I am not a type theory expert (or even enthusiast) by any means, but I imagine something that will be very useful like so: the intersection types of callees interact in interesting ways with the type of the caller (taking into account ordering). For example, the type of the caller can be: "this function locks A, then B, then unlocks B, then unlocks A". This type can arbitrarily intersect other effect types, such as: "this function writes to socket X" and also "mutates variable V".

I will be able to define a type "locks A" and "locks B", and the kind "locks" will be ordered, so that the type of a function that locks A and locks B, will be "locks A and then locks B". Also, I will be able to define interactions between types so that unlocking A will undo the effect, so that a function that simple locks A will have the type "locks A", but one that locks and later unlocks will have the type "holds A". Of course, these types intersect arbitrarily.