Commit aefa8fd8 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Safety commit

parent f2a17372
......@@ -8,7 +8,7 @@
......@@ -25,12 +25,13 @@
\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 $X$.
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).
Dually, if we are given a square in $\CCat$ of the form
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"]
and if $\sD$ has right Kan extension, then we obtain the cohomological base change morphism induced by $\alpha$:
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