*}
(\homcat\C,\pi_{-1}\Extcat_{\C}(-,-),\iota_{\C}\circ\eqref{eq:natiso})\to (\homcat\C,\pi_0\Extcat_{\C}^1(-,-),\iota_{\C}).
\end{equation*}
In other words, we can present the extriangulation on the topological extriangulated category~$\homcat\C$ using the Retakh spectrum.

Let~$F\colon \C\to \D$ be an exact functor of exact quasi-categories. By \Cref{prop:nattransExt} we have a natural transformation
\begin{equation*}
\Extcat_{\C}(-,-)\Longrightarrow \Extcat_{\mathscr{D}}(F-,F-)
\end{equation*}
of functors~$\C\op\times\C\to\Sp$. We can thus induce a natural transformation
\begin{equation*}
\eta\colon\pi_{-1}\Extcat_{\C}(-,-)\Longrightarrow \pi_{0}\Extcat^1_{\mathscr{D}}(F-,F-)
\end{equation*}
of functors~$(\homcat\C)\op\times\homcat\C\to\Ab$. Pushing this natural transformation down to the homotopy category ensures that the functor
\begin{equation*}
\homcat F\colon \homcat \C\to \homcat\D
\end{equation*}
is extriangulated.
Indeed, by \Cref{exfunclem}, we have a functor
\begin{equation*}
\homcat F_{\eta}\colon \pi_{-1}\Extcat_{\C}\mhyphen\! \Ext \to \pi_{-1}\Extcat_{\C}\mhyphen\! \Ext,
\end{equation*}
since the vertical maps in the diagram
\begin{center}
\begin{tikzcd}
\pi_{-1}\Extcat_{\C}\mhyphen\! \Ext \arrow[r,"\homcat F_{\eta}"]\arrow[d] & \pi_{-1}\Extcat_{\D}\mhyphen\! \Ext\arrow[d]\\
\seq_{\homcat\C} \arrow[r,"\seq_{F}"] & \seq_{\homcat\D}
\end{tikzcd}
\end{center}
sends an equivalence class of extensions~$[A\mono E\epi B]$ to itself, it is easy to see that the diagram commutes, and consequently that~$\homcat F$ is an extriangulated functor.

Klemenc' embedding theorem (restated here as \Cref{thm:Kle20.1}), as well as the discussion above, proves that any small topological extriangulated category embeds into a triangulated category.

As a new contribution, we generalise \Cref{NP20.4.22}. The spectrum objects~$\Extcat_{\C}(B,A)$ do not only determine a natural extriangulation on~$\homcat \C$. They also induce the higher extension functors, recently defined by Gorsky--Nakaoka--Palu \cite[Definition~3.1]{GNP21}. We review their definition first.

Let~$F$ and~$G$ be bifunctors~$\c\op\times\c \to \Ab$. For a fixed pair of objects~$(X,Y)$, consider the bifunctor~$G(-,Y)\underset{\Z}{\otimes} F(X,-)$. The coend of this defines a product of~$G$ and~$F$.
\begin{equation*}
(G\diamond F)(X,Y) \coloneqq \int\limits^{C\in\c} G(C,Y)\underset{\Z}{\otimes} F(X,C).
\end{equation*}
Since the category of abelian groups is cocomplete, the coend of a bifunctor~$S$ on~$\c$ into~$\Ab$ can be obtained as the coequalizer of the diagram
\begin{equation}\label{eq:coenddef}
\begin{tikzcd}
{\coprod\limits_{f\colon c_1\rightarrow c_2} S(c_2,c_1)} \arrow[r,shift left,"{S(c_2,f)}"]\arrow[r,shift right,"{S(f,c_1)}"'] & \coprod\limits_{c} S(c,c),
\end{tikzcd}
\end{equation}
where the first coproduct is indexed over all morphisms in~$\c$, and the second over all its objects.

Let~$(\C,\mathbf{E},\mathfrak{s})$ be an extriangulated category. We set the zeroth extension functor to be the Hom-functor~$\c(-,-)$, and the first to be~$\mathbf{E}$.
The higher extension functors~$\mathbf{E}^{\diamond n}$ are then inductively defined inductively by
\begin{equation*}
\mathbf{E}^{\diamond n} \coloneqq \mathbf{E}^{\diamond n-1}\diamond \mathbf{E}.
\end{equation*}

\begin{theorem}\label{thm:higherext}
Let~$\C$ be an exact quasi-category.
The bifunctor~$\pi_0\Extcat^n_{\C}(-,-)$ is naturally isomorphic to the~$n$th extension functor~$\pi_{0}\Extcat^1_{\C}(-,-)^{\diamond n}$.
\end{theorem}
\begin{proof}
We proceed by induction. The base cases will be~$n=0$ and~$n=1$, both of which holding by definition.

Suppose that we have natural isomorphisms
\[
\varphi^n_{X,Y}\colon \pi_{0}\Extcat^1_{\C}(X,Y)^{\diamond n} \to \pi_0\Extcat^n_{\C}(X,Y),
\]
for some~$n\geq 1$. Applying the tensor functor gives another natural isomorphism
\begin{equation*}
\begin{tikzcd}
\pi_{0}\Extcat^1_{\C}(X,Y)^{\diamond n}\underset{\Z}{\otimes} \pi_{0}\Extcat^1_{\C}(B,A) \arrow[r,"\varphi^n_{X,Y}\otimes 1"] & \pi_0\Extcat^n_{\C}(X,Y) \underset{\Z}{\otimes} \pi_{0}\Extcat^1_{\C}(B,A).
\end{tikzcd}
\end{equation*}
Consider also the homomorphism
\begin{equation*}
\begin{tikzcd}
c_{B,A}\colon\pi_0\Extcat^n_{\C}(C,A) \underset{\Z}{\otimes} \pi_{0}\Extcat^1_{\C}(B,C)\arrow[r] & \pi_0\Extcat^{n+1}_{\C}(B,A)
\end{tikzcd}
\end{equation*}
that concatenates exact sequences.
Our strategy to complete the inductive step will be to show that~$c_{B,A}\circ (\varphi^n_{C,A}\otimes 1)$ is a coequalizer of the diagram \eqref{eq:coenddef}, with the tensor product~$\pi_0\Extcat^n_{\C}(-,A)\underset{\Z}{\otimes} \pi_{0}\Extcat^1_{\C}(B,-)$ in place of~$S$.
We simplify our notation by setting~$S=\pi_0\Extcat^n_{\C}(-,A)\underset{\Z}{\otimes} \pi_{0}\Extcat^1_{\C}(B,-)$ for the remainder of this proof.

In order to have a coequalizer, it must be the case that
\begin{equation}\label{eq:iscofork}
c_{B,A}\circ (\varphi^n_{C,A}\otimes 1)\circ S(c_2,f) = c_{B,A}\circ (\varphi^n_{C,A}\otimes 1)\circ S(f,c_1).
\end{equation}
In other words, it should be shown that
\begin{equation*}
c_{B,A}\circ (\varphi^n_{C,A}\otimes 1)
\end{equation*}
is a cofork of the diagram \eqref{eq:coenddef}.
The situation is illustrated by the solid part of the diagram
\begin{equation}\label{eq:iscoforkdia}
\begin{tikzcd}
A \arrow[r, dashed] \arrow[d, dashed,"1"] & \cdots \arrow[r, dashed] & E_{(n-1).5} \arrow[r, dashed] \arrow[d,dashed,"1"] & C' \arrow[r, dashed] \arrow[d, dashed] \arrow[dr, phantom, "\lrcorner",very near start] & C_2 \arrow[d, "f"'] \arrow[r] \arrow[dr, phantom, "\ulcorner",very near end] & F \arrow[r] \arrow[d, dashed] & B \arrow[d, dashed,"1"] \\
A \arrow[r]                           & \cdots \arrow[r]         & E_{(n-1).5} \arrow[r]                   & E_n \arrow[r]                          & C_1 \arrow[r, dashed]         & F' \arrow[r, dashed]          & B
\end{tikzcd}
\end{equation}