Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to\sD_0$ for the canonical morphism to the terminal object of $\sD_0$.
Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to\sD_0$ for
the canonical morphism to the terminal object $\sD_0$ of $\oo\Cat$.
\begin{definition}\label{def:contractible}
An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to\sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}).
It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and
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@@ -96,9 +97,9 @@ We end this section with an important result on slice $\oo$\nbd{}categories (Par
Now for $0\leq k <n$, we define $\alpha_{\src_k(e_n)}$ and $\alpha_{\trgt_k(e_n)}$ as
\[
\alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}\text{ and }\alpha_{\trgt_k(e_n)}=\begin{cases}1_{\trgt_k(e_n)}, \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}.
\alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } k<n-1\\ e_n, \text{ if } k=n-1\end{cases}\text{ and }\alpha_{\trgt_k(e_n)}=\begin{cases}1_{\trgt_k(e_n)}, \text{ if } k<n-1\\ e_n, \text{ if } k=n-1.\end{cases}
\]
It is straightforward to check that this data define an oplax transformation $\alpha : \mathrm{id}_{\sD_n}\Rightarrow r\circ p$ (see \ref{paragr:formulasoplax} and Example \ref{example:natisoplax}), which proves the result.
It is straightforward to check that this data defines an oplax transformation \[\alpha : \mathrm{id}_{\sD_n}\Rightarrow r\circ p\] (see \ref{paragr:formulasoplax} and Example \ref{example:natisoplax}), which proves the result.
%% \[
%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},
%% \]
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@@ -421,7 +422,7 @@ higher than $1$.
\]
is commutative. This proves the existence part of the universal property.
Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be an $n$\nbd{}cell of $X$. Since the triangle
Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangle commute for every object $a$ of $A$ and let $x$ be an $n$\nbd{}cell of $X$. Since the triangle
\[
\begin{tikzcd}
X/f(\trgt_0(x))\ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r]& X \ar[d,"\phi'"]\\
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@@ -648,7 +649,7 @@ We now recall an important theorem due to Thomason.
\begin{remark}
It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}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 analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation.
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
Putting all the pieces together, we are now able to prove the awaited tyheorem.
is an isomorphism for every $k \leq n$. Since obviously $H_k(\iota_n\tau^{i}_{\leq n}(f))$ is also an isomorphism for $k > n$, this proves the result.
\end{proof}
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@@ -1370,7 +1370,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi
for every $0\leq k \leq n$.
\end{corollary}
\begin{proof}
From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism $\overline{\tau^{i}_{\leq n}}(\alpha_C^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
@@ -1421,7 +1421,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
Straightforward consequence of the fact that $N_n = N_{\oo}\circ\iota_n$ and the fact that the composition of left adjoints is the left adjoint of the composition.
\end{proof}
\begin{paragr}
In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$ induces, for every $\oo$\nbd{}category $C$ and every $n \geq0$, a canonical morphism of $n\Cat$
In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$ induces for every $\oo$\nbd{}category $C$ and every $n \geq0$, a canonical morphism of $n\Cat$
Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C))\to i^*(N_1(D))$ of $\Psh{\Delta_{\leq2}}$ consists of a function $F_0 : C_0\to D_0$ and a function $F_1 : C_1\to D_1$ such that
\begin{itemize}[label=-]
\begin{enumerate}[label=(\alph*)]
\item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,
\item for every $x \in C_1$, we have
\[\src(F_1(x))=F_0(\src(x)))\text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\]
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@@ -1462,8 +1462,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
\end{tikzcd}
\]
in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$.
\end{itemize}
In particular, it follows that $F_1$ is compatible with composition of $1$\nbd{}cells in an obvious sense and that for every $2$\nbd{}cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism
\end{enumerate}
In particular, it follows that $F_1$ is compatible with composition of
$1$\nbd{}cells in an obvious sense and that for every $2$\nbd{}cell $\alpha :
f \Rightarrow g$ of $C$, we have $F_1(f)=F_1(g)$. And conversely, this last
condition implies condition (c) above. This means exactly that we have a natural isomorphism
For every $\oo$\nbd{}category $C$, the canonical map of $\ho(\Ch)$
\[
\alpha^{\sing}: \sH^{\sing}(C)\to\lambda(C)
\alpha_C^{\sing}: \sH^{\sing}(C)\to\lambda(C)
\]
induces isomorphisms
\[
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@@ -1490,7 +1493,7 @@ We can now prove the important following proposition.
for $k \in\{0,1\}$.
\end{proposition}
\begin{proof}
Let $C$ be an $\oo$\nbd{}category. Recall from \ref{paragr:univmor} that the canonical morphism $\alpha^{\sing} : \sH^{\sing}(C)\to\lambda(C)$ is nothing but the image by the localization functor $\Ch\to\ho(\Ch)$ of the morphism
Let $C$ be an $\oo$\nbd{}category. Recall from \ref{paragr:univmor} that the canonical morphism $\alpha_C^{\sing} : \sH^{\sing}(C)\to\lambda(C)$ is nothing but the image by the localization functor $\Ch\to\ho(\Ch)$ of the morphism
\[
\lambda c_{\oo}N_{\oo}(C)\to\lambda(C)
\]
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@@ -1502,7 +1505,7 @@ We can now prove the important following proposition.
From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms \[H_k^{\sing}(C)\simeq H_k(\lambda(C))\] for $k \in\{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$. The result follows then from an immediate 2-out-of-3 property.
From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha_C^{\sing}$ induces isomorphisms \[H_k^{\sing}(C)\simeq H_k(\lambda(C))\] for $k \in\{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha_C^{\pol}$ induces isomorphisms $H_k^{\pol}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$. The result follows then from an immediate 2-out-of-3 property.
\end{proof}
\begin{paragr}\label{paragr:conjectureH2}
A natural question following the above proposition is:
