The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps.
After proving this, the following excerpt from a paper of Lawvere (cited below) is given.
“This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a ”basic“ definition of the concept of manifold. For example the $2$-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in $3$-space E which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x)=x/|x|$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A\times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc.” (Lawvere 1989, p.267)
Unfortunately the excerpt is not enough for me to understand neither the significance nor the idea behind the theorem, so I am looking for detailed, hand-holding explanations of as many parts of it as possible.
- How does this theorem justify bypassing the considerations of charts, atlases, etc?
- What are the details of the sphere example? How is the smooth structure specified by an open set containing it along with $x/|x|$?
- Why are all general constructions in fact functors into cauchy complete categories?
- What are some examples of general constructions and how do they follow easily? How does this approach circumvent messing with covers etc?
- What is meant by "the same for cohomology groups"?
Reference: F. William Lawvere, Qualitative distinctions between some toposes of generalized graphs, Contemporary Mathematics 92 (1989), 261-299.