Float is the levitation Quirk Nana Shimura passed down alongside One For All, letting its holder hang in the air at will. Charged by the power it travels with, it stops being a hovering trick and becomes true high-speed flight. Izuku Midoriya inherited it before it was destroyed inside Tomura Shigaraki.
An Emitter Quirk of the mobility type, Float first appears in Chapter 257 and in Episode 113 of the anime. Nana Shimura held it, Izuku Midoriya received it through the One For All succession, and it ended up forced into Tomura Shigaraki by way of blood transference before ceasing to exist entirely.
On its own the ability does one thing: it keeps the user suspended off the ground, free to hang in place or drift. Paired with the accumulated energy inside One For All, the ceiling rises sharply, and simple levitation turns into flight quick enough to cross a city.
The basic effect lines up closely with Air Walk, another Quirk built around staying aloft. Once Fa Jin and Air Force are layered on top, the result functions much like Christopher Skyline's Flight. A related curiosity sits in Nana's own family: her grandson Tenko was born with Accelerator Rings, a flight-based power meant to spin up rings whose force would shove him through the air. All For One stole it while he was an infant and folded it into several others, building the repulsion power Tomura later flew with himself.
Nana Shimura leaned on it to stay level with Gran Torino, reckoned the fastest hero of his era, riding into danger beside him by running Float and One For All together.
Izuku found the Quirk under pressure during the Paranormal Liberation War, as Tomura Shigaraki reached for every hero present with Decay. Lifting off, Izuku hauled his enemy skyward and used Blackwhip to yank his allies clear.
He kept it afterward. Against Muscular in their rematch he simply hovered out of reach, and the ability became his everyday way of crossing the city toward whatever came next. Stacking Fa Jin and the Air Force technique on top of it bought the extra speed he needed to reach the U.A. Flying Fortress, and during the fight with Tomura it kept him airborne, matching his opponent's own flight while keeping his feet clear of the ground.
The Quirk's end came inside Tomura's mind. Float was the last power driven into him, meant to break his leaden soul, and Nana handled that transfer herself while she and Izuku stood within his consciousness. The act destroyed the Quirk.

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Yes. Izuku Midoriya inherited Float along with One For All from Nana Shimura, and he first called on it during the Paranormal Liberation War to lift himself and an ally clear of Tomura Shigaraki's Decay.
On its own, Float simply lets the user hang or drift in midair. Combined with the power stored inside One For All, it becomes true flight fast enough to cross a city, which Izuku relied on to get around after inheriting it.
Float makes its anime debut in Episode 113, corresponding to the Paranormal Liberation War arc where Izuku Midoriya first uses the inherited Quirk against Tomura Shigaraki.
Float was the final power Nana Shimura transferred into Tomura Shigaraki's mind in an attempt to break through to him, and the transfer destroyed the Quirk entirely.
Float's basic hovering effect closely resembles Air Walk, another mobility Quirk built around staying aloft, and once boosted with techniques like Fa Jin and Air Force it functions much like Christopher Skyline's Flight.
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