Birkhoff-Frink representations as functors

Abstract

In an earlier article we characterized, from the viewpoint of set theory, those closure operators for which the classical result of Birkhoff and Frink, stating the equivalence between algebraic closure spaces, subalgebra lattices and algebraic lattices, holds in a many-sorted setting. In the present article we investigate, from the standpoint of category theory, the form these equivalences take when the adequate morphisms of the several different species of structures implicated in them are also taken into account. Specifically, our main aim is to provide a functorial rendering of the Birkhoff-Frink representation theorems for both single-sorted algebras and many-sorted algebras, by defining the appropriate categories and functors, covariant and contravariant, involved in the process.