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. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. We now have the abstract idea of a stack in terms of category theory. (ii) is stack if and only if it is a sheaf. (i) is a prestack if and only if it is a separated functor, 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: If the functor is an equivalence of categories, then the fibered category is a stack. If the functor is fully faithful, then the fibered category is a prestack. Now we give the definitions of prestacks and stacks using the functor we have defined earlier. If, in addition, it is also essentially surjective, then it is called an equivalence of categories. Ī functor which is both faithful and full is called fully faithful. Ī functor is essentially surjective if any object of is isomorphic to the image of some object in under. Ī functor is full if the induced map is surjective for any two objects and of. An object with descent data that is in the essential image of this functor is called effective.īefore we give the definitions of prestacks and stacks, we recall some definitions from category theory:Ī functor is faithful if the induced map is injective for any two objects and of. We can define a functor by assigning to each object of the object with descent data given by the pullback and the canonical isomorphism. Therefore we obtain a category, the category of objects with descent data, denoted. Ī morphism between two objects with descent data is a a collection of morphisms in such that. The morphisms and the are the projection morphisms. The notations and means that we are “pulling back” and from and, respectively, to. An object with descent data is a collection of objects in together with transition isomorphisms in, satisfying the cocycle condition Let be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object of. The theory of descent can be thought of as a formalization of the idea of “gluing”. 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. 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. 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). 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. Ī special kind of fibered category that we will need later on is a category fibered in groupoids. a functor we can consider it as a category fibered in sets. Īn important example of a fibered category is given by an ordinary presheaf on a category, i.e. Under the functor, the objects of which get sent to in and the morphisms of which get sent to the identity morphism in form a subcategory of called the fiber over. This means that any other morphism whose image under the functor factors as must also factor as under some unique morphism whose image under the functor is. If is a category over some other category, we say that it is a fibered category (over ) if for every object and morphism in, there is a strongly cartesian morphism in with. Given a category, we say that some other category is a category over if there is a functor from to (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View). We need first the concept of a fibered category (also spelled fibred category). 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 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.
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