A Sufficient Condition for Liftable Adjunctions between Eilenberg-Moore Categories

Abstract

This paper gives a sufficient condition for monads P, P′ and T to have an adjunction between the category of P-algebras over T-algebras and the category of P′-algebras over T-algebras. The leading example is an adjunction between the category of idempotent semirings and the category of quantales, where P is the finite powerset monad, P′ is the powerset monad, and T is the free monoid monad. The left adjoint of this leading example is given by ideal completion. Applying our result, we show that ideal completion also gives an adjunction between the category of join semilattices over T-algebras and the category of complete join semilattices over T-algebras for a general monad T satisfying certain distributive law.