Configuration Spaces in Fundamental Physics

I consider configuration spaces for $N$-body problems, gauge theories and for GR in both geometrodynamical and Ashtekar variables forms, including minisuperspace and inhomogeneous perturbations thereabout in the former case. These examples include many interesting spaces of shapes (with and without whichever of local or global notions of scale). In considering reduced configuration spaces, stratified manifolds arise. Three strategies to deal with these are `excise', `unfold' and `accept'. I show that spaces of triangles arising from various interpretations of 3-body problems already serve as model arena for all three. I furthermore argue in favour of the `accept' strategy on relational grounds. This approach requires sheaf methods (which go beyond fibre bundles and general bundles, which I contrast with sheaves and presheaves in some appendices). Sheaf methods are also required for the stratifold construct that pairs some well-behaved stratified manifolds with sheaves. I apply arguing against `excise' and `unfold' to GR's superspace and thin sandwich, and to the removal of collinear configurations in mechanics. Non-redundant configurations are also useful in providing more accurate names for various spaces and theories.