×

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

Citations:

Zbl 0754.18002
PDF BibTeX XML Cite
Full Text: DOI