On implicit operations and balanced categories

Abstract

The problem of determining whether a category is balanced (i.e., every morphism that is both an epimorphism and a monomorphism is an isomorphism) has been widely investigated, especially in algebra. For complete categories with an extremal projective generator, we characterise the property of being balanced, and the stronger property that every monomorphism is regular, in terms of implicit operations. These characterisations apply, for example, to any quasiequational class of algebras, possibly with a proper class of function symbols and unbounded arities, that admits free algebras.