category of smooth manifolds

 analysis  topology
Notation Man\Man Objects smooth manifolds Morphisms smooth maps Related Haus\Haus, Top\Top, Euclid⊔\Euclidsqcup External nLab link
Here, a smooth manifold is defined as a second-countable Hausdorff space with a smooth atlas. The dimension is locally constant, not necessarily constant.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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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\IR^{n + m} = \IR^n \times \IR^m \cong U \times V \hookrightarrow X \times Y for charts Rn≅U↪X\IR^n \cong U \hookrightarrow X and Rm≅V↪Y\IR^m \cong V \hookrightarrow Y
  • coproducts: [countable case] disjoint union with the disjoint union topology and the obvious charts

Special morphisms

Morphisms

The database stores 1 morphism based on the category of smooth manifolds.