Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completion, closure, and density relative to a monad, with examples in
  functional analysis and sheaf theory   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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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: stat