@@ -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 $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$. In particular, the canonical inclusion $i : \Delta_{\leq1}\arrow\Delta$ induces by pre-composition a functor

There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$. In particular, the canonical inclusion $i : \Delta_{\leq1}\rightarrow\Delta$ 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

@@ -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

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

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.

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