November 5, 2014
Grothendieck topologies Grothendieck topologies are an extra datum on a category. They allow us to make sense of something being locally defined. Give a formal framework for glueing problems (sheaves and stacks).
Grothendieck topologies Let s consider a topological space X. Denote by Open(X ) the category, given by open subsets U X. Morphisms are inclusions U V.
Grothendieck topologies Let s consider a topological space X. Denote by Open(X ) the category, given by open subsets U X. Morphisms are inclusions U V. Definition A (set-valued) presheaf on X is a functor F : Open(X ) op Set.
Grothendieck topologies Let s consider a topological space X. Denote by Open(X ) the category, given by open subsets U X. Morphisms are inclusions U V. Definition A (set-valued) presheaf on X is a functor F : Open(X ) op Set. To an open subset U X we associate a set F (U), as well as a restriction map ru V : F (V ) F (U) for every inclusion U V. The conditions (a) ru U = id F (U), (b) ru V r V W = r U W for triples of open subsets U V W, are satisfied.
Grothendieck topologies For a topological space Y we have a presheaf Y X U X is sent to set of functions U Y, i.e., Y X (U) = Hom Top (U, Y ). The restriction maps r V U are given by f f U,
Grothendieck topologies For a topological space Y we have a presheaf Y X U X is sent to set of functions U Y, i.e., Y X (U) = Hom Top (U, Y ). The restriction maps r V U are given by f f U,
Grothendieck topologies For a topological space Y we have a presheaf Y X U X is sent to set of functions U Y, i.e., Y X (U) = Hom Top (U, Y ). The restriction maps r V U are given by f f U,
Grothendieck topologies Let U = i I U i be an open covering, for every pair of open subsets U i, U j : U i U i U j U j. Hence, for every presheaf F we have a pair of restriction maps F (U i ) F (U i U j ) F (U j ). Taking a product over all pairs (i, j) I 2, we obtain i I F (U i) (i,j) I 2 F (U i U j ).
Grothendieck topologies Let U = i I U i be an open covering, for every pair of open subsets U i, U j : U i U i U j U j. Hence, for every presheaf F we have a pair of restriction maps F (U i ) F (U i U j ) F (U j ). Taking a product over all pairs (i, j) I 2, we obtain i I F (U i) (i,j) I 2 F (U i U j ). Definition A presheaf F is called a sheaf, if for every open subset U X, and every open covering U = i I U i, we have that F (U) i I is an equalizer diagram. F (U i ) (i,j) I 2 F (U i U j )
Grothendieck topologies What does this equalizer-business mean concretely? A presheaf is a sheaf, if and only if for every collection of local sections s i F (U i ), which satisfy r U i U i U j (s i ) = r U j U i U j (s j ), there exists a unique section s F (U), such that r U U i (s) = s i.
Grothendieck topologies Lemma The presheaf Y X is a sheaf. Concrete proof. If f i : U i Y are continuous functions, such that f i Ui U j = f j Ui U j then there is a well-defined map of sets f : U Y, sending x U to f i (x), if x U i. Since continuity is a local property, we see that f is a continuous function.
Grothendieck topologies Lemma The presheaf Y X is a sheaf. Abstract proof. We can represent U as a co-equalizer (i,j) I 2 U i U j i I U i U, i.e., as a colimit in the category Top of topological spaces. The universal property of colimits implies that Hom Top (, Y ) sends a co-equalizer to an equalizer.
Grothendieck topologies In the definition of presheaves we could replace Open X by an arbitrary category C. The sheaf Y X, extends by definition to a functor Top ophom Top(,Y ) Set Open(X ) op. Y X The only reason to prefer the category Open(X ) over C is that we have a notion of open coverings in Open(X ). This is essential to introduce sheaves. Grothendieck topologies provide a remedy for general categories.
Grothendieck topologies Definition Let C be a category. A Grothendieck topology T on C consists of a collection of sets of morphisms (called coverings) {U i U} i I for each object U C, satisfying: (a) For every isomorphism U U, the singleton {U U} is a covering. (b) Coverings are preserved by base change, i.e. if {U i U} i I is a covering, and V U a morphism in C, then {U i U V V } i I is well-defined, and a covering. (c) Given a covering {U i U}, and for each i I a covering {U ij U i } j Ji, then is a covering. {U ij U} (i,j) i I J i
Grothendieck topologies A pair (C, T ) is called a site. Grothendieck topologies were called pre-topologies. We will drop the prefix. What should be thought of as the actual topological content of a Grothendieck topology is the category of sheaves (the topos). Different Grothendieck topologies can induce equivalent topoi (hence they used to be called pre-topologies).
Grothendieck topologies For the category Open(X ), we define T (X ) to be the collection of all {U i U}, such that i I U i = U. The Grothendieck topology T (X ) can be extended to T on f Top. We say that {U i i U}i I T, if i I f (U i) = U, and each f i is an open map, which is a homeomorphism onto its image. Let Aff denote the category Rng op. Recall, that we denote the object in Aff, corresponding to the ring R by Spec R. Consider the coverings {Spec S Spec R}, where R S is a faithfully flat map of rings.
Grothendieck topologies If (C, T ) is a site, we can make sense of sheaves on C. We define a presheaf on C to be a functor C op Set. We will use the abstract coverings provided by T to make sense of the sheaf condition. We have to make sense of the intersection U i U j.
Grothendieck topologies Definition Let C be a category, and f : X Z, g : Y Z two morphisms. Consider the category of diagrams W Y X Z. If it exists, we denote the top left corner (W ) of the final object in this category by X Z Y, and call it the fibre product of the two morphisms f and g.
Grothendieck topologies For a topological space X, and inclusions of open subsets U X, V X, we have that the fibre product U X V in the category Open(X ), respectively Top, is the inclusion of the open subset U V X. Hence we replace every occurrence of intersections, in the definition of a sheaf, by fibre products.
Grothendieck topologies Definition Let C be a category. A functor F : C op Set is called a presheaf. The category of presheaves will be denoted by Pr(C). If (C, T ) is a site, a presheaf is called a sheaf, if for every {U i U} i I the diagram F (U) F (U i ) F (U i U U j ) i I (i,j) I 2 is an equalizer. We denote the full subcategory of sheaves by Sh T (C).
Grothendieck topologies The heuristics behind this definition is the same as for topological spaces. Given an abstract covering {U i U} i I of U C, and locally defined sections s i F (U i ), which agree when restricted (or pulled back) to the intersections U i U U j, then there exists a unique section s F (U), which agrees with s i over each U i.
Grothendieck topologies Sheaves for a general site seem more abstract, because maps in C are not inclusions of open subsets. We will see in the following three examples that glueing is still reasonable, even without an obvious topological interpretation of open coverings.
Grothendieck topologies Let C = Set be the category of sets. We consider the Grothendieck topology T, f which consists of all collections {U i i U}, such that i I f (U i) = U. For a set X we have the presheaf h X = Hom Set (, X ), represented by X. Then, h X is a sheaf with respect to the topology T.
Grothendieck topologies For C the category of open subsets U R n (where n is allowed to vary), and smooth maps as morphisms, f define T to consist of all sets {U i i U}i I, where each f i is a smooth submersion, and i I f (U i) = U. Every smooth manifold X gives rise to a sheaf on C, by sending U C to the set of smooth maps U X. The resulting functor Mfd Sh T (C) is an embedding of categories (i.e. fully faithful).
Recall from last week Theorem We have natural maps e 1 : T T S T, and e 2 : T T S T. The diagram of sets Hom Rng (R, S) Hom Rng (R, T ) Hom Rng (R, T S T ) is an equalizer diagram in the category of sets. I.e., the set of ring homomorphism g : R T, satisfying e 1 g = e 2 g, is in bijection with the set of ring homomorphisms f : R S.
Grothendieck topologies We choose C to be Aff = Rng op, with the topology induced by {Spec S Spec R}, with R S being a faithfully flat map of rings. For every Spec T Aff, we have the presheaf h Spec T = Hom Aff (, Spec T ) It is a sheaf by faithfully flat descent theory.
Stacks Definition A category in which every arrow is invertible is called a groupoid. Every set can be viewed as a category, with every morphism being the identity morphism of an object. Sets give rise to examples of groupoids. Visualise a groupoid as a generalised set, where every element has a possibly non-trivial group of automorphisms.
Stacks Let G be a group acting on a set X. We denote by [X /G] the so-called quotient groupoid. Its set of objects is X. A morphism x y is given by an element g G, such that g x = y. For every x X, we have Aut [X /G] (x) = G x, i.e. the stabiliser subgroup of x X.
Stacks For every topological space X we have a groupoid π 1 (X ), whose objects are given by the points x X, and morphisms are homotopy classes of paths x y. We have Aut π 1 (X )(x) = π 1 (X, x), by definition. One can show that every groupoid arises as π 1 (X ) for a topological space X. Every groupoid arises in this way. We may picture groupoids as topological spaces with vanishing higher homotopy groups.
Stacks Definition A (strict) 2-category C consists of the following data: a class of objects Obj(C), for every X, Y Obj(C) a category Hom C (X, Y ) of morphisms, for X, Y, Z Obj(C) a functor : Hom C (X, Y ) Hom C (Y, Z) Hom C (X, Z), for every X Obj(C) an object id X Hom C (X, X ), satisfying id Y f = f id X = f, for every f Hom C (X, Y ), such that associativity holds, i.e. for X, Y, Z, W Obj(C) we want the two natural functors Hom C (X, Y ) Hom C (Y, Z) Hom C (Z, W ) Hom C (X, W ) to agree.
Stacks 2-categories generalise categories, in the sense that we additionally have 2-morphisms between 1-morphisms. We denote by Cat the 2-category of categories. It s objects are categories Morphisms are given by functors C D. We denote by [F, G] the set of natural transformations between two functors F, G : C D. Recall that a natural transformation consists of a morphism η X : F (X ) G(X ) for every object X C, such that for every arrow α: X Y the diagram F (X ) F (α) F (Y ) η X η Y G(X ) G(α) G(Y ) commutes. The full 2-subcategory of groupoids will be denoted by Gpd.
Stacks 2-categories generalise categories, in the sense that we additionally have 2-morphisms between 1-morphisms. We also have two kinds of composition ( horizontal, and vertical ): X X Y Z Y
Stacks In order to define prestacks and stacks, we need to define functors between 2-categories. The naive definition, is too strict to be a good definition by today s understanding of category theory. However, it gets the job done (since every functor can be strictified), and we will therefore stick to it for now.
Stacks Definition Let C and D be 2-categories. A strict functor F : C D is given by a map between objects Obj(C) Obj(D), as well as a functor Hom C (X, Y ) Hom C (F (X ), F (Y )) for every pair of objects X, Y, which is compatible with composition. For now, we will not discuss the more general notion of a functor F : C D (often called weak, or pseudo, or 2-functor). We leave it at the remark that a functor is not required to respect composition strictly, but only up to a specified 2-isomorphism. Specifying the coherence with the remaining structures of a 2-category leads to a very long, but flexible definition. However, every functor can be strictified. Hence, strict functors suffice for our purposes.
Stacks The definition of sheaves also rested on the notion of equalizers, i.e., limits. We have to discuss limits in the 2-categorical framework of groupoids. Definition Let I be a category, and F : I Gpd a functor, which sends i I to the groupoid C i. The limit lim i I C i is defined to be the following groupoid: its objects are collections X i C i, and for every morphism α: i j in I an isomorphism φ α : F (α)(x i ) X j, such that for two composable arrows α: i j, and β : j k, we have φ β α = φ β φ α. Limits in 2-categories are often referred to as 2-limits. As one would expect, limits are characterised by a universal property (we ll discuss this in the case of fibre products later).
Stacks Definition Let C be a category. A prestack is a strict functor C op Gpd. The 2-category of prestacks will be denoted by PrSt(C). If (C, T ) is a site, we denote by St T (C) the full 2-subcategory of prestacks F, for which for every {U i U} i I the diagram a F (U) i I F (U i) (i,j) I 2 F (U ij) (i,j,k) I 2 F (U ijk) induces is a limit in the 2-category of groupoids (i.e., induces an equivalence with the limit defined in Definition 13). Here, we have denoted the fibre product U i U U j by U ij (etc). We call such a prestack a stack. The full 2-subcategory of stacks will be denoted by St T (C). a parametrised by the category, whose objects are the ordered sets {0}, {0, 1}, {0, 1, 2} N, with order-preserving maps as morphisms.
Stacks Being a stack is equivalent to the following two assertions: For every {U i U} i I, and a collection of objects X i F (U i ), and isomorphisms φ ij : X i Ui U U j Xj Ui U U j, which satisfy the cocycle condition φ ij φ jk = φ ik on U i U U j U U k, there exists an object X F (U), together with isomorphisms φ i : X Ui Xi. For every U C, and X, Y F (U), we have that the functor Hom(X, Y ): C/U Set, which sends V U to Hom(X V, Y V ) is a sheaf. 1 1 In Vistoli s convention, a prestack is required to satisfy this property.
Stacks Let s take a look at a couple of examples: Pick a topological space X, consider the category Open(X ) with its canonical Grothendieck topology. It is possible to glue set-valued sheaves on X with respect to open coverings: given U = i I U i, and a sheaf G i on U i, for each i I, and isomorphisms φ ij : G i Ui U j G j Ui U j, satisfying the cocycle condition, there is a sheaf G on U, well-defined up to a unique isomorphism, which restricts to G i on each U i.
Stacks We can reformulate this: Let F : Open(X ) op Gpd be the prestack sending U X to the groupoid of set-valued sheaves on U. Then, F is a stack.
Stacks The second example is of algebraic nature. We consider the site given by Aff = Rng op with coverings being morphisms corresponding to faithfully flat ring homomorphisms R S. We will reformulate the fact that modules satisfy faithfully flat descent, using the language of stacks. First, let s recall faithully flat descent for modules from last week.
Recall from last week Definition For a ring homomorphism R S we define a category Desc R S as the category of pairs (M S, φ), where M S is an S-module, and φ is an isomorphism of S R S-modules which satisfies the identity φ: M S R S = S R M, M S R S R S of (S R S R S)-modules. S R M R S S R S R M S (1)
Recall from last week Theorem (Faithfully flat descent) Let R S be a faithfully flat morphism of rings. The canonical functor R S : Mod(R) Desc R S is an equivalence of categories.
Stacks Let s define a map Mod : Aff op Gpd Assign to Spec R Aff op the groupoid of R-modules. The map in induced by a morphism Spec S Spec R (corresponding to R S) is given by S R. Let s pretend that Mod is a well-defined prestack. Faithfully flat descent implies that Mod is a stack.
Stacks Except that this is all wrong! We have a natural equivalence of functors T S (S R ) T R, for every composable chain of ring homomorphisms R S T. But the functors don t strictly agree! How can we fix this? We have to strictify the functor Mod.
Stacks Definition For a ring R we denote by Mod(R) the groupoid, whose objects are S-modules M(S) for every ring homomorphism R S, and isomorphisms φ S : M(S) M(R) R S, satisfying a compatibility condition, which we won t spell out here. Let Mod: Aff op Gpd denote the corresponding functor, which sends R S to the restricted collection (M(T ), φ T ) for all S T. The essential difference is that the restriction map is obtained by restricting a data to a smaller index set, which is functorial on the nose, on not just up to a natural equivalence. However, we still retain the original information, as the next lemma shows.
Stacks Lemma The groupoids Mod(R) and Mod(R) are equivalent. We have a functor F : Mod(R) Mod(R), which sends (M(S), φ S ) to M(R). Let G : Mod(R) Mod(R) be the functor, sending M Mod(R) to the tautological collection (M R S, id M R S). (F G)(M) agrees with M, by definition. Given (M(S), φ S ), the collection G(F (M(S), φ S )) is given by (M(R) R S, id M(R) R S), which is naturally equivalent to (M(S), φ S ) by means of the system of maps φ S.
Fibre products in 2-categories Next week we will define algebraic stacks. The definition of algebraicity uses fibre products of stacks. The universal property of fibre products can be formulated in an arbitrary 2-category.
Fibre products in 2-categories Definition Let C be a 2-category, an object X C is called final, if for every other object Y C, there exists a morphism φ: Y X, and for every two morphisms φ, ψ : Y X, there exists a unique invertible 2-morphism between φ and ψ.
Fibre products in 2-categories Definition For a pair of morphisms X Z and Y Z we consider the 2-category of 2-commutative diagrams W f Y g X k h Z, i.e., we have an invertible 2-morphism α: k f h g. A final object in this category of diagrams will be referred to as fibre product, we denote the respective object W in the top left corner by X Z Y.
Fibre products in 2-categories We have an explicit model for X Z Y if X, Y, and Z are groupoids, given by the groupoid, whose objects are triples (x, y, α), with x X, y Y, and k(x) α h(y) being an isomorphism between k(x) and h(y) in Z.
Fibre products in 2-categories Let be the set with one element, endowed with the trivial action of an abstract group G acting on it. The fibre product [ /G] is equivalent to the set underlying the group G. This follows directly from the last slide. We have seen that a model for the fibre product is given by triples (x, y, α) with x, y, and α an automorphism of the object [ /G] (i.e., an element of G).