Definition 14.11.1. Let $U$ be a simplicial set. We say $x$ is an $n$-simplex of $U$ to signify that $x$ is an element of $U_ n$. We say that $y$ is the $j$th face of $x$ to signify that $d^ n_ jx = y$. We say that $z$ is the $j$th degeneracy of $x$ if $z = s^ n_ jx$. A simplex is called degenerate if it is the degeneracy of another simplex.
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