Category of manifolds

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 modeled 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 modeled 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
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
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 formula_1 where formula_2 is a formula_3manifold along with a basepoint formula_4 and its morphisms are basepoint-preserving "p"-times continuously differentiable maps: e.g. formula_5 such that formula_6 The category of pointed manifolds is an example of a comma category - Man•p is exactly formula_7 where formula_8 represents an arbitrary singleton set, and the formula_9represents 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 formula_10and formula_11 with a formula_3map formula_13 between them, we can assign the vector spaces formula_14and formula_15 with a linear map between them given by the pushforward (differential): formula_16 This construction is a genuine functor because the pushforward of the identity map formula_17 is the vector space isomorphism formula_18 and the chain rule ensures that formula_19