Commit aefa8fd8 by Leonard Guetta

### Safety commit

parent f2a17372
 ... ... @@ -8,7 +8,7 @@ \maketitle \include{omegacat} %\include{omegacat} \include{homtheo} \bibliographystyle{alpha} \bibliography{memoire} ... ...
 ... ... @@ -25,12 +25,13 @@ \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{notation}[theorem]{Notation} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem*{convention}{Convention} \newtheorem*{notation}{Notation} %\newtheorem*{notation}{Notation} \newtheorem{paragr}[theorem]{} \newtheorem{example}[theorem]{Example} \newtheorem{impexample}[theorem]{Important example} ... ... @@ -115,7 +116,9 @@ \newcommand{\Psh}[1]{\ensuremath{\widehat{#1}}} %Presheaves categories \newcommand{\id}{\ensuremath{\mathrm{id}}} %identity in a category \def\colim{\mathop{\mathrm{colim}}} %colimits \def\lim{\mathop{\mathrm{lim}}} %limits \def\hocolim{\mathop{\mathrm{hocolim}}} %homotopy colimits \def\holim{\mathop{\mathrm{holim}}} %homotopy limits \newcommand{\wt}[1]{\ensuremath{\widetilde{#1}}} %A shortcut for \widetilde \newcommand{\LL}{\ensuremath{\mathbb{L}}} % A mathbb L. Useful for left derived functor \newcommand{\RR}{\ensuremath{\mathbb{R}}} % A mathbb R. Useful for right left derived functor ... ...
 ... ... @@ -336,3 +336,27 @@ It is straightforward to check that this defines an $n$-precategory. Let $(\varp p_A^* : \C(e) \to \C(A) \] is canonically isomorphic with the diagonal functor$\Delta : \C \to \C(A)$that sends an object$X$of$\C$to the constant diagram$A \to \C$with value$X$. \begin{paragr} Let$(\C,\W)$be a localizer and$A$a small category. We denote by$\C^A$the category of functors from$A$to$\C$and natural transformations between them. An arrow$\alpha : d \to d'$of$\C^A$is a \emph{pointwise weak equivalences} when$\alpha_a : d(a) \to d'(a)$belongs to$\W$for every$a \in A$. We denote by$\W_A$the class of pointwise weak equivalences. This defines a localizer$(\C^A,\W_A)$. For every$u : A \to B$morphism of$\Cat$, the functor induced by pre-composition $u^* : \C^B \to \C^A$ preserves pointwise weak equivalences. Hence, there is an induced functor between the localized categories still denoted by$u^*$: $u^* : \Ho(\C^B) \to \Ho(\C^A).$ \end{paragr} Dually, if we are given a square in$\CCat$of the form $\begin{tikzcd} A \ar[r,"f"] \ar[d,"u"'] & B \ar[d,"v"]\\ C \ar[r,"g"'] & D \ar[from=2-1,to=1-2,Rightarrow,"\alpha"] \end{tikzcd}$ and if$\sD$has right Kan extension, then we obtain the cohomological base change morphism induced by$\alpha\$:  \end{paragr}
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