Category of manifolds
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).
Manp is a concrete category
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 : Manp → 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′ : Manp → 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.
- Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
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