toto

2cat.tex

@@ -12,7 +12,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
   \[
   \src(1_{x}) = \trgt (1_{x}) = x.
   \]
-  The same vocabulary of categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a ``underlying reflexive graph'' functor
+  The vocabulary of categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a ``underlying reflexive graph'' functor
   \[
   U : \Cat \to \Rgrph,
   \]
@@ -77,7 +77,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
 
   We now review a construction of Dwyer and Kan from \cite{dwyer1980simplicial}.
 
-  Let $G$ be a reflexive graph. For every $i\leq 1$, we define the simplicial set $N^k(G)$ as the sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
+  Let $G$ be a reflexive graph. For every $k\geq 1$, we define the simplicial set $N^k(G)$ as the sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
   \[
   \begin{tikzcd}
   X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} 
@@ -134,7 +134,7 @@ From the previous proposition, we deduce the following very useful corollary.
     L(C) \ar[r,"L(\gamma)"]& L(D)
   \end{tikzcd}
   \]
-  is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.
+  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.
 \end{corollary}
 \begin{proof}
   Since the nerve $N$ induces an equivalence of op-prederivators
@@ -178,7 +178,7 @@ Actually, by working a little more, we obtain the slightly more general result b
     L(C) \ar[r,"L(\gamma)"] &L(D)
   \end{tikzcd}
   \]
-  is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.
+  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.
 \end{proposition}
 \begin{proof}
   The case where $\alpha$ or $\beta$ is injective both on objects and arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is injective on arrows. The other case being symmetric.
@@ -186,11 +186,13 @@ Actually, by working a little more, we obtain the slightly more general result b
   Let use denote by $E$ the set of objects of $B$ that lies in the image of $\beta$. For each element $x$ of $E$, we denote by $F_x$ the ``fiber'' of $x$, that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the set $E$ and each $F_x$ as discrete reflexive graphs, i.e. reflexive graphs with no non-unital arrow. Now, let $G$ be the reflexive graph defined with the following cocartesian square
   \[
   \begin{tikzcd}
-    \coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\
+    \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\
     A \ar[r] & G,
+    \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
   \end{tikzcd}
   \]
-  where the map \[ \coprod_{x \in E}F_x \to A\] is induced by the inclusion of each $F_x$ in $A$, and the map \[\coprod_{x \in E}F_x \to E\] is the only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is obtained from $A$ by identifying the objects that are identified through $\beta$. 
+  where the morphism \[ \coprod_{x \in E}F_x \to A\] is induced by the inclusion of each $F_x$ in $A$, and the morphism \[\coprod_{x \in E}F_x \to E\] is the only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is obtained from $A$ by collapsing the objects that are identified through $\beta$.
+  Notice importantly that the morphism \[ \coprod_{x \in E}F_x \to A\] is a monomorphism, i.e. injective on objects and arrows, and the morphism \[A \to G\] is injective on arrows. 
 \end{proof}
 \begin{example}[Adding a generator]
   Let $C$ be a free category, $A$ and $B$ (possibly equal) two objects of $C$ and let $C'$ be the category obtained from $C$ by adding a generator $A \to B$, i.e. defined with the following cocartesian square: