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  "body": "\n## Summary\n\nA Frölicher space is a set with a family of smooth curves into it and a family of smooth real functions on it.\nEach of the two families determines the other.\nA curve is smooth when every smooth function composed with it is smooth.\nA function is smooth when it composes smoothly with every smooth curve.\nFrölicher spaces generalize [[smooth manifolds|F45VQ47GPV]] and are used for infinite-dimensional spaces such as spaces of smooth maps.\nThey are closely related to [[diffeological spaces|7JS0JFR89E]] and form a cartesian closed category.\n\n## Explanations\n\n### L1 human\n\nOne can learn a lot about a space by testing it from two sides: with curves going into it and with functions going out of it.\nA Frölicher space fixes both families so that they match perfectly.\nA curve passes the test if every allowed function sees it as smooth, and a function passes if it sees every allowed curve as smooth.\nFor ordinary Euclidean space, the usual smooth curves and smooth functions match in this way.\n",
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