On Localization and Stabilization for Factorization Systems

If (ε, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:A→B is in ε′ if each of its pullbacks lies in ε(that is, if it is stably in ε), and is in M* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (ε′, M*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.

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