Dear Colleagues, A preprint of my paper Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory is available at http://arxiv.org/abs/1406.2361 An abstract is included below. Your comments and questions are welcome. Regards, Rory Lucyshyn-Wright ________________________________________ Rory B. B. Lucyshyn-Wright NSERC Postdoctoral Fellow Department of Mathematics and Statistics University of Ottawa http://aix1.uottawa.ca/~rlucyshy/ ________________________________________ Abstract: 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]