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Braids, twists, trace and duality in combinatory algebras

Masahito Hasegawa, Serge Lechenne

TL;DR

This work develops a geometric interpretation of untyped lambda calculus and combinatory logic inside ribbon categories, introducing ribbon combinatory algebras and the internal PRO/BOP constructions that realize braids, twists, duality and trace. A central contribution is the equivalence between reflexive objects in ribbon categories and ribbon combinatory algebras, together with two equivalent axiomatizations: as balanced extensional BC±I-algebras with a trace, and as balanced extensional BC±I-algebras equipped with duality data. The authors show that the internal PROB of a ribbon combinatory algebra is a ribbon category, and under mild conditions the ambient ribbon category is recovered; they also establish that the functor from the ambient category to the internal PROB is almost full and, under the condition $igcirc= ext{id}_I$, full and faithful. This framework yields a unified treatment of both braided lambda-calculus and framed tangles, enabling a canonical, diagrammatic reading of combinators and suggesting applications to topological quantum computation and knot-theoretic semantics. Overall, the paper lays foundational connections between combinatory algebras, trace and duality, and ribbon-category topology, with concrete constructions and clear directions for further examples and generalizations.

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.

Braids, twists, trace and duality in combinatory algebras

TL;DR

This work develops a geometric interpretation of untyped lambda calculus and combinatory logic inside ribbon categories, introducing ribbon combinatory algebras and the internal PRO/BOP constructions that realize braids, twists, duality and trace. A central contribution is the equivalence between reflexive objects in ribbon categories and ribbon combinatory algebras, together with two equivalent axiomatizations: as balanced extensional BC±I-algebras with a trace, and as balanced extensional BC±I-algebras equipped with duality data. The authors show that the internal PROB of a ribbon combinatory algebra is a ribbon category, and under mild conditions the ambient ribbon category is recovered; they also establish that the functor from the ambient category to the internal PROB is almost full and, under the condition , full and faithful. This framework yields a unified treatment of both braided lambda-calculus and framed tangles, enabling a canonical, diagrammatic reading of combinators and suggesting applications to topological quantum computation and knot-theoretic semantics. Overall, the paper lays foundational connections between combinatory algebras, trace and duality, and ribbon-category topology, with concrete constructions and clear directions for further examples and generalizations.

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.
Paper Structure (37 sections, 35 theorems, 55 equations, 2 figures)

This paper contains 37 sections, 35 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation
  3. Monoidal categories
  4. Notations.
  5. Monoidal categories
  6. PROs
  7. Braids and twists
  8. PROPs and PROBs
  9. Duality and trace
  10. Duality.
  11. Ribbon categories.
  12. Trace.
  13. Closed structure and reflexive objects
  14. Reflexive PROs
  15. Monoidal structures in combinatory algebras
  16. ...and 22 more sections

Key Result

proposition 1

Suppose that $A$ is a reflexive object in a monoidal category $\mathbb{C}$. Then $\mathbb{C}(I,A)$ is an extensional $\mathbf{BI}(\_)^\bullet$-algebra with $a\cdot b=(a\otimes b);\mathsf{app}$, $\mathit{\boldsymbol{B}}=\mathsf{cur}(\mathsf{cur}(\mathsf{cur}((A\otimes\mathsf{app});\mathsf{app})))$, $

Figures (2)

  • Figure 1: The ribbon combinatory algebra $\mathcal{A}=\mathbb{C}(I,A)$
  • Figure 2: Curry's axioms of combinatory logic

Theorems & Definitions (54)

  • definition 1
  • definition 2
  • proposition 1
  • lemma 1
  • proof
  • definition 3
  • lemma 2
  • proof
  • proposition 2
  • proposition 3
  • ...and 44 more