A finite factored set is "just" a set X with a specific choice of decomposition as a product of sets X1×…×Xn. I'm not sure what definition of phase space you're using, but for a sufficiently general definition of dynamical system (e.g. https://en.wikipedia.org/wiki/Dynamical_system#Formal_definition) I don't think that the phase space necessarily has coordinates in this way. The position / momentum phase space example is a special case, where your phase space happens to look like a product of copies of the real numbers, which is then getting back to the "factored" part of a finite factored set. I'm not convinced that there's a deep connection here, though am not very familiar with either concept so could easily be missing something here!
Thanks, that makes sense! Could you say a little about why the weak union axiom holds? I've been struggling to prove that from your definitions. I was hoping that hF(X|z,w)⊆hF(X|z) would hold, but I don't think that hF(X|z) satisfies the second condition in the definition of conditional history for hF(X|z,w).
I'm confused by the definition of conditional history, because it doesn't seem to be a generalisation of history. I would expect hF(X|∅)=hF(X), but both of the conditions in the definition of hF(X|∅) are vacuously true if E=∅. This is independent of what H is, so hF(X|∅)=∅. Am I missing something?