The grey scarf bunched around Shota Aizawa's neck is not fabric in any ordinary sense, and it is not decoration either. It is the half of Eraser Head that does the actual holding, the counterweight to a Quirk that can switch powers off but cannot lay a finger on anyone.
The Capturing Weapon is the signature tool of Shota Aizawa, later passed down to Hitoshi Shinso. Shota's version is woven from carbon fibers blended with a special metal alloy, giving the cloth-like band a strength no scarf should have. He keeps it wrapped around his neck constantly, in costume or out. An unidentified support company is credited as its origin, and it debuts in chapter 6 of the manga and episode 5 of the anime.
Shota steers the weapon deliberately, snaring targets and locking their movement down. Paired with Erasure it becomes a trap: nullify the Quirk, then bind the person before they can respond. It also covers the gap in his ability, since Mutant-type Quirk users cannot be switched off, and against them the cloth is the whole answer.
Wielding it well demanded an entire discipline. Shota built the Binding Cloth style from nothing, with no instructor to copy, and six years went into developing and mastering it. One quirk of the equipment has never been explained: it reacts to his Erasure much the way his hair does, standing up when the Quirk activates.

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Shota Aizawa steers the Capturing Weapon, a scarf-like cloth wrapped around his neck, to snare targets and lock down their movement, often pairing it with his Erasure Quirk to nullify a power first and then bind the person before they can respond.
Aizawa's Capturing Weapon is woven from carbon fibers blended with a special metal alloy, giving the cloth-like band far more strength than an ordinary scarf.
Shota Aizawa, known as Eraser Head, is the original wielder of the Capturing Weapon, and he later passes it down to his student Hitoshi Shinso.
Aizawa's Erasure Quirk cannot switch off Mutant-type Quirks, so the Capturing Weapon becomes his whole answer against those opponents, letting him restrain them physically since he cannot nullify their powers.
Shota Aizawa built the Binding Cloth style from nothing, with no instructor to learn from, spending six years developing and mastering how to wield the Capturing Weapon.
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