leaving the office

2cat.tex

@@ -47,7 +47,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
   \]
   For a graph $G$, the simplicial set $i_l(G)$ has $G_0$ as its set of $0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices degenerated for $k>2$. For future reference we put here the following lemma.
 \end{paragr}
-\begin{lemma}
+\begin{lemma}\label{lemma:monopreserved}
   The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism.
 \end{lemma}
 \begin{proof}
@@ -66,5 +66,111 @@ In this section, we review some homotopical results concerning free ($1$-)catego
   \[
   \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.
   \]
-  
+
+  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
+    \[
+  X_0 \overset{\rightarrow}{f_1} X_1 \overset{\rightarrow}{f_2} X_2 \rightarrow \cdots \rightarrow X_{n-1} \overset{\rightarrow}{f_n} X_{n} 
+  \]
+  of arrows of $L(G)$ such that the length of each $f_i$ is \emph{at most} $k$. In particular, we have
+  \[
+  N^1(G)=i_!(G)
+  \]
+  and the transfinite composition of
+  \[
+  i_!(G) = N^1(G) \hookedarrow N^2(G) \hookedarrow \cdots \hookedarrow N^{k}(G) \hookedarrow N^{k+1}(G) \hookedarrow \cdots
+  \]
+  is easily seen to be the map
+  \[
+  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
+  \]
+  where $\eta$ is the unit of the adjunction $c \dashv N$. 
 \end{paragr}
+\begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan}
+  For every $k\leq 1$, the canonical inclusion map
+  \[
+  N^{k}(G) \to N^{k+1}(G)
+  \]
+  is a weak equivalence of simplicial sets.
+\end{lemma}
+\begin{proof}
+  
+\end{proof}
+From this lemma, we deduce the following propositon.
+\begin{proposition}
+  Let $G$ be a reflexive graph. The map
+  \[
+  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
+  \]
+  where $\eta$ is the unit of the adjunction $c \dashv N$, is a weak equivalence of simplicial sets.
+\end{propostion}
+\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}
+From the previous proposition, we deduce the following very useful corollary.
+\begin{corollary}\label{cor:hmtpysquaregraph}
+  Let
+  \[
+  \begin{tikzcd}
+    A \ar[d] \ar[r] B \ar[d] \\
+    C \ar[r] D
+  \end{tikzcd}
+  \]
+  be a cocartesian square in $\RGrph$. If either the arrow $A \to B$ or the arrow $A \to C$ is a monomorphism, then the induced square
+  \[
+  \begin{tikzcd}
+    L(A) \ar[d] \ar[r] L(B) \ar[d] \\
+    L(C) \ar[r] L(D)
+  \end{tikzcd}
+  \]
+  is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.
+\end{corollary}
+\begin{proof}
+  Since the nerve $N$ induces an equivalence of op-prederivators
+  \[
+  \Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
+  \]
+  it suffices to prove that the induced square of simplicial sets
+  \[
+  \begin{tikzcd}
+    NL(A) \ar[d] \ar[r] NL(B) \ar[d] \\
+    NL(C) \ar[r] NL(D)
+  \end{tikzcd}
+  \]
+  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
+    \[
+  \begin{tikzcd}
+    i_!(A) \ar[d] \ar[r] i_!(B) \ar[d] \\
+    i_!(C) \ar[r] i_!(D).
+  \end{tikzcd}
+  \]
+  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}.
+\end{proof}
+\begin{example}[Killing a generator]
+  Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by ``killing'' $f$, i.e. defined with the following cocartesian square:
+  \[
+  \begin{tikzcd}
+    \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
+    \sD_0 \ar[r] & C'.
+  \end{tikzcd}
+  \]
+  Then, this above square is homotopy cocartesion in $\Cat$ (equipped with the Thomason weak equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$  and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$.
+\end{example}
+\begin{remark}
+    Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs. \todo{À mieux dire ?}
+  \end{remark}
+\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:
+  \[
+  \begin{tikzcd}
+        \partial\sD_1 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
+    \sD_1 \ar[r] & C'.
+  \end{tikzcd}
+  \]
+  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 $\Rgrh$. 
+\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{}). 
+\end{remark}
+