Braids, twists, trace and duality in combinatory algebras

TL;DR

This paper explores ribbon combinatory algebras, linking them to braided untyped linear lambda calculus and framed tangles, and establishes their characterization via balanced structures with trace or duality.

Contribution

It introduces ribbon combinatory algebras, providing their equational characterization and showing their correspondence with reflexive objects in ribbon categories.

Findings

Ribbon combinatory algebras interpret braided untyped linear lambda calculus.

They can be constructed from reflexive objects in ribbon categories.

Characterized as balanced algebras with trace or duality.

Abstract

We investigate a class of combinatory algebras, called ribbon combinatory algebras, in which we can interpret both the braided untyped linear lambda calculus and framed oriented tangles. Any reflexive object in a ribbon category gives rise to a ribbon combinatory algebra. Conversely, From a ribbon combinatory algebra, we can construct a ribbon category with a reflexive object, from which the combinatory algebra can be recovered. To show this, and also to give the equational characterisation of ribbon combinatory algebras, we make use of the internal PRO construction developed in Hasegawa's recent work. Interestingly, we can characterise ribbon combinatory algebras in two different ways: as balanced combinatory algebras with a trace combinator, and as balanced combinatory algebras with duality.