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 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 &rarr; 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&prime; : Manp &rarr; 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 $$(M, p_0),$$ where $$M$$ is a $$C^p$$manifold along with a basepoint $$p_0 \in M ,$$ and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. $$F: (M,p_0) \to (N,q_0),$$ such that $$F(p_0) = q_0.$$ The category of pointed manifolds is an example of a comma category - Man•p is exactly $$\scriptstyle {( \{ \bull \} \downarrow \mathbf{Man^p})},$$ where $$\{ \bull \}$$ represents an arbitrary singleton set, and the $$\downarrow$$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 $$(M, p_0)$$and $$(N, F(p_0)),$$ with a $$C^p$$map $$F: (M,p_0) \to (N,F(p_0))$$ between them, we can assign the vector spaces $$T_{p_0}M$$and $$T_{F(p_0)}N,$$ with a linear map between them given by the pushforward (differential): $$F_{*,p}:T_{p_0}M \to T_{F(p_0)}N.$$ This construction is a genuine functor because the pushforward of the identity map $$\mathbb{1}_M:M \to M$$ is the vector space isomorphism $$(\mathbb{1}_M)_{*,p_0}:T_{p_0}M \to T_{p_0}M,$$ and the chain rule ensures that $$(f\circ g)_{*,p_0} = f_{*,g(p_0)} \circ g_{*,p_0}.$$