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The reflectiveness of covering morphisms in algebra and geometry

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G. Janelidze and G. M. Kelly

Each full reflective subcategory X of a finitely-complete 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 \etaB : 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 \downarrow B \subset C/B whose objects are those f : A --> B
with f in \Theta. Many questions in mathematics can be reduced to asking
whether Cov(B) is reflective in C \downarrow B; and we give a number of
disparate conditions, each sufficient for this to be so. In this way we
recapture old results and establish new ones on the reflexion of local
homeomorphisms into coverings, on the Galois theory of commutative rings,
and on generalized central extensions of universal algebras.

Keywords: factorization system, refective subcategory, covering space, Galois theory, central extension.

1991 MSC: 08C05, 13B05, 18A32, 18A40, 18F20.

*Theory and Applications of Categories*, Vol. 3, 1997, No. 6, pp 132-159.

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