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University of Cambridge > Talks.cam > Category Theory Seminar > Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory

## Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theoryAdd to your list(s) Download to your calendar using vCal - Rory Lucyshyn-Wright. University of Cambridge
- Tuesday 20 January 2015, 14:15-15:30
- MR5, Centre for Mathematical Sciences.
If you have a question about this talk, please contact Dr Ignacio Lopez Franco. 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. This talk is part of the Category Theory Seminar series. ## This talk is included in these lists:- All CMS events
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