Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Exercise 1.1.2.14. Let $0 \leq i \leq n$ be integers. For $j \in [n] \setminus \{ i\} $, we can regard the map $\delta ^{j}$ of Exercise 1.1.2.8 as a map of simplicial sets from $\Delta ^{n-1}$ to the horn $\Lambda ^{n}_{i} \subseteq \Delta ^{n}$. Show that, for any simplicial set $S_{\bullet }$, the construction

\[ ( f: \Lambda ^{n}_{i} \rightarrow S_{\bullet } ) \mapsto \{ f \circ \delta ^{j} \} _{j \in [n] \setminus \{ i\} } \]

determines an injection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{i}, S_{\bullet } ) \rightarrow \prod _{ j \in [n] \setminus \{ i\} } S_{n-1}$, whose image consists of ‘incomplete” sequences $( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _ n)$ satisfying $d_ j(\sigma _ k) = d_{k-1}( \sigma _{j})$ for $j, k \in [n] \setminus \{ i\} $ with $j < k$.