@@ -12,7 +12,9 @@ In this section, we review some homotopical results concerning free ($1$-)catego

\[

\src(1_{x})=\trgt(1_{x})= x.

\]

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

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. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0\to G_0'$ and $f_1 : G_1\to G'_1$ are injective.

There is a ``underlying reflexive graph'' functor

\[

U : \Cat\to\Rgrph,

\]

...

...

@@ -57,7 +59,11 @@ In this section, we review some homotopical results concerning free ($1$-)catego

The functor $i_! : \Rgrph\to\Psh{\Delta}$ preserves monomorphism.

\end{lemma}

\begin{proof}

What we need to show is that, given a morphism of presheaves

\[

f : X \to Y,

\]

if $f_0 : X_0\to Y_0$ and $f_1 : X_1\to Y_1$ are monomorphisms and if all $n$-simplices of $X$ are degenerated for $n\geq2$, then $f$ is a monomorphism. A proof of this assertion is contained in \cite[Paragraph 3.4]{gabriel1967calculus}. The key argument is the Eilenberg-Zilber Lemma (Proposition 3.1 of op. cit.).

\end{proof}

\begin{paragr}

Let us denote by $N : \Psh{\Delta}\to\Cat$ (instead of $N_1$ as in Paragraph \todo{ref}) the usual nerve of categories and by $c : \Cat\to\Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain

...

...

@@ -102,10 +108,20 @@ In this section, we review some homotopical results concerning free ($1$-)catego

\[

N^{k}(G)\to N^{k+1}(G)

\]

is a weak equivalence of simplicial sets.

is a trivial cofibration of simplicial sets.

\end{lemma}

\begin{proof}

Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the union of the first and last face of the standard $(k+1)$-simplex $\Delta_{k+1}$. Notice that the canonical inclusion

\[

A_{k+1}\hookedrightarrow\Delta_{k+1}

\]

is a trivial cofibration. Now let $I_{k+1}$ be the set of chains

\[

\begin{tikzcd}

f = 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}

\end{tikzcd}

\]

where each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$

\end{proof}

From this lemma, we deduce the following propositon.

\begin{proposition}

...

...

@@ -113,10 +129,10 @@ 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.

where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial cofibration of simplicial sets.

\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 whose objects are all cofibrants \todo{ref}.

This follows from the fact that trivial cofibrations are stable by transfinite composition.

\end{proof}

From the previous proposition, we deduce the following very useful corollary.

\begin{corollary}\label{cor:hmtpysquaregraph}

...

...

@@ -237,7 +253,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\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 $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmptysquaregraph}.

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 $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.

\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 \emph{free} (1-)categories are \good{} (which we already knew since we have seen that \emph{all} (1-)categories are \good{}).

...

...

@@ -250,7 +266,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\sD_1\ar[r]& C',

\end{tikzcd}

\]

where the morphism $\sS_1\to\sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1\to\sD_1$ is injective on objects and the morphism $\sS_1\to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.

where the morphism $\sS_1\to\sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1\to\sD_1$ is injective on objects and the morphism $\sS_1\to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.

\end{example}

\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: