Category of manifolds

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

U : Man^p → 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′ : Man^p → 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.

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 ${\displaystyle (M,p_{0}),}$ where ${\displaystyle M}$ is a ${\displaystyle C^{p}}$manifold along with a basepoint ${\displaystyle p_{0}\in M,}$ and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. ${\displaystyle F:(M,p_{0})\to (N,q_{0}),}$ such that ${\displaystyle F(p_{0})=q_{0}.}$[1] The category of pointed manifolds is an example of a comma category - Man_•^p is exactly ${\displaystyle \scriptstyle {(\bullet \downarrow \mathbf {Man^{p}} )},}$ where ${\displaystyle \bullet }$ represents an arbitrary singleton set, and the ${\displaystyle \downarrow }$represents a map from that singleton to an element of Man^p, picking out a basepoint.

The tangent space construction can be viewed as a functor from Man_•^p to Vect_R as follows: given pointed manifolds ${\displaystyle (M,p_{0})}$and ${\displaystyle (N,F(p_{0})),}$ with a ${\displaystyle C^{p}}$map ${\displaystyle F:(M,p_{0})\to (N,F(p_{0}))}$ between them, we can assign the vector spaces ${\displaystyle T_{p_{0}}M}$and ${\displaystyle T_{F(p_{0})}N,}$ with a linear map between them given by the pushforward (differential): ${\displaystyle F_{*,p}:T_{p_{0}}M\to T_{F(p_{0})}N.}$ This construction is a genuine functor because the pushforward of the identity map ${\displaystyle \mathbb {1} _{M}:M\to M}$ is the vector space isomorphism[1] ${\displaystyle (\mathbb {1} _{M})_{*,p_{0}}:T_{p_{0}}M\to T_{p_{0}}M,}$ and the chain rule ensures that ${\displaystyle (f\circ g)_{*,p_{0}}=f_{*,g(p_{0})}\circ g_{*,p_{0}}.}$[1]

• Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
• Tu, Loring W. (2011). An Introduction to Manifolds. Springer New York Dordrecht Heidelberg London