Stack (descent theory)
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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.
Archetypical examples include the stack of vector bundles on topological spaces, the stack of quasi-coherent sheaves on schemes (with respect to the fpqc-topology and weaker topologies) and the stack of affine schemes on a base scheme (again with respect to the fpqc topology or a weaker one).
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 name stack (champ in the original French) together with the eventual definition appears to have been introduced by Deligne & Mumford (1969).
- Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS (36): 75–109, ISSN 1618-1913, MR0262240
- Giraud, Jean (1964), "Méthode de la descente", Société Mathématique de France. Bulletin. Supplément. Mémoire 2: viii+150, MR0190142
- Giraud, Jean (1971). Cohomologie non abélienne. Springer. ISBN 3-540-05307-7
- Grothendieck, Alexander (1959). "Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats". Séminaire Bourbaki 5 (Exposé 190): viii+150.