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  "body": "\n## Description\n\nFor each natural number $n$, let $[n]$ be the [[totally ordered|0FRXBVVR03]] set $\\set{0, 1, \\dots, n}$ with the usual order.\nThe *simplex category* $\\Delta$ has the sets $[n]$ for $n \\ge 0$ as objects.\nA morphism $[m] \\to [n]$ is an order-preserving ([[monotone|0A7KV2WB7A]]) map, that is, a map $f$ with $f(i) \\le f(j)$ whenever $i \\le j$.\nComposition is the composition of maps.\nA composite of order-preserving maps is order-preserving, and identity maps are order-preserving, so $\\Delta$ is a [[category|A2DE66937D]].\n\n## Remarks\n\nEvery morphism of $\\Delta$ is a composite of two kinds of basic maps.\nThe face map $\\delta_i \\colon [n-1] \\to [n]$, for $0 \\le i \\le n$, is the injective order-preserving map whose image misses $i$.\nThe degeneracy map $\\sigma_i \\colon [n+1] \\to [n]$, for $0 \\le i \\le n$, is the surjective order-preserving map that takes the value $i$ twice.\nFor example, the map $[2] \\to [1]$ with $0 \\mapsto 0$, $1 \\mapsto 0$, $2 \\mapsto 1$ is $\\sigma_0$.\nOne pictures $[n]$ as the standard $n$-simplex with vertices $0, \\dots, n$: face maps include a face, and degeneracy maps collapse an edge.\nFunctors from the opposite category $\\Delta^{\\mathrm{op}}$ are the [[simplicial objects|WK0A32TZHB]], and functors $\\Delta^{\\mathrm{op}} \\to \\mathbf{Set}$ are the [[simplicial sets|WW81ARB8K9]].\n\n## Explanations\n\n### L1 human\n\nThe simplex category is a small list of shapes: a point, an edge, a triangle, a tetrahedron, and so on.\nThe arrows between shapes say how one shape sits in another as a face, or how a shape can be squashed onto a smaller one.\nFor example, there are two maps $[0] \\to [1]$, which pick the two endpoints of an edge, and three maps $[1] \\to [1]$: the identity and the two constant maps.\n",
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