Commit 2cdaa8c9 authored by Leonard Guetta's avatar Leonard Guetta
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I have got to finish Lemmas 5.2.1 and 5.2.2

parent 8e07074f
......@@ -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.
%% \[
%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},
%% \]
%% and also
%% \[
%% \alpha_{\src_{n-1}(e_n)}=e_n=\alpha_{\src_{n-1}(e_n)}.
%% \]
\end{proof}
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 $n \geq 0$, we have a cocartesian square
\[
\begin{tikzcd}
\sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d,"j_n^+"]\\
\sD_n \ar[r,"j_n^-"'] & \sS_{n}.
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
\begin{lemma}\label{lemma:squarenerve}
For every $n \geq -1$, the commutative square of simplicial sets
For every $n \geq 0$, the commutative square of simplicial sets
\[
\begin{tikzcd}
N_{\oo}(\sS_n) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n+1}) \ar[d] \\
N_{\oo}(\sD_{n+1}) \ar[r] & N_{\oo}(\sS_{n+1})
N_{\oo}(\sS_{n-1}) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n}) \ar[d,"N_{\oo}(j_n^+)"] \\
N_{\oo}(\sD_{n}) \ar[r,"N_{\oo}(j_n^-)"] & N_{\oo}(\sS_{n})
\end{tikzcd}
\]
is cocartesian.
......@@ -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
\[
\overline{\int} : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th})
\]
......@@ -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.
\begin{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
\[
......
No preview for this file type
......@@ -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 $n\geq 0$ as the following amalgamated sum
Notice that for every $n\geq 0$, the following commutative square
\[
\begin{tikzcd}
\sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d]\\
\sD_n \ar[r] & \sS_{n}.
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d,"j_n^+"]\\
\sD_n \ar[r,"j_n^-"'] & \sS_{n},
% \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
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
\[
\langle \src(e_{n+1}),\trgt(e_{n+1}) \rangle : \sS^{n} \to \sD_{n+1}.
\langle \src(e_{n+1}),\trgt(e_{n+1}) \rangle : \sS_{n} \to \sD_{n+1}.
\]
\end{paragr}
\section{Free $\oo$-categories}
......@@ -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 $n \geq 0$. 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] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\E^*
\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
\[
\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 \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\
\displaystyle\coprod_{x \in \Sigma}\sS_n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*).
\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]
\end{tikzcd}
......
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