For the type IIa duality, this is much neater, since I know that you can go from the M-theory to the IIa theory by compactifying on a circle, which gives a kind of map
In any case, this map still makes sense in the case of a product, clearly, because then you can compactify "step by step", i.e. first over the fiber, then over the base
But since a generic fibration has no "gluing maps" like e.g. a vector bundle would have, I can't see how to glue these local descriptions together. Also I'm not even sure if the fibrations are guaranteed to be locally trivial
@EmilioPisanty Yeah, well, apparently one of my secondary supervisors is in America and there are links with Bristol and the company they've set up (whatever it's called now), so you never know!
(said secondary supervisor is completely unrelated)
My primary supervisor will be Anthony Laing. One of his post-docs and apparently someone from America that I've never met before will be secondary supervisors