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Factorizations, localizations, and the orthogonal subcategory problem. (English) Zbl 0553.18003
For sources in an abstract category, factorization structures (E,M) that are called locally orthogonal and that are slightly more general than the ones usually considered are defined and investigated. This allows for the inclusion of important classical examples where the class of morphisms E, is not necessarily closed under composition. In doing so, many useful factorization structures (such as (monotone, light)) from the realm of topology are abstracted to non-topological categories. General existence criteria for these factorization structures are given, and sufficient conditions for the existence of reflective hulls and for a solution of the orthogonal subcategory problem are derived from the existence of certain factorization structures. As an example of one of the many elegant results in the paper, it is shown that for any complete, cocomplete extremally well-powered category \({\mathcal K}\), every small, full subcategory of \({\mathcal K}\) has a reflective hull in \({\mathcal K}\).
Reviewer: G.E.Strecker

MSC:
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
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