In general, the more tensile strength you want in a cable made of a given material, the thicker you need to make that cable. Now sure, we can imagine whatever magical properties we want of our space elevator cable material, since no known material that could do this exists anyway. But it's far more likely that you'd need a cable that's a kilometer or more in diameter to achieve the tensile strength needed to support its weight at 35000 km of length, than it is to be a few inches wide.