Logical systems I: Lambda calculi through discreteness

TL;DR

This paper develops a 2-categorical framework for internal models of polymorphic lambda calculi, generalizing classical theorems and establishing connections with fibrational models, advancing the theoretical understanding of lambda calculus semantics.

Contribution

It introduces a 2-categorical approach to internal models for polymorphic lambda calculi and generalizes Freyd's theorem within this setting, linking to fibrational models.

Findings

Established a 2-categorical generalization of Freyd's theorem.

Proved a representation theorem connecting internal models and fibrational models.

Extended the concept of discreteness to a 2-categorical context.

Abstract

This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently (co)complete non-degenerate categories. As a simple corollary, we obtain a variant of Freyd theorem for categories internal to any tensored category. Also, with help of introduced concept of an associated category, we prove a representation theorem relating our internal models with well-studied fibrational models for polymorphism.