Commit 1466d455 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

security commit

parent 2a1fccac
......@@ -51,10 +51,10 @@ is poorly behaved. For example, \fi
\[
\gamma(X)=X
\]
for any object $X$ of $\C$.
for every object $X$ of $\C$.
The class of arrows $\W$ is said to be \emph{saturated} when we have the
property
property:
\[
f \in \W \text{ if and only if } \gamma(f) \text{ is an isomorphism. }
\]
......@@ -67,14 +67,14 @@ For later reference, we put here the following definition.
\[
\begin{tikzcd}
X \ar[r] \ar[d,"f"] & X' \ar[d,"f'"] \\
Y \ar[r] & Y' \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Y \ar[r] & Y', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
the morphism $f'$ is also a weak equivalence.
\end{definition}
\begin{paragr}
A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor
$F:\C\to\C'$ that preserves weak equivalences, i.e. such that $F(\W) \subseteq
$F:\C\to\C'$ that preserves weak equivalences, i.e.\ such that $F(\W) \subseteq
\W'$. The universal property of the localization implies that $F$ induces a
canonical functor
\[
......@@ -88,7 +88,7 @@ For later reference, we put here the following definition.
\end{tikzcd}
\]
is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of
localizers. A \emph{2-morphism of localizers} from $F$ to $G$ is simply a
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
the localization implies that there exists a canonical natural transformation
\[
......@@ -96,7 +96,7 @@ For later reference, we put here the following definition.
\ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C')
\ar[from=A,to=B,Rightarrow,"\overline{\alpha}"]\end{tikzcd}
\]
such that the 2-diagram
such that the $2$\nbd{}diagram
\[
\begin{tikzcd}[row sep=huge]
\C\ar[d,"\gamma"] \ar[r,bend left,"F",""{name=A,below}] \ar[r,bend right,"G"',""{name=B,above}] & \C'\ar[d,"\gamma'"] \ar[from=A,to=B,Rightarrow,"\alpha"] \\
......@@ -108,7 +108,7 @@ For later reference, we put here the following definition.
is commutative in an obvious sense.
\end{paragr}
\begin{remark}\label{remark:localizedfunctorobjects}
Since we always consider that for any localizer $(\C,\W)$ the categories $\C$
Since we always consider that for every localizer $(\C,\W)$ the categories $\C$
and $\ho(\C)$ have the same objects and the localization functor is the
identity on objects, it follows that for a morphism of localizer ${F : (\C,\W)
\to (\C',\W')}$, we tautologically have
......@@ -143,7 +143,7 @@ For later reference, we put here the following definition.
$F$}. Often we will abusively discard $\alpha$ and simply refer to $\LL F$
as the total left derived functor of $F$.
The notion of total right derivable functor is defined dually and denoted by
The notion of \emph{total right derivable functor} is defined dually and denoted by
$\RR F$ when it exists.
\end{paragr}
\begin{example}\label{rem:homotopicalisder}
......@@ -180,7 +180,7 @@ we shall use in the sequel.
3.1]{gonzalez2012derivability}}]\label{prop:gonz}
Let $(\C,\W)$ and $(\C',\W')$ be two localizers and
\[\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] :
G \end{tikzcd}\] an adjunction. If $G$ is absolutely totally right
G \end{tikzcd}\] be an adjunction. If $G$ is absolutely totally right
derivable with $(\RR G,\beta)$ its left derived functor and if $\RR G$ has a
left adjoint $F'$
\[\begin{tikzcd} F' : \ho(\C) \ar[r,shift left] & \ho(\C') \ar[l,shift left] :
......@@ -192,10 +192,10 @@ we shall use in the sequel.
% faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.}
\section{(op)Derivators and homotopy colimits}
\begin{notation}We denote by $\CCat$ the $2$-category of small categories and
$\CCAT$ the $2$-category of big categories. For a $2$-category
$\underline{A}$, the $2$-category obtained from $\underline{A}$ by switching
the source and targets of $1$-cells is denoted by $\underline{A}^{op}$.
\begin{notation}We denote by $\CCat$ the $2$\nbd{}category of small categories and
$\CCAT$ the $2$\nbd{}category of big categories. For a $2$\nbd{}category
$\underline{A}$, the $2$\nbd{}category obtained from $\underline{A}$ by switching
the source and targets of $1$-cells is denoted by $\underline{A}^{\op}$.
The terminal category, i.e.\ the category with only one object and no
non-trivial arrows, is canonically denoted by $e$. For a (small) category $A$,
......@@ -205,7 +205,7 @@ we shall use in the sequel.
\]
\end{notation}
\begin{definition}
An \emph{op-prederivator} is a (strict) $2$-functor
An \emph{op-prederivator} is a (strict) $2$\nbd{}functor
\[\sD : \CCat^{op} \to \CCAT.\]
More explicitly, an op-prederivator consists of the data of:
\begin{itemize}[label=-]
......@@ -262,14 +262,14 @@ we shall use in the sequel.
Note that some authors call \emph{prederivator} what we have called
\emph{op-prederivator}. The terminology we chose in the above definition is
compatible with the original one of Grothendieck, who called
\emph{prederivator} a $2$-functor from $\CCat$ to $\CCAT$ that is
contravariant at the level of $1$-cells \emph{and} at the level of $2$-cells.
\emph{prederivator} a $2$\nbd{}functor from $\CCat$ to $\CCAT$ that is
contravariant at the level of $1$-cells \emph{and} at the level of $2$\nbd{}cells.
\end{remark}
\begin{example}\label{ex:repder}
Let $\C$ be a category. For a small category $A$, we use the notation $\C(A)$
for the category $\underline{\Hom}(A,\C)$ of functors $A \to \C$ and natural
transformations between them. The correspondence $A \mapsto \C(A)$ is
$2$-functorial in an obvious sense and thus defines an op-prederivator
$2$\nbd{}functorial in an obvious sense and thus defines an op-prederivator
\begin{align*}
\C : \CCat^{op} &\to \CCAT \\
A &\mapsto \C(A)
......@@ -286,10 +286,10 @@ We now turn to the most important way of obtaining op-prederivators.
\begin{paragr}\label{paragr:homder}
Let $(\C,\W)$ be a localizer. For every small category $A$, there is a
localizer $(\C(A),\W_A)$, where $\W_A$ the class of \emph{pointwise weak
equivalences}, i.e. arrows $\alpha : d \to d'$ of $\C(A)$ such that
equivalences}, i.e.\ arrows $\alpha : d \to d'$ of $\C(A)$ such that
$\alpha_a : d(a) \to d'(a)$ belongs to $\W$ for every $a \in \Ob(A)$.
The correspondence $A \mapsto (\C(A),\W_A)$ is $2$-functorial in that every $u
The correspondence $A \mapsto (\C(A),\W_A)$ is $2$\nbd{}functorial in that every $u
: A \to B$ induces by pre-composition a morphism of localizers
\[
u^* : (\C(B),\W_B) \to (\C(A),\W_A)
......@@ -297,7 +297,7 @@ We now turn to the most important way of obtaining op-prederivators.
and every $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend
right, "v"',""{name=B,above}] & B
\ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces by pre-composition
a $2$-morphism of localizers
a $2$\nbd{}morphism of localizers
\[
\begin{tikzcd}
(\C(B),\W_B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right,
......@@ -305,10 +305,10 @@ We now turn to the most important way of obtaining op-prederivators.
\ar[from=A,to=B,Rightarrow,"\alpha^*"]
\end{tikzcd}
\]
(This last property is trivial since a $2$-morphism of localizers is simply a
(This last property is trivial since a $2$\nbd{}morphism of localizers is simply a
natural transformation between the underlying functors.) Then, by the
universal property of localization, we have for every $u : A \to B$ an induced
functor, which we still denote $u^*$,
universal property of the localization, we have for every $u : A \to B$, an induced
functor, which we still denote by $u^*$,
\[
u^* : \ho(\C(B)) \to \ho(\C(A))
\]
......@@ -349,7 +349,7 @@ We now turn to the most important way of obtaining op-prederivators.
extensions if and only if the category $\C$ has left Kan extensions along any
morphism $u : A \to B$ of $\Cat$ in the usual sense. By a standard categorical
argument, this means that the op-prederivator represented by $\C$ has left Kan
extensions if and only if $\C$ is cocomplete. Note that for any small category
extensions if and only if $\C$ is cocomplete. Note that for every small category
$A$, the functor
\[
p_A^* : \C \simeq \C(e) \to \C(A)
......@@ -362,15 +362,15 @@ We now turn to the most important way of obtaining op-prederivators.
\]
\end{example}
\begin{paragr}
A localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when the
We say that a localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when the
homotopy op-prederivator of $(\C,\W)$ has left Kan extensions. In this case,
for any small category $A$, the \emph{homotopy colimit functor of $A$-shaped
for every small category $A$, the \emph{homotopy colimit functor of $A$-shaped
diagrams} is defined as
\[
\hocolim_A := p_{A!} : \ho(\C(A)) \to \ho(\C).
\]
For an object $X$ if $\ho(\C(A))$ (which is nothing but a diagram $X : A \to
\C$, only seen ``up to weak equivalence''), the object of $\ho(\C)$
For an object $X$ of $\ho(\C(A))$ (which is nothing but a diagram $X : A \to
\C$, seen ``up to weak equivalence''), the object of $\ho(\C)$
\[
\hocolim_A(X)
\]
......@@ -402,8 +402,8 @@ We now turn to the most important way of obtaining op-prederivators.
C \ar[r,"g"'] & D \ar[from=1-2,to=2-1,Rightarrow,"\alpha"]
\end{tikzcd}
\]
be a $2$-square in $\CCat$. For any op-prederivator $\sD$, we have an induced
$2$-square:
be a $2$\nbd{}square in $\CCat$. For any op-prederivator $\sD$, we have an induced
$2$\nbd{}square:
\[
\begin{tikzcd}
\sD(A) & \sD(B) \ar[l,"f^*"'] \\
......@@ -562,13 +562,13 @@ We now turn to the most important way of obtaining op-prederivators.
\emph{strict} when $F_u$ is an identity for every $u : A \to B$.
Let $F : \sD \to \sD'$ and $G : \sD \to \sD'$ be morphisms of
op-prederivators. A \emph{$2$-morphism $\phi : F \to G$} is a modification
op-prederivators. A \emph{$2$\nbd{}morphism $\phi : F \to G$} is a modification
from $F$ to $G$. This means that $F$ consists of a natural transformation
$\phi_A : F_A \Rightarrow G_A$ for every small category $A$, and is subject
to a coherence axiom similar to the one for natural transformations.
We denote by $\PPder$ the $2$-category of op-prederivators, morphisms of
op-prederivators and $2$-morphisms of op-prederivators.
We denote by $\PPder$ the $2$\nbd{}category of op-prederivators, morphisms of
op-prederivators and $2$\nbd{}morphisms of op-prederivators.
\end{paragr}
\begin{example}
......@@ -576,7 +576,7 @@ We now turn to the most important way of obtaining op-prederivators.
denoted by $F$, at the level of op-prederivators, where for every small
category $A$, the functor $F_A : \C(A) \to \C'(A)$ is induced by
post-composition. Similarly, any natural transformation induces a
$2$-morphism at the level of represented op-prederivators.
$2$\nbd{}morphism at the level of represented op-prederivators.
\end{example}
\begin{example}
Let $F : (\C,\W) \to (\C',\W')$ be a morphism of localizers. For every small
......@@ -587,20 +587,20 @@ We now turn to the most important way of obtaining op-prederivators.
\[
\overline{F} : \Ho(\C) \to \Ho(\C').
\]
Similarly, any $2$-morphism of localizers
Similarly, any $2$\nbd{}morphism of localizers
\[
\begin{tikzcd}
(\C,\W) \ar[r,bend left,"F",""{name=A,below}] \ar[r,bend right,
"G"',""{name=B,above}] &(\C',\W') \ar[from=A,to=B,"\alpha",Rightarrow]
\end{tikzcd}
\]
induces a $2$-morphism $\overline{\alpha} : \overline{F} \Rightarrow
\overline{G}$. Altogether, we have defined a $2$-functor
induces a $2$\nbd{}morphism $\overline{\alpha} : \overline{F} \Rightarrow
\overline{G}$. Altogether, we have defined a $2$\nbd{}functor
\begin{align*}
\Loc &\to \PPder\\
(\C,\W) &\mapsto \Ho(\C),
\end{align*}
where $\Loc$ is the $2$-category of localizers.
where $\Loc$ is the $2$\nbd{}category of localizers.
\end{example}
\begin{paragr}\label{paragr:canmorphismcolimit}
Let $\sD$ and $\sD'$ be op-prederivators that admit left Kan extensions and
......@@ -649,7 +649,7 @@ We now turn to the most important way of obtaining op-prederivators.
usual sense.
\end{example}
\begin{paragr}\label{paragr:prederequivadjun}
As in any $2$-category, the notions of equivalence and adjunction make sense
As in any $2$\nbd{}category, the notions of equivalence and adjunction make sense
in $\PPder$. Precisely, we have that:
\begin{itemize}
\item[-] A morphism of op-prederivators $F : \sD \to \sD'$ is an equivalence
......@@ -658,12 +658,12 @@ We now turn to the most important way of obtaining op-prederivators.
$\mathrm{id}_{\sD}$; the morphism $G$ is a \emph{quasi-inverse} of $F$.
\item[-] A morphism of op-prederivators $F : \sD \to \sD'$ is left adjoint
to $G : \sD' \to \sD$ (and $G$ is right adjoint to $F$) when there exist
$2$-morphisms $\eta : \mathrm{id}_{\sD'} \Rightarrow GF$ and $\epsilon :
$2$\nbd{}morphisms $\eta : \mathrm{id}_{\sD'} \Rightarrow GF$ and $\epsilon :
FG \Rightarrow \mathrm{id}_{\sD}$ that satisfy the usual triangle
identities.
\end{itemize}
\end{paragr}
The following three lemmas are easy $2$-categorical routine and are left to
The following three lemmas are easy $2$\nbd{}categorical routine and are left to
the reader.
\begin{lemma}\label{lemma:dereq}
Let $F : \sD \to \sD'$ be a morphism of op-prederivators. If $F$ is an
......@@ -699,7 +699,7 @@ We now turn to the most important way of obtaining op-prederivators.
\[
\LL F : \Ho(\C) \to \Ho(\C')
\]
and a $2$-morphism of op-prederivators
and a $2$\nbd{}morphism of op-prederivators
\[
\begin{tikzcd}
\C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\
......@@ -740,7 +740,7 @@ We now turn to the most important way of obtaining op-prederivators.
\]
\end{proposition}
\begin{proof}
Let $\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$ be the $2$-morphism
Let $\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$ be the $2$\nbd{}morphism
of op-prederivators defined \emph{mutatis mutandis} as in
\ref{paragr:prelimgonzalez} but at the level of op-prederivators.
Proposition \ref{prop:gonz} gives us that for every small category $A$, the
......@@ -809,7 +809,7 @@ We now turn to the most important way of obtaining op-prederivators.
X_{(1,1)} := (1,1)^*(X).
\]
Now, since $(1,1)$ is the terminal object of $\square$, we have a canonical
$2$-triangle
$2$\nbd{}triangle
\[
\begin{tikzcd}
\ulcorner \ar[rr,"i_{\ulcorner}",""{name=A,below}] \ar[rd,"p"'] && \square\\
......@@ -817,7 +817,7 @@ We now turn to the most important way of obtaining op-prederivators.
\end{tikzcd}
\]
where we wrote $p$ instead of $p_{\ulcorner}$ for short. Hence, we have a
$2$-triangle
$2$\nbd{}triangle
\[
\begin{tikzcd}
\sD(\ulcorner) && \sD(\square) \ar[ll,"i_{\ulcorner}^*"',""{name=A,below}] \ar[dl,"{(1,1)}^*"] \\
......
No preview for this file type
......@@ -2161,7 +2161,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an $\omega$-functo
\[
\wt{F}_a : \G[\Sigma^C]_a \to \G[\Sigma^D]_{F(a)}
\]
for any $a \in C_{n}$.
for every $a \in C_{n}$.
\end{paragr}
\begin{lemma}\label{lemmafaithful} With the notations of the above paragraph, the map
\[
......@@ -2213,7 +2213,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an $\omega$-functo
\mu =(v_1,v_2,e,e') : v \to v'
\]
be an elementary move. Since, by definition,
\[\wt{F}(u)= v = v_1ev_2\]
\[\wt{F}(u)= v = v_1ev_2,\]
$u$ is necessarily of the form
\[
u=u_1\overline{e}u_2
......@@ -2248,12 +2248,12 @@ Recall from Proposition \ref{prop:conduchepractical} that for an $\omega$-functo
\wt{F}(\lambda)=\mu.
\]
All that is left to prove now is ythe existence of $\overline{e'}$ with the desired properties.
All that is left to prove now is the existence of $\overline{e'}$ with the desired properties.
\begin{description}
\item[First case:] The word $e$ is of the form
\[((x\fcomp_k y )\fcomp_k z)\]
and $e'$ is of the form
and the word $e'$ is of the form
\[(x\fcomp_k (y \fcomp_k z))\]
with $x,y,z \in \T[\Sigma^D]$.
The word $\overline{e}$ is then necessarily of the form
......@@ -2278,7 +2278,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an $\omega$-functo
\[
(x\fcomp_k(\ii_{\1^{n-1}_z}))
\]
and $e'$ is of the form
and the word $e'$ is of the form
\[
x
\]
......@@ -2296,7 +2296,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an $\omega$-functo
\[
\wt{F}(y)=\1^{n-1}_z.
\]
Then we set
Then, we set
\[
\overline{e'}:=x.
\]
......@@ -2312,7 +2312,7 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap
\[
((\ii_x) \fcomp_k (\ii_y))
\]
and $e'$ is of the form
and the word $e'$ is of the form
\[
(\ii_{x\comp_k y})
\]
......@@ -2334,7 +2334,7 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap
\[
((x\fcomp_ky)\fcomp_l(z \fcomp_k t))
\]
and $e'$ is of the form
and the word $e'$ is of the form
\[
((x\fcomp_lz)\fcomp_k(y \fcomp_l t))
\]
......@@ -2383,7 +2383,7 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap
In the proof of the previous theorem, we only used the hypothesis that $F$ is right orthogonal to $\kappa_k^n$ for any $k$ such that $0 \leq k < n-1$.
\end{remark}
\begin{lemma}\label{lemma:isomorphismgraphs}
Let $F : C \to D$ be an $\omega$-functor, $n>0$, $\Sigma^D \subseteq D_{n}$ and $\Sigma^C = F^{-1}(\Sigma^D)$. If $\tau^s_{\leq n}(F)$ is a discrete Conduché $n$\nbd{}functor, then for every $a \in C_{n}$
Let $F : C \to D$ be an $\omega$-functor, $n>0$, $\Sigma^D \subseteq D_{n}$ and $\Sigma^C := F^{-1}(\Sigma^D)$. If $\tau^s_{\leq n}(F)$ is a discrete Conduché $n$\nbd{}functor, then for every $a \in C_{n}$
\[
\wt{F}_a : \G[\Sigma^C]_a\to \G[\Sigma^D]_{F(a)}
\]
......@@ -2417,13 +2417,13 @@ If $k <n-1$, we need first to use the fact that $f$ is right orthogonal to $\kap
we conclude that $v=u'$.
\end{proof}
\begin{proposition}\label{prop:conduchenbasis}
Let $F : C \to D$ be an $\omega$-functor, $n \in \mathbb{N}$, $\Sigma^D \subseteq D_{n}$ and $\Sigma^C = F^{-1}(\Sigma^D)$. If $\tau_{\leq n}^s(f)$ is a discrete Conduché $n$\nbd{}functor, then:
Let $F : C \to D$ be an $\omega$-functor, $n \in \mathbb{N}$, $\Sigma^D \subseteq D_{n}$ and $\Sigma^C := F^{-1}(\Sigma^D)$. If $\tau_{\leq n}^s(f)$ is a discrete Conduché $n$\nbd{}functor, then:
\begin{enumerate}
\item if $\Sigma^D$ is an $n$\nbd{}basis then so is $\Sigma^C$,
\item if $F_{n} : C_{n}\to D_{n}$ is surjective and $\Sigma^C$ is an $n$\nbd{}basis then so is $\Sigma^D$.
\end{enumerate}
\end{proposition}
\begin{proof}The case $n=0$ is trivial. We know suppose that $n>0$.
\begin{proof}The case $n=0$ is trivial. We now suppose that $n>0$.
From Lemma \ref{lemma:isomorphismgraphs} we have that for every $a \in C_n$, the map
\[
\wt{F}_a : \G[\Sigma^C]_a \to \G[\Sigma^D]_{F(a)}
......@@ -2437,7 +2437,7 @@ Putting all the pieces together, we finally have the awaited proof.
\item In the case that $D$ is free, it follows immediately from the first part of the Proposition \ref{prop:conduchenbasis} that $C$ is free.
\item In the case that $C$ is free and $F_n : C_n \to D_n$ is surjective for every $n \in \mathbb{N}$, let us write $\Sigma_n^C$ for the $n$\nbd{}basis of $C$. It follows from Proposition \ref{prop:uniquebasis} and Lemma \ref{lemma:conducheindecomposable} that
\[
F^{-1}(F(\Sigma^C_n)=\Sigma_n^C.
F^{-1}(F(\Sigma^C_n))=\Sigma_n^C.
\]
Hence, we can apply the second part of Proposition \ref{prop:conduchenbasis} and $C$ is free.
\end{enumerate}
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment