Category of manifolds explained

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 → Topto 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.} The category of pointed manifolds is an example of a comma category - Man_•^p is exactly \scriptstyle {( \{ \bull \} \downarrow Man ^{p} )}, where \{ \bull \} 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 1 _{M:M} \to M is the vector space isomorphism ( 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}} .

References

• Book: Lang, Serge . [{{GBurl|dn7rBwAAQBAJ|pg=PR6}} Differential manifolds ]. Springer . 1972 . 2012 . 978-1-4684-0265-0 .
• Book: Tu, Loring W. . An introduction to manifolds . limited . 2011 . Springer . 9781441974006 . 2nd . New York . 682907530.