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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-Tsaï Chen (陳國才, ''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 R''n'' (''n'' ≥ 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'' (unicode:∘) ''f'' is a plot. Note that the domains of different plots can be subsets of R''n'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Diffeology」の詳細全文を読む スポンサード リンク
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