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.
Keywords: completion; closure; density; monad; idempotent monad; idempotent core; idempotent approximation; normed vector space; adjunction; reflective subcategory; enriched category; factorization system; orthogonal subcategory; sheaf; sheafification; Lawvere-Tierney topology; monoidal category; closed category
2010 MSC: 18A20, 18A22, 18A30, 18A32, 18A40, 18B25, 18C15, 18D15, 18D20, 18F10, 18F20, 46B04, 46B10, 46B28, 46M99
Theory and Applications of Categories, Vol. 29, 2014, No. 31, pp 896-928.