Why maximal atlas

This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes:

Let $M$ be a topological manifold. Now, even though $C^\infty$-compatibility of charts is not transitive, it is true that if two charts glue with all the charts of a given atlas, then they are compatible with each other.

Given this, one may conceivable define an equivalence relation on the atlases of a manifold, and then consider the equivalence classes. But, we do not do this. Instead we define a differentiable structure to be the maximal atlas (which being uniquely built up from a given atlas will be in its equivalence class).

Why do we do this (apart from its arguable simplicity)? Why don't we take the equivalence class of atlases in stead to be the differentiable structure?

Solution 1:

If $C$ is one of those equivalence classes, and you make the union of all the elements of $C$, you get an atlas which also belongs to $C$. It is in fact that maximal atlas of $C$. This means that nature was gracious enough to provide a canonical choice of representatives in each equivalence class, and we use it.

In practice, that an atlas be maximal means that anything that could be a coordinate chart is a coordinate chart.

Solution 2:

It comes from the same aesthetic as Bertrand Russell's famous definition of $2$ as "the class of all pairs". To be a bit more specific, in set theory and logic some people define numbers as particular sets, e.g. $2 = \{ \varnothing, \{ \varnothing \} \}$. But there are (infinitely many) other possible choices, and from a certain perspective this lack of canonicity is disturbing. (One of my favorite essays in the philosophy of mathematics takes on this issue: Paul Benacerraf's What numbers could not be.) Hence Russell's solution: define $2$ (or the cardinal number associated to any set $S$) as the proper class of all sets which have the same cardinality as $S$.

The problem with this definition is that in order to be canonical we have arranged things so that the formal definition of an arguably simple, concrete mathematical object is something big and complicated. This is exactly what is happening in the definition of an atlas as a maximal collection of coordinate charts. The study of differential topology is not the study of maximal atlases any more than arithmetic is the study of proper classes: it is not fruitful to attempt to describe all the elements of a given maximal atlas, so far as I know. (Gian-Carlo Rota wrote briefly but persuasively on this topic in his Indiscrete Thoughts: he called maximal atlases "polite fictions").

There are other ways to set up the foundations of the subject which avoid making this kind of definition. For instance, a more modern and graceful approach to geometric structures on a space is via a sheaf of functions on that space. It would also be possible to take a more categorical approach.