Category of manifolds

Short description: Category theory

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

References

• Lang, Serge (2012). Differential manifolds. Springer. ISBN 978-1-4684-0265-0. https://books.google.com/books?id=dn7rBwAAQBAJ&pg=PR6. 
• Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. ISBN 9781441974006. OCLC 682907530. https://archive.org/details/introductiontoma00lwtu_506.