Diffeomorphism groupoid of smooth manifolds

Definition

The diffeomorphism groupoid of smooth manifolds is the category

\mathbf{Man}^{\simeq}

whose objects are finite-dimensional smooth manifolds and whose morphisms are diffeomorphisms. Every morphism is invertible, so this category is a groupoid.

It is the core, or maximal subgroupoid, of $\mathbf{Man}$ . Thus it contains every object of $\mathbf{Man}$ , but only the isomorphisms among its smooth maps.

What the core forgets

Passing from $\mathbf{Man}$ to $\mathbf{Man}^{\simeq}$ preserves the question “are these manifolds diffeomorphic?” and all diffeomorphism groups. It discards noninvertible geometry: inclusions, submersions, constant maps, covering projections that are not one-sheeted, and smooth functions $M\to\mathbb R$ .

In particular, “the category whose morphisms are diffeomorphisms” is a reasonable classification groupoid, but it is not the usual category of smooth manifolds. Constructions such as pullback of smooth functions and the tangent functor are defined on the larger category.

Automorphisms

The automorphism group of $M$ in $\mathbf{Man}^{\simeq}$ is its diffeomorphism group $\operatorname{Diff}(M)$ . Distinct objects in the same connected component of this groupoid are diffeomorphic, while automorphisms record the symmetries of an individual representative.

References

Emily Riehl, Category Theory in Context, Dover, 2016. Author's edition. Relevant: categories, isomorphisms, subcategories, and groupoids.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth maps and diffeomorphisms.