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.
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.
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 spaces. 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, 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 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.
Definitions.
Abstract stacks.
A category formula_1 with a functor to a category formula_2 is called a fibered category over formula_2 if for any morphism formula_4 in formula_2 and any object formula_6 of formula_1 with image formula_8 (under the functor), there is a pullback formula_9 of formula_6 by formula_11. This means a morphism with image formula_11 such that any morphism formula_13 with image formula_14 can be factored as formula_15 by a unique morphism formula_16 in formula_1 such that the functor maps formula_18 to formula_19. The element formula_20 is called the pullback of formula_6 along formula_11 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 "Vi", elements "xi" in the fiber over "Vi", and morphisms "fji" between the restrictions of "xi" and "xj" to "Vij"="Vi"×"V""Vj" 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. 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"formula_23 "X" of stacks is representable if, for every morphism "S" formula_23 "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, 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 formula_25 is representable if and only if for any pair of morphisms of algebraic spaces formula_26, their fiber product formula_27 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 formula_28 where formula_29 is a linearly reductive algebraic group. This was recently proved to be the case: given a quasi-separated algebraic stack formula_30 locally of finite type over an algebraically closed field formula_31 whose stabilizers are affine, and formula_32 a smooth and closed point with linearly reductive stabilizer group formula_33, there exists an etale cover of the GIT quotient formula_34, where formula_35, such that the diagramformula_36is cartesian, and there exists an etale morphismformula_37inducing an isomorphism of the stabilizer groups at formula_38 and formula_39.
Examples.
Constructions with stacks.
Stack quotients.
If formula_49 is a scheme formula_54 and formula_29 is a smooth affine group scheme acting on formula_49, then there is a quotient algebraic stack formula_68, taking a scheme formula_69 to the groupoid of formula_29-torsors over the formula_71-scheme formula_8 with formula_29-equivariant maps to formula_49. Explicitly, given a space formula_49 with a formula_29-action, form the stack formula_68 which (intuitively speaking) sends a space formula_8 to the groupoid of pullback diagramsformula_79where formula_80 is a formula_29-equivariant morphism of spaces and formula_82is a principal formula_29-bundle. The morphisms in this category are just morphisms of diagrams where the arrows on the right-hand side are equal and the arrows on the left-hand side are morphisms of principal formula_29-bundles.
Classifying stacks.
A special case of this when "X" is a point gives the classifying stack "BG" of a smooth affine group scheme "G": formula_85 It is named so since the category formula_86, the fiber over "Y", is precisely the category formula_87 of principal formula_29-bundles over formula_8. Note that formula_87 itself can be considered as a stack, the moduli stack of principal "G"-bundles on "Y".
An important subexample from this construction is formula_91 which is the moduli stack of principal formula_92-bundles. Since the data of a principal formula_92-bundle is equivalent to the data of a rank formula_94 vector bundle, this is isomorphic to the moduli stack of rank formula_94 vector bundles formula_96.
Moduli stack of line bundles.
The moduli stack of line bundles is formula_97 since every line bundle is canonically isomorphic to a principal formula_98-bundle. Given a line bundle formula_99 the relative specformula_100gives a geometric line bundle. After remove the zero section, there is an associated formula_98-bundle. Conversely, from the representation formula_102, the associated line bundle can be reconstructed.
Gerbes.
A gerbe is a stack in groupoids which always has a nonempty category. for example the trivial gerbe formula_103 that assigns to each scheme the groupoid of principal formula_29-bundles over the scheme, for some group formula_29.
Relative spec and proj.
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".
Moduli stacks.
Kontsevich moduli spaces.
Another widely studied class of moduli spaces are the Kontsevich moduli spaces parameterizing the space of stable maps between curves of a fixed genus to a fixed space formula_49 whose image represents a fixed cohomology class. These moduli spaces are denotedformula_122and can have wild behavior, such as being reducible stacks whose components are non-equal dimension. For example, the moduli stack formula_123has smooth curves parametrized by an open subset formula_124. On the boundary of the moduli space, where curves may degenerate to reducible curves, there is a substack parametrizing reducible curves with a genus formula_125 component and a genus formula_126 component intersecting at one point, and the map sends the genus formula_126 curve to a point. Since all such genus formula_126 curves are parametrized by formula_129, and there is an additional formula_126 dimensional choice of where these curves intersect on the genus formula_126 curve, the boundary component has dimension formula_132.
Geometric stacks.
Weighted projective stacks.
Constructing weighted projective spaces involves taking the quotient variety of some formula_133 by a formula_98-action. In particular, the action sends a tupleformula_135and the quotient of this action gives the weighted projective space formula_136. Since this can instead be taken as a stack quotient, the weighted projective stack pg 30 isformula_137Taking the vanishing locus of a weighted polynomial in a line bundle formula_138 gives a stacky weighted projective variety.
Stacky curves.
Stacky curves, or orbicurves, can be constructed by taking the stack quotient of a morphism of curves by the monodromy group of the cover over the generic points. For example, take a projective morphismformula_139which is generically etale. The stack quotient of the domain by formula_140 gives a stacky formula_141 with stacky points that have stabilizer group formula_142 at the fifth roots of unity in the formula_143-chart. This is because these are the points where the cover ramifies.
Non-affine stack.
An example of a non-affine stack is given by the half-line with two stacky origins. This can be constructed as the colimit of two inclusion of formula_144.
Quasi-coherent sheaves on algebraic stacks.
On an algebraic stack one can construct a category of quasi-coherent sheaves similar to the category of quasi-coherent sheaves over a scheme.
A quasi-coherent sheaf is roughly one that looks locally like the sheaf of a module over a ring. The first problem is to decide what one means by "locally": this involves the choice of a Grothendieck topology, and there are many possible choices for this, all of which have some problems and none of which seem completely satisfactory. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. 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.
Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology (short for Lisse-Etale: lisse is the French term for smooth), which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. (The problem is that while one can construct a pair of adjoint functors "f"*, "f"*, as needed for a geometric morphism of topoi, the functor "f"* is not left exact in general. This problem is notorious for having caused some errors in published papers and books.) This means that constructing the pullback of a quasicoherent sheaf under a morphism of stacks requires some extra effort.
It is also possible to use finer topologies. Most reasonable "sufficiently large" Grothendieck topologies seem to lead to equivalent categories of quasi-coherent sheaves, but the larger a topology is the harder it is to handle, so one generally prefers to use smaller topologies as long as they have enough open sets. For example, the big fppf topology leads to essentially the same category of quasi-coherent sheaves as the Lis-Et topology, but has a subtle problem: the natural embedding of quasi-coherent sheaves into O"X" modules in this topology is not exact (it does not preserve kernels in general).
Other types of stack.
Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces.
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. There are several inequivalent ways of doing this. 1-sheaves are the same as sheaves, and 2-sheaves are the same as stacks. They are called higher stacks.
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). The resulting stacky objects are called derived stacks (or spectral stacks). Jacob Lurie's under-construction book "Spectral Algebraic Geometry" studies a generalization which he calls a spectral Deligne–Mumford stack. 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.)
Set-theoretical problems.
There are some minor set theoretical problems with the usual foundation of the theory of stacks, because stacks are often defined as certain functors to the category of sets and are therefore not sets. There are several ways to deal with this problem: