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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe *order complex* $\\Delta(P)$ of $P$ is the abstract [[simplicial complex|V8GZ0W1ZZ5]] with vertex set $P$ whose faces are the finite [[chains|0S8FE7ZMFM]] of $P$.\nSo the faces are the finite subsets $\\set{x_0, x_1, \\ldots, x_k} \\subseteq P$ with $x_0 \u003c x_1 \u003c \\cdots \u003c x_k$.\n\n## Remarks\n\n$\\Delta(P)$ is a simplicial complex because every subset of a chain is a chain.\nA chain with $k + 1$ elements is a face of dimension $k$, so the dimension of $\\Delta(P)$ is the length of the longest chain of $P$, when it is finite.\nTopological properties of $P$, such as its homotopy type or homology, mean those of the geometric realization of $\\Delta(P)$.\nExample: let $P = \\set{a, b, c, d}$ with $a \u003c c$, $a \u003c d$, $b \u003c c$, $b \u003c d$ and no other strict relations.\nIts nonempty chains are the four points and the four pairs $\\set{a, c}$, $\\set{a, d}$, $\\set{b, c}$, $\\set{b, d}$.\nSo $\\Delta(P)$ is a cycle of four edges, and its realization is a circle.\nNon-example: for this $P$, the family of all subsets of $P$ is a simplicial complex on the vertex set $P$.\nIt is not $\\Delta(P)$, since it contains $\\set{a, b}$, which is not a chain.\nThe order complex of a chain with $n + 1$ elements is a full $n$-simplex.\n",
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