Beyond Barr Exactness: Effective Descent Morphisms
George Janelidze, Manuela Sobral, Walter Tholen · Cambridge University Press eBooks · 2003
The general purpose of descent theory is to provide a unified treatment for various situations in algebra, geometry, and logic, where a problem on a certain base object B is first solved for an extension E of B ,and then for B itself by “descending” from E along a projection p: E → B . Of course, only “good” morphisms p will permit us to toss a “problem” back and forth between the two objects. More specifically, in Grothendieck's descent theory one asks which morphisms p allow for an algebraic description of structures over (the presumably “complicated” object) B in terms of structures over (the presumably “easier” object) E . The meaning of “structure” depends on the context, of course; for example, for a topological space X , a structureover X may be a sheaf over X (or a local homeomorphism with codomain X ), and for a ring R a structure over R may be an R -module. According to Grothendieck, the general setting for descent theory is given by a fibration Φ: D → C , so that the category of structures over the object B in C is given by the fibre Φ −1 ( B ), and descent theory then aims at a description of the fibre over B in terms of the fibre over E , with p given (see [8], [7]). In this chapter we restrict ourselves to considering the basic fibration of C (i.e., the codomain functor C 2 → C ), so the fibre of B is simply the comma category ( C ↓ B ); hence, here “structure over B ” simply means “morphism in C with codomain B”.