A parameterization process, functorially

TL;DR

This paper explores a categorical framework for the parameterization process in symbolic computation systems, showing it as a functorial construction that relates parameterized specifications to their models via natural transformations.

Contribution

It formalizes the parameterization process as a functor and the parameter passing as a natural transformation within category theory, providing a general theoretical foundation.

Findings

Parameterization process is a functor.

Parameter passing is a natural transformation.

Bijective correspondence under terminality assumptions.

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 adding a parameter as a new variable to some operations. 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 functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits.