A characterisation of algebraic exactness

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.

💡 Research Summary

The paper resolves a conjecture of Adámek, Lawvere and Rosický concerning the precise characterization of algebraically exact categories. An algebraically exact category is one that possesses all limits and colimits that appear in any variety of (possibly many‑sorted) algebras and that satisfies exactly the same equations between those operations as hold in every variety. It is known that such categories automatically have all small limits and sifted colimits, and that they can be described as the categories with small limits and sifted colimits for which the sifted‑colimit functor is continuous.

The conjecture asserted that a complete and sifted‑cocomplete category is algebraically exact precisely when it satisfies four elementary exactness conditions:

(E1) Barr‑exactness (regular epimorphisms are stable under pullback and equivalence relations are effective);
(E2) Finite limits commute with filtered colimits;
(E3) Regular epimorphisms are stable under small products;
(E4) Filtered colimits distribute over small products.

Previous work proved the conjecture only under additional hypotheses (e.g., the presence of a regular generator). This article removes those restrictions and establishes the equivalence in full generality.

The author proceeds by introducing a size‑controlled version of algebraic exactness. For any regular infinite cardinal κ, let κ′ = (Σ_{γ<κ} 2^γ)+. A “κ‑algebraically exact” category is defined to have κ‑small limits, reflexive coequalizers, and κ′‑filtered colimits, and to satisfy the κ‑restricted analogues (E1′)–(E4′) of the four conditions. The key technical device is a Kock‑Zöberlein pseudomonad S_{κ′} on the 2‑category κ‑CONTS of κ‑complete categories and κ‑continuous functors. For a κ‑complete C, S_{κ′}(C) is the closure of C under κ‑limits, reflexive coequalizers and κ′‑filtered colimits inside the presheaf category