Commit 66c67a00 authored by Leonard Guetta's avatar Leonard Guetta
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......@@ -5,9 +5,9 @@ Recall that $\sD_0$ is the terminal object of $\oo\Cat$ that for any $oo$\nbd-ca
An $\oo$\nbd-category $X$ is \emph{contractible} when there exists an object $x_0$ of $X$ and an oplax transformation
\[
\begin{tikzcd}
X \ar[r,"p_X"] \ar[rd,"1_X",""{name=A,above}]& \sD_0 \ar[d,"\langle x_0 \rangle"]\\
X \ar[r,"p_X"] \ar[rd,"1_X"',""{name=A,above}]& \sD_0 \ar[d,"\langle x_0 \rangle"]\\
&X.
\ar[from=A,to=1-2,"\alpha"]
\ar[from=A,to=1-2,"\alpha",Rightarrow]
\end{tikzcd}
\]
\end{definition}
......@@ -15,17 +15,177 @@ Recall that $\sD_0$ is the terminal object of $\oo\Cat$ that for any $oo$\nbd-ca
The previous definition admits many variations as we could use lax transformations instead oplax ones and we could change the direction of the (lax or oplax) transformation. All the results of this section could straightforwardly be adapted to all the variations of the definition of contractible $\oo$\nbd-category. Hopefully, the choice we made is consistent with the rest of the dissertation.
\end{remark}
\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:oplacloc} that $p_X : X \to \sD_0$ is a Thomason weak equivalence. In particular, we have the following corollary.
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.
\end{paragr}
\begin{corollary}
\begin{lemma}\label{lemma:hmlgycontractible}
Let $X$ be a contractible $\oo$\nbd-category. The morphism of $\ho(\Ch)$
\[
\sH(X) \to \sH(\sD_0)
\]
induced by the canonical morphism $p_X : X \to \sD_0$ is an isomorphism.
\end{corollary}
\end{lemma}
\begin{paragr}
In addition to the previous result, it is immediate to check that $\sH(\sD_0)$ is nothing but $\mathbb{Z}$ seen as an object of $\ho(\Ch)$ concentrated in degree $0$.
\end{paragr}
\begin{lemma}\label{lemma:liftingoplax}
Let
\[
\begin{tikzcd}
C' \ar[r,"f_{\epsilon}'"] \ar[d,"u"] & D' \ar[d,"v"]\\
C \ar[r,"f_{\epsilon}"] & D
\end{tikzcd}
\]
be commutative squares in $\omega\Cat$ for $\epsilon\in\{0,1\}$.
We would like now to study the polygraphic homology of contractible $\oo$\nbd-categories. In order to do so, we need some technical results.
If $C'$ is a free $\omega$-category and $v$ is a trivial fibration for the Folk model structure on $\omega\Cat$, then for any oplax transformation \[\alpha : f_0 \Rightarrow f_1,\] there is an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that
\[
v \star \alpha' = \alpha \star u.
\]
\end{lemma}
\begin{proof}
\todo{Preuve c/c de mon article. À reprendre...}
From \cite[Appendice B]{ara2016joint}, we know that given $u, v : C \to D$ two $\omega$\nobreakdash-functors, the set of oplax transformations from $u$ to $v$ is in bijection with the set of functors $\alpha : \sD_1 \otimes C \to D$ such that the diagram
\[
\begin{tikzcd}
(\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C \ar[d,"i_1 \otimes C"'] \ar[dr,"{\langle u, v \rangle}"] &\\
\sD_1 \otimes C \ar[r,"\alpha"'] & D,
\end{tikzcd}
\]
where $i_1 : \sD_0 \amalg \sD_0 \simeq \sS_0 \to \sD_1$ is the morphism introduced in \ref{paragr:defglobe}, is commutative. We use the same letter to denote an oplax transformation and the functor $\sD_1 \otimes C \to D$ associated to it.
Moreover, for an $\omega$-functor $f: B \to C$, the oplax transformation $\alpha \star f$ is represented by the functor \[\begin{tikzcd} \sD_1 \otimes B \ar[r,"\sD_1 \otimes f"] & \sD_1 \otimes C \ar[r,"\alpha"]& D\end{tikzcd}\] and for an $\omega$-functor $g : D \to E$, the oplax transformation $g \star \alpha$ is represented by the functor
\[\begin{tikzcd}\sD_1 \otimes C \ar[r,"\alpha"]& D \ar[r,"g"] & E.\end{tikzcd}\]
Using this way of representing oplax transformations, the hypotheses of the present lemma yield the following commutative square
\[
\begin{tikzcd}
(\sD_0 \amalg \sD_0)\otimes C' \ar[d,"{i_1\otimes C'}"'] \ar[rr,"{\langle f'_0, f_1' \rangle}"] && D' \ar[d,"v"] \\
\sD_1\otimes C'\ar[r,"\sD_1 \otimes u"'] & \sD_1\otimes C \ar[r,"\alpha"] & D.
\end{tikzcd}
\]
Since $i_1$ is a folk cofibration and $C'$ is cofibrant, it follows that the left vertical morphism of the previous square is a folk cofibration (see \cite[Proposition 5.1.2.7]{lucas2017cubical} or \cite{ara2019folk}). By hypothesis, $v$ is a folk trivial fibration, the above square admits a lift
\[
\alpha' : \sD_1\otimes C' \to D'.
\]
The commutativity of the two induced triangle shows what we needed to prove.
\end{proof}
In the following lemma, $\gamma_{\folk}$ is the localization functor $\oo\Cat \to \ho(\oo\Cat^{can})$.
\begin{lemma}
Let $u,v : X \to Y$ two $\oo$\nbd-functors. If there exists an oplax transformation $u\Rightarrow v$, then
\[
\sH^{\pol}(\gamma_{\folk}(u))=\sH^{\pol}(\gamma_{\folk}(v)).
\]
\end{lemma}
\begin{proof}
In the case that $X$ and $Y$ are both cofibrant with respect to the canonical model structure on $\oo\Cat$, this follows immediatly from Lemma \ref{lemma:abeloplax}.
In the general case, let
\[
p : X' \to X
\]
and
\[
q : Y' \to Y
\]
be trivial fibrations for the canonical model structure with $X'$ and $Y'$ cofibrant. Using that $q$ is a trival fibration and $X'$ is cofibrant, we know that there exists $u' : X' \to Y'$ and $v' : X' \to Y'$ such that the squares
\[
\begin{tikzcd}
X' \ar[d,"p"] \ar[r,"u'"] & Y' \ar[d,"q"] \\
X \ar[r,"u"] & Y
\end{tikzcd}
\text{ and }
\begin{tikzcd}
X' \ar[d,"p"] \ar[r,"v'"] & Y' \ar[d,"q"] \\
X \ar[r,"v"] & Y
\end{tikzcd}
\]
are commutative. From Lemma \ref{lemma:liftingoplax}, we deduce the existence of an oplax transformation $u' \Rightarrow v'$. Since $X'$ and $Y'$ are cofibrant, we have already proved that
\[\sH^{\pol}(\gamma_{\folk}(u'))=\sH^{\pol}(\gamma_{\folk}(v')).\]
The commutativity of the two previous squares and the fact that $p$ and $q$ induce isomorphisms in $\ho(\oo\Cat^{\folk})$ imply the desired result.
\end{proof}
\begin{corollary}
Let $u : X \to Y$ be a morphism of $\oo\Cat$. If $u$ is a homotopy equivalence (Paragraph \ref{paragr:hmtpyequiv}), then
\[
\sH^{\pol}(\gamma_{\folk}(u))
\]
is an isomorphism.
\end{corollary}
\todo{Expliquer p-e pourquoi le corollaire précédent est important.}
\begin{corollary}\label{cor:hmlgypolcontractible}
Let $X$ be an $\oo$-category. If $X$ is contractible, then the morphism
\[
\sH^{\pol}(\gamma_{\folk}(p_X)) : \sH^{\pol}(X) \to \sH^{\pol}(\sD_0)
\]
is an isomorphism of $\ho(\Ch)$.
\end{corollary}
We can now prove the main result of this section.
\begin{proposition}
Every contractible $\oo$\nbd-category is \good{}.
\end{proposition}
\begin{proof}
A simple calculation shows that $\sD_0$ is \good{}. Then, the result follows from Lemma \ref{cor:hmlgycontractible}, Corollary \ref{cor:hmlgypolcontractible} and the commutativity of the square
\[
\begin{tikzcd}
\sH^{\pol}(X) \ar[d] \ar[r,"\pi_X"] & \sH(X) \ar[d] \\
\sH^{\pol}(\sD_0) \ar[r] & \sH(\sD_0).
\end{tikzcd}
\]
\end{proof}
We end this section with an important result which says that slices $\oo$\nbd-category (Paragraph \ref{paragr:slices}) are contractible.
\begin{proposition}
Let $A$ be an $\oo$\nbd-category and $a_0$ an object of $A$. The $\oo$\nbd-category $A/a_0$ is contractible.
\end{proposition}
\begin{proof}
\todo{À écrire.}
\end{proof}
\section{Homology of globes and spheres}
\begin{paragr}
\todo{Rappeler définitions par récurrence des globes et sphères.}
\end{paragr}
\begin{lemma}
For every $n \in \mathbb{N}$, the $\oo$\nbd-category $\sD_n$ is contractible.
\end{lemma}
\begin{proof}
\todo{À écrire}
\end{proof}
In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}.
\begin{lemma}\label{lemma:squarenerve}
For every $n \geq -1$, the commutative square of simplicial sets
\[
\begin{tikzcd}
N_{\oo}(\sS_n) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n+1}) \ar[d] \\
N_{\oo}(\sD_{n+1}) \ar[r] & N_{\oo}(\sS_{n+1})
\end{tikzcd}
\]
is cocartesian.
\end{lemma}
\begin{proof}
\todo{À écrire}
\end{proof}
From these two lemmas, follows the important proposition below.
\begin{proposition}
For every $n \geq -1$, the $\oo$\nbd-category $\sS_n$ is \good{}.
\end{proposition}
\begin{proof}
We proceed by induction on $n$. When $n=-1$, it is trivial to check that the empty $\oo$\nbd-category is \good{}. Now, since $i_n : \sS_n \to \sD_{n+1}$ is a cofibration for the canonical model struture on $\oo\Cat$ and $\sS_{n}$ and $\sD_{n}$ are cofibrant, it follow from \todo{ref} that the cocartesian square
\begin{equation}\label{square}
\begin{tikzcd}
\sS_{n} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n+1} \\
\sD_{n+1} & \sS_{n+1}
\end{tikzcd}
\end{equation}
is homotopy cocartesian with respect to the canonical weak equivalences on $\oo\Cat$. Besides, since $N_{\oo} : \oo\Cat \to \Psh{\Delta}$ induces an equivalence of op-prederivators $\Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$ (Theorem \ref{thm:gagna}), it follows from Lemma \ref{lemma:squarenerve} that the square \eqref{square} is also homotopy cocartesian with respect to the Thomason weak equivalences on $\oo\Cat$. Then, the desired result follows from the induction hypothesis and Proposition \ref{prop:criteriongoodcat}.
\end{proof}
\begin{paragr}
The previous proposition implies what we claimed in Paragraph \ref{paragr:compcriterion}, which is that the morphism of op-prederivators
\[
\J : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th})
\]
induced by the identity functor of $\oo\Cat$, cannot be homotopy cocontinuous. Indeed, consider the ``bubble'' $B$, that is the $\oo$\nbd-category introduced in Paragraph \ref{paragr:bubble} which is \emph{not} \good{}. There is a commutative square
\[
\begin{tikzcd}
\sS_2 \ar[r] \ar[d,"i_2"] & \sD_0 \\
\sD_2 \ar[r] & B
\end{tikzcd}
\]
\end{paragr}
......@@ -408,7 +408,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
The rest of this document is devoted to (partially) answering this question. We start by giving in the next paragraph an example due to Ara and Maltsiniotis of an $\oo$-category for which the comparison map is \emph{not} an isomorphism.
\end{paragr}\fi
We begin by spelling out an example due to Ara and Maltsiniotis of an $\oo$-category which is \emph{not} \good{}. Examples of \good{} $\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd-categories.
\begin{paragr}
\begin{paragr}\label{paragr:bubble}
Let $B$ the commutative monoid $(\mathbb{N},+)$ considered as $2$-category.\footnote{The letter $B$ stands for ``bubble''.} That is,
\[
B_k=\begin{cases}\{\star\} \text{ if } k=0,1 \\ \mathbb{N} \text{ if } k=2,\end{cases}
......@@ -432,7 +432,7 @@ The previous example of non \good{} $\oo$-category also proves that triangle \re
\end{remark}\fi
We shall now proceed to give an abstract criterion to find \good{} $\oo$-categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$-categories.
\begin{paragr}
\begin{paragr}\label{paragr:compcriterion}
Both the polygraphic homology
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)
......@@ -450,7 +450,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor
\]
We will see later that $\J$ cannot be homotopy cocontinuous as if it were, every $\oo$\nbd-category would be \good{}. \todo{Mettre ref interne de où il sera montré que le morphisme n'est pas cocontinu.}
\end{paragr}
\begin{proposition}
\begin{proposition}\label{prop:criteriongoodcat}
Let $X$ be an $\oo$\nbd-category. Suppose that there exists a diagram
\[
d : I \to \oo\Cat
......@@ -461,7 +461,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor
\]
such that:
\begin{enumerate}[label=\roman*)]
\item For every $i \in \Ob(I)$, the $oo$\nbd-category $d(i)$ is \good{}.
\item For every $i \in \Ob(I)$, the $\oo$\nbd-category $d(i)$ is \good{}.
\item The canonical morphism
\[
\hocolim^{\folk}d \to X
......
......@@ -304,7 +304,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{proof}
This follows immediatly from \cite[Théorème B.11]{ara2020theoreme}.
\end{proof}
\begin{paragr}
\begin{paragr}\label{paragr:hmtpyequiv}
In particular, let us say that two $\oo$-functors $u, v : X \to Y$ are \emph{homotopic} if there exists an oplax transformations between $u$ and $v$ (in either direction). An $\oo$-functor $u : X \to Y$ is an \emph{homotopy equivalence} if there exists an $\oo$-functor $v : Y \to X$ such that $uv$ and $\mathrm{id}_Y$ are homotopic and $vu$ and $\mathrm{id}_X$ are homotopic. From Lemma \ref{lemma:oplaxloc} and the fact the class of Thomason weak equivalences is saturated \todo{Mettre la définition de saturation quelque part et le fait que les e.f. de Thomason sont saturées}, we have the following corollary.
\end{paragr}
\begin{corollary}
......@@ -337,7 +337,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\todo{Preuve à terminer}
\end{proof}
\section{Slices of $\oo$-category and Theorem $A$}
\begin{paragr}
\begin{paragr}\label{paragr:slices}
Let $A$ be an $\oo$-category and $a_0$ an object of $A$. We define the slice $\oo$-category $A/a_0$ as the following fibred product:
\[
\begin{tikzcd}
......
......@@ -12,6 +12,7 @@
\include{homtheo}
\include{hmtpy}
\include{hmlgy}
\include{contractible}
\bibliographystyle{alpha}
\bibliography{memoire}
\end{document}
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