A characterisation of algebraic exactness

TL;DR

This paper characterizes algebraically exact categories as those with specific limit and colimit properties, proving a conjecture that links algebraic exactness with Barr-exactness and colimit-product interactions.

Contribution

It proves a conjecture connecting algebraic exactness with Barr-exactness, filtered colimit and product interactions, and stability of regular epimorphisms.

Findings

Proved the conjecture relating algebraic exactness to Barr-exactness.

Established conditions under which a complete and sifted-cocomplete category is algebraically exact.

Connected algebraic exactness with the distribution of filtered colimits over small products.

Abstract

An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Ad\'amek, Lawvere and Rosick\'y: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exact, finite limits commute with filtered colimits, regular epimorphisms are stable by small products, and filtered colimits distribute over small products. We prove this conjecture.