tikz.tex

\begin{center}
    \begin{tikzpicture}
        \coordinate (a) at (-1.6, -0.1);
        \coordinate (b) at (0, -1.0);
        \coordinate (c) at (1.2, 0.4);
        \coordinate (d) at (0, 1.5);
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to node[below left] {$p$} (b);
        \draw[->, shorten >=4pt, shorten <=4pt] (b) to node[below right] {$q$} (c);
        \draw[->, shorten >=4pt, shorten <=4pt] (c) to node[above right] {$r$} (d);
        \draw[dashed, shorten >=4pt, shorten <=4pt] (a) -- (c);
        \draw[dashed, shorten >=4pt, shorten <=4pt] (a) -- (d);
        \draw[dashed, shorten >=4pt, shorten <=4pt] (b) -- (d);
        \fill (a) circle (0.07) node[left] {0};
        \fill (b) circle (0.07) node[below] {1};
        \fill (c) circle (0.07) node[right] {2};
        \fill (d) circle (0.07) node[above] {3};    \end{tikzpicture}
\end{center}
Let $g$ be a concatenation of $f$ and $r$. 
We can then fill in the dashed arrows above and get the $1$-skeleton of a tetrahedron.
\begin{center}
    \begin{tikzpicture}
        \coordinate (a) at (-1.6, -0.1);
        \coordinate (b) at (0, -1.0);
        \coordinate (c) at (1.2, 0.4);
        \coordinate (d) at (0, 1.5);
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to node[below left] {$p$} (b);
        \draw[->, shorten >=4pt, shorten <=4pt] (b) to node[below right] {$q$} (c);
        \draw[->, shorten >=4pt, shorten <=4pt] (c) to node[above right] {$r$} (d);
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to node[above left] {$f$} (c);
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to node[above left] {$g$} (d);
        \draw[->, shorten >=4pt, shorten <=4pt] (b) to node[below right] {$f'$} (d);
        \fill (a) circle (0.07) node[left] {0};
        \fill (b) circle (0.07) node[below] {1};
        \fill (c) circle (0.07) node[right] {2};
        \fill (d) circle (0.07) node[above] {3};    \end{tikzpicture}
\end{center}

If $A, B, C \in S_2$ are $2$-simplices witnessing that $f$ is a concatenation of $p, q$, that $f'$ is a concatenation of $q, r$, and that $g$ is a concatenation of $f, r$ (respectively) then we can fill this diagram in even further.
\begin{center}
    \begin{tikzpicture}
        \coordinate (a) at (-1.6, -0.1);
        \coordinate (b) at (0, -1.0);
        \coordinate (c) at (1.2, 0.4);
        \coordinate (d) at (0, 1.5);

        \fill[cornflowerblue, opacity=.6] (a) -- (b) -- (c) -- cycle;
        \fill[celadon, opacity=.6] (c) -- (d) -- (a) -- cycle;

        \draw[->, shorten >=4pt, shorten <=4pt] (a) to (c);

        \fill[bittersweet, opacity=.6] (b) -- (c) -- (d) -- cycle;
        
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to (b);
        \draw[->, shorten >=4pt, shorten <=4pt] (b) to (c);
        \draw[->, shorten >=4pt, shorten <=4pt] (c) to (d);
        \draw[->, shorten >=4pt, shorten <=4pt] (a) to (d);
        \draw[->, shorten >=4pt, shorten <=4pt] (b) to (d);
        \fill (a) circle (0.07) node[left] {0};
        \fill (b) circle (0.07) node[below] {1};
        \fill (c) circle (0.07) node[right] {2};
        \fill (d) circle (0.07) node[above] {3};

        \node[rotate=-40] at (barycentric cs:a=1.3,b=1,c=1) {$A$};
        \node[rotate=30] at (barycentric cs:b=1,c=1,d=1.4) {$B$};
        \node at (barycentric cs:a=1.6,c=1,d=1.6) {$\reflectbox{C}$};
    \end{tikzpicture}
\end{center}