A vertical fishnet tether extends from the Earth to the geostationary orbit and beyond. The tether is curved due to Coriolis force of moving cargo. The minimum mass of a steel tether is greater than the mass of the Earth! A structural material of great specific strength is needed to make this idea practical. Plastic cannot be used because it is vulnerable to space radiation and thermal fatigue. A short version of the skyhook called orbital elevator is lightweight and less vulnerable than the skyhook.
Science fiction writers sometimes call it Jacob's ladder (after the Bible, Genesis 28 v 12) or beanstalk (after the story of Jack and Beanstalk).
The maximum length of a non-tapered tether that can carry its own weight under a uniform force of one gravity is called characteristic length. The strongest commercial fiber, Spectra has the characteristic length of 315 km.
Fridrich Arturovich Tsander, "Selected Papers," Zinatne, Riga, 1978, (in Russian; manuscript written in 1910; mentions a tapered tether extending from the Moon toward the Earth).
Yuri Artsutanov, "V Kosmos na Electrovoze (in Russian, Into Space on a Train)," Komsomolskaya Pravda, July 31 1960.
John Isaacs et al, "Satellite Elongation into a True Sky-Hook," Science, Vol. 151, February 11, 1966, pp. 682-683; also Vol. 152, p 800 and Vol. 158, p. 947.
Vladimir Lvov "Sky-Hook: Old Idea," Science, Vol. 158, November 17, 1967, pp. 946-947.
Jerome Pearson, "The orbital tower: a spacecraft launcher using the Earth's rotational energy," Acta Astronautica, Vol. 2, No. 9-10, September-October 1975, pp. 785-99.
G. Polyakov, "Kosmicheskoye "Ozhereye" Zemli (in Russian, A Space "Necklace" about the Earth)," Teknika Molodezhi, No. 4, 1977, pp. 41-43 (English translation: NASA Technical Memorandum TM-75174).
Jerome Pearson, "Anchored Lunar Satellites for Cis-Lunar Transportation and Communication," European Conference on Space Settlements and Space Industries, London, England, September 20, 1977, also in Journal of the Astronautical Sciences 1978.
Jerome Pearson, "Lunar Anchored Satellite Test", AIAA/AAS Astrodynamics Conference, Palo Alto, Ca., August 7-9, 1978, AIAA Paper 78-1427.
Arthur C. Clarke, "The Space Elevator: 'Thought Experiment', or Key to the Universe?," Earth Oriented Application of Space Technology, Vol. 1, 1981, pp. 39-48.
Konrad E. Ebisch, "Skyhook: Another Space Construction Project," American Journal of Physics, Vol. 50, No 5, May 1982, pp. 467-469.
Geoffrey A. Landis and Craig Cafarelli, "The Tsiolkovski Tower Reexamined," IAF-95-V.4.07, 46th International Astronautical Congress, October 2-6, 1995, Oslo, Norway.
Skyhook page at NASA.