Definition
Category of smooth manifolds
The category whose objects are smooth manifolds and whose morphisms are smooth maps.
The category of smooth manifolds, denoted , has smooth manifolds as objects and smooth maps as morphisms. Identity maps are smooth, and composites of smooth maps are smooth, so these data form a category. The convention here allows disconnected manifolds, provided their component dimensions are globally bounded.
Categorical properties
An isomorphism in is exactly a diffeomorphism: a smooth map with a smooth inverse. The product manifold is a categorical product, and a one-point manifold is terminal.
References
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapters 1–2, smooth manifolds and smooth maps.