Preprint: Completion, closure, and density relative to a monad

Preprint: Completion, closure, and density relative to a monad

[Rory Lucyshyn-Wright]

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.




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