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”.
2-simplex
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.
3-simplex
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}

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