Relative monadicity

Abstract

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense

-functor $j:A\to E$, a

-functor $r:D\to E$ is j-monadic if and only if r admits a left j-relative adjoint and creates j-absolute colimits. This provides a refinement of the classical monadicity theorem – characterising those categories whose objects are given by those of E equipped with algebraic structure – in which the arities of the algebraic operations are valued in A. In particular, when $j=1$, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the (j-relative) monadicity of the composite of a

-functor with a (j-relatively) monadic

-functor.