I start with an Euclidean space. I drop global symmetry and I get a Riemannian manifold. I drop symmetry (isotropy) in the tangent space and I get a Finsler manifold. I drop differentiability and I get a metric space.
What happens if I start with Minkowski's spacetime? I drop global symmetry and I get a pseudo-Riemannian manifold. Can I drop symmetry of the tangent spaces? (Question) Can I drop differentiability? (Main question)
Has anyone studied categories analog to that of metric spaces or Finsler manifolds, but including a notion of causality/order/passage of time?