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?