We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

中文翻译：

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建设性集合论的同伦类型理论解释

我们介绍一个家庭（ķ,H)-对 2 ≤ 的解释ķ≤ ∞ 和 1 ≤H≤ ∞ 的建设性集合论转化为类型论，其中集合和公式被解释为同伦水平的类型ķ和H， 分别。取决于参数的值ķ和H，我们能够解释不同的理论，例如 Aczel 的 CZF 和 Myhill 的 CST。我们还在逻辑丰富类型理论的背景下定义了一个命题作为命题的解释。本文的其余部分致力于描述和分析所考虑的解释。在 prop-as-hprop 解释中有效的公式是根据唯一选择公理来表征的。我们还分析了 CST 对同伦类型理论的解释，提供了与 Aczel 解释的比较分析。这是通过在逻辑丰富的类型理论中制定两种解释的证明中使用的关键原则来完成的。最后，我们描述了一类在 (ķ, ∞)-根据 ΠΣ 选择公理的解释。