security commit

mystyle.sty

@@ -89,6 +89,8 @@
 
 \newcommand{\CCat}{\underline{\mathbf{Cat}}} %2-category of small categories
 \newcommand{\CCAT}{\underline{\mathbf{CAT}}} %2-category of big categories
+\newcommand{\PPder}{\underline{\mathbf{Pder}}} %2-category of prederivators
+
 % compositions and units
 \def\1^#1_#2{1^{(#1)}_{#2}} % for iterated units
 

recyclebin.tex

@@ -360,3 +360,82 @@ It is straightforward to check that this defines an $n$-precategory. Let $(\varp
      
      \]
   \end{paragr}
+
+
+%%% Définition transfo pseudo-naturelle
+
+   \iffalse
+    such that the following axioms are satisfied:
+    \begin{itemize}[label=-]
+    \item for every small category $A$,
+      \[
+      F_{1_A}=1_{F(A)},
+      \]
+    \item for every pair of composable arrows $A \overset{u}{\rightarrow} B \overset{v}{\rightarrow} C$ in $\CCat$,
+      \[
+      \begin{tikzcd}
+        \sD(C) \ar[d,"v^*"'] \ar[r,"F_C"] & \sD'(C) \ar[d,"v^*"]\\
+        \ar[from=1-2,to=2-1,Rightarrow,"F_v","\sim"']
+        \sD(B) \ar[d,"u^*"'] \ar[r,"F_B"] & \sD'(B) \ar[d,"u^*"]\\
+        \sD(A) \ar[r,"F_A"'] & \sD'(A)
+        \ar[from=2-2,to=3-1,Rightarrow,"F_u","\sim"']
+      \end{tikzcd}
+      =
+      \begin{tikzcd}
+        \sD(C) \ar[dd,"(vu)^*"'] \ar[r,"F_C"] & \sD'(C) \ar[dd,"(vu)^*"]\\
+        &\\
+        \sD(A) \ar[r,"F_A"'] & \sD'(A),
+        \ar[from=1-2,to=3-1,Rightarrow,"F_{vu}","\sim"']
+        \end{tikzcd}
+      \]
+    \item for every $\begin{tikzcd} A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right,"v"',""{name=B,above}] & B  \ar[from=A,to=B,Rightarrow,"\alpha"] \end{tikzcd}$ in $\CCat$,
+      \[
+      \begin{tikzcd}
+        \sD(B) \ar[d,"u^*",""{name=A,below}] \ar[r,"F_B"] & \sD'(B) \ar[d,"u^*"]\\
+        \sD(A) \ar[r,"F_A"'] & \sD'(A)
+        \ar[from=1-2,to=2-1,Rightarrow,"F_u","\sim"']
+        \ar[from=1-1,to=2-1,bend right=55,"v^*"',""{name=B,below}]
+        \ar[from=A,to=B,Rightarrow,"\alpha^*"',near start]
+      \end{tikzcd}
+      =
+            \begin{tikzcd}
+        \sD(B) \ar[d,"v^*"] \ar[r,"F_B"] & \sD'(B) \ar[d,"v^*"',""{name=D,below}]\\
+        \sD(A) \ar[r,"F_A"'] & \sD'(A).
+        \ar[from=1-2,to=2-1,Rightarrow,"F_v","\sim"']
+        \ar[from=1-2,to=2-2,bend left=55,"u^*",""{name=C,above}]
+        \ar[from=C,to=D,Rightarrow,"\alpha^*"',near start]
+        \end{tikzcd}
+      \]
+    \end{itemize}
+    \remtt{Ai-je vraiment besoin de donner la définition de transfo pseudo-naturelle ? }\fi
+
+    %%% Localization derivator
+
+    It is straightforward to check that we have the following universal property: for any op-prederivator $\sD$, the functor induced by pre-composition
+    \[
+    \gamma^* : \underline{\Hom}(\sD_{(\C,\W)},\sD) \to \underline{\Hom}(\C,\sD)
+    \]
+    is fully faithful and its essential image consists of morphisms of op-prederivators $F : \C \to \sD$ such that for every small category $A$, $F_A : \C(A) \to \sD(A)$ sends morphisms of $\W_A$ to isomorphisms of $\sD(A)$.
+
+    This universal property is a higher version of the universal property of localization seen in \ref{paragr:loc}. It follows that given two localizers $(\C,\W)$ and $(\C',\W')$ and a functor $F : \C \to \C'$, if $F$ preserves weak equivalences, then there exists a morphism of op-prederivators $ \overline{F} : \sD_{(\C,\W)} \to \sD_{(\C',\W')}$ such that the square 
+    \[
+    \begin{tikzcd}
+    \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\
+    \sD_{(\C,\W)} \ar[r,"\overline{F}"] & \sD_{(\C',\W')}
+  \end{tikzcd}
+    \]
+    is commutative.
+
+    %%% Left derived functor derivator
+
+      \begin{definition}
+    Let $(\C,\W)$ and $(\C',\W')$ be two localizers. A functor $F : \C \to \C'$ is \emph{left derivable in the sense of derivators} if there exists a morphism $\LL F : \sD_{(\C,\W)} \to \sD_{(\C',\W')}$ and a $2$-morphism
+  \[
+  \begin{tikzcd}
+    \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\
+    \sD_{(\C,\W)} \ar[r,"\LL F"'] & \sD_{(\C',\W')}
+    \arrow[from=2-1, to=1-2,"\alpha",Rightarrow]
+  \end{tikzcd}
+  \]
+    that makes $\LL F$ the \emph{right} Kan extension of $\gamma' \circ F$ along $\gamma$ in the $2$-category of op-prederivators:
+    \end{definition}