This is a joint work with Maria Manuel Clementino. Our aim is to understand the analogues of semileft exactness and the analogues of the basic ideas of JanelidzeGalois within the lax idempotent context. As it is usual, there are more than one possible approach to do so. Our first step is the study of the lifting of 2adjunctions to the comma2categories. Here, we already find two possible liftings which deserve interest. The underlying adjunction of the first lifting is the usual 1dimensional case, while the other one corresponds to the lifting of the 2monad given by the work of ClementinoLopez (lax factorization systems). We give several approaches to construct these liftings. One of them gives the lifting as a composition of 2adjunctions: one that always exists with another one that depends on the existence of pullbacks/comma objects. The second step is to verify when these liftings are Kock Zoberlein. In our context, simple lax reflections are those that induce lax idempotent 2monads on the comma 2categories, while semi left exact are those which are simple and induce a monadic 2functor between the comma 2categories. We give one possible characterization of semi left exact, which fully relies on the study of compositions of 2adjunctions. We also recover the characterization of simple lax reflections due to ClementinoLopez via this perspective. The third step is to give a sketch of analogues of the basic ideas of JanelidzeGalois. Again, there are several possibilities. Yet, for this work we stick with what might be the simplest one, avoiding the study of higher dimensional aspects of Descent. Since the importance of JanelidzeGalois relies on its great number of important examples, our final step is fundamental: to give examples of our setting. The aim of the talk is to sketch some of the ideas of our advances in each of these steps.
