The Reflectiveness of Covering Morphisms in Algebra and Geometry

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 ηB : 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 Θ 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 ⊂ C/B whose objects are those f : A → B with f ∈ Θ. Many questions in mathematics can be reduced to asking whether Cov(B) is reflective in C ↓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.