In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, compatible geometrical objects (such as
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s on
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s) 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.
Overview
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 space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s. There are inclusions:
schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks.
and give a brief introductory accounts of stacks, , and give more detailed introductions, and describes the more advanced theory.
Motivation and history
The concept of stacks has its origin in the definition of effective descent data in .
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''.
studied the Picard group of the
moduli stack of elliptic curves In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In part ...
, before stacks had been defined. Stacks were first defined by , and the term "stack" was introduced by 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 .
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
In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism \pi: X \to Y that
:(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the g ...
will not exist among schemes, but it will exist as a stack.
In the same way,
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s 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.
Definitions
Abstract stacks
A category
with a functor to a category
is called a
fibered category over
if for any morphism
in
and any object
of
with image
(under the functor), there is a pullback
of
by
. This means a morphism with image
such that any morphism
with image
can be factored as
by a unique morphism
in
such that the functor maps
to
. The element
is called the pullback of
along
and is unique up to canonical isomorphism.
The category ''c'' is called a
prestack over a category ''C'' with a
Grothendieck topology if it is fibered over ''C'' and for any object ''U'' of ''C'' and objects ''x'', ''y'' of ''c'' with image ''U'', the functor from the over category C/U to sets taking ''F'':''V''→''U'' to Hom(''F''*''x'',''F''*''y'') is a sheaf. This terminology is not consistent with the terminology for sheaves: prestacks are the analogues of separated presheaves rather than presheaves. Some authors require this as a property of stacks, rather than of prestacks.
The category ''c'' is called a stack over the category ''C'' with a Grothendieck topology if it is a prestack over ''C'' and every descent datum is effective. A descent datum consists roughly of a covering of an object ''V'' of ''C'' by a family ''V
i'', elements ''x
i'' in the fiber over ''V
i'', and morphisms ''f
ji'' between the restrictions of ''x
i'' and ''x
j'' to ''V
ij''=''V
i''×
''V''''V
j'' satisfying the compatibility condition ''f
ki'' = ''f
kjf
ji''. The descent datum is called effective if the elements ''x
i'' 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. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.
Algebraic stacks
An algebraic stack or Artin stack is a stack in groupoids ''X'' over the fppf site such that the diagonal map of ''X'' is representable and there exists a smooth surjection from (the stack associated to) a scheme to X.
A morphism ''Y''
''X'' of stacks is representable if, for every morphism ''S''
''X'' from (the stack associated to) a scheme to X, the
fiber product ''Y'' ×
''X'' ''S'' is isomorphic to (the stack associated to) an
algebraic space. The fiber product of stacks is defined using the usual
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
, and changing the requirement that diagrams commute to the requirement that they
2-commute. See also
morphism of algebraic stacks for further information.
The motivation behind the representability of the diagonal is the following: the diagonal morphism
is representable if and only if for any pair of morphisms of algebraic spaces
, their fiber product
is representable.
A
Deligne–Mumford stack is an algebraic stack ''X'' such that there is an étale surjection from a scheme to ''X''. Roughly speaking, Deligne–Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms.
Local structure of algebraic stacks
Since the inception of algebraic stacks it was expected that they are locally quotient stacks of the form