The Reflectiveness of Covering Morphisms in Algebra And Geometry (1997)
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@MISC{Janelidze97thereflectiveness,
author = {G. Janelidze and G. M. Kelly and Transmitted Peter Johnstone},
title = {The Reflectiveness of Covering Morphisms in Algebra And Geometry},
year = {1997}
}
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Abstract
. 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 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...
