Algebraic logoi

TL;DR

This paper introduces the concept of algebraic logoi, a class of categories characterized by the existence of split extension cores and geometric change-of-base functors, expanding the understanding of categorical structures in algebra.

Contribution

It generalizes the notion of normal and action cores to split extension cores in homological categories, defining and exploring algebraic logoi.

Findings

Existence of split extension cores is equivalent to geometric change-of-base functors.

Algebraic logoi include categories with small joins and are stable under common operations.

Comparison with algebraically coherent categories enhances categorical understanding.

Abstract

We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context of a homological category. We prove that, if the category is moreover well-powered with (small) joins, then the existence of split extension cores is equivalent to the condition that the change-of-base functors in the fibration of points are geometric. We call a finitely complete category that satisfies this condition an algebraic logos. We give examples of such categories, compare them with algebraically coherent ones, and study equivalent conditions as well as stability under common categorical operations.