In this article, the author endows the functor category $$[\mathbf {B}(\mathbb {Z}_2),\mathbf {Gpd}]$$[B(Z2),Gpd] with the structure of a type-theoretic fibration category with a univalent universe, using the so-called injective model structure. This gives a new model of Martin-Löf type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in another work) in the same underlying category and with the same universe, which turns out to be provably not univalent with respect to projective fibrations, provide an example of two Quillen equivalent model categories that host different models of type theory. Thus, we provide a counterexample to the model invariance problem formulated by Michael Shulman.

中文翻译：

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同伦类型论中的一个模型不变性问题

在本文中，作者赋予函子范畴 $$[\mathbf {B}(\mathbb {Z}_2),\mathbf {Gpd}]$$[B(Z2),Gpd] 一个类型的结构——具有单价宇宙的理论纤维化范畴，使用所谓的单射模型结构。这给出了 Martin-Löf 类型理论的新模型，该模型具有相依和、相依积、恒等式和单价宇宙。这个模型，连同模型（由作者在另一部作品中开发）在相同的基础类别和相同的宇宙中，结果证明在投影纤维方面不是单等的，提供了两个 Quillen 等效模型类别的例子承载不同的类型理论模型。因此，我们为 Michael Shulman 提出的模型不变性问题提供了一个反例。