We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or \(\infty \)-groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean \(\infty \)-topos.

中文翻译：

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$$\infty $$ ∞ -groupoids 的康托-施罗德-伯恩斯坦定理

我们证明了同伦类型或\(\infty \) -groupoids的 Cantor-Schröder-Bernstein 定理以下列形式成立：对于任何两种类型，如果每个类型都嵌入另一个类型，那么它们是等价的。该论证是用同伦类型理论或 Voevodsky 的单价基础 (HoTT/UF) 的语言发展起来的，并且需要经典逻辑。因此该定理适用于任何布尔值\(\infty \) -topos。