@jpp my elementary understanding is no doubt tripping us up here. As a physicist I'm used to using geometric and algebraic tools as methods for probing some aspects of spaces, say the shape of a magnetic field to interpret properties of particles, or shapes of fermionic fluids in topological insulators. I use the words "shape" and "space" as general property of physical systems. My question is: Since we are interested in describing more esoteric properties, how should I best think about the progression of these tools and the information they carry. Apologies if these questions are too vague