Generalizing the coordinate space of a manifold

Question

A manifold is, essentially, something that "looks like" $\mathbb R^n$ locally. That is, a topological space $X$ such that for every $x\in X$ there is a continuous map $\varphi$ from an open set $U_x$ containing $x$ to $\mathbb R^n$.

Suppose I want to consider a possible generalization of this concept, in particular by extending which coordinate spaces I can use. A first naive generalization would be to admit any vector space $V$ instead of $\mathbb R^n$, however, any finite dimensional $V$ is isomorphic to some $\mathbb R^n$, so this is clearly not adding anything new. A possible next attempt would be to admit an arbitrary module $M$ (perhaps over a commutative ring) as the coordinate space.

My question would be: Is this concept one that's been explored? Could it produce an interesting object of study?

Answer 1

As I said in the comments, it is not clear what topology one would put on an arbitrary module in order to do this. The point of using $\mathbb{R}^n$ is that its topology is very natural and compatible with its algebraic structure. But do note that we can use your vector space idea and extend it to infinite dimensional vector spaces; see for instance Hilbert or Banach manifolds.

Answer 2

Look up:

• Banach manifolds
• Hilbert manifolds
• Fréchet manifolds

Probably there are other generalizations of the concept of manifold out there, modeled on other spaces. You could also be interested in supermanifolds and related concepts, as well as derived algebraic geometry, where spaces are in some sense modeled by chain complexes.