Category of smooth manifolds - Imported Knowlpedia Content

Definition

The category of smooth manifolds, commonly denoted $\mathbf{Man}$ or $\mathbf{SmoothMan}$ , is the category whose objects are finite-dimensional smooth manifolds and whose morphisms $M\to N$ are smooth maps. Composition is ordinary composition of functions, and the identity morphism on $M$ is $\operatorname{id}_M$ . These operations are smooth and satisfy the category axioms. The isomorphisms in this category are exactly the diffeomorphisms. This definition fixes the finite-dimensional smooth category, not a category of infinite-dimensional or singular spaces.

Products and terminal object

The one-point manifold is a terminal object. The Cartesian product $M\times N$ , with its product smooth structure and projection maps, is the categorical product. The empty manifold is an initial object when it is admitted by the chosen manifold convention.

Functorial constructions

Many geometric operations are functorial. The tangent bundle sends a smooth map to its differential, while differential forms are contravariant under pullback. The assignment $M\mapsto C^\infty(M)$ from the algebra of smooth functions reverses arrows.

Conventions and categorical limits

The category depends on whether manifolds may have boundary, whether dimensions may vary by connected component, and whether the empty manifold is allowed. Even after fixing conventions, arbitrary set-theoretic fiber products need not be smooth manifolds; transverse fiber products are the standard well-behaved case. The basic categorical language is as in Mac Lane, Chapter I, while the smooth setting follows Lee, chapters on smooth manifolds and maps.

References

Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998. DOI record. Relevant: Chapter I, categories, functors, and natural transformations.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth manifolds, smooth maps, products, and diffeomorphisms.