Fixpoint constructions in focused orthogonality models of linear logic

TL;DR

This paper develops a categorical framework for fixpoint constructions in focused orthogonality models of linear logic, enabling the analysis of recursive types and termination properties.

Contribution

It introduces a theory of fixpoint constructions in focused orthogonality categories, including lifting theorems for initial algebras and final coalgebras, and explores domain-theoretic applications.

Findings

Focused orthogonality categories are relational fibrations.

The framework supports models with least and greatest fixpoints.

Lifting bifree algebras enables solving recursive type equations.

Abstract

Orthogonality is a notion based on the duality between programs and their environments used to determine when they can be safely combined. For instance, it is a powerful tool to establish termination properties in classical formal systems. It was given a general treatment with the concept of orthogonality category, of which numerous models of linear logic are instances, by Hyland and Schalk. This paper considers the subclass of focused orthogonalities. We develop a theory of fixpoint constructions in focused orthogonality categories. Central results are lifting theorems for initial algebras and final coalgebras. These crucially hinge on the insight that focused orthogonality categories are relational fibrations. The theory provides an axiomatic categorical framework for models of linear logic with least and greatest fixpoints of types. We further investigate domain-theoretic settings, showing how to lift bifree algebras, used to solve mixed-variance recursive type equations, to focused orthogonality categories.