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Central Extensions of Gerbes Amnon Yekutieli Department of - - PowerPoint PPT Presentation

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Central Extensions of Gerbes Amnon Yekutieli Department of Mathematics Ben Gurion University Notes available at http://www.math.bgu.ac.il/ amyekut/lectures Written 26 June 2010; corrected 11 Sep 2010 Amnon Yekutieli (BGU) Central

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Central Extensions of Gerbes

Amnon Yekutieli

Department of Mathematics Ben Gurion University Notes available at http://www.math.bgu.ac.il/∼amyekut/lectures

Written 26 June 2010; corrected 11 Sep 2010 Amnon Yekutieli (BGU) Central Extensions of Gerbes 1 / 46

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SLIDE 2

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 3

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 4

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 5

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 6

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 7

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 8

Outline

  • 1. From Groups to Groupoids
  • 2. Normal Subgroupoids and Extensions
  • 3. The Center of a Groupoid
  • 4. Geometrizing NC Groupoids: Gerbes
  • 5. Extensions of Gerbes and Obstructions
  • 6. Pronilpotent Gerbes (optional)

Lecture notes, including a bibliography list, are available online.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 2 / 46

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SLIDE 9
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 10
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 11
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 12
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 13
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 14
  • 1. From Groups to Groupoids
  • 1. From Groups to Groupoids

We begin by taking a few basic ideas from group theory and generalizing them to groupoids. Recall that a groupoid is a category G is which all morphisms are isomorphisms (i.e. they have inverses). We denote the set of objects of G by Ob(G). And for any pair of objects i, j ∈ Ob(G) we denote by G(i, j) the set of morphisms (or arrows) g : i → j. Note that for any object i, the set of arrows G(i, i) is a group. We call it the automorphism group of i. So a groupoid can be viewed as a collection of groups, with a certain interaction between them (which we shall discuss later).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 3 / 46

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SLIDE 15
  • 1. From Groups to Groupoids

Definition 1.1. Let G be a groupoid.

  • 1. G is called a nonempty groupoid if Ob(G) = ∅.
  • 2. G is called a connected groupoid if G(i, j) = ∅ for all i, j.
  • 3. If G is nonempty and connected, then we call it an NC groupoid.

In other words, a groupoid G is NC iff there is exactly one isomorphism class

  • f objects.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 4 / 46

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SLIDE 16
  • 1. From Groups to Groupoids

Definition 1.1. Let G be a groupoid.

  • 1. G is called a nonempty groupoid if Ob(G) = ∅.
  • 2. G is called a connected groupoid if G(i, j) = ∅ for all i, j.
  • 3. If G is nonempty and connected, then we call it an NC groupoid.

In other words, a groupoid G is NC iff there is exactly one isomorphism class

  • f objects.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 4 / 46

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SLIDE 17
  • 1. From Groups to Groupoids

Definition 1.1. Let G be a groupoid.

  • 1. G is called a nonempty groupoid if Ob(G) = ∅.
  • 2. G is called a connected groupoid if G(i, j) = ∅ for all i, j.
  • 3. If G is nonempty and connected, then we call it an NC groupoid.

In other words, a groupoid G is NC iff there is exactly one isomorphism class

  • f objects.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 4 / 46

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SLIDE 18
  • 1. From Groups to Groupoids

Definition 1.1. Let G be a groupoid.

  • 1. G is called a nonempty groupoid if Ob(G) = ∅.
  • 2. G is called a connected groupoid if G(i, j) = ∅ for all i, j.
  • 3. If G is nonempty and connected, then we call it an NC groupoid.

In other words, a groupoid G is NC iff there is exactly one isomorphism class

  • f objects.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 4 / 46

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SLIDE 19
  • 1. From Groups to Groupoids

Definition 1.1. Let G be a groupoid.

  • 1. G is called a nonempty groupoid if Ob(G) = ∅.
  • 2. G is called a connected groupoid if G(i, j) = ∅ for all i, j.
  • 3. If G is nonempty and connected, then we call it an NC groupoid.

In other words, a groupoid G is NC iff there is exactly one isomorphism class

  • f objects.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 4 / 46

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SLIDE 20
  • 1. From Groups to Groupoids

Figure: A connected groupoid G with Ob(G) = {0, 1}.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 5 / 46

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SLIDE 21
  • 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

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SLIDE 22
  • 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

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SLIDE 23
  • 1. From Groups to Groupoids

Figure: A morphism of groupoids F : G → H.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 7 / 46

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SLIDE 24
  • 1. From Groups to Groupoids

Example 1.2. Fix a nonzero commutative ring A and a positive integer n. Let Gn(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 Gn(A) is isomorphic to An, and therefore Gn(A) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GLn(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

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SLIDE 25
  • 1. From Groups to Groupoids

Example 1.2. Fix a nonzero commutative ring A and a positive integer n. Let Gn(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 Gn(A) is isomorphic to An, and therefore Gn(A) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GLn(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

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SLIDE 26
  • 1. From Groups to Groupoids

Example 1.2. Fix a nonzero commutative ring A and a positive integer n. Let Gn(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 Gn(A) is isomorphic to An, and therefore Gn(A) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GLn(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

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SLIDE 27
  • 1. From Groups to Groupoids

Example 1.2. Fix a nonzero commutative ring A and a positive integer n. Let Gn(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 Gn(A) is isomorphic to An, and therefore Gn(A) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GLn(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

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SLIDE 28
  • 1. From Groups to Groupoids

Example 1.2. Fix a nonzero commutative ring A and a positive integer n. Let Gn(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 Gn(A) is isomorphic to An, and therefore Gn(A) is an NC groupoid. The automorphism group of P is noncanonically isomorphic to GLn(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

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SLIDE 29
  • 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

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SLIDE 30
  • 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

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SLIDE 31
  • 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

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SLIDE 32
  • 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

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SLIDE 33
  • 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

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SLIDE 34
  • 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

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SLIDE 35
  • 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

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SLIDE 36
  • 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

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SLIDE 37
  • 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

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SLIDE 38
  • 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

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SLIDE 39
  • 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

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SLIDE 40
  • 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

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SLIDE 41
  • 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

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SLIDE 42
  • 2. Normal Subgroupoids and Extensions

Figure: A normal subgroupoid N of G.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 13 / 46

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SLIDE 43
  • 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

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SLIDE 44
  • 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

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SLIDE 45
  • 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

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SLIDE 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

slide-47
SLIDE 47
  • 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

slide-48
SLIDE 48
  • 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

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SLIDE 49
  • 2. Normal Subgroupoids and Extensions

Definition 2.3. By an extension of groupoids we mean a diagram of groupoids N − → G F − → 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

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SLIDE 50
  • 2. Normal Subgroupoids and Extensions

Definition 2.3. By an extension of groupoids we mean a diagram of groupoids N − → G F − → 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

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SLIDE 51
  • 2. Normal Subgroupoids and Extensions

Example 2.4. Let A → B be a surjective homomorphism between local commutative rings. Consider the groupoids of free modules Gn(A) and Gn(B), for some n. These are NC groupoids. Given a module P ∈ Gn(A) let F(P) := B ⊗A P ∈ Gn(B). In this way we get a morphism of groupoids F : Gn(A) → Gn(B). On automorphisms groups there is a surjection F : GLn(A) → GLn(B), and hence F is a weak epimorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46

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SLIDE 52
  • 2. Normal Subgroupoids and Extensions

Example 2.4. Let A → B be a surjective homomorphism between local commutative rings. Consider the groupoids of free modules Gn(A) and Gn(B), for some n. These are NC groupoids. Given a module P ∈ Gn(A) let F(P) := B ⊗A P ∈ Gn(B). In this way we get a morphism of groupoids F : Gn(A) → Gn(B). On automorphisms groups there is a surjection F : GLn(A) → GLn(B), and hence F is a weak epimorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46

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SLIDE 53
  • 2. Normal Subgroupoids and Extensions

Example 2.4. Let A → B be a surjective homomorphism between local commutative rings. Consider the groupoids of free modules Gn(A) and Gn(B), for some n. These are NC groupoids. Given a module P ∈ Gn(A) let F(P) := B ⊗A P ∈ Gn(B). In this way we get a morphism of groupoids F : Gn(A) → Gn(B). On automorphisms groups there is a surjection F : GLn(A) → GLn(B), and hence F is a weak epimorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46

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SLIDE 54
  • 2. Normal Subgroupoids and Extensions

Example 2.4. Let A → B be a surjective homomorphism between local commutative rings. Consider the groupoids of free modules Gn(A) and Gn(B), for some n. These are NC groupoids. Given a module P ∈ Gn(A) let F(P) := B ⊗A P ∈ Gn(B). In this way we get a morphism of groupoids F : Gn(A) → Gn(B). On automorphisms groups there is a surjection F : GLn(A) → GLn(B), and hence F is a weak epimorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46

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SLIDE 55
  • 2. Normal Subgroupoids and Extensions

Example 2.4. Let A → B be a surjective homomorphism between local commutative rings. Consider the groupoids of free modules Gn(A) and Gn(B), for some n. These are NC groupoids. Given a module P ∈ Gn(A) let F(P) := B ⊗A P ∈ Gn(B). In this way we get a morphism of groupoids F : Gn(A) → Gn(B). On automorphisms groups there is a surjection F : GLn(A) → GLn(B), and hence F is a weak epimorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 16 / 46

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SLIDE 56
  • 2. Normal Subgroupoids and Extensions

(cont.) We get an extension of NC groupoids N → Gn(A) F − → Gn(B). The kernels N(P, P) are noncanonically isomorphic to the congruence subgroup {g ∈ GLn(A) | g ≡ 1 mod I}, where I := Ker(A → B).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 17 / 46

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SLIDE 57
  • 2. Normal Subgroupoids and Extensions

(cont.) We get an extension of NC groupoids N → Gn(A) F − → Gn(B). The kernels N(P, P) are noncanonically isomorphic to the congruence subgroup {g ∈ GLn(A) | g ≡ 1 mod I}, where I := Ker(A → B).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 17 / 46

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SLIDE 58
  • 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

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SLIDE 59
  • 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

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SLIDE 60
  • 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

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SLIDE 61
  • 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

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SLIDE 62
  • 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

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SLIDE 63
  • 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

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SLIDE 64
  • 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

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SLIDE 65
  • 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

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SLIDE 66
  • 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

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SLIDE 67
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

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SLIDE 68
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

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SLIDE 69
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

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SLIDE 70
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

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SLIDE 71
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

slide-72
SLIDE 72
  • 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 restV/U : G(U) → G(V) to any inclusion of open sets V ⊂ U. The restriction homomorphisms restV/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

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SLIDE 73
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

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SLIDE 74
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

slide-75
SLIDE 75
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

slide-76
SLIDE 76
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

slide-77
SLIDE 77
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

slide-78
SLIDE 78
  • 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

restV/U : G(U) → G(V) for each inclusion of open sets V ⊂ U. Here the transitivity constraint on the morphisms restV/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

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SLIDE 79
  • 4. Geometrizing NC Groupoids: Gerbes

Figure: Open sets V ⊂ U, and the restriction morphism restV/U : G(U) → G(V) between the groupoids G(U) and G(V).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 22 / 46

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SLIDE 80
  • 4. Geometrizing NC Groupoids: Gerbes

Figure: The restriction morphism restV/U : G(U) → G(V). The objects i0 and i1 become isomorphic in G(V). A new object i2 is created in G(V).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 23 / 46

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SLIDE 81
  • 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

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SLIDE 82
  • 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

slide-83
SLIDE 83
  • 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

slide-84
SLIDE 84
  • 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

slide-85
SLIDE 85
  • 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

slide-86
SLIDE 86
  • 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

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SLIDE 87
  • 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

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SLIDE 88
  • 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

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SLIDE 89
  • 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

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SLIDE 90
  • 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

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SLIDE 91
  • 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

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SLIDE 92
  • 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

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SLIDE 93
  • 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

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SLIDE 94
  • 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

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SLIDE 95
  • 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

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SLIDE 96
  • 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

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SLIDE 97
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

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SLIDE 98
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

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SLIDE 99
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

slide-100
SLIDE 100
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

slide-101
SLIDE 101
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

slide-102
SLIDE 102
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

slide-103
SLIDE 103
  • 4. Geometrizing NC Groupoids: Gerbes

Example 4.1. Let X be an algebraic variety, with sheaf of functions OX, and let n be a positive integer. For any open set U we consider the set Gn(U) of all rank n locally free OU-modules, i.e. rank n vector bundles on U. A morphism P → Q in Gn(U) is by definition an isomorphism of OU-modules. So Gn(U) is a groupoid. The groupoid Gn(U) is nonempty, since it contains the free module On

  • U. But

it could be disconnected, since there could be nonisomorphic vector bundles

  • n U.

As we vary the open set U, we obtain a prestack of groupoids Gn on X. In fact this is a stack (the descent conditions hold). Because any P, Q ∈ Ob(Gn(U)) are locally isomorphic, it follows that Gn is a gerbe.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 27 / 46

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SLIDE 104
  • 4. Geometrizing NC Groupoids: Gerbes

Let G be a gerbe on X. A very important question is this: Is the groupoid G(X) nonempty or connected? (4.1) If the gerbe G is abelian, then there is a cohomological answer to this question: the vanishing of certain obstruction classes in the sheaf cohomology groups Hi(X, N), where N is the center of G (to be defined later), and i = 1, 2. Example 4.2. Suppose X is an algebraic variety, and c is a nonzero class in H2(X, OX). Then there is an explicit way to construct an abelian gerbe G on X with obstruction class c. (This is a classical construction; a variant of it presented in [Ye2].) So G(X) is an empty groupoid.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 28 / 46

slide-105
SLIDE 105
  • 4. Geometrizing NC Groupoids: Gerbes

Let G be a gerbe on X. A very important question is this: Is the groupoid G(X) nonempty or connected? (4.1) If the gerbe G is abelian, then there is a cohomological answer to this question: the vanishing of certain obstruction classes in the sheaf cohomology groups Hi(X, N), where N is the center of G (to be defined later), and i = 1, 2. Example 4.2. Suppose X is an algebraic variety, and c is a nonzero class in H2(X, OX). Then there is an explicit way to construct an abelian gerbe G on X with obstruction class c. (This is a classical construction; a variant of it presented in [Ye2].) So G(X) is an empty groupoid.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 28 / 46

slide-106
SLIDE 106
  • 4. Geometrizing NC Groupoids: Gerbes

Let G be a gerbe on X. A very important question is this: Is the groupoid G(X) nonempty or connected? (4.1) If the gerbe G is abelian, then there is a cohomological answer to this question: the vanishing of certain obstruction classes in the sheaf cohomology groups Hi(X, N), where N is the center of G (to be defined later), and i = 1, 2. Example 4.2. Suppose X is an algebraic variety, and c is a nonzero class in H2(X, OX). Then there is an explicit way to construct an abelian gerbe G on X with obstruction class c. (This is a classical construction; a variant of it presented in [Ye2].) So G(X) is an empty groupoid.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 28 / 46

slide-107
SLIDE 107
  • 4. Geometrizing NC Groupoids: Gerbes

Let G be a gerbe on X. A very important question is this: Is the groupoid G(X) nonempty or connected? (4.1) If the gerbe G is abelian, then there is a cohomological answer to this question: the vanishing of certain obstruction classes in the sheaf cohomology groups Hi(X, N), where N is the center of G (to be defined later), and i = 1, 2. Example 4.2. Suppose X is an algebraic variety, and c is a nonzero class in H2(X, OX). Then there is an explicit way to construct an abelian gerbe G on X with obstruction class c. (This is a classical construction; a variant of it presented in [Ye2].) So G(X) is an empty groupoid.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 28 / 46

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SLIDE 108
  • 4. Geometrizing NC Groupoids: Gerbes

For nonabelian gerbes the story is much more complicated. There is no good answer in general. See the book [Gi] on nonabelian cohomology theory. In the rest of the talk I will explain a partial answer, using central extensions

  • f gerbes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 29 / 46

slide-109
SLIDE 109
  • 4. Geometrizing NC Groupoids: Gerbes

For nonabelian gerbes the story is much more complicated. There is no good answer in general. See the book [Gi] on nonabelian cohomology theory. In the rest of the talk I will explain a partial answer, using central extensions

  • f gerbes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 29 / 46

slide-110
SLIDE 110
  • 5. Extensions of Gerbes and Obstructions
  • 5. Extensions of Gerbes and Obstructions

Definition 5.1. Let G be a gerbe on X. A normal subgroupoid of G is a subprestack of groupoids N ⊂ G such that:

◮ For every open set U the groupoid N (U) is a normal subgroupoid of

G(U).

◮ For every local object i of G, the presheaf of groups N (i, i) is a sheaf.

Warning: a normal subgroupoid N is not a gerbe!

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SLIDE 111
  • 5. Extensions of Gerbes and Obstructions
  • 5. Extensions of Gerbes and Obstructions

Definition 5.1. Let G be a gerbe on X. A normal subgroupoid of G is a subprestack of groupoids N ⊂ G such that:

◮ For every open set U the groupoid N (U) is a normal subgroupoid of

G(U).

◮ For every local object i of G, the presheaf of groups N (i, i) is a sheaf.

Warning: a normal subgroupoid N is not a gerbe!

Amnon Yekutieli (BGU) Central Extensions of Gerbes 30 / 46

slide-112
SLIDE 112
  • 5. Extensions of Gerbes and Obstructions
  • 5. Extensions of Gerbes and Obstructions

Definition 5.1. Let G be a gerbe on X. A normal subgroupoid of G is a subprestack of groupoids N ⊂ G such that:

◮ For every open set U the groupoid N (U) is a normal subgroupoid of

G(U).

◮ For every local object i of G, the presheaf of groups N (i, i) is a sheaf.

Warning: a normal subgroupoid N is not a gerbe!

Amnon Yekutieli (BGU) Central Extensions of Gerbes 30 / 46

slide-113
SLIDE 113
  • 5. Extensions of Gerbes and Obstructions
  • 5. Extensions of Gerbes and Obstructions

Definition 5.1. Let G be a gerbe on X. A normal subgroupoid of G is a subprestack of groupoids N ⊂ G such that:

◮ For every open set U the groupoid N (U) is a normal subgroupoid of

G(U).

◮ For every local object i of G, the presheaf of groups N (i, i) is a sheaf.

Warning: a normal subgroupoid N is not a gerbe!

Amnon Yekutieli (BGU) Central Extensions of Gerbes 30 / 46

slide-114
SLIDE 114
  • 5. Extensions of Gerbes and Obstructions
  • 5. Extensions of Gerbes and Obstructions

Definition 5.1. Let G be a gerbe on X. A normal subgroupoid of G is a subprestack of groupoids N ⊂ G such that:

◮ For every open set U the groupoid N (U) is a normal subgroupoid of

G(U).

◮ For every local object i of G, the presheaf of groups N (i, i) is a sheaf.

Warning: a normal subgroupoid N is not a gerbe!

Amnon Yekutieli (BGU) Central Extensions of Gerbes 30 / 46

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SLIDE 115
  • 5. Extensions of Gerbes and Obstructions

By a morphism of gerbes F : G → H we mean a collection of morphisms of groupoids F : G(U) → H(U) for all open sets U ⊂ X, that is compatible with inclusions of open sets. The morphism F : G → H is called a weak epimorphism if it locally surjective on objects and arrows. (This is the geometric version of weak epimorphism of groupoids.) Let F : G → H be a morphism of gerbes. The kernel of F is the normal subgroupoid N ⊂ G such that N (i, i) := Ker

  • F : G(i, i) → H(F(i), F(i))
  • for every local object i of G. We denote it by Ker(F).

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slide-116
SLIDE 116
  • 5. Extensions of Gerbes and Obstructions

By a morphism of gerbes F : G → H we mean a collection of morphisms of groupoids F : G(U) → H(U) for all open sets U ⊂ X, that is compatible with inclusions of open sets. The morphism F : G → H is called a weak epimorphism if it locally surjective on objects and arrows. (This is the geometric version of weak epimorphism of groupoids.) Let F : G → H be a morphism of gerbes. The kernel of F is the normal subgroupoid N ⊂ G such that N (i, i) := Ker

  • F : G(i, i) → H(F(i), F(i))
  • for every local object i of G. We denote it by Ker(F).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 31 / 46

slide-117
SLIDE 117
  • 5. Extensions of Gerbes and Obstructions

By a morphism of gerbes F : G → H we mean a collection of morphisms of groupoids F : G(U) → H(U) for all open sets U ⊂ X, that is compatible with inclusions of open sets. The morphism F : G → H is called a weak epimorphism if it locally surjective on objects and arrows. (This is the geometric version of weak epimorphism of groupoids.) Let F : G → H be a morphism of gerbes. The kernel of F is the normal subgroupoid N ⊂ G such that N (i, i) := Ker

  • F : G(i, i) → H(F(i), F(i))
  • for every local object i of G. We denote it by Ker(F).

Amnon Yekutieli (BGU) Central Extensions of Gerbes 31 / 46

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SLIDE 118
  • 5. Extensions of Gerbes and Obstructions

Definition 5.2. An extension of gerbes is a diagram N − → G F − → H

  • f morphisms of prestacks of groupoids on X, such that G and H are gerbes;

F is a weak epimorphism; N = Ker(F); and N − → G is the inclusion. In other words, this is the geometrization of the notion of extension of NC groupoids.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 32 / 46

slide-119
SLIDE 119
  • 5. Extensions of Gerbes and Obstructions

Definition 5.2. An extension of gerbes is a diagram N − → G F − → H

  • f morphisms of prestacks of groupoids on X, such that G and H are gerbes;

F is a weak epimorphism; N = Ker(F); and N − → G is the inclusion. In other words, this is the geometrization of the notion of extension of NC groupoids.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 32 / 46

slide-120
SLIDE 120
  • 5. Extensions of Gerbes and Obstructions

Example 5.3. This is a combination of Examples 2.4 and 4.1. Let A and B be sheaves of commutative rings on X, whose stalks are local rings, and let A → B be a surjective homomorphism. (For instance X is an algebraic variety, Z ⊂ X is a closed subvariety, A = OX, and B = OZ.) Let us denote by Gn(A) the gerbe of rank n locally free A-modules, and likewise Gn(B). As before there is a morphism F : Gn(A) → Gn(B). Since the homomorphism of sheaves of groups GLn(A) → GLn(B) is surjective, we get an extension of gerbes N − → Gn(A) F − → Gn(B)

  • n X.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 33 / 46

slide-121
SLIDE 121
  • 5. Extensions of Gerbes and Obstructions

Example 5.3. This is a combination of Examples 2.4 and 4.1. Let A and B be sheaves of commutative rings on X, whose stalks are local rings, and let A → B be a surjective homomorphism. (For instance X is an algebraic variety, Z ⊂ X is a closed subvariety, A = OX, and B = OZ.) Let us denote by Gn(A) the gerbe of rank n locally free A-modules, and likewise Gn(B). As before there is a morphism F : Gn(A) → Gn(B). Since the homomorphism of sheaves of groups GLn(A) → GLn(B) is surjective, we get an extension of gerbes N − → Gn(A) F − → Gn(B)

  • n X.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 33 / 46

slide-122
SLIDE 122
  • 5. Extensions of Gerbes and Obstructions

Example 5.3. This is a combination of Examples 2.4 and 4.1. Let A and B be sheaves of commutative rings on X, whose stalks are local rings, and let A → B be a surjective homomorphism. (For instance X is an algebraic variety, Z ⊂ X is a closed subvariety, A = OX, and B = OZ.) Let us denote by Gn(A) the gerbe of rank n locally free A-modules, and likewise Gn(B). As before there is a morphism F : Gn(A) → Gn(B). Since the homomorphism of sheaves of groups GLn(A) → GLn(B) is surjective, we get an extension of gerbes N − → Gn(A) F − → Gn(B)

  • n X.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 33 / 46

slide-123
SLIDE 123
  • 5. Extensions of Gerbes and Obstructions

Example 5.3. This is a combination of Examples 2.4 and 4.1. Let A and B be sheaves of commutative rings on X, whose stalks are local rings, and let A → B be a surjective homomorphism. (For instance X is an algebraic variety, Z ⊂ X is a closed subvariety, A = OX, and B = OZ.) Let us denote by Gn(A) the gerbe of rank n locally free A-modules, and likewise Gn(B). As before there is a morphism F : Gn(A) → Gn(B). Since the homomorphism of sheaves of groups GLn(A) → GLn(B) is surjective, we get an extension of gerbes N − → Gn(A) F − → Gn(B)

  • n X.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 33 / 46

slide-124
SLIDE 124
  • 5. Extensions of Gerbes and Obstructions

Example 5.3. This is a combination of Examples 2.4 and 4.1. Let A and B be sheaves of commutative rings on X, whose stalks are local rings, and let A → B be a surjective homomorphism. (For instance X is an algebraic variety, Z ⊂ X is a closed subvariety, A = OX, and B = OZ.) Let us denote by Gn(A) the gerbe of rank n locally free A-modules, and likewise Gn(B). As before there is a morphism F : Gn(A) → Gn(B). Since the homomorphism of sheaves of groups GLn(A) → GLn(B) is surjective, we get an extension of gerbes N − → Gn(A) F − → Gn(B)

  • n X.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 33 / 46

slide-125
SLIDE 125
  • 5. Extensions of Gerbes and Obstructions

Definition 5.4. Let G be a gerbe.

  • 1. The center of G is the normal subgroupoid Z(G) of G defined by

Z(G)(i, i) := Z(G(i, i)), the center of the sheaf of groups G(i, i).

  • 2. A central subgroupoid of G is a normal subgroupoid N that is contained

in Z(G). Like is the case of NC groupoids we have: Proposition 5.5. Let N be a central subgroupoid of a gerbe G. Then N can be viewed as a single sheaf of abelian groups.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 34 / 46

slide-126
SLIDE 126
  • 5. Extensions of Gerbes and Obstructions

Definition 5.4. Let G be a gerbe.

  • 1. The center of G is the normal subgroupoid Z(G) of G defined by

Z(G)(i, i) := Z(G(i, i)), the center of the sheaf of groups G(i, i).

  • 2. A central subgroupoid of G is a normal subgroupoid N that is contained

in Z(G). Like is the case of NC groupoids we have: Proposition 5.5. Let N be a central subgroupoid of a gerbe G. Then N can be viewed as a single sheaf of abelian groups.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 34 / 46

slide-127
SLIDE 127
  • 5. Extensions of Gerbes and Obstructions

Definition 5.4. Let G be a gerbe.

  • 1. The center of G is the normal subgroupoid Z(G) of G defined by

Z(G)(i, i) := Z(G(i, i)), the center of the sheaf of groups G(i, i).

  • 2. A central subgroupoid of G is a normal subgroupoid N that is contained

in Z(G). Like is the case of NC groupoids we have: Proposition 5.5. Let N be a central subgroupoid of a gerbe G. Then N can be viewed as a single sheaf of abelian groups.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 34 / 46

slide-128
SLIDE 128
  • 5. Extensions of Gerbes and Obstructions

Definition 5.4. Let G be a gerbe.

  • 1. The center of G is the normal subgroupoid Z(G) of G defined by

Z(G)(i, i) := Z(G(i, i)), the center of the sheaf of groups G(i, i).

  • 2. A central subgroupoid of G is a normal subgroupoid N that is contained

in Z(G). Like is the case of NC groupoids we have: Proposition 5.5. Let N be a central subgroupoid of a gerbe G. Then N can be viewed as a single sheaf of abelian groups.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 34 / 46

slide-129
SLIDE 129
  • 5. Extensions of Gerbes and Obstructions

Definition 5.4. Let G be a gerbe.

  • 1. The center of G is the normal subgroupoid Z(G) of G defined by

Z(G)(i, i) := Z(G(i, i)), the center of the sheaf of groups G(i, i).

  • 2. A central subgroupoid of G is a normal subgroupoid N that is contained

in Z(G). Like is the case of NC groupoids we have: Proposition 5.5. Let N be a central subgroupoid of a gerbe G. Then N can be viewed as a single sheaf of abelian groups.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 34 / 46

slide-130
SLIDE 130
  • 5. Extensions of Gerbes and Obstructions

Finally we can define the notion appearing in the title of the lecture. Definition 5.6. A central extension of gerbes is an extension of gerbes N − → G F − → H such that N is a central subgroupoid of G. By the proposition above we can write this central extension as N − → G F − → H, where N is the sheaf of abelian groups corresponding to N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 35 / 46

slide-131
SLIDE 131
  • 5. Extensions of Gerbes and Obstructions

Finally we can define the notion appearing in the title of the lecture. Definition 5.6. A central extension of gerbes is an extension of gerbes N − → G F − → H such that N is a central subgroupoid of G. By the proposition above we can write this central extension as N − → G F − → H, where N is the sheaf of abelian groups corresponding to N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 35 / 46

slide-132
SLIDE 132
  • 5. Extensions of Gerbes and Obstructions

Finally we can define the notion appearing in the title of the lecture. Definition 5.6. A central extension of gerbes is an extension of gerbes N − → G F − → H such that N is a central subgroupoid of G. By the proposition above we can write this central extension as N − → G F − → H, where N is the sheaf of abelian groups corresponding to N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 35 / 46

slide-133
SLIDE 133
  • 5. Extensions of Gerbes and Obstructions

Here is one of the main results of the paper [Ye1]. Theorem 5.7. Let N − → G F − → H be a central extension of gerbes on X, let i, j be objects of G(X), and let h : F(i) → F(j) be an isomorphism in H(X). Then there is a class cl1

F(h) ∈ H1(X, N),

called the obstruction class of h. The isomorphism h lifts to an isomorphism i → j in G(X) if and only if the

  • bstruction class cl1

F(h) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 36 / 46

slide-134
SLIDE 134
  • 5. Extensions of Gerbes and Obstructions

Here is one of the main results of the paper [Ye1]. Theorem 5.7. Let N − → G F − → H be a central extension of gerbes on X, let i, j be objects of G(X), and let h : F(i) → F(j) be an isomorphism in H(X). Then there is a class cl1

F(h) ∈ H1(X, N),

called the obstruction class of h. The isomorphism h lifts to an isomorphism i → j in G(X) if and only if the

  • bstruction class cl1

F(h) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 36 / 46

slide-135
SLIDE 135
  • 5. Extensions of Gerbes and Obstructions

Here is one of the main results of the paper [Ye1]. Theorem 5.7. Let N − → G F − → H be a central extension of gerbes on X, let i, j be objects of G(X), and let h : F(i) → F(j) be an isomorphism in H(X). Then there is a class cl1

F(h) ∈ H1(X, N),

called the obstruction class of h. The isomorphism h lifts to an isomorphism i → j in G(X) if and only if the

  • bstruction class cl1

F(h) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 36 / 46

slide-136
SLIDE 136
  • 5. Extensions of Gerbes and Obstructions

Here is one of the main results of the paper [Ye1]. Theorem 5.7. Let N − → G F − → H be a central extension of gerbes on X, let i, j be objects of G(X), and let h : F(i) → F(j) be an isomorphism in H(X). Then there is a class cl1

F(h) ∈ H1(X, N),

called the obstruction class of h. The isomorphism h lifts to an isomorphism i → j in G(X) if and only if the

  • bstruction class cl1

F(h) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 36 / 46

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SLIDE 137
  • 5. Extensions of Gerbes and Obstructions

Here is another main result from [Ye1]. Theorem 5.8. Let N − → G F − → H be a central extension of gerbes on X, and let j be an object of H(X). Then there is a class cl2

F(j) ∈ H2(X, N),

called the obstruction class of j. The object j lifts to an object of G(X) if and only if the obstruction class cl2

F(j) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 37 / 46

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SLIDE 138
  • 5. Extensions of Gerbes and Obstructions

Here is another main result from [Ye1]. Theorem 5.8. Let N − → G F − → H be a central extension of gerbes on X, and let j be an object of H(X). Then there is a class cl2

F(j) ∈ H2(X, N),

called the obstruction class of j. The object j lifts to an object of G(X) if and only if the obstruction class cl2

F(j) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 37 / 46

slide-139
SLIDE 139
  • 5. Extensions of Gerbes and Obstructions

Here is another main result from [Ye1]. Theorem 5.8. Let N − → G F − → H be a central extension of gerbes on X, and let j be an object of H(X). Then there is a class cl2

F(j) ∈ H2(X, N),

called the obstruction class of j. The object j lifts to an object of G(X) if and only if the obstruction class cl2

F(j) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 37 / 46

slide-140
SLIDE 140
  • 5. Extensions of Gerbes and Obstructions

Here is another main result from [Ye1]. Theorem 5.8. Let N − → G F − → H be a central extension of gerbes on X, and let j be an object of H(X). Then there is a class cl2

F(j) ∈ H2(X, N),

called the obstruction class of j. The object j lifts to an object of G(X) if and only if the obstruction class cl2

F(j) vanishes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 37 / 46

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SLIDE 141
  • 5. Extensions of Gerbes and Obstructions

Combining these two theorems we obtain: Corollary 5.9. Assume that the cohomology groups H1(X, N) and H2(X, N) are trivial. Then G(X) is an NC groupoid if and only if H(X) is an NC groupoid. Remark 5.10. Related notions of extensions of gerbes have appeared in other recent papers; see references [AN, Be, BX, Ro].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 38 / 46

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  • 5. Extensions of Gerbes and Obstructions

Combining these two theorems we obtain: Corollary 5.9. Assume that the cohomology groups H1(X, N) and H2(X, N) are trivial. Then G(X) is an NC groupoid if and only if H(X) is an NC groupoid. Remark 5.10. Related notions of extensions of gerbes have appeared in other recent papers; see references [AN, Be, BX, Ro].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 38 / 46

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SLIDE 143
  • 5. Extensions of Gerbes and Obstructions

Combining these two theorems we obtain: Corollary 5.9. Assume that the cohomology groups H1(X, N) and H2(X, N) are trivial. Then G(X) is an NC groupoid if and only if H(X) is an NC groupoid. Remark 5.10. Related notions of extensions of gerbes have appeared in other recent papers; see references [AN, Be, BX, Ro].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 38 / 46

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SLIDE 144
  • 5. Extensions of Gerbes and Obstructions

Combining these two theorems we obtain: Corollary 5.9. Assume that the cohomology groups H1(X, N) and H2(X, N) are trivial. Then G(X) is an NC groupoid if and only if H(X) is an NC groupoid. Remark 5.10. Related notions of extensions of gerbes have appeared in other recent papers; see references [AN, Be, BX, Ro].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 38 / 46

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SLIDE 145
  • 6. Pronilpotent Gerbes (optional)
  • 6. Pronilpotent Gerbes

In this last section I will define pronilpotent gerbes. For a pronilpotent gerbe G on a topological space X, I will give a sufficient condition for G(X) to be an NC groupoid. A morphism of gerbes E : H → H′ is called an equivalence if for every open set U the functor F : H(U) → H′(U) is an equivalence. Theorem 6.1. Let N be a normal subgroupoid of a gerbe G. Then there exists an extension of gerbes N − → G F − → H. The gerbe H is unique up to equivalence, and we denote it by G/N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 39 / 46

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SLIDE 146
  • 6. Pronilpotent Gerbes (optional)
  • 6. Pronilpotent Gerbes

In this last section I will define pronilpotent gerbes. For a pronilpotent gerbe G on a topological space X, I will give a sufficient condition for G(X) to be an NC groupoid. A morphism of gerbes E : H → H′ is called an equivalence if for every open set U the functor F : H(U) → H′(U) is an equivalence. Theorem 6.1. Let N be a normal subgroupoid of a gerbe G. Then there exists an extension of gerbes N − → G F − → H. The gerbe H is unique up to equivalence, and we denote it by G/N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 39 / 46

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SLIDE 147
  • 6. Pronilpotent Gerbes (optional)
  • 6. Pronilpotent Gerbes

In this last section I will define pronilpotent gerbes. For a pronilpotent gerbe G on a topological space X, I will give a sufficient condition for G(X) to be an NC groupoid. A morphism of gerbes E : H → H′ is called an equivalence if for every open set U the functor F : H(U) → H′(U) is an equivalence. Theorem 6.1. Let N be a normal subgroupoid of a gerbe G. Then there exists an extension of gerbes N − → G F − → H. The gerbe H is unique up to equivalence, and we denote it by G/N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 39 / 46

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SLIDE 148
  • 6. Pronilpotent Gerbes (optional)
  • 6. Pronilpotent Gerbes

In this last section I will define pronilpotent gerbes. For a pronilpotent gerbe G on a topological space X, I will give a sufficient condition for G(X) to be an NC groupoid. A morphism of gerbes E : H → H′ is called an equivalence if for every open set U the functor F : H(U) → H′(U) is an equivalence. Theorem 6.1. Let N be a normal subgroupoid of a gerbe G. Then there exists an extension of gerbes N − → G F − → H. The gerbe H is unique up to equivalence, and we denote it by G/N .

Amnon Yekutieli (BGU) Central Extensions of Gerbes 39 / 46

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  • 6. Pronilpotent Gerbes (optional)

The full normal subgroupoid of G is the normal subgroupoid N defined by taking N (i, i) := G(i, i) for every local object i of G. Definition 6.2. A central filtration on the gerbe G is a descending sequence {N p}p≥0 of normal subgroupoids of G, such that:

◮ N 0 is the full normal subgroupoid. ◮ For every p the extension of gerbes

N p/N p+1 → G/N p+1 → G/N p is central.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 40 / 46

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  • 6. Pronilpotent Gerbes (optional)

The full normal subgroupoid of G is the normal subgroupoid N defined by taking N (i, i) := G(i, i) for every local object i of G. Definition 6.2. A central filtration on the gerbe G is a descending sequence {N p}p≥0 of normal subgroupoids of G, such that:

◮ N 0 is the full normal subgroupoid. ◮ For every p the extension of gerbes

N p/N p+1 → G/N p+1 → G/N p is central.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 40 / 46

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SLIDE 151
  • 6. Pronilpotent Gerbes (optional)

The full normal subgroupoid of G is the normal subgroupoid N defined by taking N (i, i) := G(i, i) for every local object i of G. Definition 6.2. A central filtration on the gerbe G is a descending sequence {N p}p≥0 of normal subgroupoids of G, such that:

◮ N 0 is the full normal subgroupoid. ◮ For every p the extension of gerbes

N p/N p+1 → G/N p+1 → G/N p is central.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 40 / 46

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SLIDE 152
  • 6. Pronilpotent Gerbes (optional)

The full normal subgroupoid of G is the normal subgroupoid N defined by taking N (i, i) := G(i, i) for every local object i of G. Definition 6.2. A central filtration on the gerbe G is a descending sequence {N p}p≥0 of normal subgroupoids of G, such that:

◮ N 0 is the full normal subgroupoid. ◮ For every p the extension of gerbes

N p/N p+1 → G/N p+1 → G/N p is central.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 40 / 46

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  • 6. Pronilpotent Gerbes (optional)

Definition 6.3. A pronilpotent gerbe is a gerbe G, equipped with a central filtration {N p}p≥0, satisfying the following completeness condition: For every local object i of G, the canonical homomorphism of sheaves of groups G(i, i) → lim

←p G(i, i)/N p(i, i)

is an isomorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 41 / 46

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SLIDE 154
  • 6. Pronilpotent Gerbes (optional)

Definition 6.3. A pronilpotent gerbe is a gerbe G, equipped with a central filtration {N p}p≥0, satisfying the following completeness condition: For every local object i of G, the canonical homomorphism of sheaves of groups G(i, i) → lim

←p G(i, i)/N p(i, i)

is an isomorphism.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 41 / 46

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SLIDE 155
  • 6. Pronilpotent Gerbes (optional)

For every p the normal subgroupoid N p/N p+1 ⊂ G/N p+1 is central. So according to Proposition 5.5 we can consider N p/N p+1 as a sheaf of abelian groups. We denote this sheaf by Np, and we call it the p-th abelian slice of G. The next example is our reason for developing the theory of extensions of gerbes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 42 / 46

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SLIDE 156
  • 6. Pronilpotent Gerbes (optional)

For every p the normal subgroupoid N p/N p+1 ⊂ G/N p+1 is central. So according to Proposition 5.5 we can consider N p/N p+1 as a sheaf of abelian groups. We denote this sheaf by Np, and we call it the p-th abelian slice of G. The next example is our reason for developing the theory of extensions of gerbes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 42 / 46

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SLIDE 157
  • 6. Pronilpotent Gerbes (optional)

For every p the normal subgroupoid N p/N p+1 ⊂ G/N p+1 is central. So according to Proposition 5.5 we can consider N p/N p+1 as a sheaf of abelian groups. We denote this sheaf by Np, and we call it the p-th abelian slice of G. The next example is our reason for developing the theory of extensions of gerbes.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 42 / 46

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SLIDE 158
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

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SLIDE 159
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

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SLIDE 160
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

slide-161
SLIDE 161
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

slide-162
SLIDE 162
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

slide-163
SLIDE 163
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

slide-164
SLIDE 164
  • 6. Pronilpotent Gerbes (optional)

Example 6.4. Let X be a smooth algebraic variety over a field K of characteristic 0. Consider the ring of formal power series K[[]] in a variable . Let U ⊂ X be an open set. An associative deformation of OU is a sheaf A of flat complete associative K[[]]-algebras on U, with an isomorphism K ⊗K[[]] A ∼ = OU. There is a similar notion of Poisson deformation of OU. An isomorphism of deformations is called a gauge equivalence. In [Ye2] we introduced the notion of twisted deformation of OX. A twisted (associative or Poisson) deformation A consists of a gerbe G, called the gauge gerbe of A.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 43 / 46

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SLIDE 165
  • 6. Pronilpotent Gerbes (optional)

(cont.) To any open set U and local object i ∈ Ob(G(U)) we attach a deformation Ai of OU. And to any local arrow g : i → j in G we attach a gauge equivalence A(g) : Ai → Aj. All these data have to satisfy certain (rather complicated) relations. The gauge gerbe G is usually nonabelian, but it is pronilpotent. Its central filtration is called the -adic filtration. For every p, the abelian slice Np is isomorphic, as a sheaf of abelian groups, to OX.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 44 / 46

slide-166
SLIDE 166
  • 6. Pronilpotent Gerbes (optional)

(cont.) To any open set U and local object i ∈ Ob(G(U)) we attach a deformation Ai of OU. And to any local arrow g : i → j in G we attach a gauge equivalence A(g) : Ai → Aj. All these data have to satisfy certain (rather complicated) relations. The gauge gerbe G is usually nonabelian, but it is pronilpotent. Its central filtration is called the -adic filtration. For every p, the abelian slice Np is isomorphic, as a sheaf of abelian groups, to OX.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 44 / 46

slide-167
SLIDE 167
  • 6. Pronilpotent Gerbes (optional)

(cont.) To any open set U and local object i ∈ Ob(G(U)) we attach a deformation Ai of OU. And to any local arrow g : i → j in G we attach a gauge equivalence A(g) : Ai → Aj. All these data have to satisfy certain (rather complicated) relations. The gauge gerbe G is usually nonabelian, but it is pronilpotent. Its central filtration is called the -adic filtration. For every p, the abelian slice Np is isomorphic, as a sheaf of abelian groups, to OX.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 44 / 46

slide-168
SLIDE 168
  • 6. Pronilpotent Gerbes (optional)

(cont.) To any open set U and local object i ∈ Ob(G(U)) we attach a deformation Ai of OU. And to any local arrow g : i → j in G we attach a gauge equivalence A(g) : Ai → Aj. All these data have to satisfy certain (rather complicated) relations. The gauge gerbe G is usually nonabelian, but it is pronilpotent. Its central filtration is called the -adic filtration. For every p, the abelian slice Np is isomorphic, as a sheaf of abelian groups, to OX.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 44 / 46

slide-169
SLIDE 169
  • 6. Pronilpotent Gerbes (optional)

(cont.) To any open set U and local object i ∈ Ob(G(U)) we attach a deformation Ai of OU. And to any local arrow g : i → j in G we attach a gauge equivalence A(g) : Ai → Aj. All these data have to satisfy certain (rather complicated) relations. The gauge gerbe G is usually nonabelian, but it is pronilpotent. Its central filtration is called the -adic filtration. For every p, the abelian slice Np is isomorphic, as a sheaf of abelian groups, to OX.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 44 / 46

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SLIDE 170
  • 6. Pronilpotent Gerbes (optional)

The following theorem is the third main result of [Ye1]. Theorem 6.5. Let G be a pronilpotent gerbe on X, with abelian slices {Np}p≥0, and let U ⊂ X be an open set.

  • 1. If the abelian groups H2(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is nonempty.

  • 2. If the abelian groups H1(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is connected.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 45 / 46

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SLIDE 171
  • 6. Pronilpotent Gerbes (optional)

The following theorem is the third main result of [Ye1]. Theorem 6.5. Let G be a pronilpotent gerbe on X, with abelian slices {Np}p≥0, and let U ⊂ X be an open set.

  • 1. If the abelian groups H2(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is nonempty.

  • 2. If the abelian groups H1(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is connected.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 45 / 46

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SLIDE 172
  • 6. Pronilpotent Gerbes (optional)

The following theorem is the third main result of [Ye1]. Theorem 6.5. Let G be a pronilpotent gerbe on X, with abelian slices {Np}p≥0, and let U ⊂ X be an open set.

  • 1. If the abelian groups H2(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is nonempty.

  • 2. If the abelian groups H1(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is connected.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 45 / 46

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SLIDE 173
  • 6. Pronilpotent Gerbes (optional)

The following theorem is the third main result of [Ye1]. Theorem 6.5. Let G be a pronilpotent gerbe on X, with abelian slices {Np}p≥0, and let U ⊂ X be an open set.

  • 1. If the abelian groups H2(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is nonempty.

  • 2. If the abelian groups H1(U, Np) are trivial for all p ≥ 0, then the

groupoid G(U) is connected.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 45 / 46

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SLIDE 174
  • 6. Pronilpotent Gerbes (optional)

Applying the theorem to the setup of Example 6.4 we obtain: Corollary 6.6. Let X be a smooth algebraic variety in characteristic 0, and let G be the gauge gerbe of some twisted deformation. Then for any affine open set U the groupoid G(U) is nonempty and connected. This corollary plays a crucial role in the paper [Ye2].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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SLIDE 175
  • 6. Pronilpotent Gerbes (optional)

Applying the theorem to the setup of Example 6.4 we obtain: Corollary 6.6. Let X be a smooth algebraic variety in characteristic 0, and let G be the gauge gerbe of some twisted deformation. Then for any affine open set U the groupoid G(U) is nonempty and connected. This corollary plays a crucial role in the paper [Ye2].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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SLIDE 176
  • 6. Pronilpotent Gerbes (optional)

Applying the theorem to the setup of Example 6.4 we obtain: Corollary 6.6. Let X be a smooth algebraic variety in characteristic 0, and let G be the gauge gerbe of some twisted deformation. Then for any affine open set U the groupoid G(U) is nonempty and connected. This corollary plays a crucial role in the paper [Ye2].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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  • 6. Pronilpotent Gerbes (optional)

Applying the theorem to the setup of Example 6.4 we obtain: Corollary 6.6. Let X be a smooth algebraic variety in characteristic 0, and let G be the gauge gerbe of some twisted deformation. Then for any affine open set U the groupoid G(U) is nonempty and connected. This corollary plays a crucial role in the paper [Ye2].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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SLIDE 178
  • 6. Pronilpotent Gerbes (optional)

Applying the theorem to the setup of Example 6.4 we obtain: Corollary 6.6. Let X be a smooth algebraic variety in characteristic 0, and let G be the gauge gerbe of some twisted deformation. Then for any affine open set U the groupoid G(U) is nonempty and connected. This corollary plays a crucial role in the paper [Ye2].

  • END -

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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  • 6. Pronilpotent Gerbes (optional)
  • E. Aldrovandi and B. Noohi, Butterflies I: Morphisms of 2-group stacks,
  • Adv. Math. 221 (2009), 687-773.
  • C. Bertolin, Extensions of strictly commutative Picard stacks, eprint

arXiv:0906.2179v2

  • L. Breen, Théorie de Schreier supérieure, Ann. Sci. École Norm. Sup. 25
  • no. 5 (1992), 465-514.

Differentiable Stacks and Gerbes, K. Behrend and P. Xu, eprint arXiv:math/0605694v2.

  • J. Giraud, “Cohomologie non abelienne,” Grundlehren der Math. Wiss.

179, Springer (1971).

  • M. Kontsevich, Deformation quantization of algebraic varieties, Lett.
  • Math. Phys. 56 (2001), no. 3, 271-294.
  • A. Rousseau, Bicatégories monoïdales et extensions de gr-catégories,

Homology Homotopy Appl. 5 No. 1 (2003), pp. 437-547.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46

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  • 6. Pronilpotent Gerbes (optional)
  • A. Yekutieli, Central Extensions of Gerbes, Adv. Math. 225 (2010),

445-486.

  • A. Yekutieli, Twisted Deformation Quantization of Algebraic Varieties,

eprint arXiv:0905.0488 at http://arxiv.org.

Amnon Yekutieli (BGU) Central Extensions of Gerbes 46 / 46