Commit 439a60a7 authored by Leonard Guetta's avatar Leonard Guetta
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edited a few typos in chapter 2. Did not yet do what said in previous commit

parent 2ac942db
...@@ -44,7 +44,7 @@ is poorly behaved. For example, \fi ...@@ -44,7 +44,7 @@ is poorly behaved. For example, \fi
\gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D) \gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)
\] \]
is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$ is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$
that sends morphism of $\W$ to isomorphisms of $\D$. that sends the morphisms of $\W$ to isomorphisms of $\D$.
We shall always consider that $\C$ and $\ho(\C)$ have the same class of We shall always consider that $\C$ and $\ho(\C)$ have the same class of
objects and implicitly use the equality objects and implicitly use the equality
...@@ -90,7 +90,7 @@ For later reference, we put here the following definition. ...@@ -90,7 +90,7 @@ For later reference, we put here the following definition.
is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of
localizers. A \emph{$2$\nbd{}morphism of localizers} from $F$ to $G$ is simply a localizers. A \emph{$2$\nbd{}morphism of localizers} from $F$ to $G$ is simply a
natural transformation $\alpha : F \Rightarrow G$. The universal property of natural transformation $\alpha : F \Rightarrow G$. The universal property of
the localization implies that there exists a canonical natural transformation the localization implies that there exists a unique natural transformation
\[ \[
\begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}] \begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}]
\ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C') \ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C')
...@@ -109,8 +109,8 @@ For later reference, we put here the following definition. ...@@ -109,8 +109,8 @@ For later reference, we put here the following definition.
\end{paragr} \end{paragr}
\begin{remark}\label{remark:localizedfunctorobjects} \begin{remark}\label{remark:localizedfunctorobjects}
Since we always consider that for every localizer $(\C,\W)$ the categories $\C$ Since we always consider that for every localizer $(\C,\W)$ the categories $\C$
and $\ho(\C)$ have the same objects and the localization functor is the and $\ho(\C)$ have the same class of objects and the localization functor is the
identity on objects, it follows that for a morphism of localizer ${F : (\C,\W) identity on objects, it follows that for a morphism of localizers ${F : (\C,\W)
\to (\C',\W')}$, we tautologically have \to (\C',\W')}$, we tautologically have
\[ \[
\overline{F}(X)=F(X) \overline{F}(X)=F(X)
......
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