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  "body": "\n## Summary\n\nThe nerve of a small [[category|A2DE66937D]] $\\mathcal{C}$ is the [[simplicial set|WW81ARB8K9]] whose $n$-simplices are the strings of $n$ composable morphisms of $\\mathcal{C}$.\nThe construction is a fully faithful [[functor|28NQ0KRHD3]] from small categories to simplicial sets, so no information about $\\mathcal{C}$ is lost.\nIts geometric realization is the classifying space of $\\mathcal{C}$; for a group $G$ viewed as a one-object category it gives the classifying space $BG$.\nNerves are the basic examples of [[quasi-categories|9328WEVJEK]] and, for groupoids, of [[Kan complexes|FDFYCM3YYC]].\n\n## Explanations\n\n### L1 human\n\nDraw a point for each object of the category and an edge for each arrow.\nFill in a triangle for each pair of arrows that can be composed, with the composite as the third side.\nThen fill in a tetrahedron for each string of three composable arrows, and so on.\nThe resulting shape is the nerve.\nFor example, the category $0 \\to 1$ with one non-identity arrow has as nerve a single edge, the standard $1$-simplex $\\Delta^1$, plus its degenerate simplices.\n",
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