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  "body": "\n## Summary\n\nAn inverse limit, or projective limit, is the [[limit|CHS5AFJGWE]] of a system of objects indexed by a [[directed set|D9RY4667K6]].\nIts maps go from later objects to earlier ones.\nIt collects the compatible families of elements, one in each object of the system.\nThe [[p-adic integers|V6TJAFZ4MD]] and the [[profinite groups|KA0YDB6QXE]] are inverse limits of finite objects.\nThe dual notion is the [[direct limit|X9HEQ29Q49]].\n\n## Explanations\n\n### L1 human\n\nThink of a sequence of ever finer pictures of the same thing, each one able to forget detail and give the previous one.\nThe inverse limit is the object whose points are whole towers of compatible pictures.\nFor example, take the rings $\\Z/p\\Z$, $\\Z/p^2\\Z$, $\\Z/p^3\\Z$, and so on, with reduction maps between them.\nA point of the inverse limit is a sequence of residues $a_n$ modulo $p^n$, with $a_{n+1}$ reducing to $a_n$.\nThese sequences form the ring of $p$-adic integers.\n",
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