Stack (descent theory)
Encyclopedia
In mathematics
a stack is a concept used to formalise some of the main constructions of descent theory.
Descent theory is concerned with generalisations of situations where geometrical objects (such as vector bundle
s on topological space
s) 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 space
s and which are particularly useful in studying moduli space
s. 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 in the latter work.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a stack is a concept used to formalise some of the main constructions of descent theory.
Descent theory is concerned with generalisations of situations where geometrical objects (such as vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s on topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s) can be "glued together" when they are isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
(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
Fibred category
Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined...
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
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
- 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
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
and algebraic space
Algebraic space
In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory...
s and which are particularly useful in studying moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s. 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 in the latter work.
External links
- Stacks and descent on nLabNLabnLab is a wiki-lab of a novel kind, for collaborative work on mathematics, physics, and philosophy, including original research, with a focus on category theory. The nLab espouses the n-point of view : that category theory and higher category theory provide a useful unifying viewpoint for...
.