Two dimensional monadicity
arXiv:1212.5123
Abstract
The behaviour of limits of weak morphisms in 2-dimensional universal algebra is not 2-categorical in that, to fully express the behaviour that occurs, one needs to be able to quantify over strict morphisms amongst the weaker kinds. F-categories were introduced to express this interplay between strict and weak morphisms. We express doctrinal adjunction as an F-categorical lifting property and use this to give monadicity theorems, expressed using the language of F-categories, that cover each weaker kind of morphism.
v2: final journal version, some technical improvements to Section 4 which enabled the removal of unnecessary hypotheses (concerning cotensors with 2 and pullbacks) from the main results