Stack (descent Theory)

Stack (descent Theory)

In mathematics a stack is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when finite moduli spaces do not exist.

Descent theory is concerned with generalisations of situations where geometrical objects (such as vector bundles on topological spaces) can be "glued together" when they are isomorphic (in a compatible way) when restricted to intersections of the sets in an open covering of a space. In more general set-up the restrictions are replaced with general pull-backs, and fibred categories form the right framework to discuss the possibility of such "glueing". The intuitive meaning of a stack is that it is a fibred category such that "all possible glueings work". The specification of glueings requires a definition of coverings with regard to which the glueings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain glueings with respect to the Grothendieck topology.

Stacks are the underlying structure of algebraic stacks, which are a way to generalise schemes and algebraic spaces and which are particularly useful in studying moduli spaces. The concept of stacks has its origin in the definition of effective descent data in Grothendieck (1959). The theory was further developed by Grothendieck and Giraud (1964) and Giraud (1971). The term "stack" was introduced by Deligne & Mumford (1969) for the original French term "champ" meaning "field" used by Giraud (1971).