of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$. Composition is given by concatenation of chains.

of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$. Composition is given by concatenation of chains.

\end{paragr}

\end{paragr}

\begin{lemma}

\begin{lemma}

A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that

A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that

...

@@ -89,7 +89,10 @@ In this section, we review some homotopical results concerning free ($1$-)catego

...

@@ -89,7 +89,10 @@ In this section, we review some homotopical results concerning free ($1$-)catego

of arrows of $L(G)$ such that the length of each $f_i$ is \emph{at most}$k$. In particular, we have

of arrows of $L(G)$ such that

\[

\sum_{1\leq i \leq n}\ell(f_i)\leq n.

\] In particular, we have

\[

\[

N^1(G)=i_!(G)

N^1(G)=i_!(G)

\]

\]

...

@@ -113,15 +116,41 @@ In this section, we review some homotopical results concerning free ($1$-)catego

...

@@ -113,15 +116,41 @@ In this section, we review some homotopical results concerning free ($1$-)catego

\begin{proof}

\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

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}

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

\]

is a trivial cofibration. 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_{k-1}\ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1}

\end{tikzcd}

\]

of arrows of $L(G)$ such that for every $1\leq i \leq k+1$

\[

\ell(f_i)=1,

\]

\]

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

i.e.\ each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$, we define a morphism $\varphi_f : A_{k+1}\to N^{k}(G)$ in the following fashion:

\begin{itemize}

\item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of $N^{k}(G)$

which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{op},\Psh{\Delta})$ in two canonical ways.