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 ${\displaystyle (M,p_0),}$ 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 ${\displaystyle (\{\text{bull}\}\downarrow {\mathbf{Man}}^{\mathbf{p}}),}$ where ${\displaystyle \{\text{bull}\}}$ 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): ${\displaystyle F_{\ast ,p}:T_{p_0}M\to T_{F(p_0)}N.}$ 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] ${\displaystyle ({\mathbb{1}}_M{)}_{\ast ,p_0}:T_{p_0}M\to T_{p_0}M,}$ and the chain rule ensures that ${\displaystyle (f\circ g{)}_{\ast ,p_0}=f_{\ast ,g(p_0)}\circ g_{\ast ,p_0}.}$^[1]

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. 

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