Characterization of balanced algebraic theories

Characterize the algebraic theories in the sense of Lawvere whose corresponding categories of algebras are balanced, meaning that every morphism that is both epic and monic is an isomorphism.

Background

Balancedness is the categorical property that every morphism which is both an epimorphism and a monomorphism is an isomorphism. For categories of algebras, this is closely related to the question of whether epimorphisms are surjective and to the definability of implicit partial operations by explicit terms.

The paper identifies the characterization of all Lawvere algebraic theories yielding balanced categories of algebras as a longstanding unresolved problem. The paper instead studies a related construction: associating a balanced Beth companion category to a non-balanced locally finitely presentable category.

References

The question of characterising the algebraic theories (in the sense of Lawvere) such that the corresponding categories of algebras are balanced, explicitly stated in p.~57 and still open to this day, has led to the investigation of implicit partial operations.

— Beth companions of finitary essentially algebraic theories  (2609.03601 - Liberti et al., 3 Sep 2026) in Section 1, Introduction