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

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.

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 $F(p_{0})=q_{0}.$ ^[1] The category of pointed manifolds is an example of a comma category - Man_•^p is exactly $\scriptstyle {(\{\bullet \}\downarrow \mathbf {Man^{p}} )},$ where $\{\bullet \}$ represents an arbitrary singleton set, and the $\downarrow$ 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 $(M,p_{0})$ 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^[1] $(\mathbb {1} _{M})_{*,p_{0}}:T_{p_{0}}M\to T_{p_{0}}M,$ and the chain rule ensures that $(f\circ g)_{*,p_{0}}=f_{*,g(p_{0})}\circ g_{*,p_{0}}.$ ^[1]