Commit 02a5625a authored by Leonard Guetta's avatar Leonard Guetta
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Edited a lot of typos. Added a .gitignore file

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*.bbl
*.aux
*.blg
*.pls
*.fdb_latexmk
*.synctex.gz
*.toc
*.out
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*.fls
*.log
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......@@ -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 $k<n$. For $n=0$, this means that $\lambda_0(C)=\mathbb{Z}C_0$. Now let $f : C \to D$ be an $\oo$\nbd{}functor. For every $n \geq 0$, the definition of $\oo$\nbd{}functor implies that the map
for all $x,y \in C_n$ that are $k$\nbd{}composable for some $k<n$. For $n=0$,
this means that $\lambda_0(C)=\mathbb{Z}C_0$. Now let $f : C \to D$ be an
$\oo$\nbd{}functor. For every $n \geq 0$, the definition of $\oo$\nbd{}functor
implies that the linear map
\begin{align*}
\mathbb{Z}C_n &\to \mathbb{Z}D_{n}\\
x \in C_n &\mapsto f(x)
\end{align*}
induces a map
induces a linear map
\[
\lambda_n(f) : \lambda_n(C) \to \lambda_n(D).
\]
......@@ -180,7 +183,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\Hom_{n\Cat}(C,B^nG) &\to \Hom_{\Set}(C_n,\vert G \vert)\\
F &\mapsto F_n,
\end{align*}
where $\vert G \vert$ is the underlying set of $G$, is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:
is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:
\begin{enumerate}[label=(\roman*)]
\item\label{cond:comp} for every $0 \leq k <n $ and every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells of $C$, we have
\[
......@@ -224,7 +227,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\[
\mathbb{Z}\Sigma_n \to \lambda_n(C)
\]
from the previous paragraph is an isomorphism.
from the previous paragraph, is an isomorphism.
\end{lemma}
\begin{proof}
%% Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:
......@@ -273,7 +276,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
w_{\alpha}(x\comp_k y)=w_{\alpha}(x) + w_{\alpha}(y).
\]
\end{enumerate}
We can then define for each $n >0$, 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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