The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory (HoTT) research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising.

中文翻译：

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同伦型理论中的宇宙和统一性

由 Vladimir Voevodsky 提出的单价公理通常被认为是同伦类型理论 (HoTT) 研究计划中最重要的发现之一。史蒂夫·阿沃迪（Steve Awodey）说，单价体现了数学结构主义，单价可以被视为“将同一概念扩展到等价概念”。本文探讨了同伦类型理论中一元性的概念、基础和哲学地位。它将我们对 HoTT 的类型作为概念的解释扩展到了宇宙，并以这种方式提供了对单价公理的说明。我们考虑 Awodey 的非正式论点，即单价是由推理在同构下应该是不变的原则所激发的，并且我们检查是否可以给出沿着这些思路的自主和严格的证明。我们考虑面临这样一个理由的两个问题。首先，两者有区别等价和同构单价必须根据前者来表述。其次，所提出的论点不能建立单一性本身，而只能建立其较弱的版本，并且必须辅以附加原则。这篇文章认为，对 Univalence 进行自主证明的前景是有希望的。