@@ -16,7 +16,7 @@ Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the ca

%% \begin{paragr}

%% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason weak equivalence. In particular, we have the following lemma.

%% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason equivalence. In particular, we have the following lemma.

%% \end{paragr}

%% \begin{lemma}\label{lemma:hmlgycontractible}

...

...

@@ -74,22 +74,46 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para

This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}.

\end{proof}

\section{Homology of globes and spheres}

\begin{paragr}

\todo{Rappeler définitions par récurrence des globes et sphères.}

\end{paragr}

\begin{lemma}\label{lemma:globescontractible}

For every $n \in\mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is contractible.

For every $n \in\mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible.

\end{lemma}

\begin{proof}

\todo{À écrire}

Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}-cells for every $k$ such that $0\leq k<n$. These two $k$\nbd{}cells are parallel and are given by $\src_k(e_n)$ and $\trgt_k(e_n)$.

Let $r : \sD_0\to\sD_n$ be the $\oo$\nbd{}functor that points to $\trgt_0(e_n)$ (which means that $r=\langle\trgt_0(e_n)\rangle$ with the notations of \ref{paragr:defglobe}). Hence for every $k$\nbd{}cell $x$ of $\sD_n$, we have

\[

r(p(x))=\1^k_{\trgt_0(e_n)},

\]

where we wrote $p$ for the unique $\oo$\nbd{}functor $\sD_n \to\sD_0$.

Now for $0\leq k <n$, we define $\alpha_{\src_k(e_n)}$ and $\alpha_{\trgt_k(e_n)}$ as

\[

\alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}\text{ and }\alpha_{\trgt_k(e_n)}=\begin{cases}1_{\trgt_k(e_n)}, \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}.

\]

It is straightforward to check that this data define an oplax transformation $\alpha : \mathrm{id}_{\sD_n}\Rightarrow r\circ p$ (see \todo{ref}), which proves the result.

%% \[

%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},

In particular, for every $n \in\mathbb{N}$, $\sD_n$ is \good{}.

In particular, for every $n \in\mathbb{N}$, $\sD_n$ is \good{}. Recall from \ref{paragr:inclusionsphereglobe} that for every $n \geq0$, we have a cocartesian square

@@ -162,7 +186,7 @@ From these two lemmas, follows the important proposition below.

\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]

\end{tikzcd}

\]

is homotopy cocartesian with respect to Thomason equivalences, then $X$ is \good{}.

is homotopy cocartesian with respect to Thomason equivalences, then $C$ is \good{}.

\end{proposition}

\begin{proof}

Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories

...

...

@@ -443,7 +467,7 @@ Beware that in the previous corollary, we did \emph{not} suppose that $X$ was fr

We now move on to the next step needed to prove that every $1$-category is \good{}. For that purpose, let us recall a construction commonly referred to as the ``Grothendieck construction''.

\begin{paragr}

Let $A$ be a small category and $F : A \to\Cat$ a functor. We denote by $\int F$ or $\int_{a \in A}F(a)$ the category such that:

\begin{itemize}

\begin{itemize}[label=-]

\item An object of $\int F$ is a pair $(a,x)$ where $a$ is an object of $A$ and $x$ is an object of $F(a)$.

\item An arrow $(a,x)\to(a',x')$ of $\int F$ is a pair $(f,k)$ where

\[

...

...

@@ -458,10 +482,10 @@ We now move on to the next step needed to prove that every $1$-category is \good

\[

(f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).

\]

Now, every morphism of functors

Every natural transformation

\[

\begin{tikzcd}

A \ar[r,bend left,"F",""{name=A,below}]\ar[r,bend right,"G"',""{name=B, above}]&\Cat\ar[from=A,to=B,Rightarrow,"\alpha"]

A \ar[r,bend left,"F",""{name=A,below},pos=19/30]\ar[r,bend right,"G"',""{name=B, above},pos=11/20]&\Cat\ar[from=A,to=B,Rightarrow,"\alpha",pos=9/20]

\end{tikzcd}

\]

induces a functor

...

...

@@ -469,7 +493,7 @@ We now move on to the next step needed to prove that every $1$-category is \good

\int\alpha : \int F &\to\int G\\

(a,x) &\mapsto (a,\alpha_a(x)).

\end{align*}

This defines a functor

Altogether, this defines a functor

\begin{align*}

\int : \Cat(A)&\to\Cat\\

F&\mapsto\int F,

...

...

@@ -478,7 +502,7 @@ We now move on to the next step needed to prove that every $1$-category is \good

\end{paragr}

We now recall an important Theorem due to Thomason.

\begin{theorem}[Thomason]\label{thm:Thomason}

The functor $\int : \Cat(A)\to\Cat$ sends pointwise Thomason equivalence (\todo{ref}) to Thomason equivalence and the induced functor

The functor $\int : \Cat(A)\to\Cat$ sends pointwise Thomason equivalences (\ref{paragr:homder}) to Thomason equivalences and the induced functor

@@ -493,7 +517,7 @@ We now recall an important Theorem due to Thomason.

\begin{corollary}\label{cor:thomhmtpycol}

Let $A$ be a small category. The canonical map

\[

\hocolim_{a \in A}A/a \to A

\hocolim_{a \in A}(A/a)\to A

\]

induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$, is an isomorphism of $\ho(\Cat^{\Th})$.

\end{corollary}

...

...

@@ -502,24 +526,24 @@ We now recall an important Theorem due to Thomason.

\[

A/a \to\sD_0

\]

is a Thomason weak equivalence. This comes from the fact that $A/a$ is contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that $A/a$ has a terminal object.

is a Thomason equivalence. This comes from the fact that $A/a$ is oplax contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that $A/a$ has a terminal object.

In particular, the morphism of functors

\[

A/(-)\Rightarrow k_{\sD_0},

\]

where $k_{\sD_0}$ is the constant functor $A \to\Cat$ with value the terminal category $\sD_0$, is a pointwise Thomason weak equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that

where $k_{\sD_0}$ is the constant functor $A \to\Cat$ with value the terminal category $\sD_0$, is a pointwise Thomason equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that

\[

\int_{a \in A}A/a \to\int_{a \in A}k_{\sD_0}

\]

is a Thomason weak equivalence. An immediate computation shows that $\int_{a \in A}k_{\sD_0}\simeq A$. From the second part of Theorem \ref{thm:Thomason}, we have that

is a Thomason equivalence. An immediate computation shows that \[\int_{a \in A}k_{\sD_0}\simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that

\[

\hocolim_{a \in A}A/a \simeq A.

\]

A thorough analysis of all the isomorphisms involved (\todo{détailler ou ref à Maltsiniotis}) shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$.

A thorough analysis of all the isomorphisms involved (or see ) shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$.

\end{proof}

\begin{remark}

It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have $\hocolim^{\Th}_{a \in A} X/a \simeq X$. However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analoguous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.

It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have \[\hocolim^{\Th}_{a \in A}(X/a)\simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analoguous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.

\end{remark}

Putting all the pieces together, we are now able to prove the awaited Theorem.

@@ -1012,12 +1012,12 @@ Proposition \ref{prop:freeiscofibrant} to show that

there exists a free $n$\nbd{}category $C'$ such that $\tau^{i}_{\leq n}(C')=C$.

Let us write $\Sigma_k$ for the $k$\nbd{}base of $C$ with $0\leq k\leq n-1$ and

let $C'$ be the free $\oo$\nbd{}category such that:

\begin{itemize}

\begin{itemize}[label=-]

\item the $k$\nbd{}base of $C'$ is $\Sigma_k$ for every $0\leq k \leq n-1$,

\item the $n$\nbd{}base of $C'$ is the set $C_n$,

\item the $(n+1)$\nbd{}base of $C'$ is the set

\[

\{(x,y)\vert x \text{ and } y \text{ are parallel } n \text{-cells of } C\},

\{(x,y)\,\vert\, x \text{ and } y \text{ are parallel } n \text{-cells of } C\},

\]

the source (resp.\ target) of $(x,y)$ being $x$ (resp.\ $y$),

\item the $k$\nbd{}base of $C'$ is empty for $k > n$ (i.e.\ $C'$ is an $(n+1)$\nbd{}category).

...

...

@@ -1025,7 +1025,7 @@ let $C'$ be the free $\oo$\nbd{}category such that:

(For such recursive constructions of free $\oo$\nbd{}categories, see Section

\ref{section:freeoocataspolygraph}, and in particular Proposition

\ref{prop:freeonpolygraph}.)

We leave it to the reader to verify for himself that indeed $\tau^{i}_{\leq n}(C')=C$.

We invite the reader to verify for himself that indeed $\tau^{i}_{\leq n}(C')=C$.

\end{proof}

\begin{example}

Every (small) category is cofibrant for the folk model structure on $\Cat$.

...

...

@@ -1220,7 +1220,7 @@ In the following lemma, $n\Cat$ is equipped with the folk model structure and $\

The functor $\lambda_{\leq n} : n\Cat\to\Ch^{\leq n}$ is left Quillen.

\end{lemma}

\begin{proof}

Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$. We know from Proposition \ref{prop:fmsncat} that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$. What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.

Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$ as in the second part of such that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$ (which we know exist by the second part of Proposition \ref{prop:fmsncat}). What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.

% Notice that we have $\tau^{i}_{\leq n } \circ \iota_n = \mathrm{id}_{n \Cat}$ and hence

From Lemma \ref{lemma:abelianizationtruncation}, we have

@@ -246,27 +246,28 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

the canonically associated $\oo$-functor.

\end{paragr}

\begin{paragr}\label{paragr:inclusionsphereglobe}

For $n \in\mathbb{N}$, the $n$-sphere $\sS^n$ is the $n$-category such for every $k\leq n$, it has exactly two parallel $k$-cells. In other words, we have

For $n \in\mathbb{N}$, the $n$-sphere $\sS_n$ is the $n$-category such for every $k\leq n$, it has exactly two parallel $k$-cells. In other words, we have

\[

\sS^{n}=\sk_n(\sD_{n+1}),

\sS_{n}=\sk_n(\sD_{n+1}),

\]

and in particular, we have a canonical inclusion functor

\[

i_{n+1} : \sS^{n}\to\sD_{n+1}.

i_{n+1} : \sS_{n}\to\sD_{n+1}.

\]

It is also customary to define $\sS^{-1}$ to be the empty $\oo$-category and $i_{-1}$ to be the unique $\oo$-functor

It is also customary to define $\sS_{-1}$ to be the empty $\oo$-category and $i_{-1}$ to be the unique $\oo$-functor

\[

\emptyset\to\sD_0.

\]

With this definition of $\sS^{-1}$, we can also give a recursive definition of the $n$-sphere for$n\geq0$ as the following amalgamated sum

Notice that for every$n\geq0$, the following commutative square

\[

\begin{tikzcd}

\sS_{n-1}\ar[r,"i_n"]\ar[d,"i_n"]&\sD_n \ar[d]\\

\sD_n \ar[r]&\sS_{n}.

\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]

% \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]

\end{tikzcd}

\]

The two anonymous arrows of the previous square are representing each one of the two parallel $n$-cells of $\sS_n$.

where we wrote $j_n^+$ (resp.\ $j_n^-$) for the morphism $\langle\trgt(e_{n+1})\rangle : \sD_n \to\sS_n$ (resp.\ $\langle\src(e_{n+1})\rangle : \sD_n \to\sS_n$), is cocartesian.

% The two anonymous arrows of the previous square are representing each one of the two parallel $n$-cells of $\sS_n$.

Here are some pictures of the $n$-spheres in low dimension:

\[

...

...

@@ -291,7 +292,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

\]

the canonically associated $\oo$-functor. For example, the $\oo$-functor $i_n$ is nothing but

@@ -299,7 +300,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

Let $C$ be an $\oo$-category and $n \geq0$. A subset $E \subseteq C_n$ of the $n$-cells of $C$ is an \emph{$n$-basis of $C$} if the commutative square

\[

\begin{tikzcd}[column sep=huge, row sep=huge]

\displaystyle\coprod_{ x \in E}\ar[d,"\displaystyle\coprod_{x \in E} i_n"']\sS^{n-1}\ar[r,"{\langle\src(x),\trgt(x)\rangle_{x \in E}}"]&\sk_{n-1}(C)\ar[d,hook]\\

\displaystyle\coprod_{ x \in E}\ar[d,"\displaystyle\coprod_{x \in E} i_n"']\sS_{n-1}\ar[r,"{\langle\src(x),\trgt(x)\rangle_{x \in E}}"]&\sk_{n-1}(C)\ar[d,hook]\\

\displaystyle\coprod_{x \in E}\sD_n \ar[r,"\langle x \rangle_{x \in E}"]&\sk_{n}(C)

\end{tikzcd}

\]

...

...

@@ -586,7 +587,7 @@ Furthermore, this function satisfies the condition

On the other hand, any $\E=(D,\Sigma,\sigma,\tau)$, cellular extension of an $n$-category $D$, yields an $(n+1)$-category $\E^*$ defined as the following amalgamated sum:

\begin{equation}\label{squarefreecext}

\begin{tikzcd}[column sep=huge, row sep=huge]

\displaystyle\coprod_{x \in\Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"]& D \ar[d]\\

\displaystyle\coprod_{x \in\Sigma}\sS_n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"]& D \ar[d]\\

\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]

\end{tikzcd}.

...

...

@@ -643,10 +644,10 @@ We can now prove the following proposition, which is the key result of this sect

\end{proposition}

\begin{proof}

Notice first that since the map $i_{n+1} : \sS^n \to\sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1})\to\sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to\E^*$ can be identified with the canonical inclusion $\sk_n(\E^*)\to\sk_{n+1}(\E^*)=\E^*$. Hence, cocartesian square \eqref{squarefreecext} can be identified with

Notice first that since the map $i_{n+1} : \sS_n \to\sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1})\to\sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to\E^*$ can be identified with the canonical inclusion $\sk_n(\E^*)\to\sk_{n+1}(\E^*)=\E^*$. Hence, cocartesian square \eqref{squarefreecext} can be identified with