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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category.\nA small category $\\mathcal{I}$ is *cofiltered* if its [[opposite category|DAMK8XE1FR]] $\\mathcal{I}^{\\mathrm{op}}$ is [[filtered|975PMY4FG8]].\nA *pro-object* in $\\mathcal{C}$ is a functor $X \\colon \\mathcal{I} \\to \\mathcal{C}$ with $\\mathcal{I}$ small and cofiltered, written $(X_i)_{i \\in \\mathcal{I}}$.\nFor pro-objects $X \\colon \\mathcal{I} \\to \\mathcal{C}$ and $Y \\colon \\mathcal{J} \\to \\mathcal{C}$ the morphisms are\n$$\\mathrm{Pro}(\\mathcal{C})(X, Y) = \\lim_{j} \\operatorname{colim}_{i} \\mathcal{C}(X_i, Y_j).$$\nThe pro-objects with these morphisms form the category $\\mathrm{Pro}(\\mathcal{C})$.\n\n## Remarks\n\nThere is an equivalence $\\mathrm{Pro}(\\mathcal{C}) \\simeq \\mathrm{Ind}(\\mathcal{C}^{\\mathrm{op}})^{\\mathrm{op}}$ with the [[Ind-completion|1DKFWGWHH5]] of the opposite category.\nAn object $c$ gives the constant pro-object indexed by the one-point category, and this gives a fully faithful functor $\\mathcal{C} \\to \\mathrm{Pro}(\\mathcal{C})$.\nEvery pro-object is the [[limit|CHS5AFJGWE]] in $\\mathrm{Pro}(\\mathcal{C})$ of its stages.\n\nExample: in the category of finite groups, the groups $\\Z / p^n \\Z$ for $n \\ge 1$, with the reduction maps, form a pro-object.\nIts index category is the cofiltered poset $\\N$ with an arrow $m \\to n$ whenever $m \\ge n$.\nMore generally, every [[profinite group|KA0YDB6QXE]] is the limit of the pro-object of its finite quotients.\n\nNon-example: a functor from the discrete category with two objects to $\\mathcal{C}$ is not a pro-object.\nThe discrete category with two objects is not cofiltered, since its two objects have no common object mapping to both.\n",
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