Cartesian Cubical Type Theory
Cartesian Cubical Type Theory
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笛卡尔立方类型理论
DOI：
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Daniel R. Licata
中科院分区：
文献类型：
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作者：
C. Angiuli;T. Coquand;R. Harper;Daniel R. Licata
我们提出了一个立方类型理论的基础上的笛卡尔立方体类别（面，退化，对称性，对角线，但没有连接或反转）与univalent宇宙，每个包含的numerals，numerals，路径，身份，自然数，布尔，推出，胶水（等价扩展）类型。类型理论包括一个统一的Kan操作的语法描述，沿着判断等式规则定义的Kan操作的每一个类型。Kan运算使用了一组不同于Cohen，Coquand，Huber，and Mörtberg（CCHM）模型的平凡上纤化和上纤化。接下来，我们描述了一个在笛卡尔立方集的建设性模型，句法类型理论的灵感来自于这个模型，虽然我们还没有给出一个正式的解释。我们描述了一个机械化的证明，使用内部语言的立方集的风格介绍了奥顿和皮茨，胶水，双，双，路径，身份，布尔，自然数，推出类型是在这个模型中的Kan;我们还勾勒了一个证明，这种内部结构意味着外部的单叶宇宙。这种形式化方法的一个优点是，我们的结构也可以解释在立方集上的连接立方体类别，并在CCHM模型中使用的德摩根立方体类别。作为比较这些方法的第一步，我们表明，这两个Kan操作是interderivable的设置都存在（preheaves上的德摩根立方体类别，与我们的建设所需的额外cofibration）。
We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, pushout, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgemental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of trivial cofibrations and a different set of cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model in Cartesian cubical sets; the syntactic type theory is inspired by this model, though we have not yet given a formal interpretation. We describe a mechanized proof, using the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, and pushout types are Kan in this model; we also sketch a proof that this internal construction implies univalent universes externally. An advantage of this formal approach is that our construction can also be interpreted in cubical sets on the connections cube category, and on the de Morgan cube category used in the CCHM model. As a first step towards comparing these approaches, we show that the two Kan operations are interderivable in a setting where both exist (presheaves on the de Morgan cube category, with the additional cofibration required by our construction).
弗罗贝尼乌斯条件、正确性和均匀纤维化
影响因子：
0.8
作者：
Gambino N
通讯作者：
Gambino N