Codensity Lifting of Monads and its Dual

TL;DR

This paper introduces codensity lifting, a new method for lifting monads across fibrations, expanding applicability beyond previous methods, with examples across various categories and a dual approach for comonads.

Contribution

The paper presents codensity lifting as a novel technique for lifting monads along fibrations, extending previous categorical lifting methods and including a dual density lifting for comonads.

Findings

Applicable to fibrations from preorders, topological spaces, and pseudometric spaces

Characterization of liftings as limits of large diagrams

Extension of lifting techniques to algebraic operations

Abstract

We introduce a method to lift monads on the base category of a fibration to its total category. This method, which we call codensity lifting, is applicable to various fibrations which were not supported by its precursor, categorical TT-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended pseudometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We also introduce the dual method called density lifting of comonads. We next study the liftings of algebraic operations to the codensity liftings of monads. We also give a characterisation of the class of liftings of monads along posetal fibrations with fibred small meets as a limit of a certain large diagram.