Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a three-part mini course delivered at the Logic and Higher Structures workshop at CIRM-Luminy, attempt to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any -topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.

中文翻译：

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论同伦型理论的∞-topos语义

许多对同伦类型理论和单价公理的介绍都掩盖了传统基于集合的基础中这个新形式系统的语义。这篇说明性文章是作为 CIRM-Luminy 逻辑和高级结构研讨会上的三部分迷你课程的讲义编写的，试图调查现有技术，首先介绍 Voevodsky 的单价基础单纯模型，然后浏览 Shulman 的模型广泛的概括，它提供了对同伦型理论的解释，在任何情况下都具有严格的单价宇宙-拓扑。正如我们将解释的那样，这一成就是社区努力抽象和简化原始论点以及开发新推理路线的产物。