A factorization of regularity

Abstract

Earlier, the authors showed that categories with regular factorizations (in the sense of Kelly) and with regular epimorphisms closed under composition, are algebras for a KZ-doctrine R on the 2-category cat ker , of all categories and functors which preserve kernel inclusions. Here we show that the usual 2-category lex , of categories with finite limits and left exact functors, is cat ker L the 2-category of algebras for a coKZ-doctrine L on cat ker . Combined with our earlier results this shows that the 2-category reg , of regular categories and regular functors, is the 2-category of algebras for a distributive law LR → RL over the 2-category cat ker .