A categorical view of varieties and equations

TL;DR

This paper develops a categorical framework to study varieties and equations, generalizing Birkhoff's theorem to algebraic and coalgebraic categories through new concepts of subvarieties and equations.

Contribution

It introduces Birkhoff varieties in algebraic categories and generalizes the notion of equations, extending classical results to a broader categorical setting.

Findings

Birkhoff varieties are characterized as coreflexive equalizers.

Generalization of equations for subvarieties of algebraic categories.

Duality yields a characterization of cosubvarieties as varieties.

Abstract

We present a common framework to study varieties in great generality from a categorical point of view. The main application of this study is in the setting of algebraic categories, where we introduce Birkhoff varieties which are essentially subvarieties of algebraic categories, and we get a generalization of Birkhoff's variety theorem. In particular, we show that Birkhoff varieties are coreflexive equalizers. The key of this generalization is to give a more general concept of equation for subvarieties of algebraic categories. In order to get our characterization of Birkhoff varieties, we study inserters over algebraic categories, where we generalize some well-known results of algebras for finitary endofunctors over $Set$. By duality, we obtain a characterization of cosubvarieties of coalgebraic categories. Surprisingly, these cosubvarieties turn to be varieties according to our theory of varieties.