An explicit shuffle construction of the homotopy span model of differential linear logic

2026-10-07

AI Summary

This study addresses the absence of explicit constructions for cross-models of homotopy in differential linear logic. Building upon Quillen model structures, this work interprets proofs as small categories and achieves intuitive modeling by substituting original fibrant spans with independent fibrant spans. Methodologically, it integrates abstract homotopy theory with category theory and introduces shuffle structures to precisely capture the symmetry of exponential modalities. Ultimately, this research establishes a concise framework for explicit construction that successfully elucidates axiom and cut links, thereby providing a rigorous yet intuitive semantic foundation for the field.

Abstract

We give a direct and explicit construction of the homotopy span model of linear logic originally formulated by Melliès using abstract homotopy theory in the Quillen category Cat of small categories equipped with its natural model structure. Following the principles of homotopy theory, the formulas and proofs of linear logic are interpreted in the model as small categories up to categorical equivalence, the notion of weak homotopy equivalence underlying the natural Quillen model structure on Cat. Here, we explain how the original interpretation of proofs as fibrant spans can be replaced by a more liberal interpretation of proofs as separately fibrant spans. This relaxation from fibrant spans to separately fibrant spans enables us to give a simple and intuitive description of the homotopy span model, where the axiom and cut links are interpreted as identity spans, and where the symmetries of the exponential modality are captured by a shuffle structure on the interpretation of proofs.

Problem

• differential linear logic
• homotopy span model
• fibrant spans
• Quillen model structure
• shuffle construction

Innovation

• homotopy span model
• differential linear logic
• separately fibrant spans
• shuffle structure
• Quillen model structure