The Cantor–Schröder–Bernstein Theorem for \(\infty \)-groupoids

IF 0.5 4区数学

Journal of Homotopy and Related Structures Pub Date : 2021-06-28 DOI:10.1007/s40062-021-00284-6

Martín Hötzel Escardó

引用次数: 3

Abstract

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.

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\（\infty\）-群胚的Cantor–Schröder–Bernstein定理

我们证明了同伦类型或\(\infty \) -groupoids的Cantor-Schröder-Bernstein定理以以下形式成立:对于任意两个类型，如果每一个嵌入到另一个中，则它们是等价的。该论证是用同伦类型论或Voevodsky的一元基础(HoTT/UF)的语言发展起来的，并且需要经典逻辑。由此可见，该定理适用于任何布尔\(\infty \) -拓扑。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Homotopy and Related Structures Mathematics-Geometry and Topology

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期刊介绍： Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences. Journal of Homotopy and Related Structures is intended to publish papers on Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.