作为有根三价映射不变量的线性lambda项
Linear lambda terms as invariants of rooted trivalent maps
摘要：
本文的主要目的是对这种对应关系作一个简单的概念性解释（最初由Bodini，Gardy和Jacquot描述）在无边界的紧致定向曲面上的闭线性lambda项的$α$ -等价类和有根三价映射的同构类之间，作为具有自由变量上下文的线性λ项和具有自由边边界的有根三价映射之间的更一般对应关系的实例。我们首先回顾一个熟悉的图形表示的线性lambda条款，而在同一时间解释如何这样的图可以正式阅读作为一个符号的自反对象在一个对称monoidal封闭（双）范畴。从那里，对应的“容易”方向是一个简单的健忘操作，它擦除线性lambda项图上的注释，以产生一个有根的三价映射。另一个方向将线性lambda项视为其底层有根三价映射的完全不变量，通过自由边映射上的Tutte式拓扑递归重建丢失的信息。作为组合数学中的一个应用，我们使用这种分析来枚举无桥有根三价映射作为线性lambda项，不包含封闭的真子项，并得出结论，给出一个自然的四色定理的重新表述作为一个关于λ演算中的类型的声明。
英文摘要：
The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between $α$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the "easy" direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.
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