## On localization and stabilization for factorization systems.(English)Zbl 0866.18003

For any factorization system ($${\mathcal E}, {\mathcal M}$$) on a category $${\mathcal C}$$, new classes of maps are defined as follows: $$f:A\rightarrow B$$ is in $${\mathcal E}'$$ if all of its pullbacks are in $${\mathcal E}$$ (that is, it is stably in $${\mathcal E}$$); $$f$$ is in $${\mathcal M}^*$$ if some pullback of it along an effective descent map lies in $$\mathcal M$$ (that is, it is locally in $$\mathcal M$$). In many interesting cases, ($${\mathcal E}', {\mathcal M}^*$$) is another factorization system; the unique fill-in property always holds, but not all maps need factor as $$me$$ with $$m$$ in $${\mathcal M}^*$$ and $$e$$ in $${\mathcal E}'$$. In particular, the monotone-light factorization in compact Hausdorff spaces, the factorization of a field extension as a purely inseparable extension of a separable extension; and various factorization systems associated with hereditary torsion theories on abelian categories arise in this way. “Our chief aims are to give a necessary and sufficient condition, in terms of the factorization system ($$\mathcal E, \mathcal M$$), for ($${\mathcal E}', {\mathcal M}^*$$) to be a factorization system, and to work out in detail” the indicated examples.
“At the same time we wish to point out the connection with Galois theory” after G. Janelidze, [in: Category Theory, Lect. Notes Math. 1488, 157-173 (1991; Zbl 0754.18002)]. Some of the Galois theories associated with these factorization systems are new, in particular the one corresponding to monotone-light factorization, which may be seen as a Galois theory for $$C^*$$-algebras. The authors have included much background material to make the paper largely self-contained.

### MSC:

 18A32 Factorization systems, substructures, quotient structures, congruences, amalgams 12F10 Separable extensions, Galois theory 18E40 Torsion theories, radicals

Zbl 0754.18002
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