modifs légères

2cat.tex

@@ -37,7 +37,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
   Note that for a morphism of reflexive graphs $ f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units. 
 \end{remark}
 \begin{paragr}
-  There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \arrow \Delta$ induces by pre-composition a functor
+  There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition a functor
   \[
   i^* : \Psh{\Delta} \to \Rgrph,
   \]
@@ -79,7 +79,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
   \]
   and the transfinite composition of
   \[
-  i_!(G) = N^1(G) \hookedarrow N^2(G) \hookedarrow \cdots \hookedarrow N^{k}(G) \hookedarrow N^{k+1}(G) \hookedarrow \cdots
+  i_!(G) = N^1(G) I\hookrightarrow N^2(G) I\hookrightarrow \cdots I\hookrightarrow N^{k}(G) I\hookrightarrow N^{k+1}(G) I\hookrightarrow \cdots
   \]
   is easily seen to be the map
   \[
@@ -104,7 +104,7 @@ From this lemma, we deduce the following propositon.
   \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
   \]
   where $\eta$ is the unit of the adjunction $c \dashv N$, is a weak equivalence of simplicial sets.
-\end{propostion}
+\end{proposition}
 \begin{proof}
   It is an immediate consequence of Lemma \ref{lemma:dwyerkan} and the fact that filtered colimits are homotopic in a model category such that all objects are cofibrants \todo{ref}.
 \end{proof}
@@ -113,15 +113,15 @@ From the previous proposition, we deduce the following very useful corollary.
   Let
   \[
   \begin{tikzcd}
-    A \ar[d,"\alpha"] \ar[r,"\beta"] B \ar[d,"\delta"] \\
-    C \ar[r,"\gamma"] D
+    A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
+    C \ar[r,"\gamma"]& D
   \end{tikzcd}
   \]
-  be a cocartesian square in $\RGrph$. If either $\alpha$ or $\beta$ is a monomorphism, then the induced square
+  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a monomorphism, then the induced square
   \[
   \begin{tikzcd}
-    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] L(B) \ar[d,"L(\delta)"] \\
-    L(C) \ar[r,"L(\gamma)"] L(D)
+    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
+    L(C) \ar[r,"L(\gamma)"]& L(D)
   \end{tikzcd}
   \]
   is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.
@@ -134,15 +134,15 @@ From the previous proposition, we deduce the following very useful corollary.
   it suffices to prove that the induced square of simplicial sets
   \[
   \begin{tikzcd}
-    NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"] NL(B) \ar[d,"NL(\delta)"] \\
-    NL(C) \ar[r,"NL(\gamma)"] NL(D)
+    NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
+    NL(C) \ar[r,"NL(\gamma)"]& NL(D)
   \end{tikzcd}
   \]
   is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to the square of simplicial sets
     \[
   \begin{tikzcd}
-    i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] i_!(B) \ar[d,"i_!(\delta)"] \\
-    i_!(C) \ar[r,"i_!(\gamma)"] i_!(D).
+    i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
+    i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
   \end{tikzcd}
   \]
   This square is cocartesian because $i_!$ is a left adjoint and since $i_!$ preserves monomorphism (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and \todo{ref}.
@@ -152,20 +152,20 @@ Actually, by working a little more, we obtain the slightly more general result b
   Let
   \[
   \begin{tikzcd}
-    A \ar[d,"\alpha"] \ar[r,"\beta"] B \ar[d,"\delta"] \\
-    C \ar[r,"\gamma"] D
+    A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
+    C \ar[r,"\gamma"]& D
   \end{tikzcd}
   \]
-  be a cocartesian square in $\RGrph$. Suppose that both following conditions are satisfied
-  \begin{itemize}[label=\alph*)]
+  be a cocartesian square in $\Rgrph$. Suppose that both following conditions are satisfied
+  \begin{enumerate}[label=\alph*)]
   \item Either $\alpha$ or $\beta$ is injective on objects.
     \item Either $\alpha$ or $\beta$ is injective on morphisms.
-  \end{itemize}
+  \end{enumerate}
   Then, the square
   \[
   \begin{tikzcd}
-    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] L(B) \ar[d,"L(\delta)"] \\
-    L(C) \ar[r,"L(\gamma)"] L(D)
+    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
+    L(C) \ar[r,"L(\gamma)"] &L(D)
   \end{tikzcd}
   \]
   is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.
@@ -181,7 +181,7 @@ Actually, by working a little more, we obtain the slightly more general result b
     \sD_1 \ar[r] & C'.
   \end{tikzcd}
   \]
-  Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrh$. 
+  Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrph$. 
 \end{example}
 \begin{remark}
   Since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example yields another proof that free (1-)categories are \good{} (which we already knew since we have seen that all (1-)categories are \good{}). 
@@ -190,7 +190,7 @@ Actually, by working a little more, we obtain the slightly more general result b
   Let $C$ be a free category and $f,g : A \to B$ parallel generating arrows of $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by ``identifying'' $f$ and $g$, i.e. defined with the following cocartesian square
   \[
   \begin{tikzcd}
-    \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] C \ar[d] \\
+    \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
     \sD_1  \ar[r] & C',  
   \end{tikzcd}
   \]
@@ -201,7 +201,7 @@ Actually, by working a little more, we obtain the slightly more general result b
   where $F_2$ is the free monoid with two generators, seen as a category. In particular, it is free and notice that the map on the left comes from a monomorphism of reflexive graphs. Now, this factorization yields a factorization of our cocartesian square into two cocartesian squares
   \[
   \begin{tikzcd}
-    
+    a
   \end{tikzcd}
   \]
 \end{example}