类型化Lambda演算与Cartesian闭范畴的极大性

笛卡尔范畴的极大性

The Maximality of Cartesian Categories

本文证明了在自由范畴的语言中，若用任意新的等式扩充它们，则使一个范畴坍缩为一个前序，在这个意义上，为范畴所设的箭头之间的等式是极大的。一个类似的结果也适用于具有二进制乘积的范畴，它可能缺少一个终对象。的证明是基于一个连贯性的结果carbohydrate类别，这是有关的模型理论的方法规范化。

Algebraic models of simple type theories: a polynomial approach

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed $λ$-calculi, the computational $λ$-calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into…

微分范畴中的切丛

Tangent bundles in differential lambda-categories

微分类范畴是由布恰雷利等人引入的，作为埃尔哈德和雷尼尔的微分类范畴演算的简单类型化版本的模型。微分微分范畴是Blute等人的一个闭微分范畴，其中的微分算子与闭结构相容。我们证明了任何微分范畴都有一个正则强交换单子，它的构造类似于光滑流形范畴中切丛的构造。本文的大部分结果在任意的Carnival微分范畴中仍然有效。我们的重点是微分类范畴的动机是预期的应用程序的理论发展在本说明中的设计和语义的类演算扩展的推进操作。

Curry-Howard对应的推广

A Generalization of the Curry-Howard Correspondence

我们提出了在（Lambek 1972，1974）中开发的演绎系统演算的一个变体，并给出了Curry-Howard-Lambek定理的一个推广，该定理给出了类型化的类演算范畴与利用广义范畴理论的Carnival闭范畴和指数保持态射范畴之间的等价性（Schoenbaum 2016）。我们讨论了潜在的应用和扩展。

Cartesian closed bicategories: type theory and coherence

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 dif…

克隆、闭范畴与组合逻辑

Clones, closed categories, and combinatory logic

本文用多元结构阐释简单类型λ演算的语义，引入外延SK-克隆并证明其可靠完备，进而证明不含产品的简单类型λ演算是SK-范畴的内部语言。