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  "body": "\n## Summary\n\nThe order complex of a [[partially ordered set|0QV2JS9RJ3]] $P$ is a [[simplicial complex|V8GZ0W1ZZ5]] with the elements of $P$ as vertices.\nIts faces are the finite [[chains|0S8FE7ZMFM]] of $P$.\nIt turns a poset into a topological space, its [[geometric realization|TG6WHHR7CF]].\nSo one can speak of the homotopy type and the homology of a poset.\nThis is the basis of poset topology.\nFor example, take an open interval of a finite poset.\nBy a theorem of Philip Hall, the reduced Euler characteristic of its order complex equals the value of the [[Möbius function|2Y75S8NG5B]] on that interval.\nThe order complex of the face poset of a simplicial complex is its [[barycentric subdivision|J5DZEYCEX2]].\n\n## Explanations\n\n### L1 human\n\nDraw a point for each element of the poset.\nJoin two points by an edge when they are comparable, fill in a triangle for each chain of three elements, a tetrahedron for each chain of four, and so on.\nThe resulting shape is the order complex.\nFor a chain $a \u003c b \u003c c$ the shape is a full triangle.\nFor the poset with $a, b \u003c c, d$ and no other relations, the four edges $ac$, $ad$, $bc$, $bd$ form a square loop, which is a circle.\n",
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