category of smooth manifolds

Here, a smooth manifold is assumed to be finite-dimensional, Hausdorff, and second-countable.

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There are 4 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

Special objects

  • terminal object: singleton manifold of dimension 00
  • initial object: empty manifold
  • products: [finite case] direct products X×YX \times Y with the product topology and the charts Rn+m=Rn×Rm≅U×V↪X×Y\mathbb{R}^{n + m} = \mathbb{R}^n \times \mathbb{R}^m \cong U \times V \hookrightarrow X \times Y for charts Rn≅U↪X\mathbb{R}^n \cong U \hookrightarrow X and Rm≅V↪Y\mathbb{R}^m \cong V \hookrightarrow Y
  • coproducts: [countable case] disjoint union with the disjoint union topology and the obvious charts

Special morphisms

  • isomorphisms: diffeomorphisms
  • monomorphisms: injective smooth maps
  • epimorphisms: smooth maps with dense image
  • regular monomorphisms:
  • regular epimorphisms: