security commit

main.tex

@@ -8,9 +8,9 @@
 
 \maketitle
 
-\include{omegacat}
+%\include{omegacat}
 \include{homtheo}
-\include{hmtpy}
+%\include{hmtpy}
 \bibliographystyle{alpha}
 \bibliography{memoire}
 \end{document}

mystyle.sty

@@ -10,7 +10,7 @@
 % Maths packages
 
 \usepackage{amsmath,amssymb,amsthm}
-
+\usepackage{mathtools}
 \usepackage{tikz-cd}
 
 % List
@@ -131,8 +131,8 @@
 \newcommand{\RR}{\ensuremath{\mathbb{R}}} % A  mathbb R. Useful for right left derived functor
 \newcommand{\Fib}{\ensuremath{\mathrm{Fib}}} % A mathrm "Fib". Used to denote fibrations in a model category.
 \newcommand{\Cof}{\ensuremath{\mathrm{Cof}}} % A mathrm "Cof". Used to denote cofibrations in a model category.
-\newcommand{\Ho}{\ensuremath{\mathrm{Ho}}} %Useful for the homotopy category
-
+\newcommand{\Ho}{\ensuremath{\mathcal{H}\mathrm{o}}} %homotopy derivator
+\newcommand{\ho}{\ensuremath{\mathrm{ho}}} %homotopy category
 % Orientals
 
 \newcommand{\Or}{\ensuremath{\mathcal{O}}} % for orientals

recyclebin.tex

@@ -439,3 +439,30 @@ It is straightforward to check that this defines an $n$-precategory. Let $(\varp
   \]
     that makes $\LL F$ the \emph{right} Kan extension of $\gamma' \circ F$ along $\gamma$ in the $2$-category of op-prederivators:
     \end{definition}
+
+      %% Old version of representable op-prederivator
+
+      \iffalse
+\begin{example}\label{ex:repder}
+  Let $\C$ be a category. For $A$ a small category, let $\C(A)$ be the category of functors $A \to \C$ and natural transformations between them.
+  This canonically defines an op-prederivator
+  \begin{align*}
+    \C : \CCat^{op} &\to \CCAT \\
+    A &\mapsto \C(A)
+  \end{align*}
+  where for any $u : A \to B$ in $\CCat$, the functor
+  \[
+  u^* : \C(A) \to \C(B)
+  \]
+  is induced from $u$ by pre-composition, and similarly for $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ in $\CCat$, the natural transformation
+  \[
+  \begin{tikzcd}
+    \C(B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & \C(A) \ar[from=A,to=B,Rightarrow,"\alpha^*"]
+    \end{tikzcd}
+  \]
+  is induced by pre-composition. This op-prederivator is sometimes referred to as the op-prederivator \emph{represented} by $\C$. Notice that for the terminal category $e$, we have a canonical isomorphism
+  \[
+  \C(e) \simeq \C
+  \]
+  that we shall use without further reference.
+\end{example}\fi