How many adjunctions give rise to the same monad?

TL;DR

This paper investigates the relationship between adjunctions and monads, providing methods to classify all adjunctions that produce a given monad, with explicit computations for free modules over Dedekind domains.

Contribution

It introduces a class of well-behaved adjunctions and develops techniques to compute all adjunctions corresponding to a monad, including explicit examples and a uniqueness criterion.

Findings

Developed methods to classify adjunctions for a given monad

Explicitly computed homological presentations for free modules over Dedekind domains

Proved a criterion for uniqueness of adjunctions producing a monad

Abstract

Given an adjoint pair of functors $F,G$, the composite $GF$ naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free $A$-module monad on the category of sets, for $A$ a Dedekind domain. We also prove a criterion, reminiscent of Beck's monadicity theorem, for when there is essentially (in a precise sense) only a single adjunction that gives rise to a given monad.