FREE ALGEBRAS—ARE THEY CATEGORICALLY DEFINABLE?

Abstract

It is proven that, in general, the free algebras of an equational class, considered as an abstract category, are not definable strictly in the language of categories. As a concrete counterexample, a categorical equivalence between the categories of 2-rings and 3-rings is constructed without the axiom of choice. Isomorphism follows, as well as the non-correspondence of free algebras. An assortment of similar negative results and two open questions close the paper.