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Sheaves and Stacks November 5, 2014 Sheaves and Stacks Sheaves and Stacks 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). 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 . 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 . Definition A (set-valued) presheaf on X is a functor F : Open(X )op ! Set. 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 . 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), V as well as a restriction map rU : F (V ) ! F (U) for every inclusion U ⊂ V . The conditions U (a) rU = idF (U), V W W (b) rU ◦ rV = rU for triples of open subsets U ⊂ V ⊂ W , are satisfied. Sheaves and Stacks 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) = HomTop(U; Y ): V The restriction maps rU are given by f 7! f jU ; Sheaves and Stacks 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) = HomTop(U; Y ): V The restriction maps rU are given by f 7! f jU ; Sheaves and Stacks 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) = HomTop(U; Y ): V The restriction maps rU are given by f 7! f jU ; Sheaves and Stacks Grothendieck topologies S Let U = i2I Ui be an open covering, for every pair of open subsets Ui , Uj : Ui - Ui \ Uj ,! Uj : Hence, for every presheaf F we have a pair of restriction maps F (Ui ) ! F (Ui \ Uj ) F (Uj ): Taking a product over all pairs (i; j) 2 I 2, we obtain Q Q i2I F (Ui ) ⇒ (i;j)2I 2 F (Ui \ Uj ): Sheaves and Stacks Grothendieck topologies S Let U = i2I Ui be an open covering, for every pair of open subsets Ui , Uj : Ui - Ui \ Uj ,! Uj : Hence, for every presheaf F we have a pair of restriction maps F (Ui ) ! F (Ui \ Uj ) F (Uj ): Taking a product over all pairs (i; j) 2 I 2, we obtain Q Q i2I F (Ui ) ⇒ (i;j)2I 2 F (Ui \ Uj ): Definition A presheaf F is called a sheaf, if for every open subset U ⊂ X , and S every open covering U = i2I Ui , we have that Y Y F (U) ! F (Ui ) ⇒ F (Ui \ Uj ) i2I (i;j)2I 2 is an equalizer diagram. Sheaves and Stacks Grothendieck topologies What does this equalizer-business mean concretely? A presheaf is a sheaf, if and only if for every collection of local sections si 2 F (Ui ), U which satisfy r Ui (s ) = r j (s ), Ui \Uj i Ui \Uj j there exists a unique section s 2 F (U), such that r U (s) = s . Ui i Sheaves and Stacks Grothendieck topologies Lemma The presheaf Y X is a sheaf. Concrete proof. If fi : Ui ! Y are continuous functions, such that fi jUi \Uj = fj jUi \Uj then there is a well-defined map of sets f : U ! Y , sending x 2 U to fi (x), if x 2 Ui . Since continuity is a local property, we see that f is a continuous function. Sheaves and Stacks Grothendieck topologies Lemma The presheaf Y X is a sheaf. Abstract proof. We can represent U as a co-equalizer ` ` (i;j)2I 2 Ui \ Uj ⇒ i2I Ui ! U; i.e., as a colimit in the category Top of topological spaces. The universal property of colimits implies that HomTop(−; Y ) sends a co-equalizer to an equalizer. Sheaves and Stacks Grothendieck topologies In the definition of presheaves we could replace OpenX by an arbitrary category C. The sheaf Y X , extends by definition to a functor Hom (−;Y ) Topop Top / Set O 9 Y X Open(X )op: 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. Sheaves and Stacks Grothendieck topologies Definition Let C be a category. A Grothendieck topology T on C consists of a collection of sets of morphisms (called coverings) fUi ! Ugi2I for each object U 2 C, satisfying: (a) For every isomorphism U0 ! U, the singleton fU0 ! Ug is a covering. (b) Coverings are preserved by base change, i.e. if fUi ! Ugi2I is a covering, and V ! U a morphism in C, then fUi ×U V ! V gi2I is well-defined, and a covering. (c) Given a covering fUi ! Ug, and for each i 2 I a covering fUij ! Ui gj2Ji , then fU ! Ug Q ij (i;j)2 i2I Ji is a covering. Sheaves and Stacks 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). Sheaves and Stacks Grothendieck topologies For the category Open(X ), we define T (X ) to be the S collection of all fUi ⊂ Ug, such that i2I Ui = U. The Grothendieck topology T (X ) can be extended to T on fi S Top. We say that fUi −! Ugi2I 2 T , if i2I f (Ui ) = U, and each fi is an open map, which is a homeomorphism onto its image. Let Aff denote the category Rngop. Recall, that we denote the object in Aff, corresponding to the ring R by Spec R. Consider the coverings fSpec S ! Spec Rg, where R ! S is a faithfully flat map of rings. Sheaves and Stacks 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 Cop ! 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 Ui \ Uj . Sheaves and Stacks 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. Sheaves and Stacks 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. Sheaves and Stacks Grothendieck topologies Definition Let C be a category. A functor F :Cop ! 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 fUi ! Ugi2I the diagram Y Y F (U) ! F (Ui ) ⇒ F (Ui ×U Uj ) i2I (i;j)2I 2 is an equalizer. We denote the full subcategory of sheaves by ShT (C). Sheaves and Stacks Grothendieck topologies The heuristics behind this definition is the same as for topological spaces. Given an abstract covering fUi ! Ugi2I of U 2 C, and locally defined sections si 2 F (Ui ), which agree when restricted (or pulled back) to the "intersections" Ui ×U Uj , then there exists a unique section s 2 F (U), which agrees with si over each Ui . Sheaves and Stacks 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. Sheaves and Stacks Grothendieck topologies Let C = Set be the category of sets. We consider the Grothendieck topology T , fi which consists of all collections fUi −! Ug, such that S i2I f (Ui ) = U. For a set X we have the presheaf hX = HomSet(−; X ), represented by X . Then, hX is a sheaf with respect to the topology T . Sheaves and Stacks Grothendieck topologies n For C the category of open subsets U ⊂ R (where n is allowed to vary), and smooth maps as morphisms, fi define T to consist of all sets fUi −! Ugi2I , S where each fi is a smooth submersion, and i2I f (Ui ) = U. Every smooth manifold X gives rise to a sheaf on C, by sending U 2 C to the set of smooth maps U ! X . The resulting functor Mfd ! ShT (C) is an embedding of categories (i.e. fully faithful). Sheaves and Stacks Recall from last week Theorem We have natural maps e1 : T ! T ⊗S T , and e2 : T ! T ⊗S T .
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