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Functorial Factorization, Wellpointedness and Separability
"... A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and f ..."
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A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordantdissonant and inseparableseparable.
The Reflectiveness of Covering Morphisms in Algebra And Geometry
, 1997
"... Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies ..."
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Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass  or rather a subcategory  of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...
Cofibrantly generated natural weak factorisation systems
, 2007
"... There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with r ..."
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There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C]. In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.’s. We then construct them by a method which is reminiscent of Quillen’s small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of mapswithstructure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on. 1
Preprint Number 15–11 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS
"... Abstract: For a particular class of Galois structures, we prove that the normal extensions are precisely those extensions that are “locally ” split epic and trivial, and we use this to prove a “Galois theorem ” for normal extensions. Furthermore, we interpret the normalisation functor as a Kan exten ..."
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Abstract: For a particular class of Galois structures, we prove that the normal extensions are precisely those extensions that are “locally ” split epic and trivial, and we use this to prove a “Galois theorem ” for normal extensions. Furthermore, we interpret the normalisation functor as a Kan extension of the trivialisation functor.