There are the following (and more) types of geometric cell complexes:
- 1) The geometric realization of a simplicial set
- 2) CW-complexes
- 3) The geometric realization of an abstract simplicial complex
- 4) A geometric simplicial complex
What are the differences? Which of these classes of spaces include each other, and what are examples which demonstrate that these classes don't coincide? Of course I am only interested in the (not weak!) homotopy types.
I hope that 3),4) are the same, and that these are a special case of 1), which is also a special case of 2).
If some inclusion doesn't hold for trivial reasons (say not every CW-complex is the geometric realization of a simplicial complex), what assumptions should we add to get a better comparision (for example what about regular CW-complexes)?