Linear lambda terms as invariants of rooted trivalent maps

The main aim of the article is to give a simple and conceptual account for the correspondence 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, and explain how these diagrams may be interpreted formally as 1-cells in a symmetric monoidal closed bicategory equipped with a reflexive object. 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 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, we show how to use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed subterms, and conclude with a natural reformulation of the Four Color Theorem as a statement about lambda calculus.