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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a small [[category|A2DE66937D]], and let $\\Delta$ be the [[simplex category|FRFYHJGAN3]].\nView each object $[n]$ of $\\Delta$ as the category with objects $0, 1, \\dots, n$ and exactly one morphism $i \\to j$ when $i \\le j$.\nThe *nerve* of $\\mathcal{C}$ is the [[simplicial set|WW81ARB8K9]] $N\\mathcal{C}$ with\n$$N\\mathcal{C}_n = \\set{\\text{functors } [n] \\to \\mathcal{C}},$$\nwhere an order-preserving map $\\theta \\colon [m] \\to [n]$ acts by precomposition, sending $F \\colon [n] \\to \\mathcal{C}$ to $F \\circ \\theta$.\n\n## Remarks\n\nA [[functor|28NQ0KRHD3]] $[n] \\to \\mathcal{C}$ is the same as a string $c_0 \\xrightarrow{f_1} c_1 \\to \\cdots \\xrightarrow{f_n} c_n$ of composable morphisms.\nSo $N\\mathcal{C}_0$ is the set of objects and $N\\mathcal{C}_1$ the set of morphisms.\nThe face map $d_i$ for $0 \u003c i \u003c n$ composes $f_{i+1} \\circ f_i$, while $d_0$ drops $f_1$ and $d_n$ drops $f_n$; the degeneracy $s_i$ inserts an identity at $c_i$.\nExample: if $\\mathcal{C}$ has one object and only the identity morphism, then $N\\mathcal{C}_n$ has one element for each $n$, and $N\\mathcal{C}$ is the point $\\Delta^0$.\nNon-example: the boundary $\\partial \\Delta^2$ consists of three edges forming a triangle, with no 2-simplex.\nIt is not the nerve of any category, because in a nerve each pair of composable edges is filled by a 2-simplex.\nA simplicial set is isomorphic to a nerve exactly when every inner horn $\\Lambda^n_k \\to X$ with $0 \u003c k \u003c n$ has a unique filler.\n",
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