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  "body": "\n## Definition\n\nFor $n \\ge 0$ let $[n] = \\set{0, 1, \\dots, n}$, and let $\\Delta^n$ be the standard $n$-simplex simplicial set, whose $m$-simplices are the order-preserving maps $[m] \\to [n]$.\nLet $n \\ge 1$ and $0 \\le k \\le n$.\nThe *horn* $\\Lambda^n_k$ is the simplicial subset of $\\Delta^n$ whose $m$-simplices are the order-preserving maps $\\alpha \\colon [m] \\to [n]$ such that\n$$\\alpha([m]) \\cup \\set{k} \\ne [n].$$\nThe horn is *inner* if $0 \u003c k \u003c n$ and *outer* if $k = 0$ or $k = n$.\n\n## Remarks\n\nEquivalently, $\\Lambda^n_k$ 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 $i$.\nIts geometric realization is the boundary of the $n$-simplex with the interior of the face opposite the vertex $k$ removed.\n\nExample: $\\Lambda^2_1$ consists of the edges $\\set{0, 1}$ and $\\set{1, 2}$ and their vertices.\nThe 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 $\\Lambda^2_1$.\nThe edge $\\set{0, 1}$ has image $\\set{0, 1}$, and $\\set{0, 1} \\cup \\set{1} \\ne [2]$, so it is in $\\Lambda^2_1$.\n\nNon-example: the boundary $\\partial \\Delta^2$ contains all three edges, so it is not a horn.\nThe 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 $k = 2$.\n\nA 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.\n",
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