### Edited a lot of typos. Added a .gitignore file

parent 65ac5cc0
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 *.bbl *.aux *.blg *.pls *.fdb_latexmk *.synctex.gz *.toc *.out auto *.fls *.log *.pdf *.bak ! main.pdf \ No newline at end of file
 ... ... @@ -113,12 +113,15 @@ We write $\Ab$ for the category of abelian groups and for an abelian group $G$, $x \comp_k y \sim x+y$ for all $x,y \in C_n$ that are $k$\nbd{}composable for some $k0$, a map $w_n : C_n \to \mathbb{Z}\Sigma_n$ with the formula We can then define for each $n \geq 0$, a map $w_n : C_n \to \mathbb{Z}\Sigma_n$ with the formula $w_n(x)=\sum_{\alpha \in \Sigma_n}w_{\alpha}(x)\cdot \alpha$ for every $x \in C_n$. ... ...
 ... ... @@ -45,7 +45,7 @@ \begin{description} \item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composition of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cells appearing exactly once in the composite. \end{description} Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight (see \ref{paragr:weight}) of the $(n-1)$\nbd{}cells corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$. Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight (\ref{paragr:weight}) of the $(n-1)$\nbd{}cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$. Here are some pictures in low dimension: $\Or_0 = \langle 0 \rangle, ... ... @@ -231,7 +231,7 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with \begin{remark} Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on n\Cat with \W^{\Th}_n as the weak equivalences. For n=1, this was established by Thomason \cite{thomason1980cat}, and for n=2, by Ara and Maltsiniotis \cite{ara2014vers}. For n>3, the existence of such a model structure is conjectured but not yet established. \end{remark} By definition, for all 1 \leq n \leq m \leq \omega, the canonical inclusion \[n\Cat \hookrightarrow m\Cat$ sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivators $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$. By definition, for all $1 \leq n \leq m \leq \omega$, the canonical inclusion $n\Cat \hookrightarrow m\Cat$ sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivators $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th}).$ \begin{proposition}\label{prop:nthomeqder} For all $1 \leq n \leq m \leq \omega$, the canonical morphism $... ... @@ -334,8 +334,8 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with \item for all 0\leq k < n, for all n\nbd{}cells x and y of X that are k-composable, \[ \begin{multlined} \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}\comp_k\alpha_y\right)}\\ {\comp_{k+1}\left(\alpha_{\trgt_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}. \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(y)}\comp_1\cdots\comp_{k-1}\alpha_{\src_{k-1}(y)}\comp_k\alpha_y\right)}\\ {\comp_{k+1}\left(\alpha_x \comp_k\alpha_{\trgt_{k-1}(x)}\comp_{k-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}. \end{multlined}$ \end{enumerate} ... ... @@ -398,7 +398,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \begin{definition}\label{def:oplaxhmtpyequiv} An $\oo$\nbd{}functor $u : C \to D$ is an \emph{oplax homotopy equivalence} if there exists an $\oo$\nbd{}functor $v : D \to C$ such that $u\circ v$ is oplax homotopic to $\mathrm{id}_D$ and $v\circ u$ is oplax homotopic to $\mathrm{id}_C$. \end{definition} Recall that we write $\gamma^{\Th} : \oo\Cat \to \ho(\oo\Cat^{\Th})$ the localization functor with respect to the Thomason equivalences. Recall that we write $\gamma^{\Th} : \oo\Cat \to \ho(\oo\Cat^{\Th})$ for the localization functor with respect to the Thomason equivalences. \begin{lemma}\label{lemma:oplaxloc} Let $u, v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $\alpha : u \Rightarrow v$, then $\gamma^{\Th}(u)=\gamma^{\Th}(v)$. \end{lemma} ... ... @@ -464,10 +464,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \begin{tikzcd} \sD_1\otimes A \ar[r,"\sD_1\otimes u"] \ar[d,"\sD_1\otimes i"] & \sD_1 \otimes A' \ar[d,"\sD_1 \otimes i'"] \ar[dd,bend left=75,"p\otimes i'"] \\ \sD_1\otimes B \ar[d,"\alpha"] \ar[r,"\sD_1 \otimes v"] & \sD_1 \otimes B' \ar[d,"\alpha'",dashed ] \\ \sD_1 \otimes B \ar[r,"v"] & \sD_1 \otimes B'. B \ar[r,"v"] & B'. \end{tikzcd} \] The existence of $\alpha' : \sD_1 \otimes B' \to B'$ that makes the whole diagram commutes follows from the fact that the functor $\sD_1 \otimes \shortminus$ preserves colimits. In particular, we have $\alpha' \circ (\sD_1 \otimes i') = p \otimes i'$. The existence of $\alpha' : \sD_1 \otimes B' \to B'$ that makes the whole diagram commutes follows from the fact that the functor $\sD_1 \otimes \shortminus$ preserves colimits. In particular, we have $\alpha' \circ (\sD_1 \otimes i') = p \otimes i'.$ Now, notice that for every $\oo$\nbd{}category $C$, the maps $... ... @@ -575,7 +575,9 @@ For later reference, we put here the following trivial but important lemma, whos \end{example} \begin{theorem}\label{thm:folkms} There exists a cofibrantly generated model structure on \omega\Cat such that the weak equivalences are the equivalences of \oo\nbd{}categories, and the set \{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\} (see \ref{paragr:defglobe}) is a set of generating cofibrations. There exists a cofibrantly generated model structure on \omega\Cat whose weak equivalences are the equivalences of \oo\nbd{}categories, and whose cofibrations are generated by the set \{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\} (see \ref{paragr:defglobe}). \end{theorem} \begin{proof} This is the main result of \cite{lafont2010folk}. ... ... @@ -748,7 +750,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen & (x_2,a_{3}) \end{pmatrix}}:&{\begin{tikzcd}[column sep=small] x_0 \ar[rr,"x_1"] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2", shorten <=1em, shorten >=1em]\end{tikzcd}\; \overset{a_3}{\Lleftarrow} \; \begin{tikzcd}[column sep=small] x_0\ar[rr,bend left=50,"x_1",pos=11/20,""{name=toto,below}] \ar[rr,"x_1'"description,""{name=titi,above}] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2'", shorten <=1em, shorten >=1em] \ar[from=toto,to=titi,Rightarrow,"x_2",pos=1/5]\end{tikzcd}} \end{tabular} \item The source, target of the n\nbd{}cell (a,x) are given by the matrices: \item The source and target of the n\nbd{}cell (a,x) are given by the matrices: \[ s(x,a)=\begin{pmatrix} \begin{matrix} ... ... @@ -764,8 +766,8 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') \end{matrix} & (x'_{n-1},a'_{n}). \end{pmatrix} & (x'_{n-1},a'_{n}) \end{pmatrix}.$ % It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$ \item The unit of the $n$\nbd{}cell $(a,x)$ is given by the table: ... ... @@ -776,7 +778,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') & (x_n,a_{n+1}) \end{matrix} & (1_{x_n},1_{a_{n+1}}) \end{pmatrix} \end{pmatrix}. \] \item The composition of $n$\nbd{}cells $(x,a)$ and $(y,b)$ such that $\src_k(y,b)=\trgt_k(a,x)$, is given by the table: $... ... @@ -935,7 +937,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen \end{enumerate} \end{proof} \begin{paragr} The name folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its \oo\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one. \begin{paragr} The name folk Theorem A'' is an explicit reference to Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its \oo\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one. \end{paragr} \begin{theorem}[Ara and Maltsiniotis' Theorem A] Let \[ ... ...  ... ... @@ -43,7 +43,7 @@ is poorly behaved. For example, \fi \[ \gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)$ is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$ is fully faithful and its essential image consists of those functors $F~:~\C~\to~\D$ that send the morphisms of $\W$ to isomorphisms of $\D$. We shall always consider that $\C$ and $\ho(\C)$ have the same class of ... ... @@ -89,7 +89,7 @@ For later reference, we put here the following definition. \] is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of localizers. A \emph{$2$\nbd{}morphism of localizers} from $F$ to $G$ is simply a natural transformation $\alpha : F \Rightarrow G$. The universal property of a natural transformation ${\alpha : F \Rightarrow G}$. The universal property of the localization implies that there exists a unique natural transformation $\begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}] ... ... @@ -147,9 +147,8 @@ For later reference, we put here the following definition. denoted by \RR F when it exists. \end{paragr} \begin{example}\label{rem:homotopicalisder} Let (\C,\W) and (\C',\W') be two localizers and F: \C \to \C' be a functor. If F preserves weak equivalences (i.e.\ it is a morphism of localizers) then the universal property of localization implies that F is Let F : (\C,\W) \to (\C',\W') a morphism of localizers. The universal property of the localization implies that F is absolutely totally left and right derivable and \LL F \simeq \RR F \simeq \overline{F}. \end{example} ... ... @@ -158,7 +157,7 @@ we shall use in the sequel. \begin{paragr}\label{paragr:prelimgonzalez} Let (\C,\W) and (\C',\W') be two localizers and let \begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd} be an adjunction whose unit is denoted by \eta. Suppose that G is totally right derivable whose unit is denoted by \eta. Suppose that the functor G is totally right derivable with (\RR G,\beta) its total right derived functor and suppose that \RR G has a left adjoint F' : \ho(\C) \to \ho(\C'); the co-unit of this last adjunction being denoted by \epsilon'. All this data induces a natural ... ... @@ -396,7 +395,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. \[ \hocolim_A(X) \to \colim_A(X).$ This comparison map will be of great importance in the sequel. This canonical morphism will be of great importance in the sequel. \end{paragr} \begin{paragr} Let ... ... @@ -683,8 +682,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. \end{lemma} \begin{lemma}\label{lemma:ladjcocontinuous} Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions and $F : \sD \to \sD'$ a morphism of op\nbd{}prederivators. If $F$ is left adjoint (of a morphism $G : \sD' \to \sD$), then it is cocontinuous. and let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. If $F$ is a left adjoint, then it is cocontinuous. \end{lemma} We end this section with a generalization of the notion of localization in the context of op\nbd{}prederivators. ... ... @@ -824,7 +823,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. &e\ar[ru,"{(1,1)}"']&, \ar[from=A,to=2-2,Rightarrow,"\alpha"] \end{tikzcd} \] where we wrote $p$ instead of $p_{\ulcorner}$ for short. Hence, we have a where we wrote $p$ instead of $p_{\ulcorner}$ for short and where $\alpha$ is the unique such natural transformation. Hence, we have a $2$\nbd{}triangle $\begin{tikzcd} ... ... @@ -921,7 +921,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. D \ar[r,"w"]&E \ar[r,"x"] & F \end{tikzcd}$ be a commutative diagram in $\C$. If the square on the left is cocartesian be a commutative diagram in $\C$. If the square on the left is cocartesian, then the outer square is cocartesian if and only if the right square is cocartesian. \end{lemma} ... ...
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