Commit 23674342 authored by Leonard Guetta's avatar Leonard Guetta
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intro relue par chait

parent 514e1420
......@@ -1174,7 +1174,7 @@ following proposition.
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or
$n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point.
If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a
$K(\mathbb{Z},2)$\nbd{}space.
$K(\mathbb{Z},2)$.
\end{proposition}
\section{Zoology of $2$-categories: more examples}
As a warm-up, let us begin with an example which is direct consequence of the
......@@ -1482,7 +1482,7 @@ Let us now get into more sophisticated examples.
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
$K(\mathbb{Z},2)$-space (\ref{paragr:bubble}), this proves that $P$ have
$K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ have
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 above
dimension $2$, which means that $P$ is \emph{not} \good{}.
......
......@@ -641,7 +641,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\[
H^{\pol}_k(C)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}
\]
On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism).
On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism).
\end{paragr}
Another consequence of the above counter-example is the following result, which we claimed in \ref{paragr:polhmlgythomeq}. Recall that given a morphism $u : C \to D$ of $\oo\Cat$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$.
\begin{proposition}\label{prop:polhmlgynotinvariant}
......@@ -1130,7 +1130,7 @@ We now turn to truncations of chain complexes.
\]
The isomorphism being obviously induced by the unit map $K \to \iota_n\tau^{i}_{\leq n}(K)$.
\end{proof}
As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:truncationhomotopical}.
As a consequence of this lemma, we have the analogous of Proposition \ref{prop:truncationhomotopical}.
\begin{proposition}
The functor $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq n}$ sends weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.
\end{proposition}
......@@ -1216,7 +1216,7 @@ is commutative.
\]
Then, it follows from an argument similar to the proof of Lemma \ref{lemma:nfunctortomonoid} (see also the proof of Lemma \ref{lemma:adjlambdasusp}) that this last set is naturally isomorphic to the set of functions $f_n : C_n \to G$ such that:
\begin{itemize}[label=-]
\item for every $0 \leq k <n $ and every pair $(x,y)$ of $k$-composables $n$\nbd{}cells of $C$, we have
\item for every $0 \leq k <n $ and every pair $(x,y)$ of $k$-composable $n$\nbd{}cells of $C$, we have
\[
f(x \comp_k y) = f(x) + f(y),
\]
......@@ -1285,7 +1285,7 @@ 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 morhism $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ 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 $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
\[
\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).
\]
......@@ -1444,7 +1444,7 @@ Finally, we obtain the result we were aiming for.
\[
H_{2p}^{\sing}(B^2\mathbb{N}) \not\simeq H^{\pol}_{2p}(B^2\mathbb{N})
\]
for every $p \geq 2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq 3$, the (nerve of the) $\oo$\nbd{}category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$-space. In particular, we have
for every $p \geq 2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq 3$, the (nerve of the) $\oo$\nbd{}category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$. In particular, we have
\[
H_{2p+3}^{\sing}(B^{2p +1}\mathbb{N})\simeq \mathbb{Z}/{2\mathbb{Z}}
\]
......
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\documentclass[12pt,a4paper,draft]{report}
\usepackage[final]{hyperref}
\usepackage{mystyle}
\iffalse
......@@ -11,7 +13,7 @@
\fi
%%% For line numbering (used for proodreading purposes)
\usepackage[pagewise]{lineno}
\usepackage[pagewise,displaymath, mathlines]{lineno}
\linenumbers
......
......@@ -11,7 +11,8 @@
% url, references
\usepackage{url}
%\usepackage{url}
% Maths packages
......@@ -54,18 +55,21 @@
\newtheorem*{definition*}{Definition} %Unnumbered definition
% customed theoremstyle for theorem style with name as argument
\declaretheoremstyle[
spaceabove=\topsep, spacebelow=\topsep,
headfont=\normalfont\bfseries,
notefont=\bfseries, notebraces={}{},
bodyfont=\normalfont,
postheadspace=0.5em,
name={\ignorespaces},
numbered=no,
headpunct=.]
{mystyle}
\declaretheorem[style=mystyle]{named}
% \declaretheoremstyle[
% spaceabove=\topsep, spacebelow=\topsep,
% headfont=\normalfont\bfseries,
% notefont=\bfseries, notebraces={}{},
% bodyfont=\normalfont,
% postheadspace=0.5em,
% name={\ignorespaces},
% numbered=no,
% headpunct=.]
% {mystyle}
% \declaretheorem[style=mystyle]{named}
\newtheoremstyle{mystyle}{}{}{}{}{\bfseries}{.}{.5em}{\thmnote{#3}#1}
\theoremstyle{mystyle}
\newtheorem*{named}{}
%%%%%% MACROS %%%%%%%
......
......@@ -632,7 +632,7 @@ Furthermore, this function satisfies the condition
is injective.
\end{lemma}
\begin{proof}
A thorough reading of thet echniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function
A thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function
\[
w_{x} : (\E^*)_{n+1} \to \mathbb{N}
\]
......@@ -1855,9 +1855,9 @@ We end this section with yet another characterisation of $n$-basis of $\oo$\nbd{
Let $n>0$, $C$ be an $n$\nbd{}category with an $n$\nbd{}basis $\Sigma$ and
$\alpha \in \Sigma$. It is immediate to check from the definition of
elementary moves that for two equivalent well formed words $u \sim u'$ of
$\T[\Sigma]$, the number of occurences of $\cc_{\alpha}$ in $u$ and $u'$ are
$\T[\Sigma]$, the number of occurrences of $\cc_{\alpha}$ in $u$ and $u'$ are
the same. In particular, for any $a \in C_n$, we can define the integer
$w_{\alpha}(a)$ to be the number of occurences of $\cc_{\alpha}$ in any well
$w_{\alpha}(a)$ to be the number of occurrences of $\cc_{\alpha}$ in any well
formed word $u$ such that $\rho_{\sigma}(u)=a$. An immediate induction using
the properties of $\rho_{\Sigma}$ shows that this function $w_{\alpha} : C_n \to \mathbb{N}$ is the same
as the one whose existence was established in Proposition
......
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