A factorization of regularity (Q2573482)

In any category, the kernel of a morphism \(f: X\to A\) is the class \(\ker(f)\) of parallel morphisms \((x, y): T\rightrightarrows X\) satisfying \(fx =fy\). A functor \(F: {\mathcal C}\to{\mathcal D}\) is said to preserve kernel inclusions whenever for any span (\(f: X\to A\), \(g: X\to B\)) in \({\mathcal C}\), if \(\ker(f)\subset\ker(g)\) then \(\ker(F(f))\subset\ker(F(g))\). The 2-category \({\mathcal C} at_{\ker}\) has categories as objects, kernel inclusion preserving functors as morphisms, and natural transformations as 2-cells. The authors show that the category \({\mathcal L}ex\) of categories with finite limits and left exact functors is the 2-category \({\mathcal C} at^{{\mathcal L}}_{\ker}\) of algebras for a \(\text{co}\mathcal{KZ}\)-doctrine \({\mathcal L}\) on \({\mathcal C} at_{\ker}\). Combined with a preveous result, they show that the 2-category \({\mathcal R} eg\) of regular categories and regular functors, is the 2-category of algebras for a distributive law over the category \({\mathcal C} at_{\ker}\).