Algebraically coherent categories

TL;DR

This paper introduces the concept of algebraically coherent categories, exploring their properties, examples, and implications in semi-abelian categories, including strong protomodularity and normality of commutators.

Contribution

It defines algebraically coherent categories via change-of-base functors, provides examples, and studies their properties and consequences in semi-abelian contexts.

Findings

Examples include coherent categories and categories of interest

Algebraically coherent categories exhibit strong protomodularity

Normality of Higgins commutators holds in these categories

Abstract

We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give examples of categories satisfying this condition; for instance, coherent categories, categories of interest in the sense of Orzech, and (compact) Hausdorff algebras over a semi-abelian algebraically coherent theory. We study equivalent conditions in the context of semi-abelian categories, as well as some of its consequences: including amongst others, strong protomodularity, and normality of Higgins commutators for normal subobjects, and in the varietal case, fibre-wise algebraic cartesian closedness.