Stadyshell
Statite Dynamic Shell, aka, Stapleton Dyson Shell
The Standard Story
The continued increase of the human population, expanding into the solar system and later to other star systems, is a common assumption in 20th century science fiction. People move into orbiting space habitats at the L4 and L5, terraform Mars, etc. When planetary systems are exhausted, the expansion continues into more space habitats orbiting the sun, perhaps manufactured from the asteroids, Mercury, and Venus. Some propose dismantling the Earth to build more habitats, eventually capturing most of the energy from the Sun.
Nikolai Kardashev characterized future supercivilizations by the energy they capture and use.
A type I supercivilization captures all the power reaching their planet from their star for artificial uses. For the Earth, that is 174 petawatts, 1.74e17 watts. Plants intercept about 20 petawatts (2e16 W), and convert 200 terawatts (2e14 W) into material animals (and now biofuel-guzzling machinery) can eat. Current global artificial power production is about 15 terawatts (1.5e13 W).
A type II supercivilization captures all the power emitted by their star. Our sun emits 380 yottawatts (3.8e26 W) visible light, UV, and infrared, 2 billion times more power than the earth intercepts, and 25 trillion times more power than human artificial power production.
A type III supercivilization captures all the power emitted by all the stars in their galaxy. Our galaxy emits perhaps 40 trillion yottawatts, (4e37 W), more than two heptillion (2e24) times more power than human artificial power production.
So, a supercivilization capturing all the power emitted by the Sun with a shell of habitats would be a Type II supercivilization.
Olaf Stapleton wrote about type II and type III supercivilizations in his 1937 science fantasy novel Starmaker. Freeman Dyson wrote about how such supercivilizations might appear as highly luminous, compact infrared sources in 1960, Search for Artificial Stellar Sources of Infrared Radiation.