Algebraic Spaces and Stacks

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category \mathcal{C}, we say that some other category \mathcal{S} is a category over \mathcal{C} if there is a functor p from \mathcal{S} to \mathcal{C} (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If \mathcal{S} is a category over some other category \mathcal{C}, we say that it is a fibered category (over \mathcal{C}) if for every object U=p(x) and morphism f: V\rightarrow U in \mathcal{C}, there is a strongly cartesian morphism \phi: f^{*}x\rightarrow x in \mathcal{S} with f=p(\phi).

This means that any other morphism \psi: z\rightarrow x whose image p(\psi) under the functor p factors as p(\psi)=p(\phi)\circ h must also factor as \psi=\phi\circ \theta under some unique morphism \theta: z\rightarrow f^{*}x whose image under the functor p is h. We refer to f^{*}x as the pullback of x along f.

Under the functor p, the objects of \mathcal{S} which get sent to U in \mathcal{C} and the morphisms of \mathcal{S} which get sent to the identity morphism i_{U} in \mathcal{C} form a subcategory of \mathcal{S} called the fiber over U. We will also write it as \mathcal{S}_{U}.

An important example of a fibered category is given by an ordinary presheaf on a category \mathcal{C}, i.e. a functor F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set}); we can consider it as a category fibered in sets \mathcal{S}_{F}\rightarrow\mathcal{C}.

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let \mathcal{U}=\{f_{i}:U_{i}\rightarrow U\} be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object U of \mathcal{C}. An object with descent data is a collection of objects X_{i} in \mathcal{S}_{U} together with transition isomorphisms \varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j} in \mathcal{S}_{U_{i}\times_{U}U_{j}}, satisfying the cocycle condition

\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}

The morphisms \text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i} and the \text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j} are the projection morphisms. The notations \text{pr}_{0}^{*}X_{i} and \text{pr}_{1}^{*}X_{j} means that we are “pulling back” X_{i} and X_{j} from \mathcal{S}_{U_{i}} and \mathcal{S}_{U_{j}}, respectively, to \mathcal{S}_{U_{i}\times_{U}U_{j}}.

A morphism between two objects with descent data is a a collection of morphisms \psi_{i}:X_{i}\rightarrow X'_{i} in \mathcal{S}_{U_{i}} such that \varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}. Therefore we obtain a category, the category of objects with descent data, denoted \mathcal{DD}(\mathcal{U}).

We can define a functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) by assigning to each object X of \mathcal{S}_{U} the object with descent data given by the pullback f_{i}^{*}X and the canonical isomorphism \text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X. An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor F:\mathcal{A}\rightarrow\mathcal{B} is faithful if the induced map \text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y)) is injective for any two objects x and y of \mathcal{A}.

A functor F:\mathcal{A}\rightarrow\mathcal{B} is full if the induced map \text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y)) is surjective for any two objects x and y of \mathcal{A}.

A functor F:\mathcal{A}\rightarrow\mathcal{B} is essentially surjective if any object y of \mathcal{B} is isomorphic to the image F(x) of some object x in \mathcal{A} under F.

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) we have defined earlier.

If the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) is fully faithful, then the fibered category \mathcal{S}\rightarrow\mathcal{C} is a prestack.

If the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) is an equivalence of categories, then the fibered category \mathcal{S}\rightarrow\mathcal{C} is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) F is a prestack if and only if it is a separated functor,

(ii) F is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from \mathcal{C} to \textbf{Sets} which is naturally isomorphic to the functor which assigns to any object X the set of morphisms \text{Hom}(X,U), for some fixed object U of \mathcal{C}.

A representable presheaf is a contravariant functor from \mathcal{C} to \textbf{Sets} which is naturally isomorphic to the functor which assigns to any object X the set of morphisms \text{Hom}(U,X), for some fixed object U of \mathcal{C}. If \mathcal{C} is the category of schemes, the latter functor is also called the functor of points of the object U.

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points h_{U} of a scheme U may be identified with U itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets} be functors, and let a:F\rightarrow G and b:H\rightarrow G be natural transformations between these functors. Then the fiber product F\times_{a,G,b}H is a functor from \mathcal{C}^{\text{op}} to \textbf{Sets}, and is given by the formula

\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)

for any object X of \mathcal{C}.

Let F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets} be functors. We say that a natural transformation a:F\rightarrow G is representable, or that F is relatively representable over G if for every U\in\text{Ob}(\mathcal{C}) and any \xi\in G(U) the functor h_{U}\times_{G}F is representable.

We now let (\text{Sch}/S)_{\text{fppf}} be the site (a category with a Grothendieck topology –  see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of S-schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme S is a presheaf

\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}

with the following properties

(1) The presheaf F is a sheaf.

(2) The diagonal morphism F\rightarrow F\times F is representable.

(3) There exists a scheme U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and a map h_{U}\rightarrow F which is surjective, and etale (This is often written simply as U\rightarrow F). The scheme U is also called an atlas.

The diagonal morphism being representable implies that the natural transformation h_{U}\rightarrow F is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have U is the disjoint union of affine schemes U_{i} and where the map h_{U}\rightarrow F is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let \mathcal{C} be a category. A category fibered in groupoids p:\mathcal{S}\rightarrow\mathcal{C} is called representable if there exists an object X of \mathcal{C} and an equivalence j:\mathcal{S}\rightarrow \mathcal{C}/X (The notation \mathcal{C}/X signifies a slice category, whose objects are morphisms f:U\rightarrow X in \mathcal{C}, and whose morphisms are morphisms h:U\rightarrow V in \mathcal{C} such that f=g\circ h, where g:U\rightarrow X).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let \mathcal{X} be a category fibered in groupoids over (\text{Sch}/S)_{\text{fppf}}. Then \mathcal{X} is representable by a scheme if there exists a scheme U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and an equivalence j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}} of categories over (\text{Sch}/S)_{\text{fppf}}.

A category fibered in groupoids p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}} is representable by an algebraic space over S if there exists an algebraic space F over S and an equivalence j:\mathcal{X}\rightarrow \mathcal{S}_{F} of categories over (\text{Sch}/S)_{\text{fppf}}.

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let F:\mathcal{X}\rightarrow\mathcal{S} and G:\mathcal{Y}\rightarrow\mathcal{S} be morphisms of categories over \mathcal{C}. The fiber product \mathcal{X}\times_{\mathcal{S}}\mathcal{Y} is given by the following description:

(1) an object of \mathcal{X}\times_{\mathcal{S}}\mathcal{Y} is a quadruple (U,x,y,f), where U\in\text{Ob}(\mathcal{C}), x\in\text{Ob}(\mathcal{X}_{U}), y\in\text{Ob}(\mathcal{Y}_{U}), and f : F(x)\rightarrow G(y) is an isomorphism in \mathcal{S}_{U},

(2) a morphism (U,x,y,f) \rightarrow (U',x',y',f') is given by a pair (a,b), where a:x\rightarrow x' is a morphism in X, and b:y\rightarrow y' is a morphism in Y such that a and b induce the same morphism U\rightarrow U', and f'\circ F(a)=G(b)\circ f.

Let S be a scheme. A morphism f:\mathcal{X}\rightarrow \mathcal{Y} of categories fibered in groupoids over (\text{Sch}/S)_{\text{fppf}} is called representable by algebraic spaces if for any U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and any y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y} the category fibered in groupoids

\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}

over (\text{Sch}/U)_{\text{fppf}} is representable by an algebraic space over U.

An algebraic stack (or Artin stack) over a scheme S is a category

\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}

with the following properties:

(1) The category \mathcal{X} is a stack in groupoids over (\text{Sch}/S)_{\text{fppf}} .

(2) The diagonal \Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X} is representable by algebraic spaces.

(3) There exists a scheme U\in\text{Ob}((\text{Sch/S})_{\text{fppf}}) and a morphism (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} which is surjective and smooth (This is often written simply as U\rightarrow\mathcal{X}). Again, the scheme U is called an atlas.

If the morphism (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by \mathcal{M}_{1,1} – the reason for this notation will become clear later). A family of elliptic curves over some “base space” B is a fibration \pi:X\rightarrow B with a section O:B\rightarrow X such that the fiber \pi^{-1}(b) over any point b of B is an elliptic curve with origin O(b).

Ideally what we want is to be able to obtain every family X\rightarrow B by pulling back a “universal object” E\rightarrow\mathcal{M}_{1,1} via the map B\rightarrow\mathcal{M}_{1,1}. This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by M_{1,1}). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus g with \nu marked points, denoted \mathcal{M}_{g,\nu}. The moduli stack of elliptic curves is simply the special case \mathcal{M}_{1,1}. Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.

References:

Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

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One thought on “Algebraic Spaces and Stacks

  1. Pingback: Adjoint Functors and Monads | Theories and Theorems

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