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.

中文翻译：

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笛卡尔封闭二分类：类型理论和连贯性

在本论文中，我将简单类型的 lambda 演算和笛卡尔闭范畴之间的 Curry--Howard--Lambek 对应关系提升到双分类设置，然后使用结果类型理论来证明笛卡尔闭双范畴的相干结果。笛卡尔封闭二分类---配备类似弱乘积和指数的2类“高达同构”---出现在逻辑、分类代数和博弈语义中。我表明，在一个集合上的自由笛卡尔封闭双分类中，任何平行的一对 1 单元之间最多有一个 2 单元，因此——就计算的难度而言——带来笛卡尔封闭双分类的数据下降到熟悉的笛卡尔封闭类别级别。事实上，我从两个方面证明了这个结果。第一个论点与权力密切相关” 具有灵活双极限的双范畴的相干定理。对于第二个，这是本文的核心关注点，证明策略有两个部分：类型理论的构建，以及它满足我称为“局部一致性”的规范化形式的证明。我使用通用代数的（多重）抽象克隆的新颖概括，从代数原理中综合了类型理论，称为“双克隆”。使用 M. Fiore 对归一化评估的分类分析的双分类处理，然后我证明了一个归一化结果，该结果包含笛卡尔封闭双分类的相干定理。与之前的双类别相干结果相比，该论点不依赖于重写或使用米田嵌入进行严格化的理论。