Enriched Categorical Semantics for Intuitionistic Subexponential Linear Logic (ISELL): Initial-Model Completeness

Abstract

We give a \(\mathcal {V}\)-enriched categorical semantics for Intuitionistic Subexponential Linear Logic (ISELL), with string-diagram and numbered equational proofs in the main text for all one-step cut-reduction schemes. A model is a \(\mathcal {V}\)-enriched symmetric monoidal closed category (SMCC) equipped with a family of subexponential coalgebra modalities \((!^{a})_{a\in (\mathbb {E},\preceq )}\) and, whenever \(a\preceq b\), monoidal comonad morphisms \(\theta ^{b\Rightarrow a}:!^{b}\Rightarrow !^{a}\) that preserve all available comonoid structure \((\Delta ,e)\). This preservation yields an explicit “promotion prism” identity that semantically accounts for the subexponential side-condition. We build an initial enriched syntactic model by generators-and-relations and deduce completeness. We also prove a separation result showing that models without preserving comparisons cannot validate the same promotion behaviour.