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 C2 → 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”.