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Construction (The Horn $\Lambda ^{n}_{i}$). Suppose we are given a pair of integers $0 \leq i \leq n$ with $n > 0$. We define a simplicial set $\Lambda ^{n}_{i}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ by the formula

\[ ( \Lambda ^{n}_{i} )([m]) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}([m], [n]): [n] \nsubseteq \alpha ([m] ) \cup \{ i \} \} . \]

We regard $\Lambda ^{n}_{i}$ as a simplicial subset of the boundary $\operatorname{\partial \Delta }^{n} \subseteq \Delta ^ n$. We will refer to $\Lambda ^{n}_{i}$ as the $i$th horn in $\Delta ^{n}$. We will say that $\Lambda ^{n}_{i}$ is an inner horn if $0 < i < n$, and an outer horn if $i = 0$ or $i = n$.