Geometrical Meaning of Matching Faces

Let $X$ be a simplicial set. The elements $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ are said to be matching faces with respect to $i$ if $d_jx_k=d_kx_{j+1}$ for $j\geq k$ and $k,j+1\neq i$ .

Geometrically, matching faces are faces that “match” along lower-dimensional faces. In other words, they are “adjacent”.

In the 2-simplex, let $x_0=v_1v_2$ , $x_1=v_0v_2$ , $x_2=v_0v_1$ . Then $x_0, x_2$ are matching faces with respect to 1, since $d_1x_0=v_1=d_0x_2$ .

In the 3-simplex, let $x_0=v_1v_2v_3$ , $x_1=v_0v_1v_2$ , $x_2=v_0v_1v_3$ , $x_3=v_0v_1v_2$ . Then $x_0$ , $x_2$ , $x_3$ are matching faces with respect to 1, since the following hold:
$\begin{aligned} d_1x_0&=v_1v_3=d_0x_2\\ d_2x_0&=v_1v_2=d_0x_3\\ d_2x_2&=v_0v_1=d_2x_3 \end{aligned}$