Commit 27596721 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Edited a lot of typos. This should almost be it.

parent f3c9baf3
...@@ -18,8 +18,8 @@ In this section, we review some homotopical results on free ...@@ -18,8 +18,8 @@ In this section, we review some homotopical results on free
The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or
\emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells},
arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A
\emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 : \emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 \colon
G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and
units in an obvious sense. This defines the category $\Rgrph$ of reflexive units in an obvious sense. This defines the category $\Rgrph$ of reflexive
graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they
are the morphisms $f : G \to G'$ that are injective on objects and on arrows, are the morphisms $f : G \to G'$ that are injective on objects and on arrows,
...@@ -264,9 +264,9 @@ From the previous proposition, we deduce the following very useful corollary. ...@@ -264,9 +264,9 @@ From the previous proposition, we deduce the following very useful corollary.
\] \]
This square is cocartesian because $i_!$ is a left adjoint. Since This square is cocartesian because $i_!$ is a left adjoint. Since
$i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the
result follows from the fact that the monomorphisms are the result follows from Lemma \ref{lemma:hmtpycocartesianreedy} and the fact that
cofibrations of the standard Quillen model structure on simplicial the monomorphisms are the cofibrations of the standard Quillen model structure on simplicial
sets and from Lemma \ref{lemma:hmtpycocartesianreedy}. sets.
\end{proof} \end{proof}
\begin{paragr} \begin{paragr}
By working a little more, we obtain the more general result stated By working a little more, we obtain the more general result stated
...@@ -342,7 +342,8 @@ From the previous proposition, we deduce the following very useful corollary. ...@@ -342,7 +342,8 @@ From the previous proposition, we deduce the following very useful corollary.
\] \]
where the arrow $E \to B$ is the canonical inclusion. Hence, by universal where the arrow $E \to B$ is the canonical inclusion. Hence, by universal
property, the dotted arrow exists and makes the whole diagram commute. A property, the dotted arrow exists and makes the whole diagram commute. A
thorough verification easily shows that the morphism $G \to B$ is a thorough verification easily shows that, because $\beta$ is quasi-injective on
arrows, the morphism $G \to B$ is a
monomorphism of $\Rgrph$. monomorphism of $\Rgrph$.
By forming successive cocartesian squares and combining with the square By forming successive cocartesian squares and combining with the square
...@@ -482,10 +483,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition ...@@ -482,10 +483,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\geq 0$, we use the notations \geq 0$, we use the notations
\begin{align*} \begin{align*}
X_{n,m} &:= X([n],[m]) \\ X_{n,m} &:= X([n],[m]) \\
\partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\ \partial_i^v &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\
\partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\ \partial_j^h &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\
s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\ s_i^v &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\
s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}. s_j^h&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}.
\end{align*} \end{align*}
The maps $\partial_i^h$ and $s_i^h$ will be referred to as the The maps $\partial_i^h$ and $s_i^h$ will be referred to as the
\emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and \emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and
...@@ -548,8 +549,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition ...@@ -548,8 +549,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
definition, $\delta^*$ induces a morphism of op-prederivators definition, $\delta^*$ induces a morphism of op-prederivators
\[ \[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
\Ho(\Psh{\Delta}). \Ho(\Psh{\Delta}),
\] \]
where $\Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}})$ is the homotopy
op-prederivator of $\Psh{\Delta\times \Delta}$ equipped with diagonal weak equivalences.
Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category
of bisimplicial sets can be equipped with a model structure whose weak of bisimplicial sets can be equipped with a model structure whose weak
equivalences are the diagonal weak equivalences and whose fibrations are the equivalences are the diagonal weak equivalences and whose fibrations are the
...@@ -1120,15 +1123,15 @@ of $2$\nbd{}categories. ...@@ -1120,15 +1123,15 @@ of $2$\nbd{}categories.
is \emph{not} \good{} (see \ref{paragr:bubble}). is \emph{not} \good{} (see \ref{paragr:bubble}).
\end{paragr} \end{paragr}
\begin{paragr} \begin{paragr}
For $n\geq 0$, we write $\Delta_n$ for the linear order ${0 \leq \cdots \leq For $n\geq 0$, we write $\Delta_n$ for the linear order $\{0 \leq \cdots \leq
n}$ seen as a small category. Let $i : \Delta_1 \to \Delta_n$ be the unique n\}$ seen as a small category. Let $i : \Delta_1 \to \Delta_n$ be the unique
functor such that functor such that
\[ \[
i(0)=0 \text{ and } i(1)=n. i(0)=0 \text{ and } i(1)=n.
\] \]
\end{paragr} \end{paragr}
\begin{lemma}\label{lemma:istrngdefrtract} \begin{lemma}\label{lemma:istrngdefrtract}
For $n\neq0$, the functor $i : \Delta_1 \to \Delta_n$ is a strong deformation For $n\neq0$, the functor $i : \Delta_1 \to \Delta_n$ is a strong oplax deformation
retract (\ref{paragr:defrtract}). retract (\ref{paragr:defrtract}).
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
...@@ -1156,7 +1159,7 @@ of $2$\nbd{}categories. ...@@ -1156,7 +1159,7 @@ of $2$\nbd{}categories.
\] \]
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$\nbd{}functor that sends the unique where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$\nbd{}functor that sends the unique
non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating
$2$\nbd{}cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong $2$\nbd{}cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong oplax
deformation retract and thus, a co-universal Thomason equivalence (Lemma deformation retract and thus, a co-universal Thomason equivalence (Lemma
\ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to
A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is
...@@ -1531,11 +1534,15 @@ Let us now get into more sophisticated examples. ...@@ -1531,11 +1534,15 @@ Let us now get into more sophisticated examples.
\item $G(A')=A$, \item $G(A')=A$,
\item $G(\gamma)=\alpha\comp_1\beta$. \item $G(\gamma)=\alpha\comp_1\beta$.
\end{itemize} \end{itemize}
Notice that we have $F\circ G = \mathrm{id}_{P'}$, which means that $P'$ is a Notice that we have $F\circ G = \mathrm{id}_{P'}$ and that we have an oplax
retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of transformation \[h \colon \mathrm{id}_{P} \Rightarrow G \circ F\] defined as
$\sH^{\sing}(P')$ and since $P'$ has the homotopy type of a \begin{itemize}[label=-]
$K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ has \item $h_A:=1_A$,
non-trivial singular homology groups in all even dimension. But since it is a \item $h_f:=\alpha$.
\end{itemize}
Hence, $F$ is a Thomason equivalence (Proposition \ref{prop:oplaxhmtpyisthom}) and $P$ has the homotopy type of a
$K(\mathbb{Z},2)$ (see \ref{paragr:bubble}). In particular, it has non-trivial
singular homology groups in all even dimension; but since it is a
free $2$\nbd{}category, all its polygraphic homology groups are trivial strictly above free $2$\nbd{}category, all its polygraphic homology groups are trivial strictly above
dimension $2$, which means that $P$ is \emph{not} \good{}. dimension $2$, which means that $P$ is \emph{not} \good{}.
\end{paragr} \end{paragr}
...@@ -1703,7 +1710,7 @@ Now let $\sS_2$ be labelled as ...@@ -1703,7 +1710,7 @@ Now let $\sS_2$ be labelled as
$\sS_2$. Notice that we have a cocartesian square $\sS_2$. Notice that we have a cocartesian square
\begin{equation}\label{square:bouquet} \begin{equation}\label{square:bouquet}
\begin{tikzcd} \begin{tikzcd}
\sD_1 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] & \sD_2 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] &
P' \ar[d] \\ P' \ar[d] \\
P'' \ar[r] & P, \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] P'' \ar[r] & P, \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd} \end{tikzcd}
...@@ -1712,19 +1719,19 @@ Now let $\sS_2$ be labelled as ...@@ -1712,19 +1719,19 @@ Now let $\sS_2$ be labelled as
show that the square induced by the nerve show that the square induced by the nerve
\[ \[
\begin{tikzcd} \begin{tikzcd}
N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta N_{\oo}(\sD_2) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta
\rangle"] & \rangle"] &
N_{\oo}(P') \ar[d] \\ N_{\oo}(P') \ar[d] \\
N_{\oo}(P'') \ar[r] & N_{\oo}(P) N_{\oo}(P'') \ar[r] & N_{\oo}(P)
\end{tikzcd} \end{tikzcd}
\] \]
is also cocartesian. Since $\langle \beta \rangle : \sD_1 \to P'$ and $\langle is also cocartesian. Since $\langle \beta \rangle : \sD_2 \to P'$ and $\langle
\beta \rangle : \sD_1 \to P''$ are monomorphisms and $N_{\oo}$ preserves \beta \rangle : \sD_2 \to P''$ are monomorphisms and $N_{\oo}$ preserves
monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that
square \eqref{square:bouquet} is Thomason homotopy cocartesian and in square \eqref{square:bouquet} is Thomason homotopy cocartesian and in
particular that $P$ has the homotopy type of a bouquet of two particular that $P$ has the homotopy type of a bouquet of two
$2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are free and \good{} and since $2$\nbd{}spheres. Since $\sD_2$, $P'$ and $P''$ are free and \good{} and since
$\langle \beta \rangle : \sD_1 \to P'$ and $\langle \beta \rangle : \sD_1 \to $\langle \beta \rangle : \sD_2 \to P'$ and $\langle \beta \rangle : \sD_2 \to
P''$ are folk cofibrations, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}). P''$ are folk cofibrations, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}).
\end{paragr} \end{paragr}
\begin{paragr} \begin{paragr}
...@@ -1905,7 +1912,7 @@ Now let $\sS_2$ be labelled as ...@@ -1905,7 +1912,7 @@ Now let $\sS_2$ be labelled as
\item $(1_f,\cdots,1_f)$, \item $(1_f,\cdots,1_f)$,
\item $(1_g,\cdots,1_g)$, \item $(1_g,\cdots,1_g)$,
\end{itemize} \end{itemize}
whose generating arrows from $B$ to $C$ are $k$\nbd{}tuple of one of the whose arrows from $B$ to $C$ are $k$\nbd{}tuple of one of the
following form following form
\begin{itemize}[label=-] \begin{itemize}[label=-]
\item $(1_h,\cdots,1_h,\gamma,1_i,\cdots,1_i)$, \item $(1_h,\cdots,1_h,\gamma,1_i,\cdots,1_i)$,
...@@ -1950,9 +1957,9 @@ homotopy type of the torus. ...@@ -1950,9 +1957,9 @@ homotopy type of the torus.
f \cdots fg\cdots g f \cdots fg\cdots g
\] \]
where $f$ is repeated $n$ times and $g$ is repeated $m$ times. where $f$ is repeated $n$ times and $g$ is repeated $m$ times.
Recall that the category $B^1(\mathbb{N}\times\mathbb{N})$ is nothing but the Recall that we write $B^1(\mathbb{N}\times\mathbb{N})$ for the
monoid $\mathbb{N}\times\mathbb{N}$ considered as a category with only one monoid $\mathbb{N}\times\mathbb{N}$ considered as a category with only one
object, and let $F : P \to B^1(\mathbb{N}\times\mathbb{N})$ be the unique object, and let \[F : P \to B^1(\mathbb{N}\times\mathbb{N})\] be the unique
$2$\nbd{}functor such that: $2$\nbd{}functor such that:
\begin{itemize}[label=-] \begin{itemize}[label=-]
\item $F(f)=(1,0)$ and $F(g)=(0,1)$, \item $F(f)=(1,0)$ and $F(g)=(0,1)$,
...@@ -2073,7 +2080,7 @@ homotopy type of the torus. ...@@ -2073,7 +2080,7 @@ homotopy type of the torus.
\begin{itemize}[label=-] \begin{itemize}[label=-]
\item $x$ is not a unit, \item $x$ is not a unit,
\item $\src_0(x)=\trgt_0(x)$, \item $\src_0(x)=\trgt_0(x)$,
\item $\trgt(x)=\src(x)=1_{\src_0(x)}$. \item $\trgt_1(x)=\src_1(x)=1_{\src_0(x)}$.
\end{itemize} \end{itemize}
\end{definition} \end{definition}
\begin{paragr} \begin{paragr}
......
...@@ -2,7 +2,8 @@ ...@@ -2,7 +2,8 @@
consequences} consequences}
\chaptermark{Contractible $\omega$-categories and consequences} \chaptermark{Contractible $\omega$-categories and consequences}
\section{Contractible \texorpdfstring{$\oo$}{ω}-categories} \section{Contractible \texorpdfstring{$\oo$}{ω}-categories}
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} \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}). 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}).
...@@ -60,7 +61,7 @@ Consider the commutative square ...@@ -60,7 +61,7 @@ Consider the commutative square
\[ \[
\begin{tikzcd} \begin{tikzcd}
\sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\ \sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0). \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sD_0}"] & \sH^{\sing}(\sD_0).
\end{tikzcd} \end{tikzcd}
\] \]
It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and
...@@ -96,9 +97,9 @@ We end this section with an important result on slice $\oo$\nbd{}categories (Par ...@@ -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 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.
%% \[ %% \[
%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)}, %% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},
%% \] %% \]
...@@ -421,7 +422,7 @@ higher than $1$. ...@@ -421,7 +422,7 @@ higher than $1$.
\] \]
is commutative. This proves the existence part of the universal property. 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} \begin{tikzcd}
X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\ X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\
...@@ -648,7 +649,7 @@ We now recall an important theorem due to Thomason. ...@@ -648,7 +649,7 @@ We now recall an important theorem due to Thomason.
\begin{remark} \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. 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} \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.
\begin{theorem}\label{thm:categoriesaregood} \begin{theorem}\label{thm:categoriesaregood}
Every $1$\nbd{}category is \good{}. Every $1$\nbd{}category is \good{}.
\end{theorem} \end{theorem}
......
...@@ -1222,7 +1222,7 @@ As a consequence of this lemma, we have the analogous of Proposition \ref{prop:t ...@@ -1222,7 +1222,7 @@ As a consequence of this lemma, we have the analogous of Proposition \ref{prop:t
\] \]
where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. It follows from Lemma \ref{lemma:unitajdcomp} that where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. It follows from Lemma \ref{lemma:unitajdcomp} that
\[ \[
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K')) H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to H_k(\iota_n\tau^{i}_{\leq n}(K'))
\] \]
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. 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} \end{proof}
...@@ -1370,7 +1370,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi ...@@ -1370,7 +1370,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi
for every $0 \leq k \leq n$. for every $0 \leq k \leq n$.
\end{corollary} \end{corollary}
\begin{proof} \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
\[ \[
\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)). \LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).
\] \]
...@@ -1421,7 +1421,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f ...@@ -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. 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} \end{proof}
\begin{paragr} \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 $n \geq 0$, 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 $n \geq 0$, a canonical morphism of $n\Cat$
\[ \[
c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C), c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C),
\] \]
...@@ -1449,7 +1449,7 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t ...@@ -1449,7 +1449,7 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
&\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))). &\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
\end{align*} \end{align*}
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 $\Psh{\Delta_{\leq 2}}$ consists of a function $F_0 : C_0 \to D_0$ and a function $F_1 : C_1 \to D_1$ such that 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 $\Psh{\Delta_{\leq 2}}$ 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 $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,
\item for every $x \in C_1$, we have \item for every $x \in C_1$, we have
\[\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\] \[\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\]
...@@ -1462,8 +1462,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t ...@@ -1462,8 +1462,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
\end{tikzcd} \end{tikzcd}
\] \]
in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$. in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$.
\end{itemize} \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 $F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism 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
\[ \[
\Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D). \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D).
\] \]
...@@ -1481,7 +1484,7 @@ We can now prove the important following proposition. ...@@ -1481,7 +1484,7 @@ We can now prove the important following proposition.
\begin{proposition}\label{prop:singhmlgylowdimension} \begin{proposition}\label{prop:singhmlgylowdimension}
For every $\oo$\nbd{}category $C$, the canonical map of $\ho(\Ch)$ 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 induces isomorphisms
\[ \[
...@@ -1490,7 +1493,7 @@ We can now prove the important following proposition. ...@@ -1490,7 +1493,7 @@ We can now prove the important following proposition.
for $k \in \{0,1\}$. for $k \in \{0,1\}$.
\end{proposition} \end{proposition}
\begin{proof} \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
\[ \[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C) \lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\] \]
...@@ -1502,7 +1505,7 @@ We can now prove the important following proposition. ...@@ -1502,7 +1505,7 @@ We can now prove the important following proposition.
\[ \[
\tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C). \tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C).
\] \]
This means exactly that the image by $\overline{\tau^{i}_{\leq 1}}$ of $\alpha^{\sing}$ is an isomorphism, which is what we wanted to prove. This means exactly that the image by $\overline{\tau^{i}_{\leq 1}}$ of $\alpha_C^{\sing}$ is an isomorphism, which is what we wanted to prove.
\end{proof} \end{proof}
Finally, we obtain the result we were aiming for. Finally, we obtain the result we were aiming for.
\begin{proposition}\label{prop:comphmlgylowdimension} \begin{proposition}\label{prop:comphmlgylowdimension}
...@@ -1520,11 +1523,11 @@ Finally, we obtain the result we were aiming for. ...@@ -1520,11 +1523,11 @@ Finally, we obtain the result we were aiming for.
Let $C$ be an $\oo$\nbd{}category and consider the following commutative triangle of $\ho(\Ch)$ Let $C$ be an $\oo$\nbd{}category and consider the following commutative triangle of $\ho(\Ch)$
\[ \[
\begin{tikzcd}[column sep=tiny] \begin{tikzcd}[column sep=tiny]
\sH^{\sing}(C) \ar[rd,"\alpha^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha^{\pol}"] \\ \sH^{\sing}(C) \ar[rd,"\alpha_C^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha_C^{\pol}"] \\
&\lambda(C)&. &\lambda(C)&.
\end{tikzcd} \end{tikzcd}
\] \]
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} \end{proof}
\begin{paragr}\label{paragr:conjectureH2} \begin{paragr}\label{paragr:conjectureH2}
A natural question following the above proposition is: A natural question following the above proposition is:
......
No preview for this file type
...@@ -9,7 +9,7 @@ ...@@ -9,7 +9,7 @@
publisher={Narnia} publisher={Narnia}
} }
@article{ara2014vers, @article{ara2014vers,
title={Vers une structure de cat{\'e}gorie de mod{\`e}les {\`a} la {T}homason sur la cat{\'e}gorie des n\nbd{}cat{\'e}gories strictes}, title={Vers une structure de cat{\'e}gorie de mod{\`e}les {\`a} la {T}homason sur la cat{\'e}gorie des $n$\nbd{}cat{\'e}gories strictes},
author={Ara, Dimitri and Maltsiniotis, Georges}, author={Ara, Dimitri and Maltsiniotis, Georges},
journal={Advances in Mathematics}, journal={Advances in Mathematics},
volume={259}, volume={259},
...@@ -40,7 +40,7 @@ ...@@ -40,7 +40,7 @@
year={2020} year={2020}
} }
@article{ara2020theoreme, @article{ara2020theoreme,
title={Un th{\'e}or{\`e}me {A} de {Q}uillen pour les $\infty$-cat{\'e}gories strictes {II} : la preuve $\infty$-cat{\'e}gorique}, title={Un th{\'e}or{\`e}me {A} de {Q}uillen pour les $\infty$\nbd{}cat{\'e}gories strictes {II} : la preuve $\infty$-cat{\'e}gorique},
author={Ara, Dimitri and Maltsiniotis, Georges}, author={Ara, Dimitri and Maltsiniotis, Georges},
volume={4}, volume={4},
number={1}, number={1},
......
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