How many adjunctions give rise to the same 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.