Stack (mathematics)

In mathematics a stack or 2-sheaf 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 fine moduli spaces do not exist.Grothendieck's letter to Serre, 1959 Nov 5. In mathematics a stack or 2-sheaf 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 fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be 'glued together' within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that 'all possible gluings work'. The specification of gluings requires a definition of coverings with regard to which the gluings 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 gluings with respect to the Grothendieck topology. Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks. Edidin (2003) and Fantechi (2001) give a brief introductory accounts of stacks, Gómez (2001), Olsson (2007) and Vistoli (2005) give more detailed introductions, and Laumon & Moret-Bailly (2000) describes the more advanced theory. The concept of stacks has its origin in the definition of effective descent data in Grothendieck (1959).In a 1959 letter to Serre, Grothendieck observed that a fundamental obstruction to constructing good moduli spaces is the existence of automorphisms. A major motivation for stacks is that if a moduli space for some problem does not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack. Mumford (1965) studied the Picard group of the moduli stack of elliptic curves, before stacks had been defined. Stacks were first defined by Giraud (1966, 1971), and the term 'stack' was introduced by Deligne & Mumford (1969) for the original French term 'champ' meaning 'field'. In this paper they also introduced Deligne–Mumford stacks, which they called algebraic stacks, though the term 'algebraic stack' now usually refers to the more general Artin stacks introduced by Artin (1974). When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Constructions of moduli spaces often proceed by first constructing a larger space parametrizing the objects in question, and then quotienting by group action to account for objects with automorphisms which have been overcounted. A category c {displaystyle c} with a functor to a category C {displaystyle C} is called a fibered category over C {displaystyle C} if for any morphism F : X → Y {displaystyle F:X o Y} in C {displaystyle C} and any object y {displaystyle y} of c {displaystyle c} with image Y {displaystyle Y} (under the functor), there is a pullback f : x → y {displaystyle f:x o y} of y {displaystyle y} by F {displaystyle F} . This means a morphism with image F {displaystyle F} such that any morphism g : z → y {displaystyle g:z o y} with image G = F ∘ H {displaystyle G=Fcirc H} can be factored as g = f ∘ h {displaystyle g=fcirc h} by a unique morphism h : z → x {displaystyle h:z o x} in c {displaystyle c} such that the functor maps h {displaystyle h} to H {displaystyle H} . The element x = F ∗ y {displaystyle x=F^{*}y} is called the pullback of y {displaystyle y} along F {displaystyle F} and is unique up to canonical isomorphism.