Disjoint union of smooth manifolds - Imported Knowlpedia Content

Definition

For a finite or countable family of smooth manifolds $(M_i)_{i\in I}$ , their smooth disjoint union $\bigsqcup_iM_i$ is the set-theoretic disjoint union with the disjoint-union topology and the smooth atlas formed by all component charts. Each canonical inclusion $M_i\hookrightarrow\bigsqcup_iM_i$ is a smooth open embedding. A map $f:\bigsqcup_iM_i\to N$ is smooth exactly when every restriction $f|_{M_i}$ is smooth. Hence the disjoint union, with these inclusions, is the coproduct in the category of smooth manifolds whenever it remains an object of the chosen category.

Universal property

Given smooth maps $f_i:M_i\to N$ , there is a unique set map $f:\bigsqcup_iM_i\to N$ whose restriction to $M_i$ is $f_i$ . Because smoothness is checked componentwise, this map is smooth. This proves the coproduct universal property directly and explains why no compatibility conditions between different components are required.

Categorical structure

The empty manifold is the coproduct of the empty family and is an initial object when the convention admits it. Binary disjoint union gives the smooth-manifold category a symmetric monoidal operation whose unit is the empty manifold. Associativity, commutativity, and unit identifications are canonical diffeomorphisms.

Countability and dimension conventions

Under the usual Hausdorff, second-countable convention, an uncountable disjoint union of nonempty manifolds is not second countable, so arbitrary coproducts need not exist. If an $n$ -manifold must have one fixed dimension, all nonempty $M_i$ must have that dimension. A convention allowing dimension to vary by connected component admits mixed-dimensional disjoint unions. These restrictions come from the ambient definition of manifold, not from the componentwise smooth atlas Lee, chapter on smooth manifolds.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: the foundational conventions for smooth manifolds, atlases, and smooth maps.
Saunders Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998. DOI record. Relevant: coproducts and symmetric monoidal categories.