#
Completion, closure, and density relative to a monad, with examples in
functional analysis and sheaf theory

##
Rory B. B. Lucyshyn-Wright

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.

Published 2014-12-19.

http://www.tac.mta.ca/tac/volumes/29/31/29-31.pdf

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