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 $\mathbb V$-functor $j \colon A \to E$, a $\mathbb V$-functor $r \colon D \to E$ 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 $\mathbb V$-functor with a ($j$-relatively) monadic $\mathbb V$-functor.