If (e, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:A→B is in e′ if each of its pullbacks lies in e(that is, if it is stably in e), 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 (e′, 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.

On Localization and Stabilization for Factorization Systems

We show, for a monoidal closed category V =( V0, ⊗ ,I ), that the category V -Cat of small V -categories is locally λ-presentable if V0 is so, and that it is locally λ-bounded if the closed category V is so, meaning that V0 is locally λ-bounded and that a side condition involving the monoidal structure is satisfied. Many important properties of a monoidal category V are inherited by the category V -Cat of small V -categories. For instance, if V is symmetric monoidal, V -Cat has a canonical symmetric monoidal structure, as was observed already in (4). Much later (7, Remark 5.2), it was realized that if V is only braided monoidal then V -Cat still has a canonical monoidal structure, although it need not have a braiding unless the braiding on V is in fact a symmetry. Similarly, it is straightforward to show that V -Cat is monoidal closed when V is closed and complete, and that V -Cat is complete when V is so. All of these results are essentially routine; the less trivial fact that V -Cat is cocomplete when V is so was first proved in (11).

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V -cat is locally presentable or locally bounded if v is so

Torsion theories in homological categories

Torsion theories and radicals in normal categories

An algebraic approach to subframe logics. Intuitionistic case

This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category.

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Reflective subcategories, localizations and factorizationa systems

Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hebert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.

C. Cassidy

Papers

Reflective subcategories, localizations and factorizationa systems

Reflective subcategories, localizations and factorization systems: Corrigenda