Commit 6f5ac846 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

security commit

parent 110441cc
......@@ -8,9 +8,9 @@
......@@ -10,7 +10,7 @@
% Maths packages
% 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
......@@ -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:
%% Old version of representable op-prederivator
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
\C : \CCat^{op} &\to \CCAT \\
A &\mapsto \C(A)
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
\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^*"]
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.
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