### 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 $0 \leq k This diff is collapsed. No preview for this file type  \documentclass[12pt,a4paper,draft]{report} \usepackage[final]{hyperref} \usepackage{mystyle} \iffalse ... ... @@ -11,7 +13,7 @@ \fi %%% For line numbering (used for proodreading purposes) \usepackage[pagewise]{lineno} \usepackage[pagewise,displaymath, mathlines]{lineno} \linenumbers ... ...  ... ... @@ -11,7 +11,8 @@ % url, references \usepackage{url} %\usepackage{url} % Maths packages ... ... @@ -54,18 +55,21 @@ \newtheorem*{definition*}{Definition} %Unnumbered definition % customed theoremstyle for theorem style with name as argument \declaretheoremstyle[ spaceabove=\topsep, spacebelow=\topsep, headfont=\normalfont\bfseries, notefont=\bfseries, notebraces={}{}, bodyfont=\normalfont, postheadspace=0.5em, name={\ignorespaces}, numbered=no, headpunct=.] {mystyle} \declaretheorem[style=mystyle]{named} % \declaretheoremstyle[ % spaceabove=\topsep, spacebelow=\topsep, % headfont=\normalfont\bfseries, % notefont=\bfseries, notebraces={}{}, % bodyfont=\normalfont, % postheadspace=0.5em, % name={\ignorespaces}, % numbered=no, % headpunct=.] % {mystyle} % \declaretheorem[style=mystyle]{named} \newtheoremstyle{mystyle}{}{}{}{}{\bfseries}{.}{.5em}{\thmnote{#3}#1} \theoremstyle{mystyle} \newtheorem*{named}{} %%%%%% MACROS %%%%%%% ... ...  ... ... @@ -632,7 +632,7 @@ Furthermore, this function satisfies the condition is injective. \end{lemma} \begin{proof} A thorough reading of thet echniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining$\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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