Lax comma $2$-categories and admissible $2$-functors
This paper is a contribution towards a two dimensional extension of the basic ideas and results of Janelidze-Galois theory. In the present paper, we give a suitable counterpart notion to that of \textit{absolute admissible Galois structure} for the lax idempotent context, compatible with the context of \textit{lax orthogonal factorization systems}. As part of this work, we study lax comma $2$-categories, giving analogue results to the basic properties of the usual comma categories. We show that each morphism of a $2$-category induces a $2$-adjunction between lax comma $2$-categories and comma $2$-categories, playing the role of the usual \textit{change of base functors}. With these induced $2$-adjunctions, we are able to show that each $2$-adjunction induces $2$-adjunctions between lax comma $2$-categories and comma $2$-categories, which are our analogues of the usual lifting to the comma categories used in Janelidze-Galois theory. We give sufficient conditions under which these liftings are $2$-premonadic and induce a lax idempotent $2$-monad, which corresponds to our notion of $2$-admissible $2$-functor. In order to carry out this work, we analyse when a composition of $2$-adjunctions is a lax idempotent $2$-monad, and when it is $2$-premonadic. We give then examples of our $2$-admissible $2$-functors (and, in particular, simple $2$-functors), specially using a result that says that all admissible ($2$-)functors in the classical sense are also $2$-admissible (and hence simple as well). We finish the paper relating coequalizers in lax comma $2$-categories and Kan extensions.