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  "body": "\n## Definition\n\nLet $\\Delta$ be the [[simplex category|FRFYHJGAN3]].\nA *simplicial set* is a functor $X \\colon \\Delta^{\\mathrm{op}} \\to \\mathbf{Set}$, that is, a [[presheaf|AND2WHZC8P]] on $\\Delta$.\nThe elements of $X_n = X([n])$ are the $n$-simplices of $X$.\nA map of simplicial sets is a natural transformation.\nThus a simplicial set is a [[simplicial object|WK0A32TZHB]] in the category of sets.\n\n## Remarks\n\nConcretely, a simplicial set is a family of sets $X_n$ with face maps $d_i \\colon X_n \\to X_{n-1}$ and degeneracy maps $s_i \\colon X_n \\to X_{n+1}$, $0 \\le i \\le n$, that satisfy the simplicial identities.\nAn $n$-simplex is degenerate when it lies in the image of some $s_i$.\nExample: the standard $n$-simplex $\\Delta^n = \\Hom_{\\Delta}(-, [n])$ is a representable functor, so it is a simplicial set; its nondegenerate $m$-simplices are the injective order-preserving maps $[m] \\to [n]$.\nNon-example: a family of sets $X_n$ with only face maps and no degeneracy maps is not a simplicial set; such data form a semi-simplicial set.\nThe category of simplicial sets is often written $\\mathbf{sSet}$.\n",
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