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  "body": "\n## Summary\n\nA diffeology on a set declares which maps from open subsets of Euclidean spaces into the set count as smooth; these maps are called plots.\nA set with a diffeology is a diffeological space.\nEvery [[smooth manifold|F45VQ47GPV]] is a diffeological space, but so are quotients, subsets and spaces of smooth maps, which are often not manifolds.\nDiffeological spaces form a [[cartesian closed category|BFBNQ36QS8]] with all limits and colimits, so constructions that leave the world of manifolds stay inside this one.\n\n## Explanations\n\n### L1 human\n\nOn a manifold, one knows which curves and surfaces inside it are smooth.\nA diffeology takes this knowledge as the definition: a space is \"smooth\" once one says which parametrized families of points in it are smooth.\nFor instance, the quotient of the torus by a line of irrational slope has a useless topology, since its only open sets are the empty set and everything.\nYet its diffeology still remembers a lot of geometry.\nThis makes diffeology a good language for singular and infinite-dimensional spaces.\n",
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