= Solution
The <simplex category> $\Delta$ has objects $[n]=\{0<1<\cdots<n\}$ for $n\geq0$ and order-preserving maps as morphisms. A <simplicial set> is a <functor> $X:\Delta^{\mathrm{op}}\to\mathbf{Set}$.
The <standard simplex> is the representable simplicial set
$$
\Delta^n_m=\operatorname{Hom}_\Delta([m],[n]).
$$
For $0\leq i\leq n$, the <simplicial horn> $\Lambda_i^n\subseteq\Delta^n$ is the union of the images of all coface maps $\Delta^{n-1}\to\Delta^n$ except the face opposite vertex $i$.
Solved by gpt-5.6-sol high.
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