From Principal 2-Bundles to Bundle Gerbes

In this section we define the 2-functor

E

M : 2-Bun_Γ(M) ^// Grb_Γ(M).

Definition of

E

M on objects

LetP be a principal Γ-2-bundle overM, with projectionπ:P ^// M and right action R of Γ on P. The first ingredient of the Γ-bundle gerbe

E

M(P) is the surjective submersion π:P₀ ^// M. The second ingredient is a principal Γ-bundleP overP₀^[2]. We put

P :=P1×Γ0.

Bundle projection, anchor and Γ-action are given, respectively, by χ(ρ, g) := (t(ρ), R(s(ρ), g⁻¹)) , α(ρ, g) :=g

and (ρ, g)◦γ := (R(ρ,id_g⁻¹ ·γ), s(γ)). (3.14) These definitions are motivated by Remark 3.7.4 below.

Lemma 3.7.3. This defines a principal Γ-bundle over P₀^[2].

Proof. First we check thatχ:P ^// P₀^[2] is a surjective submersion. Since the functor τ = (id, R) is a weak equivalence, we know from Theorem 3.2.23 that

f : (P0×Γ0)τ×t×tP₁^{[2] //} P₀^[2] : (p, g, ρ1, ρ2) ^// (s(ρ1), s(ρ2)) is a surjective submersion. Now consider the smooth surjective map

g : (P₀×Γ₀)_τ×t×tP₁^{[2] //} P₁×Γ₀ : (p, g, ρ₁, ρ₂) ^// (ρ⁻¹₁ ◦R(ρ₂,id_g⁻¹), g⁻¹).

We have χ◦g =f; thus, χ is a surjective submersion. Next we check that we have defined an action. Suppose (ρ, g)∈P andγ ∈Γ₁such thatα(ρ, g) =g =t(γ). Then, α((ρ, g)◦γ) =s(γ). Moreover, suppose γ₁, γ₂ ∈Γ₁ with t(γ₁) = g and t(γ₂) = s(γ₁).

Then,

((ρ, g)◦γ1)◦γ2 = (R(ρ,id_g⁻¹ ·γ1), s(γ1))◦γ2

= (R(ρ,id_g⁻¹ ·γ₁·id_s(γ1)⁻¹ ·γ₂), s(γ₂)) = (ρ, g)◦(γ₁◦γ₂), where we have used thatγ1◦γ2 =γ1·id_s(γ1)⁻¹·γ2 in any 2-group. It remains to check that the smooth map

˜

τ :P _α×_tΓ₁ ^// P _χ×_χP : ((ρ, g), γ) ^// ((ρ, g),(ρ, g)◦γ)

Equivalence between Bundle Gerbes and 2-Bundles 111 is a diffeomorphism. For this purpose, we consider the diagram

P₁^[2]

s×t

(P₀×Γ₀)×(P₀×Γ₀) _τ×τ ^//P₀^[2]× P₀^[2]

(3.15)

and claim that (a) N₁ := P _α×_t Γ₁ is a pullback of (3.15), (b) N₂ := P _χ×_χ P is a pullback of (3.15), and (c) that the unique map N1 // N2 is ˜τ. Thus, ˜τ is a diffeomorphism.

In order to prove claim (a) we use again that the functor τ = (id, R) is a weak equivalence, so that by Theorem 3.2.23 the triple (P₁×Γ₁, τ, s×t) is a pullback of (3.15). We consider the smooth map

ξ:N₁ ^// P₁×Γ₁ : ((ρ, g), γ) ^// (R(ρ,id_g⁻¹), γ)

which is a diffeomorphism because (ρ, γ) ^// ((R(ρ,id_t(γ)), t(γ)), γ) is a smooth map which is inverse to ξ. Thus, putting f₁ := τ ◦ξ and g₁ := (s×t)◦ξ we see that (N1, f1, g1) is a pullback of (3.15). In order to prove claim (b), we put

f₂((ρ₁, g₁),(ρ₂, g₂)) := (R(ρ₁,id_g⁻¹

1 ), ρ₂)

g₂((ρ₁, g₁),(ρ₂, g₂)) := (R(s(ρ), g⁻¹₁ ), g₂, R(t(ρ₁), g₁⁻¹), g₁),

and it is straightforward to check that the cone (N₂, f₂, g₂) makes (3.15) commutative.

The triple (N₂, f₂, g₂) is also universal: in order to see this supposeN⁰ is any smooth manifold with smooth maps f⁰ : N^{0 //} P₁^[2] and g⁰ : N^{0 //} (P₀×Γ₀)×(P₀ ×Γ₀) so that (3.15) is commutative. For n ∈ N⁰, we write f⁰(n) = (ρ₁, ρ₂) and g⁰(n) = (p₁, g₁, p₂, g₂). Then, σ(n) := ((R(ρ₁,id_g⁻¹

2 ), g₂),(ρ₂, g₁)) defines a smooth map σ : N^{0 //} P χ×χP. One checks that f2◦σ =f⁰ and g2◦σ=g⁰, and that σ is the only smooth map satisfying these equations. This proves that (N₂, f₂, g₂) is a pullback.

We are left with claim (c). Here one only has to check that τ : N₁ ^// N₂ satisfies f2 =f1◦τ and g2 =g1◦τ.

Remark 3.7.4. The smooth functor τ = (id, R) : P × Γ ^// P ×_M P is a weak equivalence, and so has a canonical inverse anafunctor τ⁻¹ (Remark 3.2.24). The anafunctor

P₀^{[2] ι //}P ×_M P ^{c //}P ×_M P ^{τ−1 //}P ×Γ ^{pr2 //}Γ,

where c is the functor that switches the factors, corresponds to a principal Γ-bundle over P₀^[2] that is canonically isomorphic to the bundle P defined above.

112 Four Equivalent Versions of Non-Abelian Gerbes It remains to provide the bundle gerbe product

µ:π₂₃^∗ P ⊗π₁₂^∗ P ^// π₁₃^∗ P, which we define by the formula

µ((ρ₂₃, g₂₃),(ρ₁₂, g₁₂)) := (ρ₁₂◦R(ρ₂₃,id_g12), g₂₃g₁₂). (3.16) Lemma 3.7.5. Formula (3.16) defines an associative isomorphism µ : π₂₃^∗ P ⊗ π₁₂^∗ P ^// π^∗₁₃P of principal Γ-bundles over P₀^[3].

Proof. First of all, we recall from Example 3.2.31 (b) that an element in the tensor product π^∗₂₃P ⊗π^∗₁₂P is represented by a triple (p₂₃, p₁₂, γ) where p₂₃, p₁₂ ∈ P with π₁(χ(p₂₃)) =π₂(χ(p₁₂)), andα(p₂₃)·α(p₁₂) =t(γ). In (3.16) we refer to triples where γ = id_g23g12, and this definition extends to triples with general γ ∈Γ₁ by employing the equivalence relation

(p₁, p₂, γ)∼(p₁◦(γ·id_α(p2)⁻¹), p₂,id_s(γ)). (3.17) The complete formula for µis then

µ((ρ₂₃, g₂₃),(ρ₁₂, g₁₂), γ) = (ρ₁₂◦R(ρ₂₃,id_g⁻¹

23 ·γ), s(γ)). (3.18) Next we check that (3.18) is well-defined under the equivalence relation (3.17):

µ(((ρ₂₃, g₂₃),(ρ₁₂, g₁₂), γ))

= (ρ₁₂◦R(ρ₂₃,id_g⁻¹

23 ·γ), s(γ))

= (ρ₁₂◦R(ρ₂₃◦R(id_R(s(ρ

23),g⁻¹₂₃), γ·id_g⁻¹

12),id_g12), s(γ))

=µ((ρ₂₃◦R(id_R(s(ρ

23),g₂₃⁻¹), γ·id_g⁻¹

12), s(γ)g₁₂⁻¹),(ρ₁₂, g₁₂),id_s(γ)))

=µ(((ρ₂₃, g₂₃)◦(γ ·id_g⁻¹

12),(ρ₁₂, g₁₂),id_s(γ))).

Now we have shown that µis a well-defined map from π^∗₂₃P ⊗π₁₂^∗ P to π^∗₁₃P, and it remains to prove that it is a bundle morphism. Checking that it preserves fibres and anchors is straightforward. It remains to check that (3.18) preserves the Γ-action.

We calculate

µ(((ρ₂₃, g₂₃),(ρ₁₂, g₁₂), γ)◦˜γ)

=µ((ρ₂₃, g₂₃),(ρ₁₂, g₁₂), γ◦γ)˜

= (ρ₂₃◦R(ρ₁₂,id_g12·i(γ◦γ)), s(˜˜ γ))

= (ρ₂₃◦R(R(ρ₁₂,id_g12), i(γ)◦i(˜γ)), s(˜γ))

= (ρ23◦R(R(ρ12,idg12), i(γ)))◦R(id_R(s(ρ12),g), i(˜γ)), s(˜γ))

= (ρ₂₃◦R(ρ₁₂,id_g12·i(γ))◦R(id_R(s(ρ12),g), i(˜γ)), s(˜γ))

= (ρ₂₃◦R(ρ₁₂,id_g12·i(γ)), s(γ))◦γ˜

=µ((ρ₂₃, g₂₃),(ρ₁₂, g₁₂), γ)◦˜γ.

Equivalence between Bundle Gerbes and 2-Bundles 113 Summarizing, µis a morphism of Γ-bundles overP₀^[3]. The associativity of µfollows directly from the definitions.

Definition of

E

M on 1-morphisms

We define a 1-morphism

E

M(F) :

E

M(P) ^//

E

M(P⁰) between Γ-bundle gerbes from a 1-morphism F : P ^// P⁰ between principal Γ-2-bundles. The refinement of the surjective submersions π : P ^// M and π⁰ : P^{0 //} M is the fibre product Z :=

P₀×_M P₀⁰. Its principal Γ-bundle has the total space Q:=F ×Γ₀,

and its projection, anchor and Γ-action are given, respectively, by χ(f, g) := (α_l(f), R(α_r(f), g⁻¹)), α(f, g) :=g

and (f, g)◦γ := (ρ(f,id_g⁻¹ ·γ), s(γ)), (3.19) where ρ : F ×Γ1 // F denotes the Γ1-action on F that comes from the given Γ-equivariant structure on F (see Appendix 3.8.1).

Lemma 3.7.6. This defines a principal Γ-bundle Q over Z.

Proof. We show first the the projectionχ:Q ^// Z is a surjective submersion. Since the functorτ⁰ :P⁰×Γ ^// P ×_MP is a weak equivalence, we have by Theorem 3.2.23 a pullback

X ^//

ξ

(P₀⁰ ×Γ₀)_R×_t(P₁⁰ ×_M P₁⁰)

s◦pr₂

F _π⁰◦α_l(f)×_π⁰ P₀^{0 //}P₀⁰ ×_M P₀⁰

along the bottom map (f, p⁰) ^// (α_r(f), p⁰), which is well-defined because the ana-functor F preserves the projections to M (see Remark 3.6.6 (b)). In particular, the map ξ is a surjective submersion. It is easy to see that the smooth map

k :X ^// F ×Γ₀ : ((f, p⁰),(p⁰₀, g, ρ,ρ))˜ ^// (f◦ρ⁻¹◦R( ˜ρ,id_g⁻¹), g⁻¹) is surjective. Now we consider the commutative diagram

X

ξ

k //F ×Γ₀

χ

F _π⁰◦α_l(f)×_π⁰ P₀

αl×id //P₀×_M P₀⁰.

114 Four Equivalent Versions of Non-Abelian Gerbes The surjectivity ofk and the fact thatξandα_l×id are surjective submersions shows that χ is one, too.

Next, one checks (similarly as in the proof of Lemma 3.7.3) that the Γ-action on Qdefined above is well-defined and preserves the projection. Then it remains to check that the smooth map

ξ : Q_α×_tΓ₁ ^// Q×_P0×MP⁰

0 Q: (f, g, γ) ^// (f, g, ρ(f,id_g⁻¹ ·γ), s(γ))

is a diffeomorphism. An inverse map is given as follows. For a given element (f₁, g₁, f₂, g₂) on the right hand side, we have α_l(f₁) = α_l(f₂), so that there ex-ists a unique element ρ⁰ ∈ P₁⁰ such that f1◦ρ⁰ =f2. One calculates that (ρ⁰, g2) and (id_αr(f1), g₁) are elements of the principal Γ-bundleP⁰×Γ₀ overP₀^0[2] of Lemma 3.7.3.

Thus, there exists a unique element γ ∈ Γ₁ such that (ρ⁰, g₂) = (id_αr(f1), g₁) ◦γ.

Clearly, t(γ) = g₁ and s(γ) = g₂, and we have ρ⁰ = R(id_αr(f1),id_g⁻¹

1 ·γ). We de-fine ξ⁻¹(f1, g1, f2, g2) := (f1, g1, γ). The calculation that ξ⁻¹ is an inverse for ξ uses property (ii) of Definition 3.8.1 for the action ρ, and is left to the reader.

The next step in the definition of the 1-morphism

E

^(F) is to define the bundle morphism

β :P⁰⊗ζ₁^∗Q ^// ζ₂^∗Q⊗P

over Z ×M Z. We use the notation of Example 3.2.31 (b) for elements of tensor products of principal Γ-bundles; in this notation, the morphism β in the fibre over a point ((p₁, p⁰₁),(p₂, p⁰₂))∈Z×_M Z is given by

β : ((ρ⁰, g⁰),(f, g), γ) ^// (( ˜f , g⁰gh),( ˜ρ, h⁻¹), γ),

where h∈Γ0 and ˜ρ∈ P₁⁰ are chosen such that s( ˜ρ) =R(p2, h⁻¹) and t( ˜ρ) = p1, and f˜:=ρ( ˜ρ⁻¹ ◦f ◦R(ρ⁰,idg),idh). (3.20) Lemma 3.7.7. This defines an isomorphism between principal Γ-bundles.

Proof. The existence of choices of ˜ρ, hfollows because the functorτ⁰ :P⁰×Γ ^// P⁰×_M P⁰ is smoothly essentially surjective (Theorem 3.2.23); in particular, one can choose them locally in a smooth way. We claim that the equivalence relation on ζ₂^∗Q⊗P identifies different choices; thus, we have a well-defined smooth map. In order to prove this claim, we assume other choices ˜ρ⁰, h⁰. The pairs ( ˜ρ, h⁻¹) and ( ˜ρ⁰, h⁰⁻¹) are elements in the principal Γ-bundle P⁰ over P₀⁰ ×_M P₀⁰ and sit over the same fibre;

thus, there exists a unique ˜γ ∈ Γ₁ such that ( ˜ρ, h⁻¹)◦˜γ = ( ˜ρ⁰, h⁰⁻¹), in particular, R( ˜ρ,id_h·γ) = ˜˜ ρ⁰. Now we have

(( ˜f , g⁰gh),( ˜ρ, h⁻¹), γ) = (( ˜f , g⁰gh),( ˜ρ, h⁻¹),(id_t(γ)·i(˜γ)·˜γ)◦γ)

∼(( ˜f , g⁰gh)◦(idt(γ)·i(˜γ)),( ˜ρ, h⁻¹)◦γ, γ)˜

Equivalence between Bundle Gerbes and 2-Bundles 115 so that it suffices to calculate

( ˜f , g⁰gh)◦(id_t(γ)·i(˜γ)) = (ρ( ˜f ,id_h⁻¹ ·i(˜γ)), g⁰gh⁰)

= (ρ( ˜ρ⁻¹◦f◦R(ρ⁰,idg), i(˜γ)), g⁰gh⁰)

= (ρ(R( ˜ρ⁻¹, i(˜γ)·id_h⁰⁻¹)◦f ◦R(ρ⁰,id_g),id_h⁰), g⁰gh⁰), where the last step uses the compatibility condition for ρ from Definition 3.8.1 (ii).

In any 2-group, we have i(˜γ)·id_s(˜γ) = (id_t(˜γ)⁻¹ ·γ)˜ ⁻¹, in which case the last line is exactly the formula (3.20) for the pair ( ˜ρ⁰, h⁰).

Next we check thatβ is well-defined under the equivalence relation on the tensor product P⁰⊗ζ₁^∗Q. We have

x:= ((ρ⁰, g⁰),(f, g),(γ₁·γ₂)◦γ)∼((ρ⁰, g⁰)◦γ₁,(f, g)◦γ₂, γ) =: x⁰

for γ₁, γ₂ ∈Γ₁ such that t(γ₁) = g⁰, t(γ₂) = g and s(γ₁)s(γ₂) =t(γ). Taking advan-tage of the fact that we can make the same choice of ( ˜ρ, h) for both representatives x and x⁰, it is straightforward to show that β(x) =β(x⁰). Finally, it is obvious from the definition of β that it is anchor-preserving and Γ-equivariant.

In order to show that the triple (Z, Q, β) defines a 1-morphism between bundle gerbes, it remains to verify that the bundle isomorphismβ is compatible the with the bundle gerbe productsµ₁andµ₂in the sense of diagram (3.8). This is straightforward to do and left for the reader.

Definition of

E

M on 2-morphisms, compositors and unitors

Let F1, F2 : P ^// P⁰ be 1-morphisms between principal Γ-bundles over M, and let η:F ⁺³ Gbe a 2-morphism. Between the Γ-bundlesQ₁ andQ₂, which live over the same common refinement Z =P₀×_M P₀⁰, we find immediately the smooth map

η:Q1 // Q2 : (f1, g) ^// (η(f1), g)

which is easily verified to be a bundle morphism. Its compatibility with the bundle morphisms β₁ and β₂ in the sense of the simplified diagram (3.11) is also easy to check. Thus, we have defined a 2-morphism

E

M(η) :

E

M(F₁) ⁺³

E

M(F₂).

The compositor for 1-morphisms F₁ : P ^// P⁰ and F₂ : P^{0 //} P⁰⁰ is a bundle gerbe 2-morphism

c_F1,F2 :

E

M(F₂◦F₁) ^//

E

M(F₂)◦

E

M(F₁).

Employing the above constructions, the 1-morphism

E

M(F₂◦F₁) is defined on the common refinementZ₁₂ :=P₀×_MP₀⁰⁰and has the Γ-bundleQ₁₂ = (F₁×P₀⁰F₂)/P₁⁰×Γ₀, whereas the 1-morphism

E

M(F₂)◦

E

M(F₁) is defined on the common refinement Z := P₀ ×_M P₀⁰ ×_M P₀⁰⁰ and has the Γ-bundle Q₂ ⊗Q₁ with Q_k = F_k ×Γ₀. The

116 Four Equivalent Versions of Non-Abelian Gerbes compositor c_F1,F2 is defined over the refinementZ with the obvious refinement maps pr₁₃ : Z ^// Z₁₂ and id : Z ^// Z making diagram (3.10) commutative. It is thus a bundle morphism c_F1,F2 : pr^∗₁₃Q₁₂ ^// Q₂⊗Q₁. For elements in a tensor product of Γ-bundles we use the notation of Example 3.2.31 (b). Then, we define c_F1,F2 by

((p, p⁰, p⁰⁰),(f₁, f₂, g)) ^// ((ρ₂( ˜ρ⁻¹◦f₂,id_h), gh),(f₁◦ρ, h˜ ⁻¹),id_g), (3.21) where h∈Γ0 and ˜ρ:R(p⁰, h⁻¹) ^// αr(f1) =αl(f2) are chosen in the same way as in the proof of Lemma 3.7.7. The assignment (3.21) does not depend on the choices of h and ˜ρ, and also not on the choice of the representative (f₁, f₂) in (F₁×P₀⁰ F₂)/P₁⁰. It is obvious that (3.21) is anchor-preserving, and its Γ-equivariance can be seen by choosing ( ˜ρ, h) in order to compute c_F1,F2((p, p⁰, p⁰⁰),(f₁, f₂, g)) and ( ˜ρ⁰, h) with

˜

ρ⁰ :=R( ˜ρ,id_g⁻¹ ·γ⁻¹) in order to compute c_F1,F2(((p, p⁰, p⁰⁰),(f₁, f₂, g))◦γ). In order to complete the construction of the bundle gerbe 2-morphismc_F1,F2 we have to prove that the bundle morphismc_F1,F2 is compatible with the isomorphismsβ₁₂of

E

M(F₂◦ F₁) and (id⊗β₁)◦(β₂⊗id) of

E

M(F₂)◦

E

M(F₁) in the sense of diagram (3.11). We start with an element ((ρ⁰⁰, g⁰⁰),(f₁₂, g))∈

E

M(P⁰⁰)⊗ζ₁^∗Q₁₂, where f₁₂= (f₁, f₂). We have

β₁₂((ρ⁰⁰, g⁰⁰),(f₁₂, g)) = (ff₁₂, g⁰⁰gh,ρ, h˜ ⁻¹)

upon choosing ( ˜ρ, h) as required in the definition of

E

M(F₂ ◦F₁). Writing ff₁₂ = ( ˜f1,f˜2) further we have

(ζ₂^∗c_F1,F2 ⊗id)((ff₁₂, g⁰⁰gh,ρ, h˜ ⁻¹)) = (ρ₂( ˜ρ⁻¹₂ ◦f˜₂,id_h2), g⁰⁰ghh₂,f˜₁◦ρ˜₂, h⁻¹₂ ,ρ, h˜ ⁻¹) (3.22) upon choosing appropriate ( ˜ρ₂, h₂) as required in the definition ofc_F1,F2. This is the result of the clockwise composition of diagram (3.11). Counter-clockwise, we first get

(id⊗ζ₁^∗cF1,F2)((ρ⁰⁰, g⁰⁰),(f12, g)) = (ρ⁰⁰, g⁰⁰, f⁰⁰, gh1, f⁰, h⁻¹₁ )

for choices ( ˜ρ₁, h₁), where f⁰⁰ := ρ₂( ˜ρ⁻¹₁ ◦f₂,id_h1) and f⁰ :=f₁◦ρ˜₁. Next we apply the isomorphism β₂ of

E

M(F₂) and get

(β₂⊗id)(ρ⁰⁰, g⁰⁰, f⁰⁰, gh₁, f₁⁰, h⁻¹₁ ) = (ff⁰⁰, g⁰⁰ghh₂,ρ,ˆ ˆh⁻¹, f₁⁰, h⁻¹₁ )

where we have used the choices ( ˆρ,h) defined by ˆˆ ρ:=R( ˜ρ⁻¹₁ , h₁)◦R( ˜ρ₂, h⁻¹h₁) and ˆh:=h⁻¹₁ hh₂. The last step is to apply the isomorphism β₁ of

E

M(F₂) which gives

(id⊗β₁)(ff⁰⁰, g⁰⁰ghh₂,ρ,ˆ hˆ⁻¹, f₁⁰, h⁻¹₁ ) = (ff⁰⁰, g⁰⁰ghh₂,fe⁰, h⁻¹₂ ,ρ, h˜ ⁻¹), (3.23) where we have used the choices ( ˜ρ, h) from above. Comparing (3.22) and (3.23), we have obviously coincidence in all but the first and the third component. For these remaining factors, coincidence follows from the definitions of the various variables.

Equivalence between Bundle Gerbes and 2-Bundles 117 Finally, we have to construct unitors. The unitor for a principal Γ-2-bundle P over M is a bundle gerbe 2-morphism

uP :

E

M(idP) ⁺³ id_EM(P).

Abstractly, one can associate to id_EM(P) the 1-morphism id^{F P}_E

M(P) constructed in the proof of Lemma 3.5.18, and then notice that id^{F P}_EM(P) and

E

M(idP) are canonically 2-isomorphic. In more concrete terms, the unitor uP has the refinement W := P₀^[3]

with the surjective submersions r := pr₁₂ and r⁰ := pr₃ to the refinements Z = P₀^[2]

and Z⁰ = P₀ of the 1-morphisms

E

M(id_P) and id_EM(P), respectively. The relevant maps x_W and y_W are pr₁₃ and pr₂₃, respectively. The principal Γ-bundle of the 1-morphism id_EM(P) is the trivial bundle Q⁰ = I₁. We claim that the principal Γ-bundle Q of

E

M(id_P) is the bundle P of the bundle gerbe

E

M(P). Indeed, the formulae (3.19) reduce for the identity anafunctor idP to those of (3.14). Now, the bundle isomorphism of the unitor uP is

y_W^∗ P ⊗r^∗Q= pr^∗₂₃P ⊗pr^∗₁₂P ^{µ //}pr^∗₁₃P ∼=r^0∗Q⁰⊗x^∗_WP,

whereµis the bundle gerbe product of

E

M(P). The commutativity of diagram (3.9) follows from the associativity of µ.

Proposition 3.7.8. The assignments

E

M for objects, 1-morphisms and 2-morphisms, together with the compositors and unitors defined above, define a 2-functor

E

M : 2-Bun_Γ(M) ^// Grb_Γ(M).

Proof. A list of axioms for a 2-functor with the same conventions as we use here can be found in [SW08, Appendix A]. The first axiom requires that the 2-functor

E

M

respects the vertical composition of 2-morphisms – this follows immediately from the definition.

The second axiom requires that the compositors respect the horizontal compo-sition of 2-morphisms. To see this, let F₁, F₁⁰ : P ^// P⁰ and F₂, F₂⁰ : P^{0 //} P⁰⁰ be 1-morphisms between principal Γ-2-bundles, and letη₁ :F₁ ⁺³F₁⁰ and η₂ :F₂ ⁺³ F₂⁰ be 2-morphisms. Then, the diagram

E

M(F₂◦F₁)

cF1,F2

EM(η1◦η₂) +3

E

M(F₂⁰ ◦F₁⁰)

c_F0 1,F0

2

E

M(F₂)◦

E

M(F₁)

EM(η1)◦EM(η2) +3

E

M(F₂⁰)◦

E

M(F₁⁰) has to commute. Indeed, in order to computec_F1,F2 andc_F⁰

1,F₂⁰ one can make the same choice of ( ˜ρ, h), because the transformations η and η₂ preserve the anchors. Then,

118 Four Equivalent Versions of Non-Abelian Gerbes commutativity follows from the fact that η₁ and η₂ commute with the groupoid actions and the Γ₁-action according to Definition 3.8.1.

The third axiom describes the compatibility of the compositors with the compo-sition of 1-morphisms in the sense that the diagram

E

M(F₃◦F₂◦F₁) ^{cF2◦F1,F3 +3}

cF3◦F2,F1

E

M(F₃)◦

E

M(F₂◦F₁)

id◦c_F2,F1

E

M(F₃◦F₂)◦

E

M(F₁)

cF3,F2◦id +3

E

M(F₃)◦

E

M(F₂)◦

E

M(F₁).

is commutative. In order to verify this, one starts with an element (f₁, f₂, f₃, g) in

E

M(F₃ ◦F₂ ◦F₁). In order to go clockwise, one chooses pairs ( ˜ρ_12,3, h_12,3) and ( ˜ρ_1,2, h_1,2) and gets from the definitions

CW = ((ρ₃( ˜ρ⁻¹_12,3◦f₃,id_h12,3), gh_12,3),(ρ₂( ˜ρ⁻¹_1,2◦f₂◦ρ˜_12,3,id_h1,2), h⁻¹_12,3h_1,2),(f₁◦ρ˜_1,2, h⁻¹_1,2)).

Counter-clockwise, one can choose firstly again the pair ( ˜ρ_1,2, h_1,2) and then the pair ( ˜ρ2,3, h2,3) with ˜ρ2,3 =R( ˜ρ12,3,idh1,2) and h2,3 =h⁻¹_1,2h12,3. Then, one gets

CCW = ((ρ₃( ˜ρ⁻¹_2,3◦ρ₃(f₃,id_h1,2),id_h2,3), gh_1,2h_2,3),

(ρ₂( ˜ρ⁻¹_1,2◦f₂,id_h1,2)◦ρ˜_2,3, h⁻¹_2,3),(f₁◦ρ˜_1,2, h⁻¹_1,2)), where one has to use formula (3.33) for the Γ₁-action on the composition of equi-variant anafunctors. Using the definitions of h_2,3 and ˜ρ_2,3 as well as the axiom of Definition 3.8.1 (ii) one can show that CW = CCW.

The fourth and last axiom requires that compositors and unitors are compatible with each other in the sense that for each 1-morphismF :P ^// P⁰ the 2-morphisms

E

M(F)∼=

E

M(F ◦idP)^cidP,F+3

E

M(F)◦

E

M(idP) ^id◦uP+3

E

M(F)◦id_EM(P) ∼=

E

M(F) and

E

M(F)∼=

E

M(idP⁰ ◦F)^cF,idP0+3

E

M(idP⁰)◦

E

M(F)^uP0◦id+3id_EM(P⁰₎◦

E

M(F)∼=

E

M(F) are the identity 2-morphisms. We prove this for the first one and leave the second as an exercise. Using the definitions, we see that the 2-morphism has the refinement W :=P₀×_MP₀×_MP₀⁰ withr = pr₁₃and r⁰ = pr₂₃. The maps x_W :W ^// P₀×_MP₀ and y_W :W ^// P₀⁰×_MP₀⁰ are pr₁₂ and ∆◦pr₃, respectively, where ∆ is the diagonal map. Its bundle morphism is a morphism

ϕ: pr^∗₁₃Q ^// pr^∗₂₃Q⊗pr^∗₁₂P,

Equivalence between Bundle Gerbes and 2-Bundles 119 whereQ=F×Γ₀is the principal Γ-bundle of

E

M(F), andP =P₁×Γ₀is the principal Γ-bundle of

E

M(P). Over a point (p₁, p₂, p⁰) and (f, g)∈pr^∗₁₃Q, i.e. α_l(f) =p₁ and R(α_r(f), g⁻¹) =p⁰, the bundle morphism ϕis given by

(f, g) ^// (ρ( ˜ρ⁻¹◦f,id_h), gh,ρ, h˜ ⁻¹),

where h ∈ Γ₀, and ˜ρ ∈ P₁ with s( ˜ρ) = R(p₂, h⁻¹) and t( ˜ρ) = α_l(f). We have to compare (W, ϕ) with the identity 2-morphism of

E

M(F), which has the refinement Z with r=r⁰ = id and the identity bundle morphism. According to the equivalence relation on bundle gerbe 2-morphisms we have to evaluate ϕ over a point w ∈ W with r(w) =r⁰(w), i.e. w is of the formw= (p, p, p⁰). Here we can choose h= 1 and

˜

ρ= id_p, in which case we haveϕ(f, g) = ((f, g),(id_p,1)). This is indeed the identity on Q.

Properties of the 2-functor

E

M

For the proof of Theorem 3.7.1 we provide the following two statements.

Lemma 3.7.9. The 2-functor

E

M is fully faithful on Hom-categories.

Proof. Let P,P⁰ be principal Γ-2-bundles over M, let F₁, F₂ : P ^// P⁰ be 1-mor-phisms. By Lemma 3.5.18 every 2-morphism η :

E

M(F₁) ⁺³

E

M(F₂) can be rep-resented by one whose refinement is P₀ ×_M P₀⁰, so that its bundle isomorphism is η:Q₁ ^// Q₂, whereQ_k :=F_k×Γ for k = 1,2. We can read of a map η :F₁ ^// F₂, and it is easy to see that this is a 2-morphismη :F₁ ⁺³ F₂. This procedure is clearly inverse to the 2-functor

E

M on 2-morphisms.

Proposition 3.7.10. The 2-functors

E

M form a 1-morphism between pre-2-stacks.

Proof. For a smooth map f :M ^// N, we have to look at the diagram 2-Bun_Γ(N)

EN

f^∗ //2-Bun_Γ(M)

EM

Grb_Γ(N)

f^∗ //Grb_Γ(M)

of 2-functors. For P a principal Γ-2-bundle over N, the Γ-bundle gerbe

E

M(f^∗P) has the surjective submersion pr₁ : Y := M ×_N P₀ ^// M, the principal Γ-bundle P :=M×_NP₁×Γ₀ overY^[2], and a bundle gerbe productµdefined as in (3.16) that ignores theM-factor. On the other hand, the Γ-bundle gerbef^∗

E

N(P) has the same surjective submersion, and – up to canonical identifications between fibre products – the same Γ-bundle and the same bundle gerbe product. These canonical identi-fications make up a pseudonatural transformation that renders the above diagram commutative.

120 Four Equivalent Versions of Non-Abelian Gerbes

Im Dokument Higher Categorical Structures in Geometry - General Theory and Applications to Quantum Field Theory (Seite 132-142)