Let be a simplicial set. The elements are said to be matching faces with respect to if for and .

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

In the 2-simplex, let , , . Then are matching faces with respect to 1, since .

In the 3-simplex, let , , , . Then , , are matching faces with respect to 1, since the following hold:

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