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".
It turns out that the general language for describing these coverings is that of a Grothendieck topology.
La conclusion pratique à laquelle je suis arrivé dès maintenant, c'est que chaque fois que en vertu de mes critères, une variété de modules (ou plutôt, un schéma de modules) pour la classification des variations (globales, ou infinitésimales) de certaines structures (variétés complètes non singulières, fibrés vectoriels, etc.)
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.
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 descent datum consists roughly of a covering of an object V of C by a family Vi, elements xi in the fiber over Vi, and morphisms fji between the restrictions of xi and xj to Vij=Vi×VVj satisfying the compatibility condition fki = fkjfji.
The descent datum is called effective if the elements xi are essentially the pullbacks of an element x with image V. A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images of objects of C) are groupoids.
locally of finite type over an algebraically closed field
a smooth and closed point with linearly reductive stabilizer group
A special case of this when X is a point gives the classifying stack BG of a smooth affine group scheme G:
If A is a quasi-coherent sheaf of algebras in an algebraic stack X over a scheme S, then there is a stack Spec(A) generalizing the construction of the spectrum Spec(A) of a commutative ring A.
An object of Spec(A) is given by an S-scheme T, an object x of X(T), and a morphism of sheaves of algebras from x*(A) to the coordinate ring O(T) of T. If A is a quasi-coherent sheaf of graded algebras in an algebraic stack X over a scheme S, then there is a stack Proj(A) generalizing the construction of the projective scheme Proj(A) of a graded ring A.
and can have wild behavior, such as being reducible stacks whose components are non-equal dimension.
component intersecting at one point, and the map sends the genus
Constructing weighted projective spaces involves taking the quotient variety of some
and the quotient of this action gives the weighted projective space
Taking the vanishing locus of a weighted polynomial in a line bundle
[citation needed] An example of a non-affine stack is given by the half-line with two stacky origins.
For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves.
This problem is notorious for having caused some errors in published papers and books.
[5]) This means that constructing the pullback of a quasicoherent sheaf under a morphism of stacks requires some extra effort.
More generally one can define the notion of an n-sheaf or n–1 stack, which is roughly a sort of sheaf taking values in n–1 categories.
A very similar and analogous extension is to develop the stack theory on non-discrete objects (i.e., a space is really a spectrum in algebraic topology).
By definition, it is a ringed ∞-topos that is étale-locally the étale spectrum of an E∞-ring (this notion subsumes that of a derived scheme, at least in characteristic zero.)