This is a joint work with Maria Manuel Clementino. Our aim is to understand the analogues of semi-left exactness and the analogues of the basic ideas of Janelidze-Galois 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 2-adjunctions to the comma-2-categories. Here, we already find two possible liftings which deserve interest. The underlying adjunction of the first lifting is the usual 1-dimensional case, while the other one corresponds to the lifting of the 2-monad given by the work of Clementino-Lopez (lax factorization systems). We give several approaches to construct these liftings. One of them gives the lifting as a composition of 2-adjunctions: 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 2-monads on the comma 2-categories, while semi left exact are those which are simple and induce a monadic 2-functor between the comma 2-categories.
We give one possible characterization of semi left exact, which fully relies on the study of compositions of 2-adjunctions. We also recover the characterization of simple lax reflections due to Clementino-Lopez via this perspective.
The third step is to give a sketch of analogues of the basic ideas of Janelidze-Galois. 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 Janelidze-Galois 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.