Commit 02a5625a by Leonard Guetta

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

parent 65ac5cc0
.gitignore 0 → 100644
 *.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 $G$, ... @@ -113,12 +113,15 @@ We write $\Ab$ for the category of abelian groups and for an abelian group $G$, $\[ x \comp_k y \sim x+y x \comp_k y \sim x+y$ \] 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 $w_n(x)=\sum_{\alpha \in \Sigma_n}w_{\alpha}(x)\cdot \alpha$ $w_n(x)=\sum_{\alpha \in \Sigma_n}w_{\alpha}(x)\cdot \alpha$ for every $x \in C_n$. for every $x \in C_n$. ... ...
 ... @@ -45,7 +45,7 @@ ... @@ -45,7 +45,7 @@ \begin{description} \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. \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} \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: Here are some pictures in low dimension: $\[ \Or_0 = \langle 0 \rangle, \Or_0 = \langle 0 \rangle, ... @@ -231,7 +231,7 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with ... @@ -231,7 +231,7 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with \begin{remark} \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. 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} \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} \begin{proposition}\label{prop:nthomeqder} For all $1 \leq n \leq m \leq \omega$, the canonical morphism 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 ... @@ -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, \item for all 0\leq k < n, for all n\nbd{}cells x and y of X that are k-composable, \[ \[ \begin{multlined} \begin{multlined} \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}\comp_k\alpha_y\right)}\\ \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(y)}\comp_1\cdots\comp_{k-1}\alpha_{\src_{k-1}(y)}\comp_k\alpha_y\right)}\\ {\comp_{k+1}\left(\alpha_{\trgt_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}. {\comp_{k+1}\left(\alpha_x \comp_k\alpha_{\trgt_{k-1}(x)}\comp_{k-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}. \end{multlined} \end{multlined}$ \] \end{enumerate} \end{enumerate} ... @@ -398,7 +398,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with ... @@ -398,7 +398,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \begin{definition}\label{def:oplaxhmtpyequiv} \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 $\mathrm{id}_C$. 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} \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} \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 $\gamma^{\Th}(u)=\gamma^{\Th}(v)$. 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} \end{lemma} ... @@ -464,10 +464,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with ... @@ -464,10 +464,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \begin{tikzcd} \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 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[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} \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 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 ... @@ -575,7 +575,9 @@ For later reference, we put here the following trivial but important lemma, whos \end{example} \end{example} \begin{theorem}\label{thm:folkms} \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} \end{theorem} \begin{proof} \begin{proof} This is the main result of \cite{lafont2010folk}. 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 ... @@ -748,7 +750,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen & (x_2,a_{3}) & (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{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} \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: \[ \[ s(x,a)=\begin{pmatrix} s(x,a)=\begin{pmatrix} \begin{matrix} \begin{matrix} ... @@ -764,8 +766,8 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen ... @@ -764,8 +766,8 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') \end{matrix} \end{matrix} & (x'_{n-1},a'_{n}). & (x'_{n-1},a'_{n}) \end{pmatrix} \end{pmatrix}.$ \] % It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$ % 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: \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 ... @@ -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}) (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') & (x_n,a_{n+1}) \end{matrix} \end{matrix} & (1_{x_n},1_{a_{n+1}}) & (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: \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 ... @@ -935,7 +937,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen \end{enumerate} \end{enumerate} \end{proof} \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} \end{paragr} \begin{theorem}[Ara and Maltsiniotis' Theorem A] Let \begin{theorem}[Ara and Maltsiniotis' Theorem A] Let \[ \[ ... ...  ... @@ -43,7 +43,7 @@ is poorly behaved. For example, \fi ... @@ -43,7 +43,7 @@ is poorly behaved. For example, \fi \[ \[ \gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D) \gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)$ \] 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 $\D$. that send the morphisms of $\W$ to isomorphisms of $\D$. We shall always consider that $\C$ and $\ho(\C)$ have the same class of 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. ... @@ -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 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 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 the localization implies that there exists a unique natural transformation $\[ \begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}] \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. ... @@ -147,9 +147,8 @@ For later reference, we put here the following definition. denoted by \RR F when it exists. denoted by \RR F when it exists. \end{paragr} \end{paragr} \begin{example}\label{rem:homotopicalisder} \begin{example}\label{rem:homotopicalisder} Let (\C,\W) and (\C',\W') be two localizers and F: \C \to \C' be a Let F : (\C,\W) \to (\C',\W') a morphism of localizers. The universal functor. If F preserves weak equivalences (i.e.\ it is a morphism of property of the localization implies that F is localizers) then the universal property of localization implies that F is absolutely totally left and right derivable and \LL F \simeq \RR F \simeq absolutely totally left and right derivable and \LL F \simeq \RR F \simeq \overline{F}. \overline{F}. \end{example} \end{example} ... @@ -158,7 +157,7 @@ we shall use in the sequel. ... @@ -158,7 +157,7 @@ we shall use in the sequel. \begin{paragr}\label{paragr:prelimgonzalez} \begin{paragr}\label{paragr:prelimgonzalez} Let (\C,\W) and (\C',\W') be two localizers and let \begin{tikzcd} F : \C 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 \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 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 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 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. ... @@ -396,7 +395,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. \[ \[ \hocolim_A(X) \to \colim_A(X). \hocolim_A(X) \to \colim_A(X).$ \] This comparison map will be of great importance in the sequel. This canonical morphism will be of great importance in the sequel. \end{paragr} \end{paragr} \begin{paragr} \begin{paragr} Let Let ... @@ -683,8 +682,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. ... @@ -683,8 +682,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. \end{lemma} \end{lemma} \begin{lemma}\label{lemma:ladjcocontinuous} \begin{lemma}\label{lemma:ladjcocontinuous} Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions 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 and let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. If $F$ is a left adjoint (of a morphism $G : \sD' \to \sD$), then it is cocontinuous. adjoint, then it is cocontinuous. \end{lemma} \end{lemma} We end this section with a generalization of the notion of localization in the We end this section with a generalization of the notion of localization in the context of op\nbd{}prederivators. context of op\nbd{}prederivators. ... @@ -824,7 +823,8 @@ We now turn to the most important way of obtaining 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"] &e\ar[ru,"{(1,1)}"']&, \ar[from=A,to=2-2,Rightarrow,"\alpha"] \end{tikzcd} \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 $2$\nbd{}triangle $\[ \begin{tikzcd} \begin{tikzcd} ... @@ -921,7 +921,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. ... @@ -921,7 +921,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. D \ar[r,"w"]&E \ar[r,"x"] & F D \ar[r,"w"]&E \ar[r,"x"] & F \end{tikzcd} \end{tikzcd}$ \] 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 then the outer square is cocartesian if and only if the right square is cocartesian. cocartesian. \end{lemma} \end{lemma} ... ...
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