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  "body": "\n## Summary\n\nA horn is the union of all the codimension-one faces of a standard simplex except one.\nIn the theory of simplicial sets, horns are the shapes against which one tests extension properties.\nA Kan complex is a simplicial set in which every horn can be filled.\nA quasi-category is one in which every inner horn can be filled.\n\n## Explanations\n\n### L1 human\n\nTake a triangle and keep only two of its three edges.\nThe two edges form a corner shaped like the letter V, which is the horn $\\Lambda^2_k$, where $k$ is the vertex where the two edges meet.\nFilling the horn means finding the third edge and the triangle itself.\nIn a space, a horn of paths can always be filled.\nTwo paths that meet at a point compose to a third path, and a triangle witnesses the composition.\n",
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