He proposed that a single hair-like 20-ton 'seed' cable[failed verification] be deployed in the traditional way, giving a very lightweight elevator with very little lifting capacity.
Then, progressively heavier cables would be pulled up from the ground along it, repeatedly strengthening it until the elevator reaches the required mass and strength.
By increasing the thickness of the cable from the ground a very quick (exponential) build-up of a new elevator may be performed (it helps that no active climbers are needed, and power is applied mechanically.)
The space manufacturing design would use a carbonaceous asteroid or near-Earth object for mining its material and producing a carbon nanotube cable.
The cable seeding design could be rendered infeasible in case the material strength is considerably lower than was projected by Brad Edwards.