  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.
