Codiscrete cofibrations vs iterated discrete fibrations for (∞,ℓ)-congruences

I will present an approach to the exactness properties characterising (∞,ℓ)-topoi based purely on cartesian-enriched weighted colimits, with the role of categorified congruences played by internal (ℓ-1)-categories whose underlying (ℓ-1)-graphs are iterated discrete two-sided fibrations. To compare it to that based on lax colimits (and internal categories whose underlying graphs are (ℓ-2)-categorical two-sided fibrations), the crucial point will be to relate both structures to codiscrete two-sided cofibrations, using the expressivity of generalised kernels. The construction can then also be used to understand the internal structure of classifying fibrations.