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In mathematics, the category of manifolds, often denoted Man''p'', is the category whose objects are manifolds of smoothness class ''C''''p'' and whose morphisms are ''p''-times continuously differentiable maps. This is a category because the composition of two ''C''''p'' maps is again continuous and of class ''C''''p''. One is often interested only in ''C''''p''-manifolds modelled on spaces in a fixed category ''A'', and the category of such manifolds is denoted Man''p''(''A''). Similarly, the category of ''C''''p''-manifolds modelled on a fixed space ''E'' is denoted Man''p''(''E''). One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Man''ω''. ==Man''p'' is a concrete category== Like many categories, the category Man''p'' is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a ''C''''p''-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor :''U'' : Man''p'' → Top to the category of topological spaces which assigns to each manifold the underlying topological space the underlying set and to each ''p''-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor :''U''′ : Man''p'' → Set to the category of sets which assigns to each manifold the underlying set and to each ''p''-times continuously differentiable function the underlying function. 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「Category of manifolds」の詳細全文を読む スポンサード リンク
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