A parameterization process as a categorical construction

TL;DR

This paper models the parameterization process in symbolic computation systems within a categorical framework, showing it as a free functor and natural transformation, and explores the bijection between parameters and models.

Contribution

It introduces a categorical formalization of the parameterization process, connecting it to free functors and natural transformations, and analyzes its properties using advanced categorical concepts.

Findings

Parameterization process is a free functor.

Parameter passing is a natural transformation.

Bijection between parameters and models under certain conditions.

Abstract

The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by transforming some operations into parameterized operations, which depend on one additional variable called the parameter. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a free functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits.