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  "body": "\n## Definition\n\nLet $K$ be a simplicial set, with $n$-simplices $K_n$ and structure maps $\\theta^* \\colon K_n \\to K_m$ for order-preserving $\\theta \\colon [m] \\to [n]$.\nLet $\\abs{\\Delta^n}$ be the standard topological $n$-simplex, and let $\\theta_* \\colon \\abs{\\Delta^m} \\to \\abs{\\Delta^n}$ be the affine map sending the vertex $e_i$ to $e_{\\theta(i)}$.\nGive each set $K_n$ the discrete topology.\nThe *geometric realization* of $K$ is the [[quotient space|0V8046R0P9]]\n$$\\abs{K} = \\Bigl( \\bigsqcup_{n \\ge 0} K_n \\times \\abs{\\Delta^n} \\Bigr) \\Big/ \\sim$$\nof the [[disjoint union|A486VR787Z]], where $\\sim$ is the equivalence relation generated by\n$$(\\theta^* x, t) \\sim (x, \\theta_* t) \\quad \\text{for all } \\theta \\colon [m] \\to [n], \\ x \\in K_n, \\ t \\in \\abs{\\Delta^m}.$$\n\n## Remarks\n\nA simplicial map $f \\colon K \\to L$ induces a continuous map $\\abs{f} \\colon \\abs{K} \\to \\abs{L}$, so realization is a functor.\nIt is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor: continuous maps $\\abs{K} \\to X$ correspond to simplicial maps $K \\to \\operatorname{Sing} X$.\nEach point of $\\abs{K}$ has a unique representative $(x, t)$ with $x$ nondegenerate and $t$ in the interior of $\\abs{\\Delta^n}$.\nIt follows that $\\abs{K}$ is a [[CW complex|5W95QFHXPR]] with one $n$-cell for each nondegenerate $n$-simplex.\n\nExample: for the standard simplex $\\Delta^n$, the realization $\\abs{\\Delta^n}$ is the topological $n$-simplex, since every simplex of $\\Delta^n$ is $\\theta^*$ of the top simplex $\\id_{[n]}$.\n\nExample: the quotient $\\Delta^1 / \\partial \\Delta^1$ has one nondegenerate $0$-simplex and one nondegenerate $1$-simplex whose two faces are equal, so its realization is a circle.\n\nNon-example: the disjoint union $\\bigsqcup_n K_n \\times \\abs{\\Delta^n}$ without the relation $\\sim$ is not the realization; for $K = \\Delta^1$ it has infinitely many components, while $\\abs{\\Delta^1}$ is an interval.\n\nSome authors use the category of compactly generated spaces, where the realization functor preserves finite products.\n",
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