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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], and let $\\Delta$ be the [[simplex category|FRFYHJGAN3]] with [[opposite category|DAMK8XE1FR]] $\\Delta^{\\mathrm{op}}$.\nA *simplicial object* in $\\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $X \\colon \\Delta^{\\mathrm{op}} \\to \\mathcal{C}$.\nOne writes $X_n$ for $X([n])$.\nA morphism of simplicial objects $X \\to Y$ is a [[natural transformation|KYCPCC57FT]] from $X$ to $Y$.\n\n## Remarks\n\nEquivalently, a simplicial object is a family of objects $X_n$, $n \\ge 0$, with face maps $d_i \\colon X_n \\to X_{n-1}$ and degeneracy maps $s_i \\colon X_n \\to X_{n+1}$ for $0 \\le i \\le n$, subject to the simplicial identities:\n$$d_i d_j = d_{j-1} d_i \\ (i \u003c j), \\qquad s_i s_j = s_{j+1} s_i \\ (i \\le j),$$\n$$d_i s_j = s_{j-1} d_i \\ (i \u003c j), \\qquad d_j s_j = d_{j+1} s_j = \\id, \\qquad d_i s_j = s_j d_{i-1} \\ (i \u003e j+1).$$\nHere $d_i = X(\\delta_i)$ and $s_i = X(\\sigma_i)$ for the face and degeneracy maps $\\delta_i$, $\\sigma_i$ of $\\Delta$.\nExample: for an object $A$, the constant functor with $X_n = A$ and every $d_i$, $s_i$ equal to $\\id_A$ satisfies all identities, so it is a simplicial object.\nNon-example: a covariant functor $\\Delta \\to \\mathcal{C}$ is not a simplicial object, because its maps go from $X_{n-1}$ to $X_n$ for faces; such functors are called cosimplicial objects.\nSimplicial objects in $\\mathbf{Set}$ are [[simplicial sets|WW81ARB8K9]].\n",
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