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  "body": "\n## Definition\n\nLet $(I, \\le)$ be a [[directed set|D9RY4667K6]] and let $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn *inverse system* in $\\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \\in I$ with [[morphisms|859F0TFVA3]] $f_{ij} \\colon A_j \\to A_i$ for all $i \\le j$, such that:\n\n1. $f_{ii} = \\id_{A_i}$ for all $i$.\n2. $f_{ij} \\circ f_{jk} = f_{ik}$ for all $i \\le j \\le k$.\n\nAn *inverse limit* of this system is an object $A$ with morphisms $\\pi_i \\colon A \\to A_i$ such that:\n\n1. $f_{ij} \\circ \\pi_j = \\pi_i$ for all $i \\le j$.\n2. For every object $B$ with morphisms $g_i \\colon B \\to A_i$ such that $f_{ij} \\circ g_j = g_i$ for all $i \\le j$, there is a unique morphism $u \\colon B \\to A$ with $\\pi_i \\circ u = g_i$ for all $i$.\n\n## Remarks\n\nThe inverse limit is written $\\varprojlim A_i$.\nIt is the [[limit|CHS5AFJGWE]] of the diagram given by the system, and it is unique up to a unique isomorphism.\nIn sets, groups, rings and modules it always exists.\nThere it is the set of families $(a_i) \\in \\prod_i A_i$ with $f_{ij}(a_j) = a_i$ for all $i \\le j$, with the projections as $\\pi_i$.\n\nExample: the rings $\\Z/p^n\\Z$ for $n \\ge 1$, with the reduction maps, form an inverse system over $\\N$.\nIts inverse limit is the ring $\\Z_p$ of $p$-adic integers.\nClause 1 holds because reducing a compatible sequence at level $n+1$ gives its entry at level $n$.\n\nNon-example: $\\Z$ with the reduction maps $\\Z \\to \\Z/p^n\\Z$ satisfies clause 1 but not clause 2.\nTaking $B = \\Z_p$, a morphism $u \\colon \\Z_p \\to \\Z$ with $\\pi_n \\circ u$ equal to reduction would have to be injective, which is impossible since $\\Z_p$ is uncountable.\n",
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