Filtered colimit elimination from Birkhoff's variety theorem

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 $\mathscr{A}$.