Algebraic models of simple type theories: a polynomial approach

TL;DR

This paper introduces algebraic models for simple type theories using polynomial endofunctors, extending universal algebra to include variable binding and providing a unified semantic framework.

Contribution

It develops a novel algebraic framework for simple type theories, incorporating variable binding and substitution via polynomial endofunctors and structured categories.

Findings

Models of simple type theories in presheaf categories

Construction of initial models representing syntax

Sound and complete semantics in structured cartesian multicategories

Abstract

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed $\lambda$-calculi, the computational $\lambda$-calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed $\lambda$-calculus and cartesian-closed categories, to arbitrary simple type theories.