一元弱范畴作为类型论的模型

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2022-06-27 DOI:10.1017/s0960129522000172
Thibaut Benjamin
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引用次数: 2

摘要

众所周知,弱$\omega$类别很难定义,因为它们的公理非常复杂。基于赋予细胞的不同形状,已经探索了各种方法。有趣的是,同伦型理论包含了球状环境中弱$\omega$-群胚的定义,因为每个类型都有这样的结构。从这句话开始,Brunerie可以提取出球状弱$\omega$-群胚的定义,公式化为一种类型理论。通过完善其规则,Finster和Mimram定义了一个称为$\mathsf{CaTT}$的类型理论,其模型是弱$\omega$类别。在这里,我们将这种方法推广到单模态弱$\omega$-范畴。基于它们应该等价于只有一个0-单元的弱$\omega$-类别的原理,我们能够导出类型论$\mathsf{MCaTT}$,其模型是单模弱$\omega$-类。这需要改变理论的规则,以便对唯一的0单元所携带的信息进行编码。通过定义我们的类型论$\mathsf{MCaTT}$和类型论$\athsf{CaTT}$之间的一对翻译,表明了所得类型论的正确性。我们的主要贡献是通过详细分析模型的概念如何与两种类型理论的结构规则相互作用,表明这些翻译将我们的类型理论模型与类型理论模型$\mathsf{CaTT}$联系起来,该模型由只有一个0单元的$\omega$类别组成。
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Monoidal weak ω-categories as models of a type theory
Weak $\omega$ -categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak $\omega$ -groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak $\omega$ -groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called $\mathsf{CaTT}$ , whose models are weak $\omega$ -categories. Here, we generalize this approach to monoidal weak $\omega$ -categories. Based on the principle that they should be equivalent to weak $\omega$ -categories with only one 0-cell, we are able to derive a type theory $\mathsf{MCaTT}$ whose models are monoidal weak $\omega$ -categories. This requires changing the rules of the theory in order to encode the information carried by the unique 0-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory $\mathsf{MCaTT}$ and the type theory $\mathsf{CaTT}$ . Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory $\mathsf{CaTT}$ consisting of $\omega$ -categories with only one 0-cell by analyzing in details how the notion of models interact with the structural rules of both type theories.
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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