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  "body": "\n## Definition\n\nLet $(I, \\le)$ be a [[directed set|D9RY4667K6]] and let $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *direct system* in $\\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \\in I$ with [[morphisms|859F0TFVA3]] $f_{ij} \\colon A_i \\to A_j$ for all $i \\le j$, such that:\n\n1. $f_{ii} = \\id_{A_i}$ for all $i$.\n2. $f_{jk} \\circ f_{ij} = f_{ik}$ for all $i \\le j \\le k$.\n\nA *direct limit* of this system is an object $A$ with morphisms $\\phi_i \\colon A_i \\to A$ such that:\n\n1. $\\phi_j \\circ f_{ij} = \\phi_i$ for all $i \\le j$.\n2. For every object $B$ with morphisms $g_i \\colon A_i \\to B$ such that $g_j \\circ f_{ij} = g_i$ for all $i \\le j$, there is a unique morphism $u \\colon A \\to B$ with $u \\circ \\phi_i = g_i$ for all $i$.\n\n## Remarks\n\nThe direct limit is written $\\varinjlim A_i$.\nIt is the [[colimit|BJS1CGTKQ5]] 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 disjoint union of the $A_i$ divided by a relation: $a_i \\in A_i$ and $a_j \\in A_j$ are identified when $f_{ik}(a_i) = f_{jk}(a_j)$ for some $k \\ge i, j$.\n\nExample: let $A_n = \\Z$ for $n \\in \\N$, and let $f_{n,n+1}$ be multiplication by $2$.\nThe maps $\\phi_n \\colon \\Z \\to \\Z[1/2]$, $\\phi_n(a) = a/2^n$, make $\\Z[1/2]$ a direct limit.\nClause 1 holds because $\\phi_{n+1}(2a) = 2a/2^{n+1} = \\phi_n(a)$.\n\nNon-example: the maps $\\phi_n(a) = a/2^n$ into $\\Q$ satisfy clause 1 but not clause 2.\nFor $B = \\Z[1/2]$ with the maps above, there is no group homomorphism $u \\colon \\Q \\to \\Z[1/2]$ with $u(a/2^n) = a/2^n$ for all $a$ and $n$.\nIndeed $u(1/3)$ would be an element $x$ with $3x = 1$, and $\\Z[1/2]$ has no such element.\n",
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