BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Completion\, closure\, and density relative to a monad\, with exam
 ples in functional analysis and sheaf theory - Rory Lucyshyn-Wright. Unive
 rsity of Cambridge
DTSTART:20150120T141500Z
DTEND:20150120T153000Z
UID:TALK57489@talks.cam.ac.uk
CONTACT:Dr Ignacio Lopez Franco
DESCRIPTION:Given a monad T on a suitable enriched category B equipped wit
 h a\nproper factorization system (E\,M)\, we define notions of T-completio
 n\,\nT-closure\, and T-density.  We show that not only the familiar notion
 s\nof completion\, closure\, and density in normed vector spaces\, but als
 o\nthe notions of sheafification\, closure\, and density with respect to a
 \nLawvere-Tierney topology\, are instances of the given abstract notions.\
 nThe process of T-completion is equally the enriched idempotent monad\nass
 ociated to T (which we call the idempotent core of T)\, and we show\nthat 
 it exists as soon as every morphism in B factors as a T-dense\nmorphism fo
 llowed by a T-closed M-embedding.  The latter hypothesis is\nsatisfied as 
 soon as B has certain pullbacks as well as wide\nintersections of M-embedd
 ings.  Hence the resulting theorem on the\nexistence of the idempotent cor
 e of an enriched monad entails Fakir's\nexistence result in the non-enrich
 ed case\, as well as adjoint functor\nfactorization results of Applegate-T
 ierney and Day.
LOCATION:MR5\, Centre for Mathematical Sciences
END:VEVENT
END:VCALENDAR