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.

One is often interested only in "C"^"p"-manifolds modelled 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 modelled 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 the underlying set 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" → Setto the category of sets which assigns to each manifold the underlying set and to each "p"-times continuously differentiable function the underlying function.

References

*