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  "body": "\n## Summary\n\nA functor $U \\colon \\mathcal{D} \\to \\mathcal{C}$ is monadic when it has a left adjoint and $\\mathcal{D}$ is, up to equivalence, the category of algebras of the resulting [[monad|FF6XXA6D1A]].\nSo the objects of $\\mathcal{D}$ are exactly objects of $\\mathcal{C}$ with algebraic structure described by the monad.\nThe forgetful functors from groups, rings, modules and compact Hausdorff spaces to sets are monadic.\nBeck's monadicity theorem characterizes monadic functors.\n\n## Explanations\n\n### L1 human\n\nA forgetful functor forgets structure, for example it sends a group to its underlying set.\nIt is monadic when the forgotten structure can be fully recovered as a way of evaluating formal expressions.\nGroups are monadic over sets: a group is a set with a consistent way to evaluate words in its elements.\nTopological spaces are not monadic over sets, since a topology is not an algebraic structure of this kind.\n",
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