Combinatory Completeness in Structured Multicategories

TL;DR

This paper introduces a general framework for combinatory completeness in structured multicategories, providing new characterizations of applicative systems and their polynomial notions, including classical combinatory algebras.

Contribution

It systematically develops a unified approach to combinatory completeness using faithful cartesian clubs, linking various structured multicategories and applicative systems.

Findings

Classical combinatory algebras characterized as combinatory complete applicative systems

New notions of polynomial over applicative systems derived from structured multicategories

Framework unifies different notions of combinatory completeness

Abstract

We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club determines a notion of structured multicategory, with the different notions of structured multicategory obtained in this way giving different notions of polynomial over an applicative system, which in turn give different notions of combinatory completeness. We obtain the classical characterisation of combinatory algebras as combinatory complete applicative systems as a specific instance.