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Functorial Factorization, Wellpointedness and Separability
"... A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and f ..."
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A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordantdissonant and inseparableseparable.
COMPLETION, CLOSURE, AND DENSITY RELATIVE TO A MONAD, WITH EXAMPLES IN FUNCTIONAL ANALYSIS AND SHEAF THEORY
"... Abstract. Given a monad T on a suitable enriched category B equipped with a proper factorization system (E,M), we define notions of Tcompletion, Tclosure, and Tdensity. We show that not only the familiar notions of completion, closure, and density in normed vector spaces, but also the notions of ..."
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Abstract. Given a monad T on a suitable enriched category B equipped with a proper factorization system (E,M), we define notions of Tcompletion, Tclosure, and Tdensity. 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 LawvereTierney topology, are instances of the given abstract notions. The process of Tcompletion 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 Tdense morphism followed by a Tclosed Membedding. The latter hypothesis is satisfied as soon as B has certain pullbacks as well as wide intersections of Membeddings. Hence the resulting theorem on the existence of the idempotent core of an enriched monad entails Fakir’s existence result in the nonenriched case, as well as adjoint functor factorization results of ApplegateTierney and Day. 1.
with examples in functional analysis
"... closure, and density relative to a monad, ..."
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