Generalization of manifold Is there a generalization of the concept of manifold that captures the following idea:
Consider a sphere that instead of being made of a smooth material is actually made up of a mesh of thin wire. Now for certain beings living on the sphere the world appears flat and 2D, unware that they are actually living on a mesh, but for certain other smaller beings, the world appears to be 1D most of the time (because of the wire mesh).

One thing to look at is foliations (and laminations), which are decompositions of manifolds into lower-dimension manifolds. While there is no "mesh" because each lower-dimension manifold has another lower-dimension manifold in any neighborhood, there is still a lower-dimensionality that is something like what you seek. (When you're looking at surfaces in a $3$-manifold, you can also look at the one-dimensional transversals.) See, e.g., H. B. Lawson, Foliations, Bulletin of the AMS 80:3 (1974), 369–418, MR 0343289 (49 #8031).
