Commit 302b5a50 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

modifs légères

parent 36687578
......@@ -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}
......
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