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’s Galois theory. In the present paper, we give a suitable counterpart notion to that of absolute admissible Galois structure for the lax idempotent context, compatible with the context of 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 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’s 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), especially using a result that says that all admissible (2-)functors in the classical sense are also 2-admissible (and hence simple as well).