Category of Manifolds

In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous.

One is often interested only in Cp-manifolds modelled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modelled on a fixed space E is denoted Manp(E).

One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.