A two-dimensional Birkhoff's theorem

TL;DR

This paper extends Birkhoff's variety theorem to enriched category settings, providing a characterization of equational subcategories via their closure properties in algebraic categories.

Contribution

It introduces an analogue of Birkhoff's theorem for enriched categories, broadening the theorem's applicability beyond classical universal algebra.

Findings

Characterization of equational subcategories in enriched categories

Closure properties used to identify subcategories

Extension of Birkhoff's theorem to new categorical contexts

Abstract

Birkhoff's variety theorem from universal algebra characterises equational subcategories of varieties. We give an analogue of Birkhoff's theorem in the setting of enrichment in categories. For a suitable notion of an equational subcategory we characterise these subcategories by their closure properties in the ambient algebraic category.