Preface:
Note that although this is a proof verification question, I'd also be happy with a link to a reference outlining a proof of the following statement. As far as I can tell this fact has not been proven on MSE, though it is referenced in two other questions here and here.
My motivation here is that I need this fact to prove that the latching space $L_nX$ for a simplicial set $X$ is the set of degenerate $n$-simplices, (specifically that the canonical map $L_nX\to X$ induced by the degeneracies is injective) and I realized that I'd never gone and proved this.
The statement: If $X$ is a simplicial set, $x\in X_n$ is an $n$-simplex, then there is a unique non-degenerate simplex $y\in X_m$ such that $x=s_{i_1}s_{i_2}\cdots s_{i_{n-m}}y$ for some indices $i_k$.
My proof.
I'm not really sure if this is standard terminology, but I will say a simplex $x$ is a degeneracy of a simplex $y$ if $x$ can be obtained from $y$ by a sequence of degeneracy maps $s_i$. In particular, this induces a partial order on the set of all simplices in $X$. (Where $y\le x$ if $x$ is a degeneracy of $y$).
In this language then, we want to show that the set $\{y: y\le x\}$ has a unique minimal element, since an element of a downward closed subset of this poset is minimal if and only if it is nondegenerate.
Note that this poset is nonempty, since it always contains $x$, and therefore has minimal elements, since chains are bounded in length by the dimension of $x$.
Suppose then that $y_1$ and $y_2$ are both minimal elements. For notational convenience, I'll use capital letters for multi-indices, so $s_I=s_{i_1}\circ\cdots\circ s_{i_k}$ where $I=(i_1,\ldots,i_k)$ is a sequence of indices.
We can write
$x=s_Jy_2$ for some sequence of indices $J$.
Now note that if $y\le x$, then this implies that $y$ is a face of $x$, since every surjective map $[n]\to [k]$ of posets admits a section.
Therefore we also have $y_1=d_{I}x$ for some sequence of indices $I$. Thus
$$y_1= d_{I}x=d_{I}s_{J}y_2$$
By the simplicial identities, we can commute a sequence of face maps past a sequence of degeneracy maps to get
$$y_1=s_{J'}d_{I'}y_2$$
for some sequences $J'$ and $I'$. However, by assumption, $y_1$ is minimal (and thus nondegenerate), so $J'$ must be empty. Therefore
$y_1=d_{I'}y_2$, or $y_1$ is a face of $y_2$. However, by symmetry, we must also have $y_2$ is a face of $y_1$, and therefore $y_1=y_2$, as desired.
$\blacksquare$
End note:
While not directly related to my question, the first question linked above, is a really interesting proof verification question purporting to show that the sequence of degeneracies $s_{i_{n-m}}\cdots s_{i_1}$ is unique assuming that the indices are in ascending order. The proof seems a bit sketchy to me around the claim $j_{n+1}\in \{i_{n+1},i_{n+1}+1\}$, but I'm still very much new to simplicial sets. I figured I'd give the question a fresh shout out here, in hopes that someone more experienced might be able to either give a reference to a proof, or provide a counterexample. (The other question linked above is actually the same underlying question, but a bit less precisely phrased imho.)
Best Answer
Proposition 4.8 in Friedman's "Elementary illustrated introduction" is this result, and a proof is given.