Commit dde283d0 by Leonard Guetta

### security commit

parent aefa8fd8
 ... ... @@ -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 ... ...
 ... ... @@ -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}
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