Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, the Polish Circle has the same sheaf cohomology with constant coefficients as $S^1$, and both spaces also have equivalent categories of covering spaces. So one could ask if in such a model structure, the polish circle would be weakly equivalent to an ordinary one.
Edit: I would be happy with an answer for any nice category of topological spaces which contains the Polish circle, e.g. compactly generated weak Hausdorff spaces. Please ignore the following paragraph.
Also, sheaf theory is defined on rather general topological spaces, so it would be nice if the underlying category of topological spaces would be much more general than, say, compactly generated weak Hausdorff spaces.