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  "body": "\n## Summary\n\nA pro-object in a category $\\mathcal{C}$ is a formal cofiltered limit of objects of $\\mathcal{C}$.\nPro-objects form the category $\\mathrm{Pro}(\\mathcal{C})$, which is the dual of the [[Ind-completion|1DKFWGWHH5]] of the opposite category.\nThey appear in algebraic geometry, in shape theory and in the étale homotopy type.\nThere a limit that does not exist in $\\mathcal{C}$ is kept as a formal system.\nThe category of profinite sets, that is of Stone spaces, is equivalent to the pro-objects in finite sets.\n\n## Explanations\n\n### L1 human\n\nA pro-object is a system of ever finer approximations, kept as a whole instead of being replaced by its limit.\nThe finite groups $\\Z / p^n \\Z$, linked by the reduction maps, form a pro-object in finite groups.\nIts limit is the group of $p$-adic integers, but the pro-object remembers the finite stages themselves.\n",
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