Varieties defined by natural transformations

TL;DR

This paper introduces a new way to define algebraic varieties using natural transformations, establishing their equivalence to traditional equational class definitions, and explores conditions for free algebra existence.

Contribution

It presents a novel approach to defining algebraic varieties via natural transformations and proves their equivalence to classical equational classes, including conditions for free algebra existence.

Findings

Varieties can be characterized by pairs of natural transformations.

Equivalence between natural transformation-based varieties and equational classes is established.

Existence of free algebras is proved under accessibility assumptions.

Abstract

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to the one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility.