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 $F(p_0) = q_0.$ The category of pointed manifolds is an example of a comma category - Man_•^p is exactly $\scriptstyle ,$ where $\$ 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 $T_M$and $T_N,$ with a linear map between them given by the pushforward (differential): $F_:T_M \to T_N.$ This construction is a genuine functor because the pushforward of the identity map $\mathbb_M:M \to M$ is the vector space isomorphism $(\mathbb_M)_:T_M \to T_M,$ and the chain rule ensures that $(f\circ g)_ = f_ \circ g_.$

References

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Manifolds
Manifolds

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