Английская Википедия:Category of manifolds
Шаблон:Short description 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 modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled 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
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 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.
Pointed manifolds and the tangent space functor
It is often convenient or necessary to work with the category of manifolds along with a distinguished point: Man•p analogous to Top• - the category of pointed spaces. The objects of Man•p are pairs <math>(M, p_0),</math> where <math>M</math> is a <math>C^p</math>manifold along with a basepoint <math>p_0 \in M ,</math> and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. <math>F: (M,p_0) \to (N,q_0),</math> such that <math>F(p_0) = q_0.</math>[1] The category of pointed manifolds is an example of a comma category - Man•p is exactly <math>\scriptstyle {( \{ \bull \} \downarrow \mathbf{Man^p})},</math> where <math>\{ \bull \}</math> represents an arbitrary singleton set, and the <math>\downarrow</math>represents a map from that singleton to an element of Manp, picking out a basepoint.
The tangent space construction can be viewed as a functor from Man•p to VectR as follows: given pointed manifolds <math>(M, p_0)</math>and <math>(N, F(p_0)),</math> with a <math>C^p</math>map <math>F: (M,p_0) \to (N,F(p_0))</math> between them, we can assign the vector spaces <math>T_{p_0}M</math>and <math>T_{F(p_0)}N,</math> with a linear map between them given by the pushforward (differential): <math>F_{*,p}:T_{p_0}M \to T_{F(p_0)}N.</math> This construction is a genuine functor because the pushforward of the identity map <math>\mathbb{1}_M:M \to M</math> is the vector space isomorphism[1] <math>(\mathbb{1}_M)_{*,p_0}:T_{p_0}M \to T_{p_0}M,</math> and the chain rule ensures that <math>(f\circ g)_{*,p_0} = f_{*,g(p_0)} \circ g_{*,p_0}.</math>[1]
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