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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?

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I have no knowledge about this, but I think there is also a Finsler approach for Lorentzian geometry, because people doing general relativity using Finsler spacetimes –  Marcel Bischoff May 8 '11 at 9:04
I'm sure others will be able to give a more precise reference, but I seem to recall that a causal structure on a smooth manifold is equivalent to specifying a conformal structure. I'm not sure how much differentiability is required, though: I suspect smoothness might be too strong. There is also a discrete approach to quantum gravity based on the notion of causal sets, in which one does not even have a (topologucal) manifold. –  José Figueroa-O'Farrill May 8 '11 at 9:38
I've briefly heard about the concept of a "Causal Space". but I do not know enough about it to know if this is what you are asking. I believe the paper is called "On the structure of causal spaces," however, so feel free to take a look. –  David Carchedi May 8 '11 at 10:50
Are you looking for something like a directed topological space? If so, you could start at ncatlab.org/nlab/show/directed+topological+space and follow the references. –  Loop Space May 9 '11 at 7:02

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