Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic.

For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are there something like "differential locale" or "differential pointless topology"?

Solution 1:

Yes, absolutely. The approach I know about is synthetic differential geometry, which begins with thinking of manifolds not via their underlying locally Euclidean topology (this is analytic differential geometry!) but via their smooth functions, in particular those to and from the line $R$ ($R$ behaves differently enough from $\mathbb{R}$ that it's a good idea to use the different notation.)

In one development of the theory, we start with $R$ the formal dual to $C^\infty(\mathbb{R})$ the smooth functions on the classical line, and can get lots of interesting generalized manifolds by setting $M=M(I)\subset R^n$ to be the formal dual of $C^\infty(\mathbb{R}^n)/I$ where $I$ is some ideal-usually with some good analytic properties. If we conceptualize $M$ as the zero locus of functions in $I$, as in algebraic geometry, then we have for instance $D$, the dual of $C^\infty(\mathbb{R})/(x^2)=\{a+b\epsilon,\epsilon^2=0\}$, given as $\{d\in R:d^2=0\}$. As in pointless topology, it's possible to interpret points in this context, as maps from a singleton to $M$. By dualizing, these are just maps from $C^\infty(M)\to C^\infty(*)=\mathbb{R}$. Now the only ring homomorphism from $C^\infty(D)\to \mathbb{R}$ is the evaluation at zero, $a+b\epsilon\mapsto a$. In other words, $0$ is the only point of $D$. But $D$ is not nearly isomorphic to the singleton $*$: their algebras of smooth functions don't even have the same dimension over $\mathbb{R}$!

This behavior of $D$, in fact, is the first key to the whole theory, as it lets us define differentiation by restricting maps on $R$ to maps on $D$ and setting the derivative to be $b$ as in $a+b\epsilon$. So, as in pointless topology, points don't tell nearly the whole story in SDG.