1. From Groups to Groupoids A morphism of groupoids F : G → H is, by definition, a functor between these categories. Note that given a morphism of groupoids F : G → H and an object i ∈ Ob ( G ) , there is a group homomorphism F : G ( i , i ) → H ( F ( i ) , F ( i )) between the corresponding automorphism groups. Amnon Yekutieli (BGU) Central Extensions of Gerbes 6 / 46
1. From Groups to Groupoids A morphism of groupoids F : G → H is, by definition, a functor between these categories. Note that given a morphism of groupoids F : G → H and an object i ∈ Ob ( G ) , there is a group homomorphism F : G ( i , i ) → H ( F ( i ) , F ( i )) between the corresponding automorphism groups. Amnon Yekutieli (BGU) Central Extensions of Gerbes 6 / 46
1. From Groups to Groupoids Figure: A morphism of groupoids F : G → H . Amnon Yekutieli (BGU) Central Extensions of Gerbes 7 / 46
1. From Groups to Groupoids Fix a nonzero commutative ring A and a positive integer n . Example 1.2. Let G n ( A ) be the category whose objects are the free A -modules of rank n , and the morphisms are A -linear isomorphisms. This is a groupoid. Any object P of G n ( A ) is isomorphic to A n , and therefore G n ( A ) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GL n ( A ) . Like in this example, for us a groupoid is usually a bookkeeping device. It serves to enumerate certain objects and the isomorphisms between them. Amnon Yekutieli (BGU) Central Extensions of Gerbes 8 / 46
1. From Groups to Groupoids Fix a nonzero commutative ring A and a positive integer n . Example 1.2. Let G n ( A ) be the category whose objects are the free A -modules of rank n , and the morphisms are A -linear isomorphisms. This is a groupoid. Any object P of G n ( A ) is isomorphic to A n , and therefore G n ( A ) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GL n ( A ) . Like in this example, for us a groupoid is usually a bookkeeping device. It serves to enumerate certain objects and the isomorphisms between them. Amnon Yekutieli (BGU) Central Extensions of Gerbes 8 / 46
1. From Groups to Groupoids Fix a nonzero commutative ring A and a positive integer n . Example 1.2. Let G n ( A ) be the category whose objects are the free A -modules of rank n , and the morphisms are A -linear isomorphisms. This is a groupoid. Any object P of G n ( A ) is isomorphic to A n , and therefore G n ( A ) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GL n ( A ) . Like in this example, for us a groupoid is usually a bookkeeping device. It serves to enumerate certain objects and the isomorphisms between them. Amnon Yekutieli (BGU) Central Extensions of Gerbes 8 / 46
1. From Groups to Groupoids Fix a nonzero commutative ring A and a positive integer n . Example 1.2. Let G n ( A ) be the category whose objects are the free A -modules of rank n , and the morphisms are A -linear isomorphisms. This is a groupoid. Any object P of G n ( A ) is isomorphic to A n , and therefore G n ( A ) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GL n ( A ) . Like in this example, for us a groupoid is usually a bookkeeping device. It serves to enumerate certain objects and the isomorphisms between them. Amnon Yekutieli (BGU) Central Extensions of Gerbes 8 / 46
1. From Groups to Groupoids Fix a nonzero commutative ring A and a positive integer n . Example 1.2. Let G n ( A ) be the category whose objects are the free A -modules of rank n , and the morphisms are A -linear isomorphisms. This is a groupoid. Any object P of G n ( A ) is isomorphic to A n , and therefore G n ( A ) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GL n ( A ) . Like in this example, for us a groupoid is usually a bookkeeping device. It serves to enumerate certain objects and the isomorphisms between them. Amnon Yekutieli (BGU) Central Extensions of Gerbes 8 / 46
1. From Groups to Groupoids Definition 1.3. A morphism of groupoids F : G → H is called a weak epimorphism if it satisfies these conditions: ( ∗ ) F is surjective on isomorphism classes of objects. ( ∗∗ ) F is surjective on sets of arrows. This means that for any i , j ∈ Ob ( G ) the function F : G ( i , j ) → H ( F ( i ) , F ( j )) is surjective. Note that condition ( ∗ ) is automatically satisfied when G and H are NC groupoids. Amnon Yekutieli (BGU) Central Extensions of Gerbes 9 / 46
1. From Groups to Groupoids Definition 1.3. A morphism of groupoids F : G → H is called a weak epimorphism if it satisfies these conditions: ( ∗ ) F is surjective on isomorphism classes of objects. ( ∗∗ ) F is surjective on sets of arrows. This means that for any i , j ∈ Ob ( G ) the function F : G ( i , j ) → H ( F ( i ) , F ( j )) is surjective. Note that condition ( ∗ ) is automatically satisfied when G and H are NC groupoids. Amnon Yekutieli (BGU) Central Extensions of Gerbes 9 / 46
1. From Groups to Groupoids Definition 1.3. A morphism of groupoids F : G → H is called a weak epimorphism if it satisfies these conditions: ( ∗ ) F is surjective on isomorphism classes of objects. ( ∗∗ ) F is surjective on sets of arrows. This means that for any i , j ∈ Ob ( G ) the function F : G ( i , j ) → H ( F ( i ) , F ( j )) is surjective. Note that condition ( ∗ ) is automatically satisfied when G and H are NC groupoids. Amnon Yekutieli (BGU) Central Extensions of Gerbes 9 / 46
1. From Groups to Groupoids Definition 1.3. A morphism of groupoids F : G → H is called a weak epimorphism if it satisfies these conditions: ( ∗ ) F is surjective on isomorphism classes of objects. ( ∗∗ ) F is surjective on sets of arrows. This means that for any i , j ∈ Ob ( G ) the function F : G ( i , j ) → H ( F ( i ) , F ( j )) is surjective. Note that condition ( ∗ ) is automatically satisfied when G and H are NC groupoids. Amnon Yekutieli (BGU) Central Extensions of Gerbes 9 / 46
2. Normal Subgroupoids and Extensions 2. Normal Subgroupoids and Extensions Let G be a groupoid. Given an arrow g : i → j in G , there is an induced group isomorphism Ad ( g ) : G ( i , i ) → G ( j , j ) , namely Ad ( g )( h ) := g ◦ h ◦ g − 1 . Amnon Yekutieli (BGU) Central Extensions of Gerbes 10 / 46
2. Normal Subgroupoids and Extensions 2. Normal Subgroupoids and Extensions Let G be a groupoid. Given an arrow g : i → j in G , there is an induced group isomorphism Ad ( g ) : G ( i , i ) → G ( j , j ) , namely Ad ( g )( h ) := g ◦ h ◦ g − 1 . Amnon Yekutieli (BGU) Central Extensions of Gerbes 10 / 46
2. Normal Subgroupoids and Extensions 2. Normal Subgroupoids and Extensions Let G be a groupoid. Given an arrow g : i → j in G , there is an induced group isomorphism Ad ( g ) : G ( i , i ) → G ( j , j ) , namely Ad ( g )( h ) := g ◦ h ◦ g − 1 . Amnon Yekutieli (BGU) Central Extensions of Gerbes 10 / 46
2. Normal Subgroupoids and Extensions Figure: An arrow g : 0 → 1, and the induced group isomorphism Ad ( g ) : G ( 0 , 0 ) → G ( 1 , 1 ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 11 / 46
2. Normal Subgroupoids and Extensions Definition 2.1. Let G be a groupoid. A normal subgroupoid of G is a subgroupoid N ⊂ G with these properties: (i) N has the same objects as G ; i.e. Ob ( N ) = Ob ( G ) . (ii) N is totally disconnected; i.e. N ( i , j ) = ∅ whenever i � = j . (iii) For any arrow g : i → j in G there is equality Ad ( g )( N ( i , i )) = N ( j , j ) , between these subgroups of G ( j , j ) . In particular, for any i the subgroup N ( i , i ) is normal in G ( i , i ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 12 / 46
2. Normal Subgroupoids and Extensions Definition 2.1. Let G be a groupoid. A normal subgroupoid of G is a subgroupoid N ⊂ G with these properties: (i) N has the same objects as G ; i.e. Ob ( N ) = Ob ( G ) . (ii) N is totally disconnected; i.e. N ( i , j ) = ∅ whenever i � = j . (iii) For any arrow g : i → j in G there is equality Ad ( g )( N ( i , i )) = N ( j , j ) , between these subgroups of G ( j , j ) . In particular, for any i the subgroup N ( i , i ) is normal in G ( i , i ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 12 / 46
2. Normal Subgroupoids and Extensions Definition 2.1. Let G be a groupoid. A normal subgroupoid of G is a subgroupoid N ⊂ G with these properties: (i) N has the same objects as G ; i.e. Ob ( N ) = Ob ( G ) . (ii) N is totally disconnected; i.e. N ( i , j ) = ∅ whenever i � = j . (iii) For any arrow g : i → j in G there is equality Ad ( g )( N ( i , i )) = N ( j , j ) , between these subgroups of G ( j , j ) . In particular, for any i the subgroup N ( i , i ) is normal in G ( i , i ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 12 / 46
2. Normal Subgroupoids and Extensions Definition 2.1. Let G be a groupoid. A normal subgroupoid of G is a subgroupoid N ⊂ G with these properties: (i) N has the same objects as G ; i.e. Ob ( N ) = Ob ( G ) . (ii) N is totally disconnected; i.e. N ( i , j ) = ∅ whenever i � = j . (iii) For any arrow g : i → j in G there is equality Ad ( g )( N ( i , i )) = N ( j , j ) , between these subgroups of G ( j , j ) . In particular, for any i the subgroup N ( i , i ) is normal in G ( i , i ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 12 / 46
2. Normal Subgroupoids and Extensions Definition 2.1. Let G be a groupoid. A normal subgroupoid of G is a subgroupoid N ⊂ G with these properties: (i) N has the same objects as G ; i.e. Ob ( N ) = Ob ( G ) . (ii) N is totally disconnected; i.e. N ( i , j ) = ∅ whenever i � = j . (iii) For any arrow g : i → j in G there is equality Ad ( g )( N ( i , i )) = N ( j , j ) , between these subgroups of G ( j , j ) . In particular, for any i the subgroup N ( i , i ) is normal in G ( i , i ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 12 / 46
2. Normal Subgroupoids and Extensions Figure: A normal subgroupoid N of G . Amnon Yekutieli (BGU) Central Extensions of Gerbes 13 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Suppose F : G → H is a morphism of groupoids. We can define a subgroupoid N ⊂ G as follows. The set of objects is Ob ( N ) := Ob ( G ) . If i , j are distinct objects of G we let N ( i , j ) := ∅ . And we let N ( i , i ) := Ker � � F : G ( i , i ) → G ( F ( i ) , F ( i )) . It is not hard to verify that N is a normal subgroupoid of G . Definition 2.2. The groupoid N above is called the kernel of F , and is denoted by Ker ( F ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 14 / 46
2. Normal Subgroupoids and Extensions Definition 2.3. By an extension of groupoids we mean a diagram of groupoids → G F N − − → H in which F : G → H is a weak epimorphism, N = Ker ( F ) , and N → G is the inclusion. In the next slide we will see an example of such an extension. Amnon Yekutieli (BGU) Central Extensions of Gerbes 15 / 46
2. Normal Subgroupoids and Extensions Definition 2.3. By an extension of groupoids we mean a diagram of groupoids → G F N − − → H in which F : G → H is a weak epimorphism, N = Ker ( F ) , and N → G is the inclusion. In the next slide we will see an example of such an extension. Amnon Yekutieli (BGU) Central Extensions of Gerbes 15 / 46
2. Normal Subgroupoids and Extensions Let A → B be a surjective homomorphism between local Example 2.4. commutative rings. Consider the groupoids of free modules G n ( A ) and G n ( B ) , for some n . These are NC groupoids. Given a module P ∈ G n ( A ) let F ( P ) := B ⊗ A P ∈ G n ( B ) . In this way we get a morphism of groupoids F : G n ( A ) → G n ( B ) . On automorphisms groups there is a surjection F : GL n ( A ) → GL n ( B ) , and hence F is a weak epimorphism. Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46
2. Normal Subgroupoids and Extensions Let A → B be a surjective homomorphism between local Example 2.4. commutative rings. Consider the groupoids of free modules G n ( A ) and G n ( B ) , for some n . These are NC groupoids. Given a module P ∈ G n ( A ) let F ( P ) := B ⊗ A P ∈ G n ( B ) . In this way we get a morphism of groupoids F : G n ( A ) → G n ( B ) . On automorphisms groups there is a surjection F : GL n ( A ) → GL n ( B ) , and hence F is a weak epimorphism. Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46
2. Normal Subgroupoids and Extensions Let A → B be a surjective homomorphism between local Example 2.4. commutative rings. Consider the groupoids of free modules G n ( A ) and G n ( B ) , for some n . These are NC groupoids. Given a module P ∈ G n ( A ) let F ( P ) := B ⊗ A P ∈ G n ( B ) . In this way we get a morphism of groupoids F : G n ( A ) → G n ( B ) . On automorphisms groups there is a surjection F : GL n ( A ) → GL n ( B ) , and hence F is a weak epimorphism. Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46
2. Normal Subgroupoids and Extensions Let A → B be a surjective homomorphism between local Example 2.4. commutative rings. Consider the groupoids of free modules G n ( A ) and G n ( B ) , for some n . These are NC groupoids. Given a module P ∈ G n ( A ) let F ( P ) := B ⊗ A P ∈ G n ( B ) . In this way we get a morphism of groupoids F : G n ( A ) → G n ( B ) . On automorphisms groups there is a surjection F : GL n ( A ) → GL n ( B ) , and hence F is a weak epimorphism. Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46
2. Normal Subgroupoids and Extensions Let A → B be a surjective homomorphism between local Example 2.4. commutative rings. Consider the groupoids of free modules G n ( A ) and G n ( B ) , for some n . These are NC groupoids. Given a module P ∈ G n ( A ) let F ( P ) := B ⊗ A P ∈ G n ( B ) . In this way we get a morphism of groupoids F : G n ( A ) → G n ( B ) . On automorphisms groups there is a surjection F : GL n ( A ) → GL n ( B ) , and hence F is a weak epimorphism. Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46
2. Normal Subgroupoids and Extensions (cont.) We get an extension of NC groupoids N → G n ( A ) F − → G n ( B ) . The kernels N ( P , P ) are noncanonically isomorphic to the congruence subgroup { g ∈ GL n ( A ) | g ≡ 1 mod I } , where I := Ker ( A → B ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 17 / 46
2. Normal Subgroupoids and Extensions (cont.) We get an extension of NC groupoids N → G n ( A ) F − → G n ( B ) . The kernels N ( P , P ) are noncanonically isomorphic to the congruence subgroup { g ∈ GL n ( A ) | g ≡ 1 mod I } , where I := Ker ( A → B ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 17 / 46
3. The Center of a Groupoid 3. The Center of a Groupoid We denote the center of a group G by Z ( G ) . Definition 3.1. Let G be a groupoid. 1. The center of G is the normal subgroupoid Z ( G ) with Z ( G )( i , i ) := Z ( G ( i , i )) for all i ∈ Ob ( G ) . 2. A central subgroupoid of G is a normal subgroupoid N that is contained in Z ( G ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 18 / 46
3. The Center of a Groupoid 3. The Center of a Groupoid We denote the center of a group G by Z ( G ) . Definition 3.1. Let G be a groupoid. 1. The center of G is the normal subgroupoid Z ( G ) with Z ( G )( i , i ) := Z ( G ( i , i )) for all i ∈ Ob ( G ) . 2. A central subgroupoid of G is a normal subgroupoid N that is contained in Z ( G ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 18 / 46
3. The Center of a Groupoid 3. The Center of a Groupoid We denote the center of a group G by Z ( G ) . Definition 3.1. Let G be a groupoid. 1. The center of G is the normal subgroupoid Z ( G ) with Z ( G )( i , i ) := Z ( G ( i , i )) for all i ∈ Ob ( G ) . 2. A central subgroupoid of G is a normal subgroupoid N that is contained in Z ( G ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 18 / 46
3. The Center of a Groupoid 3. The Center of a Groupoid We denote the center of a group G by Z ( G ) . Definition 3.1. Let G be a groupoid. 1. The center of G is the normal subgroupoid Z ( G ) with Z ( G )( i , i ) := Z ( G ( i , i )) for all i ∈ Ob ( G ) . 2. A central subgroupoid of G is a normal subgroupoid N that is contained in Z ( G ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 18 / 46
3. The Center of a Groupoid 3. The Center of a Groupoid We denote the center of a group G by Z ( G ) . Definition 3.1. Let G be a groupoid. 1. The center of G is the normal subgroupoid Z ( G ) with Z ( G )( i , i ) := Z ( G ( i , i )) for all i ∈ Ob ( G ) . 2. A central subgroupoid of G is a normal subgroupoid N that is contained in Z ( G ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 18 / 46
3. The Center of a Groupoid Let G be an NC groupoid, and N a central subgroupoid of G . Let g , g ′ : i → j be arrows in G . It is easy to see that the group isomorphisms Ad ( g ) , Ad ( g ′ ) : G ( i , i ) → G ( j , j ) differ by an inner automorphism of G ( j , j ) . Therefore they induce the same group isomorphism N ( i , i ) → N ( j , j ) . By identifying the groups N ( i , i ) in this canonical way, we can view the central groupoid N as a single abelian group, say N . Amnon Yekutieli (BGU) Central Extensions of Gerbes 19 / 46
3. The Center of a Groupoid Let G be an NC groupoid, and N a central subgroupoid of G . Let g , g ′ : i → j be arrows in G . It is easy to see that the group isomorphisms Ad ( g ) , Ad ( g ′ ) : G ( i , i ) → G ( j , j ) differ by an inner automorphism of G ( j , j ) . Therefore they induce the same group isomorphism N ( i , i ) → N ( j , j ) . By identifying the groups N ( i , i ) in this canonical way, we can view the central groupoid N as a single abelian group, say N . Amnon Yekutieli (BGU) Central Extensions of Gerbes 19 / 46
3. The Center of a Groupoid Let G be an NC groupoid, and N a central subgroupoid of G . Let g , g ′ : i → j be arrows in G . It is easy to see that the group isomorphisms Ad ( g ) , Ad ( g ′ ) : G ( i , i ) → G ( j , j ) differ by an inner automorphism of G ( j , j ) . Therefore they induce the same group isomorphism N ( i , i ) → N ( j , j ) . By identifying the groups N ( i , i ) in this canonical way, we can view the central groupoid N as a single abelian group, say N . Amnon Yekutieli (BGU) Central Extensions of Gerbes 19 / 46
3. The Center of a Groupoid Let G be an NC groupoid, and N a central subgroupoid of G . Let g , g ′ : i → j be arrows in G . It is easy to see that the group isomorphisms Ad ( g ) , Ad ( g ′ ) : G ( i , i ) → G ( j , j ) differ by an inner automorphism of G ( j , j ) . Therefore they induce the same group isomorphism N ( i , i ) → N ( j , j ) . By identifying the groups N ( i , i ) in this canonical way, we can view the central groupoid N as a single abelian group, say N . Amnon Yekutieli (BGU) Central Extensions of Gerbes 19 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes 4. Geometrizing NC Groupoids: Gerbes Fix a topological space X . The geometric version of a group G is a sheaf of groups G on X . There is an intermediate notion: presheaf of groups. Recall that a presheaf of groups G is the assignment of a group G ( U ) to any open set U ⊂ X , and a group homomorphism rest V / U : G ( U ) → G ( V ) to any inclusion of open sets V ⊂ U . The restriction homomorphisms rest V / U must satisfy an obvious transitivity condition. A presheaf G is called a sheaf if it satisfies the familiar sheaf condition, also called the descent condition. Amnon Yekutieli (BGU) Central Extensions of Gerbes 20 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Now let’s try to generalize this geometrization procedure to groupoids. A prestack of groupoids G on X consists of the following data: ◮ A groupoid G ( U ) for each open set U . ◮ A restriction morphism rest V / U : G ( U ) → G ( V ) for each inclusion of open sets V ⊂ U . Here the transitivity constraint on the morphisms rest V / U must be relaxed; but I prefer to skip the details. See illustration on next two slides. Amnon Yekutieli (BGU) Central Extensions of Gerbes 21 / 46
4. Geometrizing NC Groupoids: Gerbes Figure: Open sets V ⊂ U , and the restriction morphism rest V / U : G ( U ) → G ( V ) between the groupoids G ( U ) and G ( V ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 22 / 46
4. Geometrizing NC Groupoids: Gerbes Figure: The restriction morphism rest V / U : G ( U ) → G ( V ) . The objects i 0 and i 1 become isomorphic in G ( V ) . A new object i 2 is created in G ( V ) . Amnon Yekutieli (BGU) Central Extensions of Gerbes 23 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes An object i ∈ Ob ( G ( U )) , for some open set U , is called a local object of G . A morphism g : i → j between local objects is called a local arrow. To any pair i , j of such local objects, the restriction morphisms give rise to a presheaf of sets G ( i , j ) on U , called the presheaf of arrows. Let G be a prestack of groupoids on X . We say that G is a stack of groupoids if it satisfies these two conditions: (i) Descent for arrows. (ii) Descent for objects. Condition (i) says that the presheaves of arrows G ( i , j ) are sheaves. Condition (ii) says, roughly, that the “presheaves of sets” U �→ Ob ( G ( U )) are sheaves. Amnon Yekutieli (BGU) Central Extensions of Gerbes 24 / 46
4. Geometrizing NC Groupoids: Gerbes Making all these definitions precise is actually quite difficult. It is necessary to use concepts such as “2-category” and “pseudofunctor”. Full details can be found in [Ye1]. Amnon Yekutieli (BGU) Central Extensions of Gerbes 25 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes The geometric version of an NC groupoid is a gerbe. A gerbe on X is a stack of groupoids satisfying the following two conditions: ◮ G is locally nonempty. ◮ G is locally connected. Let me explain the first condition. It says that any point x ∈ X has some open neighborhood U such that the groupoid G ( U ) is nonempty. The second condition is a geometric version of “connected groupoid”. It is good to see an example now. Amnon Yekutieli (BGU) Central Extensions of Gerbes 26 / 46
4. Geometrizing NC Groupoids: Gerbes Let X be an algebraic variety, with sheaf of functions O X , and Example 4.1. let n be a positive integer. For any open set U we consider the set G n ( U ) of all rank n locally free O U -modules, i.e. rank n vector bundles on U . A morphism P → Q in G n ( U ) is by definition an isomorphism of O U -modules. So G n ( U ) is a groupoid. The groupoid G n ( U ) is nonempty, since it contains the free module O n U . But it could be disconnected, since there could be nonisomorphic vector bundles on U . As we vary the open set U , we obtain a prestack of groupoids G n on X . In fact this is a stack (the descent conditions hold). Because any P , Q ∈ Ob ( G n ( U )) are locally isomorphic, it follows that G n is a gerbe. Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46
4. Geometrizing NC Groupoids: Gerbes Let X be an algebraic variety, with sheaf of functions O X , and Example 4.1. let n be a positive integer. For any open set U we consider the set G n ( U ) of all rank n locally free O U -modules, i.e. rank n vector bundles on U . A morphism P → Q in G n ( U ) is by definition an isomorphism of O U -modules. So G n ( U ) is a groupoid. The groupoid G n ( U ) is nonempty, since it contains the free module O n U . But it could be disconnected, since there could be nonisomorphic vector bundles on U . As we vary the open set U , we obtain a prestack of groupoids G n on X . In fact this is a stack (the descent conditions hold). Because any P , Q ∈ Ob ( G n ( U )) are locally isomorphic, it follows that G n is a gerbe. Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46
4. Geometrizing NC Groupoids: Gerbes Let X be an algebraic variety, with sheaf of functions O X , and Example 4.1. let n be a positive integer. For any open set U we consider the set G n ( U ) of all rank n locally free O U -modules, i.e. rank n vector bundles on U . A morphism P → Q in G n ( U ) is by definition an isomorphism of O U -modules. So G n ( U ) is a groupoid. The groupoid G n ( U ) is nonempty, since it contains the free module O n U . But it could be disconnected, since there could be nonisomorphic vector bundles on U . As we vary the open set U , we obtain a prestack of groupoids G n on X . In fact this is a stack (the descent conditions hold). Because any P , Q ∈ Ob ( G n ( U )) are locally isomorphic, it follows that G n is a gerbe. Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46
4. Geometrizing NC Groupoids: Gerbes Let X be an algebraic variety, with sheaf of functions O X , and Example 4.1. let n be a positive integer. For any open set U we consider the set G n ( U ) of all rank n locally free O U -modules, i.e. rank n vector bundles on U . A morphism P → Q in G n ( U ) is by definition an isomorphism of O U -modules. So G n ( U ) is a groupoid. The groupoid G n ( U ) is nonempty, since it contains the free module O n U . But it could be disconnected, since there could be nonisomorphic vector bundles on U . As we vary the open set U , we obtain a prestack of groupoids G n on X . In fact this is a stack (the descent conditions hold). Because any P , Q ∈ Ob ( G n ( U )) are locally isomorphic, it follows that G n is a gerbe. Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46
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