A new preprint "On localization and stabilization for factorization systems", by Carboni, Janelidze, Kelly, and Pare', is available in our ftp site at the address maths.su.oz.au (= 129.78.68.2), in the directory sydcat/papers/kelly, under the titles cjkp.dvi or cjkp.ps; there is also cjkp.tex, but that requires two macros - namely diagrams.tex and kluwer.sty. The paper contains new ideas, but also self-contained introductions to several areas with which some may be unfamiliar: namely factorization systems, descent theory, Galois theory, Eilenberg's monotone-light factorization for maps between compact hausdorff spaces, hereditary torsion theories for abelian categories, the category of finite families of objects of a given category, and the (separable, purely-inseparable) factorization for field extensions. What ways are there of constructing a factorization system (E, M) on a category C ? One simple one is to start with a full reflective subcategory X of C, and to take E to consist of the maps inverted by the reflexion. Of course, this (E, M) doesn't have E pullback-stable except in the special case where X is a LOCALIZATION of C. We examine another general process, which leads to an (E, M) with E stable when it succeeds. We start from ANY factorization system (E, M), and define new classes thus: a map lies in E' if EACH of its pullbacks lies in E; and it lies in M* if SOME pullback of it along an effective descent map lies in M. Note the connexion with Galois theory, in Janelidze's categorical formulation of it: if the (E, M) we begin with arises as above from a reflective full subcategory, the class M* consists of what Janelidze calls the COVERINGS (or, in some contexts, the CENTRAL EXTENSIONS). It is not always the case that (E', M*) is a factorization system; we give necessary and sufficient conditions for it to be so, and apply these to three major examples: Eilenberg's factorization above, certain factorizations connected to hereditary torsion theories, and a new factorization system for finite-dimensional algebras over a field that generalizes (separable, purely-inseparable) factorization for field extensions. Regards to all - Max Kelly. ps: For those without electronic access, a limited number of printed copies will shortly be available; please request them soon, so that we can alert the printer. Max Kelly.
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Max Kelly