What captures our intuitive notion of faces, edges, and vertices? This answer suggests that laypeople's intuitive notion of the meaning of these words is consistent with the following claims:


*

*A cube has 6 faces, 12 edges, 8 vertices.

*A cylinder has 3 faces, 2 edges, 0 vertices.

*A cone has 2 faces, 1 edge, 1 vertex.

*A sphere has 1 face, 0 edges, 0 vertices.


What formal, mathematical definition best captures this intuition and is consistent with the above claims? For example, the following was suggested in the comments:

I think the definition of the tangent space at a point via equivalence
  classes of smooth curves works in this situation, and then I believe
  it's true that the naive count of faces, edges, etc. counts connected
  components of the subspaces of points whose tangent spaces have the
  relevant dimensions.

Is this a good approach? If so, does it have an existing name and literature? Are there any other approaches?
 A: Partial answer: From generalizations of manifolds:

Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. Roughly
speaking, it is a space which locally looks like the quotients of some
simple space (e.g. Euclidean space) by the actions of various finite
groups. The singularities correspond to fixed points of the group
actions, and the actions must be compatible in a certain sense.
Stratified space: A "stratified space" is a space that can be divided into pieces ("strata"), with each stratum a manifold, with the
strata fitting together in prescribed ways (formally, a filtration by
closed subsets). There are various technical definitions, notably a
Whitney stratified space (see Whitney conditions) for smooth manifolds
and a topologically stratified space for topological manifolds.
CW-complexes: A CW complex
is a topological space formed by gluing disks of different
dimensionality together. In general the resulting space is singular,
and hence not a manifold. However, they are of central interest in
algebraic topology, especially in homotopy theory, as they are easy to
compute with and singularities are not a concern.

From topologically stratified space:

In topology, a branch of mathematics, a topologically stratified space
is a space X that has been decomposed into pieces called strata; these
strata are manifolds and are required to fit together in a certain
way. Topologically stratified spaces provide a purely topological
setting for the study of singularities analogous to the more
differential-geometric theory of Whitney.
Basic examples of stratified spaces include manifolds with boundary
(top dimension and codimension 1 boundary) and manifolds with corners
(top dimension, codimension 1 boundary, codimension 2 corners).

