Two dimensional monadicity

TL;DR

This paper explores the limits of weak morphisms in 2-dimensional universal algebra, introducing F-categories to better understand the interplay between strict and weak morphisms and establishing monadicity theorems.

Contribution

It introduces F-categories to analyze the behaviour of limits of weak morphisms and formulates monadicity theorems within this framework.

Findings

F-categories effectively model the interplay between strict and weak morphisms.

Doctrinal adjunction is characterized as an F-categorical lifting property.

Monadicity theorems are established for various kinds of morphisms.

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.