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
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},
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 $f_0 :
G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and
\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
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
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.
\]
This square is cocartesian because $i_!$ is a left adjoint. Since
$i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the
result follows from the fact that the monomorphisms are the
cofibrations of the standard Quillen model structure on simplicial
sets and from Lemma \ref{lemma:hmtpycocartesianreedy}.
result follows from Lemma \ref{lemma:hmtpycocartesianreedy} and the fact that
the monomorphisms are the cofibrations of the standard Quillen model structure on simplicial
sets.
\end{proof}
\begin{paragr}
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.
\]
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
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 $\Rgrph$.
By forming successive cocartesian squares and combining with the square
......@@ -482,10 +483,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\geq 0$, we use the notations
\begin{align*}
X_{n,m} &:= X([n],[m]) \\
\partial_i^h &:=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}\\
s_i^h &:=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}.
\partial_i^v &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\
\partial_j^h &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\
s_i^v &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\
s_j^h&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}.
\end{align*}
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
......@@ -548,8 +549,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
definition, $\delta^*$ induces a morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
\Ho(\Psh{\Delta}).
\Ho(\Psh{\Delta}),
\]
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
of bisimplicial sets can be equipped with a model structure whose weak
equivalences are the diagonal weak equivalences and whose fibrations are the
......@@ -1120,15 +1123,15 @@ of $2$\nbd{}categories.
is \emph{not} \good{} (see \ref{paragr:bubble}).
\end{paragr}
\begin{paragr}
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
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
functor such that
\[
i(0)=0 \text{ and } i(1)=n.
\]
\end{paragr}
\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}).
\end{lemma}
\begin{proof}
......@@ -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
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
\ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to
A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is
......@@ -1531,11 +1534,15 @@ Let us now get into more sophisticated examples.
\item $G(A')=A$,
\item $G(\gamma)=\alpha\comp_1\beta$.
\end{itemize}
Notice that we have $F\circ G = \mathrm{id}_{P'}$, which means that $P'$ is a
retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of
$\sH^{\sing}(P')$ and since $P'$ has the homotopy type of a
$K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ has
non-trivial singular homology groups in all even dimension. But since it is a
Notice that we have $F\circ G = \mathrm{id}_{P'}$ and that we have an oplax
transformation \[h \colon \mathrm{id}_{P} \Rightarrow G \circ F\] defined as
\begin{itemize}[label=-]
\item $h_A:=1_A$,
\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
dimension $2$, which means that $P$ is \emph{not} \good{}.
\end{paragr}
......@@ -1703,7 +1710,7 @@ Now let $\sS_2$ be labelled as
$\sS_2$. Notice that we have a cocartesian square
\begin{equation}\label{square:bouquet}
\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[r] & P, \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
......@@ -1712,19 +1719,19 @@ Now let $\sS_2$ be labelled as
show that the square induced by the nerve
\[
\begin{tikzcd}
N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta
N_{\oo}(\sD_2) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta
\rangle"] &
N_{\oo}(P') \ar[d] \\
N_{\oo}(P'') \ar[r] & N_{\oo}(P)
\end{tikzcd}
\]
is also cocartesian. Since $\langle \beta \rangle : \sD_1 \to P'$ and $\langle
\beta \rangle : \sD_1 \to P''$ are monomorphisms and $N_{\oo}$ preserves
is also cocartesian. Since $\langle \beta \rangle : \sD_2 \to P'$ and $\langle
\beta \rangle : \sD_2 \to P''$ are monomorphisms and $N_{\oo}$ preserves
monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that
square \eqref{square:bouquet} is Thomason homotopy cocartesian and in
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
$\langle \beta \rangle : \sD_1 \to P'$ and $\langle \beta \rangle : \sD_1 \to
$2$\nbd{}spheres. Since $\sD_2$, $P'$ and $P''$ are free and \good{} and since
$\langle \beta \rangle : \sD_2 \to P'$ and $\langle \beta \rangle : \sD_2 \to
P''$ are folk cofibrations, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}).
\end{paragr}
\begin{paragr}
......@@ -1905,7 +1912,7 @@ Now let $\sS_2$ be labelled as
\item $(1_f,\cdots,1_f)$,
\item $(1_g,\cdots,1_g)$,
\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
\begin{itemize}[label=-]
\item $(1_h,\cdots,1_h,\gamma,1_i,\cdots,1_i)$,
......@@ -1950,9 +1957,9 @@ homotopy type of the torus.
f \cdots fg\cdots g
\]
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
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:
\begin{itemize}[label=-]
\item $F(f)=(1,0)$ and $F(g)=(0,1)$,
......@@ -2073,7 +2080,7 @@ homotopy type of the torus.
\begin{itemize}[label=-]
\item $x$ is not a unit,
\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{definition}
\begin{paragr}
......
......@@ -2,7 +2,8 @@
consequences}
\chaptermark{Contractible $\omega$-categories and consequences}
\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}
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
\[
\begin{tikzcd}
\sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0).
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sD_0}"] & \sH^{\sing}(\sD_0).
\end{tikzcd}
\]
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
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.
%% \[
%% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},
%% \]
......@@ -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'"] \\
......@@ -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.
\begin{theorem}\label{thm:categoriesaregood}
Every $1$\nbd{}category is \good{}.
\end{theorem}
......
......@@ -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
\[
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K'))
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to H_k(\iota_n\tau^{i}_{\leq n}(K'))
\]
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.
\end{proof}
......@@ -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
\[
\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).
\]
......@@ -1421,7 +1421,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
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 $n \geq 0$, 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 $n\Cat$
\[
c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C),
\]
......@@ -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))).
\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
\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
\[\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\]
......@@ -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
\[
\Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D).
\]
......@@ -1481,7 +1484,7 @@ We can now prove the important following proposition.
\begin{proposition}\label{prop:singhmlgylowdimension}
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
\[
......@@ -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
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
......@@ -1502,7 +1505,7 @@ We can now prove the important following proposition.
\[
\tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C).
\]
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}
Finally, we obtain the result we were aiming for.
\begin{proposition}\label{prop:comphmlgylowdimension}
......@@ -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 $\ho(\Ch)$
\[
\begin{tikzcd}[column sep=tiny]
\sH^{\sing}(C) \ar[rd,"\alpha^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha^{\pol}"] \\
\sH^{\sing}(C) \ar[rd,"\alpha_C^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha_C^{\pol}"] \\
&\lambda(C)&.
\end{tikzcd}
\]
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}
\begin{paragr}\label{paragr:conjectureH2}
A natural question following the above proposition is:
......
No preview for this file type
......@@ -9,7 +9,7 @@
publisher={Narnia}
}
@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},
journal={Advances in Mathematics},
volume={259},
......@@ -40,7 +40,7 @@
year={2020}
}
@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},
volume={4},
number={1},
......
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