in mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. in some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold. the concept was first introduced by jean-marie souriau in the 1980s and developed first by his students paul donato (homogeneous spaces and coverings) and patrick iglesias (diffeological fiber bundles, higher homotopy etc.), later by other people. a related idea was introduced by kuo-tsaÃ¯ chen (é³åæ, chen guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots. ==definition== if x is a set, a diffeology on x is a set of maps, called plots, from open subsets of rn (n â¥ 0) to x such that the following hold: * every constant map is a plot. * for a given map, if every point in the domain has a neighborhood such that restricting the map to this neighborhood is a plot, then the map itself is a plot. * if p is a plot, and f is a smooth function from an open subset of some real vector space into the domain of p, then the composition p  f is a plot. note that the domains of different plots can be subsets of rn for different values of n. a set together with a diffeology is called a diffeological space. a map between diffeological spaces is called differentiable if and only if composing it with every plot of the first space is a plot of the second space. it is a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable. the diffeological spaces, together with differentiable maps as morphisms, form a category. the isomorphisms in this category are just the diffeomorphisms defined above. the category of diffeological spaces is closed under many categorical operations. a diffeological space has the d-topology: the finest topology such that all plots are continuous. if y is a subset of the diffeological space x, then y is itself a diffeological space in a natural way: the plots of y are those plots of x whose images are subsets of y. if x is a diffeological space and ~ is some equivalence relation on x, then the quotient set x/~ has the diffeology generated by all compositions of plots of x with the projection from x to x/~. this is called the quotient diffeology. note that the quotient d-topology is the d-topology of the quotient diffeology, and that this topology may be trivial without the diffeology being trivial. a cartan de rham calculus can be developed in the framework of diffeology, as well as fiber bundles, homotopy etc. ==smooth manifolds== [[diffe
Chinese_mathematical_discoveries Differential_geometry Functions_and_mappings