Horn

Remarks

Equivalently, $\Lambda^n_k$ is the union of the faces $d^i \Delta^{n-1} \subseteq \Delta^n$ for $i \ne k$, where $d^i$ is the face that omits the vertex $i$. Its geometric realization is the boundary of the $n$-simplex with the interior of the face opposite the vertex $k$ removed.

Example: $\Lambda^2_1$ consists of the edges $\set{0, 1}$ and $\set{1, 2}$ and their vertices. The edge $\set{0, 2}$ is the map $[1] \to [2]$ with image $\set{0, 2}$, and $\set{0, 2} \cup \set{1} = [2]$, so it is not in $\Lambda^2_1$. The edge $\set{0, 1}$ has image $\set{0, 1}$, and $\set{0, 1} \cup \set{1} \ne [2]$, so it is in $\Lambda^2_1$.

Non-example: the boundary $\partial \Delta^2$ contains all three edges, so it is not a horn. The edge $\set{0, 2}$ fails the condition for $k = 1$, the edge $\set{1, 2}$ for $k = 0$, and the edge $\set{0, 1}$ for $k = 2$.

A map $\Lambda^n_k \to X$ to a simplicial set $X$ is the same as a family of $n$ simplices $x_i \in X_{n-1}$, $i \ne k$, that agree on their common faces.