Sure, one built with imaginary and probably impossible materials, ignoring petty little details such as the non-equivalence of the accelerating inertial reference frames at the ground and top (not to mention all of the way up).
Here's a hint for you (and everybody else that proposes this absurdity):
a) Equatorial speed relative to center of Earth: 460 meters/second or just over 1000 mph. Around 1.35 x the speed of sound.
b) Speed at geosync orbit at R \approx 6 R_e: 6 x 460 = 2.78 km/sec.
Or if you prefer energy:
c) Total mechanical energy of 1 kg object sitting at "rest" at equator: = GMm/R_e + 1/2 m v^2 (using v from a) above): -64 MJ/kg
d) TME of same object sitting at "rest" in geosync orbit \approx -10 MJ/kg
And the worst of them all, angular momentum:
e) Total z-directed angular momentum of 1 kg on equator = mR_e v (from a): 2.9 x 10^9 kg-m^2/sec
f) Ditto at geosync (36x larger): 105 x 10^9 kg-m^2/sec
So, to lift something up from the surface to geosync, one has to:
1) Increase its tangential speed -- tangent to great circles around the axis -- by a factor of 6 as it rises.
2) Increase its mechanical energy by 50 million Joules per kilogram of payload.
3) Increase its angular momentum by well over 100 BILLION kg-m^2/sec. This is done by means of the same torque required to increase its tangential speed.
Now, let's ignore all the picky details, such as how to make a cable that can support its own weight hanging to the ground from geosync orbit -- where if we not unreasonably insist on it having a specific gravity around 1 (same as water) then we need the weight of a cable well over 5 R_e long (at least, if you want to be able to apply tangential force via the same cable) so let's START with 5 R_e -- in round numbers 5 x 6.4 x 10^6 = 32 million meters long (yes, that is million). Figuring the top tension is a bit tricky and involves an integral, but the result of the integral is that the tension at the top is the change in potential energy per kilogram times the density times the cross sectional area: GM_e\rho \Delta A (1/R_e - 1/6R_e) = 50 MJ/kg x 1000 kg/m^3 \Delta A.
This is 50 x 10^9 Newtons times the cross sectional area (in m^2), and is most easily expressed by dividing out the area to get:
Requirement: 50 GPa minimum tensile (yield!) strength
The highest tensile yield strength observed in any material (so far) is less than 5 GPa. Carbon nanotubes have a tensile strength reported as high as 63 GPa, but this is not a yield strength and could not support a sustained load, certainly not safely. So far, then, we are (honestly) well over an order of magnitude short of the required yield tensile strength required for a cable to JUST support its own weight "hanging" from the vicinity of geosync orbit. One cannot force a lower orbit USING such a cable, and higher orbits (with more cable) that are still geosync require more tension and make little sense. But hey, this is science fiction, let's PRETEND that we can make carbon nanotube cables 32 million meters long that have a yield strength of (what the hell) 500 GPa -- our 50 plus a generous margin of safety. Let's not worry about what we are going to wrap our cable around as far as pulleys go at the ends, how we will build bearings etc -- heck, that's just "engineering". Heck, anybody can design a pulley that can support a 64+ million meter long cable (looped, remember! -- oops, there goes a factor of 2 of our ten already!) that has been looped and joined "perfectly" without the slightest defect that might lower the yield strength to (say) 5 GPa and lead to catastrophe! Engineering is just drawing a picture! Why worry about what actual material you might make it out of that the cable won't cut right through (if it is thin) or crush (if it is thick) or wear out in short order as it operates?
Let's just imagine how it might WORK. After all, the whole idea is that you send a payload up on one side of the loop at the same time you send some counterweight down on the other, adding energy as needed with the bottom pulley.
Still, we have to add all of that velocity, energy, and angular momentum to the mass WHILE IT RISES! This means that the cable has to exert: a) enough vertical force on the payload mass to overcome local gravity. This comes out of the total tension budget but hey, we assumed that we could build a cable with enough surplus to be able to lift several times the weight of the cable, however thick or thin we make it. We just make the cable thick enough that the cable plus payload are WELL within this utterly imaginative, fictitious, limit. b) enough TANGENTIAL force to be able to exert the required torque on the payload as it rises! And this is a very, very serious problem, one that it will not be easy to overcome with science fiction.
For one thing, this utterly rules out "straight" cables. Even if you JUST run the loop WITHOUT payload, the rising cable will (apparently) deflect to antispinward/west in order to build up enough of a bow for the tension in the cable to be able to provide the necessary torque on the cable itself. The bigger the bow, the better the angle, but the longer the cable (and what do you do with all of that cable when you AREN'T running the loop? Hmmmm....) the greater the static tension, operating or not. The smaller the bow, the flatter the angle, the bigger the tension needed to provide the transverse acceleration/torque as well as lift the payload.
In the meantime, the descending cable is going TOO FAST for the ground. It balloons out spinward/east as it descends. The operating loop doesn't look like two vertical lines, then. It looks like two enormous arcs, stretched in opposite directions!
Lifting the mass has two more really interesting effects. One is that the cable pulls the geosync end DOWN relative to static tension equilibrium when you load it. The loops have to come from somewhere, and if the station stretches the cables "tight" at equilibrium, the opening of the loops has to reduce the vertical span of the loop, period. Moving it down moves it out of geostationary stasis -- the station ITSELF starts to deflect to spinward as it is going too fast for a lower circle at constant angular velocity (there are lovely demonstrations of this involving coin spirals in shopping malls or figure skaters). As the mass rises, though, it pulls backwards on the station. It is basically undergoing a "slow" inelastic collision with the station! It started with the wrong angular momentum (too low, from Earth's surface). The Earth's surface cannot ADD angular momentum to the space station -- it moved forward because it was being conserved when we turned on the loop and it moved to a lower circle! But as the particle rises, it really DOES reduce the angular momentum of the space station as it increases the angular momentum of the payload. In fact, overall, it is sort of like you fired the payload at the station in the direction opposite its motion at 2 km/sec, and the mass then collides with the station and sticks!
It doesn't happen that fast and it may not release heat during the collision (assuming our motors do their work) but as far as angular momentum is concerned, that's what happens. The rising mass pulls antispinwards on BOTH the mass AND the space station, but the space station is not fixed.
At the end of the day, if the station was in geosynchronous static tension equilibrium before you sent up your payload, after you've sent an unbalanced load up it is not, basically because strings cannot exert any shear stress on the space station and can exert tangential forces on the mass only by pulling the station down to a different radius. And this is all in the quasi-static limit, ignoring the WAVES that any loaded motion would induce on the loop and just what happens to the slack that develops in the loop.
In the end, you have to add energy and angular momentum back to the station plus platform in order to re-achieve geosync orbit. To do that, you are right back to using rockets (or still more science fiction).
Let's face it. The "space elevator" concept is deeply flawed, IN ADDITION to the fact that it is currently literally impossible to build from the material science point of view. We haven't even touched on the problems building an attractive trillion dollar target in the sky that a laser or physical projectile can trivially destroy by just creating defects in the cable, in the case of a laser quite possibly from hundreds of miles away or elsewhere in orbit, or its vulnerability to space junk all along its considerable length. Stories that utilize it always seem to forget the coriolis (pseudo) "force" and the problems with inducing waves on rotating loops where one end of the loop is free floating in an either system rotating (of necessity) with a constant angular velocity. If we're going to push sci-fi, let's contemplate land-based electromagnetic mass drivers (which are in principle feasible, if still pretty absurd in cost and engineering "details").