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  "body": "\n## Definition\n\nFor $n \\ge 1$ and $0 \\le k \\le n$, the *horn* $\\Lambda^n_k$ is the simplicial subset of the standard $n$-simplex $\\Delta^n$ generated by all faces $d_i \\Delta^n$ with $i \\ne k$.\nA [[simplicial set|WW81ARB8K9]] $X$ is a *Kan complex* if for all $n \\ge 1$ and $0 \\le k \\le n$, every map of simplicial sets $\\Lambda^n_k \\to X$ extends to a map $\\Delta^n \\to X$.\n\n## Remarks\n\nEquivalently, $X$ is a Kan complex when the map $X \\to \\Delta^0$ is a Kan fibration.\nA map $\\Lambda^n_k \\to X$ is the same as a list of simplices $x_i \\in X_{n-1}$ for $i \\ne k$ with $d_i x_j = d_{j-1} x_i$ whenever $i \u003c j$ and $i, j \\ne k$.\nA filler is an $x \\in X_n$ with $d_i x = x_i$ for $i \\ne k$.\n\nExample: for a topological space $T$, the [[singular simplicial set|RWBMXQW6S5]] $\\mathrm{Sing}(T)$ is a Kan complex.\nA map $\\Lambda^n_k \\to \\mathrm{Sing}(T)$ is a continuous map from the geometric horn $\\abs{\\Lambda^n_k}$ to $T$, and $\\abs{\\Lambda^n_k}$ is a retract of the topological simplex $\\abs{\\Delta^n}$, so the map extends.\n\nExample: the [[nerve|RV0MKJ3CQ6]] of a [[groupoid|FH3C0T643F]] is a Kan complex.\n\nNon-example: $\\Delta^1$, the nerve of the poset $0 \u003c 1$, is not a Kan complex.\nThe map $\\Lambda^2_0 \\to \\Delta^1$ sending the vertices $0, 1, 2$ to $0, 1, 0$ is defined, since the edges $0 \\to 1$ and $0 \\to 2$ go to $0 \\le 1$ and $0 \\le 0$.\nA filler would be an order-preserving map $[2] \\to [1]$ with values $0, 1, 0$, which does not exist.\nMore generally, the nerve of a category is a Kan complex if and only if the category is a groupoid.\n",
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