Beyond Topology?

 Thinking about classification of phases of matter along with the pragmatist standard of never reaching the answer the easy natural question occurred to me `what next?' Is it possible to have distinct phases of matter with the same topology?

One --not the only-- way to generalize this is to ask is it possible to get even `more global' than topology?

One way to answer this is to apply the axiom of infinite divisibility to the topological invariants like genus, meaning that we can/should/must consider the possibility of topological invariants being real numbers.

This way we are talking about a genuinely differential topology...

First we make topological invariants real and hyperreal (differentials), then we will have a differential topology proper. Then if we recall $$\int\text{Differential Geometry}=\text{Topology},$$ (Gauss-Bonnet theorem) We might similarly say $$\int\text{Differential Topology}=\text{`Globaller than Topology'}$$

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.133903