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 $k,j+1\neq i$.
My question is what is the geometric intuition behind the term matching faces? Why the terminology "matching", which suggests "similarity" of some sort?
Thanks for enlightenment!
It's exactly as if $x_k$ were the $k$th face of some $n$-simplex $y$ (i.e. $x_k = d_k y$), except that the full simplex $y$ may not necessarily exist. As Zhen Lin suggest in the comments, drawing a picture is quite helpful. For example when $n = 2$ and $i = 0$, this looks like this:
For $n=3$ it looks like a (hollow) tetrahedron with the face opposite to the vertex $i$ removed; the $x_k$ are the remaining faces (with $x_k$ being opposite to the vertex $k$). When the simplex $y$ exists for all combinations of matching faces, the simplicial set is said to be a Kan complex, or fibrant.