Filtered colimit elimination from Birkhoff's variety theorem

TL;DR

This paper investigates conditions under which filtered colimit closure can be eliminated from Birkhoff's variety theorem, focusing on specific algebraic theories and categories where this simplification is possible.

Contribution

It introduces a sufficient condition based on a noetherian-like property of categories that ensures filtered colimit elimination in the generalized Birkhoff's theorem.

Findings

Filtered colimit elimination holds under certain noetherian-like conditions.

The study extends Birkhoff's theorem to broader algebraic contexts without colimit closure.

Provides a criterion for when filtered colimits can be omitted in algebraic characterizations.

Abstract

Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon "filtered colimit elimination," and study a sufficient condition for it. We show that if a locally finitely presentable category $\mathscr{A}$ satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to $\mathscr{A}$.