Proposition 1.1.4.13. Let $n$ be a positive integer. For every simplicial set $S_{\bullet }$, the map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n}, S_{\bullet } ) \rightarrow (S_{n-1})^{n+1} \quad \quad f \mapsto \{ f \circ \delta ^{k}_{n} \} _{0 \leq k \leq n} \]
is an injection, whose image consists of those tuples of $( \sigma _0, \sigma _1, \cdots , \sigma _ n )$ of $(n-1)$-simplices of $S$ which satisfy the identity $d^{n-1}_ i(\sigma _ j) = d^{n-1}_{j-1}( \sigma _{i})$ for $0 \leq i < j \leq n$.
Proof of Proposition 1.1.4.13.
Let $w: \coprod _{0 \leq k \leq n} \Delta ^{n-1} \rightarrow \operatorname{\partial \Delta }^{n}$ be the map given on the $k$th summand by $\delta ^{k}_{n}$. To prove the first assertion of Proposition 1.1.4.13, we must show that $w$ is an epimorphism of simplicial sets: that is, it is surjective on $m$-simplices for each $m \geq 0$. In fact, we can be a bit more precise. Let $\alpha $ be an $m$-simplex of $\Delta ^{n}$, which we identify with a nondecreasing function from $[m]$ to $[n]$. Then $\alpha $ belongs to the boundary $\operatorname{\partial \Delta }^{n}$ if and only if it is not surjective: that is, if and only if there exists some integer $0 \leq i \leq n$ such that $\alpha $ factors through $[n] \setminus \{ i\} $. In this case, there is a unique $m$-simplex $\beta _{i}$ which belongs to the $i$th summand of $\coprod _{0 \leq k \leq n} \Delta ^{n-1}$ and satisfies $w( \beta _ i ) = \alpha $.
For every integer $0 \leq k \leq n$, let $u_{k}: \coprod _{ 0 \leq i < k } \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ be the map given on the $i$th summand by $\delta ^{i}_{n-1}$, and let $v_{k}: \coprod _{ k < j \leq n} \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ by the map given on the $j$th summand by $\delta ^{j-1}_{n-1}$. Passing to the coproduct over $k$ and reindexing, we obtain a pair of maps
\[ (u,v): \coprod _{0 \leq i < j \leq n} \Delta ^{n-2} \rightrightarrows \coprod _{0 \leq k \leq n} \Delta ^{n-1}. \]
Let $\operatorname{Coeq}(u,v)_{\bullet }$ denote the coequalizer of $u$ and $v$ in the category of simplicial sets. The morphism $w$ satisfies $w \circ u = w \circ v$ (see Remark 1.1.1.7), and therefore factors uniquely through a map $\overline{w}: \operatorname{Coeq}(u,v)_{\bullet } \rightarrow \operatorname{\partial \Delta }^{n}$. Proposition 1.1.4.13 asserts that $\overline{w}$ is an isomorphism of simplicial sets: that is, for every integer $m \geq 0$, it induces a bijection from $\operatorname{Coeq}(u,v)_{m}$ to the set of $m$-simplices of $\operatorname{\partial \Delta }^{n}$. The surjectivity of this map was established above. To prove injectivity, it will suffice to observe that if $\alpha : [m] \rightarrow [n]$ is as above and we are given two elements $i,j \in [n]$ which do not belong to the image of $\alpha $, then $\beta _{i}$ and $\beta _{j}$ have the same image in $\operatorname{Coeq}(u,v)_{\bullet }$. If $i = j$, this is automatic; we may therefore assume without loss of generality that $i < j$. In this case, the desired result follows from the observation that we can write $\beta _ j = u(\gamma )$ and $\beta _ i = v(\gamma )$, where $\gamma $ is the $m$-simplex of the $(i,j)$th summand of $\coprod _{ 0 \leq i < j \leq n} \Delta ^{n-2}$ corresponding to the nondecreasing function $[m] \xrightarrow {\alpha } [n] \setminus \{ i < j \} \simeq [n-2]$.
$\square$