Cartesian closed bicategories: type theory and coherence
Cartesian closed bicategories: type theory and coherence
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笛卡尔封闭双范畴：类型理论和连贯性
DOI：
10.17863/cam.55080
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
P. Saville
中科院分区：
文献类型：
--
作者：
P. Saville
在本文中，我将单型Lambda演算与笛卡尔闭范畴之间的Curry-Howard-Lambek对应提升到双范畴，然后利用所得到的类型理论证明了笛卡尔闭双范畴的一个相干性结果。笛卡尔闭合双范畴-配备有类似弱乘积和指数的2-范畴-出现在逻辑、范畴代数和游戏语义学中。我证明了在一个集合上的自由笛卡尔闭双范畴中的任何平行的一对1-胞格之间至多有一个2-胞格，因此-就计算的难度而言-将笛卡儿闭双范畴的数据降低到笛卡尔闭范畴的熟悉水平。
事实上，我用两种方式证明了这一结果。第一个论点与具有灵活胆界的双范畴的Power相干性定理密切相关。对于第二部分，也就是本文的重点，证明策略包括两个部分：类型理论的构建，以及它满足一种我称之为“局部连贯”的规范化形式的证明。我从代数原理中综合了类型理论，使用了一种新的泛代数(多分类)抽象克隆的泛化，称为“双克隆”。使用M.Fiore的按赋值归一化的范畴分析的双范畴处理，然后我证明了一个归一化结果，该结果需要笛卡儿闭双范畴的连贯性定理。与之前关于双范畴的连贯结果不同，该论点不依赖于使用Yoneda嵌入的重写或严格的理论。在此过程中，我证明了一系列公认的范畴理论结果的双范畴推广，提出了双范畴粘合的概念，并对民俗结果进行了双范畴化，为粘合范畴是笛卡儿闭合提供了充分条件。
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories.
In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call "local coherence". I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called "biclones". Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I then prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. Along the way I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed.
通过评估类型化 lambda 演算进行标准化的语义分析
影响因子：
0.5
作者：
Fiore M
通讯作者：
Fiore M