Algebraic logoi

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