  Given a monad T on a suitable enriched category B equipped with a proper
factorization system (E,M), we define notions of T-completion, T-closure, and
T-density. We show that not only the familiar notions of completion, closure,
and density in normed vector spaces, but also the notions of sheafification,
closure, and density with respect to a Lawvere-Tierney topology, are instances
of the given abstract notions. The process of T-completion is equally the
enriched idempotent monad associated to T (which we call the idempotent core of
T), and we show that it exists as soon as every morphism in B factors as a
T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is
satisfied as soon as B has certain pullbacks as well as wide intersections of
M-embeddings. Hence the resulting theorem on the existence of the idempotent
core of an enriched monad entails Fakir's existence result in the non-enriched
case, as well as adjoint functor factorization results of Applegate-Tierney and
Day.
