Category of manifolds

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In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp.

One is often interested only in Cp-manifolds modelled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modelled on a fixed space E is denoted Manp(E).

One may also speak of the category of smooth manifolds, Man, or the category of analytic manifolds, Manω.

Manp is a concrete category[edit]

Like many categories, the category Manp 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 Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor

U : ManpTop

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′ : ManpSet

to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.

References[edit]

  • Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.