CONNECTIVE STRUCTURES ON

GERBES AND TWO-VECTOR BUNDLES

Notes by John Rognes

March 6th 2009

Gerbes and 2-vector bundles

Let V be a bipermutative groupoid of finite-dimensional complex vector spaces, under direct sum and tensor product. For example, we may take V to have objects ℂ^{n} for n ≥ 0, so that ℂ^{n} has automorphism group GL_{n}(ℂ)

Recall from Brylinski, Section 5.2:

Definition. Let X be a smooth manifold, perhaps infinite-dimensional. A smooth ℂ^{∗}-gerbe over X is a sheaf of groupoids G that is locally equivalent to the sheaf of smooth functions to the groupoid GL_{1}(V), with objects the 1-dimensional complex vector spaces and isomorphisms the complex linear isomorphisms between these.

In more detail:

(P1) for each local homeomorphism f: Y → X there is a groupoid G(Y) =

G(Y ⟶^{f} X),

(P2) for every further local homeomorphism g: Z → Y there is a functor

g^{−1} : G(Y) → G(Z) ,

(P3) and for every further local homeomorphism h: W → Z there is a natural

isomorphism θ_{g,h}: h^{−1}g^{−1} ⟶^{≅} (gh)^{−1} of functors G(Y) → G(W), such that (P4) for each further local homeomorphism k: T → W the identity θ_{gh,k}θ_{h,k} =

θ_{g,hk}θ_{g,h} holds. This makes G a presheaf of groupoids.

For G to be a sheaf of groupoids, two descent conditions must hold:

(D1) For each local homeomorphism f: Y → X and each pair of objects P, Q ∈ G(Y), the assignment taking a further local homeomorphism g: Z → Y to the morphism set G(Z)(g^{−1}P, g^{−1}Q) is a sheaf of sets on Y, denoted \underline{G(Y)}(P, Q).

(D2) For each open subset V ⊆ X, each surjective local homeomorphism f: Y → V, and each object P ∈ G(Y) equipped with a descent datum, that is, an

isomorphism ϕ: p_{2}^{−1}P ⟶^{≅} p_{1}^{−1}P in G(Y ×_{X} Y) such that p_{12}^{−1}ϕ ∘ p_{23}^{−1}ϕ ≡

p_{13}^{−1}ϕ (modulo θ’s), there exists an object Q ∈ G(V) and an isomorphism ψ: f^{−1}(Q) ⟶^{≅} P in G(Y), such that ϕ ≡ (p_{1}^{−1}ψ)(p_{2}^{−1}ψ)^{−1} (modulo θ’s).

Typeset by AMS-TEX

For G to be a smooth ℂ^{∗}-gerbe, three gerbe conditions must hold:

(G3) There is a surjective local homeomorphism f: Y → X such that G(Y) is

non-empty. (G2) For any local homeomorphism f: Y → X and any two objects P and Q in

G(Y), there exists a surjective local homeomorphism g: Z → Y such that g^{−1}P and g^{−1}Q are isomorphic in G(Z).

(G1) Given any local homeomorphism f: Y → X and any object P of G(Y), there is a preferred isomorphism from the sheaf \underline{G(Y)}(P, P) of automorphisms of P to the sheaf \underline{ℂ}_{Y}^{∗} of smooth ℂ^{∗}-valued functions.

((Omitted: When are two gerbes equivalent?)

Here is a definition from Baas–Dundas–Rognes.

Definition. By an ordered open cover (U, I) of a topological space X, we mean a partially ordered indexing set I, and an open subset U_{α} ⊆ X for each α ∈ I, so that for each x ∈ X the partial ordering on I restricts to a total (= linear) ordering on the subset I_{x} = {α ∈ I | x ∈ U_{α}}. It follows that for any subset J ⊆ I of indices with ⋂_{α∈J} U_{α} ≠ ∅, J is totally ordered.

The following defines a charted 2-vector bundle of rank 1. We use the abbreviations U_{αβ} = U_{α} ∩U_{β}, U_{αβγ} = U_{α} ∩U_{β} ∩U_{γ}, etc. See also Murray’s paper on bundle gerbes.

Definition. Let X be a topological space, with an ordered open cover (U, I). A charted 2-line bundle L over X consists of

(1) a complex line bundle L^{αβ} over U_{αβ} for each pair α < β in I, and

(2) an isomorphism

ϕ^{αβγ}: L^{αβ} ⊗ L^{βγ} ⟶^{≅} L^{αγ}

of complex line bundles over U_{αβγ} for each triple α < β < γ in I, such that (3) the diagram

\xymatrix{L^{αβ} ⊗ (L^{βγ} ⊗ L^{γδ}) \ar[rr]^{\underline{α}} \ar[d]_{id⊗ϕ^{βγδ}} &  & (L^{αβ} ⊗ L^{βγ}) ⊗ L^{γδ} \ar[d]^{ϕ^{αβγ}⊗id} \\ L^{αβ} ⊗ L^{βδ} \ar[r]_{ϕ^{αβδ}} & L^{αδ} & L^{αγ} ⊗ L^{γδ} \ar[l]^{ϕ^{αγδ}}}

of line bundle isomorphisms over U_{αβγδ} commutes for each chain α < β < γ < δ in I, where \underline{α} is the coherent natural associativity isomorphism.

The L^{αβ} are the gluing line bundles and the ϕ^{αβγ} are the coherence isomorphisms of the charted 2-line bundle.

If X is a smooth manifold, each L^{αβ} is smooth line bundle, and each ϕ^{αβγ} is a smooth isomorphism, we get a smooth charted 2-line bundle. This notion is closely related to that of a bundle gerbe.

((Omitted: When are two 2-line bundles equivalent?)

Here is the general definition, for n ≥ 1.

Definition. Let X be a topological space, with an ordered open cover (U, I). A charted 2-vector bundle E of rank n over X consists of

(1) an n × n matrix

E^{αβ} = (E_{ij}^{αβ})_{i,j=1}^{n}

of complex vector bundles over U_{αβ}, with dimension matrix dim(E^{αβ}) of determinant ±1 everywhere, for each pair α < β in I, and

(2) an n × n matrix

ϕ^{αβγ} = (ϕ_{ik}^{αβγ})_{i,k=1}^{n}: E^{αβ} · E^{βγ} ⟶^{≅} E^{αγ}

of vector bundle isomorphisms over U_{αβγ}, where

(E^{αβ} · E^{βγ})_{ik} = ⨁_{j=1}^{n} E_{ij}^{αβ} ⊗ E_{jk}^{βγ}

for i, k = 1, …, n, for each triple α < β < γ in I, such that

(3) the diagram

\xymatrix{E^{αβ} · (E^{βγ} · E^{γδ}) \ar[rr]^{\underline{α}} \ar[d]_{id·ϕ^{βγδ}} &  & (E^{αβ} · E^{βγ}) · E^{γδ} \ar[d]^{ϕ^{αβγ}·id} \\ E^{αβ} · E^{βδ} \ar[r]_{ϕ^{αβδ}} & E^{αδ} & E^{αγ} · E^{γδ} \ar[l]^{ϕ^{αγδ}}}

of n × n matrices of vector bundle isomorphisms over U_{αβγδ} commutes for each chain α < β < γ < δ in I, where \underline{α} is the coherent natural associativity isomorphism.

The E^{αβ} are the gluing bundle matrices and the ϕ^{αβγ} are the coherence isomorphism matrices of the charted 2-vector bundle.

If X is a smooth manifold, each E_{ij}^{αβ} is smooth vector bundle, and each ϕ_{ik}^{αβγ} is a smooth isomorphism, we get a smooth charted 2-vector bundle.

The following construction is adapted from Brylinski’s Definition and Proposition 7.2.1, and the proof of surjectivity in his Theorem 5.2.8.

Proposition. To each smooth charted 2-line bundle L over X there is a naturally associated smooth ℂ^{∗}-gerbe G over X.

Proof. We work in the smooth category. Let f: Y → X be a local homeomorphism, and let Y_{α} = f^{−1}(U_{α}), Y_{αβ} = f^{−1}(U_{αβ}), etc. be open subsets of Y. Let G(Y) =

G(Y ⟶^{f} X) be the groupoid with objects consisting of (1) a complex line bundle P^{α} over Y_{α} for each α ∈ I,

(2) an isomorphism

ξ^{αβ}: L^{αβ} ⊗ P^{β} ⟶^{≅} P^{α}

of complex line bundles over Y_{αβ}, for each pair α < β in I, such that (3) the diagram

\xymatrix{L^{αβ} ⊗ (L^{βγ} ⊗ P^{γ}) \ar[rr]^{\underline{α}} \ar[d]_{id⊗ξ^{βγ}} &  & (L^{αβ} ⊗ L^{βγ}) ⊗ P^{γ} \ar[d]^{ϕ^{αβγ}⊗id} \\ L^{αβ} ⊗ P^{β} \ar[r]_{ξ^{αβ}} & P^{α} & L^{αγ} ⊗ P^{γ} \ar[l]^{ξ^{αγ}}}

of isomorphisms over U_{αβγ} commutes for each triple α < β < γ in I. The isomorphisms in G(Y) from (P^{α}, ξ^{αβ}) to (Q^{α}, η^{αβ}) consist of

(1) isomorphisms ψ^{α}: P^{α} → Q^{α} over Y_{α} for each α ∈ I, such that

(2) the diagram

\xymatrix{L^{αβ} ⊗ P^{β} \ar[r]^{ξ^{αβ}} \ar[d]_{id⊗ψ^{β}} & P^{α} \ar[d]^{ψ^{α}} \\ L^{αβ} ⊗ Q^{β} \ar[r]_{η^{αβ}} & Q^{α}}

of isomorphisms over Y_{αβ} commutes for each pair α < β in I. For each local homeomorphism g: Z → Y the functor g^{−1} : G(Y) → G(Z) is

given on objects by pullback of the line bundles P^{α} along Z_{α} → Y_{α} and of the isomorphisms ξ^{αβ} along Z_{αβ} → Y_{αβ}, while on isomorphisms it is given by pullback of the isomorphisms ψ^{α} along Z_{α} → Y_{α}.

This accounts for (P1) and (P2). We omit to discuss (P3), (P4), (D1), (D2), (G3) and (G2).

The automorphisms of P = (P^{α}, ξ^{αβ}) consist of isomorphisms ψ^{α}: P^{α} → P^{α} over Y_{α}, for each α ∈ I, which amounts to multiplication by a (smooth) function a^{α}: Y_{α} → ℂ^{∗}, and these satisfy a^{α} = a^{β} over Y_{αβ}, hence these functions glue together to a (smooth) ℂ^{∗}-valued function a on Y. More generally this works for the pullback over any local homeomorphism g: Z → Y, so we get the isomorphism of sheaves between \underline{G(Y)}(P, P) and \underline{ℂ}_{Y}^{∗}, required in (G1). □

Definition. Let GL_{n}(V) be the (smooth) groupoid with objects the n × n matrices V = (V_{ij})_{i,j=1}^{n} of complex vector spaces with dimension matrix dim(V) = dim(V_{ij})_{i,j=1}^{n} of determinant ±1, and morphisms n × n matrices ϕ: V → W of complex linear isomorphisms between these. Matrix multiplication V · W with

(V · W)_{ik} = ⨁_{j=1}^{n} V_{ij} ⊗ W_{jk}

defines a monoidal structure on GL_{n}(V).

There is one isomorphism class of objects in GL_{n}(V), for each n × n matrix D = (d_{ij})_{i,j=1}^{n} with non-negative integer entries and determinant ±1. These matrices form the monoid GL_{n}(ℕ_{0}) = Mat_{n}(ℕ_{0}) ∩ GL_{n}(ℤ), whose invertible elements are exactly the permutation matrices Σ_{n} ⊆ GL_{n}(ℕ_{0}).

The automorphisms in GL_{n}(V) of the object ℂ^{D} = (ℂ^{d_{ij}})_{i,j=1}^{n} form the product group

GL_{D}(ℂ) = ∏_{i,j=1}^{n} GL_{d_{ij}}(ℂ) .

For example, if D is a permutation matrix, GL_{D}(ℂ) ≅ GL_{1}(ℂ)^{n} = (ℂ^{∗})^{n}.

((Let ∥D∥ = ∑_{i,j=1}^{n} d_{ij}. Then ∥D∥ ≥ n, with equality if and only if D is a permutation matrix. If A · B = C then ∥B∥ ≤ ∥C∥, with equality if and only if A is a permutation matrix. Hence a given matrix D can be factored in at most finitely many ways, up to permutation matrices (= units).))

Proposition. To each smooth charted 2-vector bundle E of rank n over X there is a naturally associated sheaf of groupoids P over X with a right action by the monoidal sheaf of smooth functions to GL_{n}(V).

((Can say more about local structure, under further hypotheses on the ordered open cover.))

Proof. We work in the smooth category. Let f: Y → X be a local homeomorphism, and let Y_{α} = f^{−1}(U_{α}), Y_{αβ} = f^{−1}(U_{αβ}), etc. be open subsets of Y. Let P(Y) =

P(Y ⟶^{f} X) be the groupoid with objects consisting of

(1) an n × n matrix P^{α} = (P_{ij}^{α})_{i,j=1}^{n} of complex vector bundles over Y_{α}, with dimension matrix dim(P^{α}) of determinant ±1 everywhere, for each α ∈ I, and

(2) an n × n matrix

ξ^{αβ} = (ξ_{ik}^{αβ})_{i,k=1}^{n}: E^{αβ} · P^{β} ⟶^{≅} P^{α}

of vector bundle isomorphisms over Y_{αβ}, for each pair α < β in I, such that (3) the diagram

\xymatrix{E^{αβ} · (E^{βγ} · P^{γ}) \ar[rr]^{\underline{α}} \ar[d]_{id·ξ^{βγ}} &  & (E^{αβ} · E^{βγ}) · P^{γ} \ar[d]^{ϕ^{αβγ}·id} \\ E^{αβ} · P^{β} \ar[r]_{ξ^{αβ}} & P^{α} & E^{αγ} · P^{γ} \ar[l]^{ξ^{αγ}}}

of n × n matrices of vector bundle isomorphisms over U_{αβγ} commutes for each triple α < β < γ in I.

The isomorphisms in P(Y) from (P^{α}, ξ^{αβ}) to (Q^{α}, η^{αβ}) consist of

(1) an n × n matrix

ψ^{α} = (ψ_{ij}^{α})_{i,j=1}^{n}: P^{α} → Q^{α}

of vector bundle isomorphisms over Y_{α}, for each α ∈ I, such that

(2) the diagram

\xymatrix{E^{αβ} · P^{β} \ar[r]^{ξ^{αβ}} \ar[d]_{id·ψ^{β}} & P^{α} \ar[d]^{ψ^{α}} \\ E^{αβ} · Q^{β} \ar[r]_{η^{αβ}} & Q^{α}}

of n × n matrices of vector bundle isomorphisms over Y_{αβ} commutes, for each pair α < β in I.

For each local homeomorphism g: Z → Y the functor g^{−1} : P(Y) → P(Z) is given on objects by pullback of the vector bundle matrix P^{α} along Z_{α} → Y_{α} and of the isomorphism matrix ξ^{αβ} along Z_{αβ} → Y_{αβ}, while on isomorphisms it is given by pullback of the isomorphism matrix ψ^{α} along Z_{α} → Y_{α}.

This accounts for (P1) and (P2). The remaining presheaf conditions (P3) and (P4), and the sheaf conditions (D1) and (D2) should be clear.

The value of the sheaf of smooth functions to GL_{n}(V) at Y can be viewed as the groupoid of n × n matrices R = (R_{ij})_{i,j=1}^{n} of complex vector bundles over Y, with dim(R) of determinant ±1, and n × n matrices ζ: R → S of vector bundle isomorphisms between these. The right action of R on an object (P^{α}, ξ^{αβ}) gives the object consisting of

(1) the n × n matrix P^{α} · R of vector bundles over Y_{α}, for each α ∈ I, and (2) the n × n matrix of composite vector bundle isomorphisms

E^{αβ} · (P^{β} · R) ⟶^{\underline{α}} (E^{αβ} · P^{β}) · R ⟶^{ξ^{αβ}·id} P^{α} · R

over Y_{αβ}, for each pair α < β in I.

The right action of ζ: R → S on an isomorphism (ψ^{α}) from (P^{α}, ξ^{αβ}) to (Q^{α}, η^{αβ}) is the collection of isomorphisms

ψ^{α} · ζ: P^{α} · R ⟶^{≅} Q^{α} · S

for α ∈ I. □

When does P have objects locally?

Proposition. Suppose that x ∈ X has an open neighborhood V such that the subset I_{V} = {α ∈ I | V ∩ U_{α} ≠ ∅} of I is finite and totally ordered. Suppose also that the V_{α} = V ∩ U_{α} for α ∈ I_{V}, as well as all of their intersections, are contractible. ((Need a condition about extension to closures, to ensure that the union of two trivial bundles along a contractible open subspace is again trivial.)) Then P(V) is not the empty category, hence has objects.

((The missing condition might be that X is a normal space (for the Tietze extension theorem), the gluing bundle matrices E^{αβ} are defined over the closed intersections Ū_{α} ∩ Ū_{β}, and that the coherence isomorphism matrices ϕ^{αβγ} are defined over the closed intersections Ū_{α} ∩ Ū_{β} ∩ Ū_{γ}, with the usual compatibility. Furthermore, the V̄ ∩ Ū_{α} and all of their intersections, should be contractible.))

Proof. Write I_{V} = {α < ⋯ < ω}. For all β < γ in I_{V} we (should) know that E^{βγ} is trivial over the contractible open set V_{βγ}. Choose an n × n matrix of trivial vector bundles P^{ω} over V_{ω}. As the last step of a descending induction over I_{V}, we shall assume that we have chosen

(1) matrices of trivial vector bundles P^{β} over V_{β} for all β with α < β ≤ ω, and (2) matrices of isomorphisms ξ^{βγ}: E^{βγ} · P^{γ} → P^{β} over V_{βγ} for all β and γ with

α < β < γ ≤ ω, subject to the usual compatibility condition over V_{βγδ} for all α < β < γ < δ ≤ ω. Then we can

(1) form the matrix of trivial vector bundles E^{αβ} · P^{β} over V_{αβ}, for each β with α < β ≤ ω, (2) these are related by a chain of matrices of isomorphisms

\xymatrix{E^{αβ} · (E^{βγ} · P^{γ}) \ar[r]^{\underline{α}} \ar[d]_{id·ξ^{βγ}} & (E^{αβ} · E^{βγ}) · P^{γ} \ar[d]^{ϕ^{αβγ}·id} \\ E^{αβ} · P^{β} & E^{αγ} · P^{γ}}

over V_{αβγ}, for all β and γ with α < β < γ ≤ ω, and

(3) these chains of isomorphisms are strictly compatible over V_{αβγδ}, for all

α < β < γ < δ ≤ ω. Hence we can find a matrix of vector bundles P̃^{α} over

Ṽ_{α} = ⋃_{α<β≤ω} V_{αβ} ⊆ V_{α}

with isomorphisms ξ^{αβ}: E^{αβ} · P^{β} → P̃^{α} over V_{αβ} for all α < β ≤ ω, such that the completed diagram

\xymatrix{E^{αβ} · (E^{βγ} · P^{γ}) \ar[rr]^{\underline{α}} \ar[d]_{id·ξ^{βγ}} &  & (E^{αβ} · E^{βγ}) · P^{γ} \ar[d]^{ϕ^{αβγ}·id} \\ E^{αβ} · P^{β} \ar[r]_{ξ^{αβ}} & P̃^{α} & E^{αγ} · P^{γ} \ar[l]^{ξ^{αγ}}}

commutes over V_{αβγ}.

((Now we argue that P̃^{α} over Ṽ_{α} is a matrix of trivial bundles, hence can be extended to a matrix P^{α} over V_{α} of trivial bundles.)) □

When are objects of P locally isomorphic?

((Objects (P^{α}, ξ^{αβ}) and (Q^{α}, η^{αβ}) in P(V) can only be isomorphic if they have the same dimension matrices dim(P^{α}) = dim(Q^{α}) over V_{α} = V ∩ U_{α} for all α. Assuming this, we can build a local isomorphism for small V, assuming that isomorphisms can be extended over inclusions Ṽ_{α} ⊆ V_{α}, as in the previous result.))

Connective structures on gerbes

Let q ≥ 0 and let Y be a smooth manifold. The assignment taking a local homeomorphism g: Z → Y to the complex vector space A^{q}(Z; ℂ) = Ω^{q}(Z)⊗ℂ of complex alternating q-forms on Z, defines a complex vector space sheaf \underline{A}_{Y,ℂ}^{q} over Y. For each complex line bundle P over Y, the set of connections ∇ on P forms an affine space (a single free orbit) under A^{1}(Y; ℂ). For a more general complex vector bundle E, the connections form an affine space under the 1-forms in the endomorphism bundle of E, but when E = P is a line bundle, the endomorphism bundle is a trivial ℂ^{1}-bundle. Passing to restrictions along the local homeomorphisms g: Z → Y, the collection of connections over the various Z, denoted Co(P), forms a torsor under \underline{A}_{Y,ℂ}^{1}. If ψ: P ⟶^{≅} Q is an isomorphism of complex line bundles over Y, there is an induced isomorphism ψ_{∗}: Co(P) → Co(Q) that takes a connection ∇ on P to the connection ψ_{∗}(∇) on Q, determined on smooth sections s in Q → Y by

ψ_{∗}(∇)_{w}(s) = ψ(∇_{v}(ψ^{−1}s))

when T ψ: T P → T Q takes the tangent vector v to w. When ψ is the automorphism given by multiplication by a smooth function a: Y → ℂ^{∗}, we can compute that

ψ_{∗}(∇)_{w}(s) = a · ∇_{v}(a^{−1} · s)
= a · a^{−1} · ∇_{w}(s) + a · d(a^{−1})(w) · s
= ∇_{w}(s) − a^{−1} da(w) · s

so that ψ_{∗}(∇) = ∇ − a^{−1} da = ∇ − d log a. In a ℂ^{∗}-gerbe, the local groupoids are non-canonically equivalent to the groupoid of complex line bundles, and a connective structure specifies corresponding torsors under the complex vector space sheaf \underline{A}_{Y,ℂ}^{1}.

Definition. Let G be a smooth ℂ^{∗}-gerbe over X. A connective structure Co on G consists of

(1) a rule that for each local homeomorphism f: Y → X and each object P ∈ G(Y) associates an \underline{A}_{Y,ℂ}^{1}-torsor Co(P),

(2) an isomorphism

co^{g}: g^{−1}Co(P) ⟶^{≅} Co(g^{−1}P)

of \underline{A}_{Z,ℂ}^{1}-torsors for each further local homeomorphism g: Z → Y, and

(3) an isomorphism

ψ_{∗}: Co(P) → Co(Q)

of \underline{A}_{Y,ℂ}^{1}-torsors for each isomorphism ψ: P → Q in G(Y). These are required to be functorial and compatible, in the sense that

(4) co^{gh} ≡ co^{h} ∘ co^{g} (modulo canonical isomorphisms and θ’s) for further local homeomorphisms h: W → Z,

(5) (ψ′ ∘ ψ)_{∗} = ψ′_{∗} ∘ ψ_{∗} for further isomorphisms ψ′: Q → R in G(Y), and

(6) the diagram

\xymatrix{g^{−1}Co(P) \ar[r]^{g^{−1}(ψ_{∗})} \ar[d]_{co^{g}} & g^{−1}Co(Q) \ar[d]^{co^{g}} \\ Co(g^{−1}P) \ar[r]_{(g^{−1}ψ)_{∗}} & Co(g^{−1}Q)}

commutes. Finally, the isomorphisms ψ_{∗} behave as expected for gauge automorphisms, meaning that

(7) when ψ: P → P is the automorphism induced by multiplication by a smooth function a: Y → ℂ^{∗}, we have

ψ_{∗}(∇) = ∇ − a^{−1} da

for all ∇ in the \underline{A}_{Y,ℂ}^{1}-torsor Co(P).

See Brylinski’s Definition 5.3.1. He writes α_{g} for our co^{g}, and labels (6) as (R2) and (7) as (R1). The following combines Brylinski’s Proposition 5.3.2 and 5.3.6.

Proposition. Let G be a smooth ℂ^{∗}-gerbe over a smooth manifold X. Then G admits connective structures, and the isomorphism classes of torsors under \underline{A}_{X,ℂ}^{1} act freely and transitively on the isomorphism classes of connective structures on G.

Definition. Let X be a smooth manifold, with an ordered open cover (U, I). Let L → X be a smooth charted 2-line bundle, with gluing line bundles L^{αβ} and coherence isomorphisms ϕ^{αβγ}. A connective structure ∇ on L is

(1) a connection ∇^{αβ} on L^{αβ} over U_{αβ}, for each α < β in I, such that

(2) the isomorphism ϕ^{αβγ}: L^{αβ} ⊗L^{βγ} ⟶^{≅} L^{αγ} over U_{αβγ} takes the tensor product connection ∇^{αβ} + ∇^{βγ} on L^{αβ} ⊗ L^{βγ} to the connection ∇^{αγ} on L^{αγ}, so that

(ϕ^{αβγ})_{∗}(∇^{αβ} + ∇^{βγ}) = ∇^{αγ}

over U_{αβγ}. For sections s in L^{αβ} and t in L^{βγ}, we have

(∇^{αβ} + ∇^{βγ})(s ⊗ t) = ∇^{αβ}(s) ⊗ t + s ⊗ ∇^{βγ}(t)

as 1-forms of sections in L^{αβ} ⊗ L^{βγ} over U_{αβγ}.

The following is close to the first part of the proof of Brylinski’s Proposition 5.3.2. For an ordered open cover (U, I) of X and an abelian sheaf \underline{A} over X, the ordered Čech complex consists of the abelian groups

Č^{p}(U; \underline{A}) = ∏_{α_{0}<···<α_{p}} A(U_{α_{0}...α_{p}}) .

An ordered Čech p-cochain η associates to each sequence α_{0} < ⋯ < α_{p} an element η(α_{0}, …, α_{p}) ∈ A(U_{α_{0}...α_{p}}). Its coboundary dη is the ordered Čech (p + 1)- cochain that associates to each sequence α_{0} < ⋯ < α_{p+1} the alternating sum ∑_{i=0}^{p+1}(−1)^{i}η(α_{0}, …, α̂_{i}, …, α_{p+1}), interpreted via various restriction maps to lie in A(U_{α_{0}...α_{p+1}}). Its cohomology is denoted Ȟ^{∗}(U; \underline{A}). The colimit over all ordered open covers (U, I) defines the Čech cohomology Ȟ^{∗}(X; \underline{A}).

Proposition. Let L be a smooth charted 2-line bundle over a smooth manifold X. There exists a connective structure ∇ on L.

Proof I. First choose arbitrary connections ˜∇^{αβ} on L^{αβ} for each α < β in I. Then

(ϕ^{αβγ})_{∗}( ˜∇^{αβ} + ˜∇^{βγ}) = ˜∇^{αγ} + ω^{αβγ}

for a unique complex 1-form ω^{αβγ} on U_{αβγ}. Letting α < β < γ vary, we get a Čech 2-cochain ω in Č^{2}(U; \underline{A}_{X,ℂ}^{1}). By condition (3) in the definition of a charted 2-line bundle, this Čech 2-cochain is a Čech 2-cocycle.

As discussed in Brylinski’s Section 1.4, the sheaves \underline{A}_{X,ℂ}^{q} are soft, so that the restriction maps A^{q}(X; ℂ) → A^{q}(Z; ℂ) are surjective for all closed Z ⊆ X, hence acyclic. So for reasonable X, Ȟ^{p}(U; \underline{A}_{X,ℂ}^{q}) ≅ H^{p}(U; \underline{A}_{X,ℂ}^{q}) = 0 for p > 0. Murray gives a more direct proof if this fact. In particular, Ȟ^{2}(U; \underline{A}_{X,ℂ}^{1}) = 0, so the Čech 2-cocycle ω is a Čech 2-coboundary. ((Does it matter that we consider “ordered” Čech cochains?))

Hence there exists a Čech 1-cochain η with dη = ω. This η consists of complex 1-forms η^{αβ} over U_{αβ} for each α < β, such that η^{βγ} − η^{αγ} + η^{αβ} = ω^{αβγ} over U_{αβγ}. Let ∇^{αβ} = ˜∇^{αβ} − η^{αβ} be a new connection on L^{αβ}, for all α < β in I. Then

(ϕ^{αβγ})_{∗}(∇^{αβ} + ∇^{βγ}) = ∇^{αγ} ,

so ∇ is a connective structure on L. □

Proof II. First choose 1-forms η^{αβ} on U_{α}β for all α < β with α + 1 = β, so that β is the successor of α. For α + 1 = β and β + 1 = γ the 1-form η^{αγ} is determined on U_{αβγ} from the relation

η^{βγ} − η^{αγ} + η^{αβ} = ω^{αβγ} .

We (should then be able to) choose an extension of η^{αγ} over U_{αβγ} ⊆ U_{αγ}. For α + 1 = β, β + 1 = γ and γ + 1 = δ, the 1-form η^{αδ} is determined on U_{αβδ} by

η^{βδ} − η^{αδ} + η^{αβ} = ω^{αβδ}

and on U_{αγδ} by

η^{γδ} − η^{αδ} + η^{αγ} = ω^{αγδ} .

These 1-forms agree on U_{αβγδ}, as a consequence of the relation

ω^{βγδ} − ω^{αγδ} + ω^{αβδ} − ω^{αβγ} = 0 m

so they define a 1-form on Ũ_{αδ} = U_{αβδ} ∪ U_{αγδ}. We (should then be able to) choose an extension of η^{αδ} over Ũ_{αδ} ⊆ U_{αδ}.

Continuing like this, we determine η^{αβ} for all α < β where α + k = β for finite k. □

(Classify the choices of connective structure.)

Proposition. Let ∇ and ¯∇ be two connective structures on L over X. Then ((ETC))

Proof. There are unique complex 1-forms η^{αβ} on U_{αβ} so that ¯∇^{αβ} = ∇^{αβ} + η^{αβ}. This defines a Čech 1-cochain η. Since both ∇ and ¯∇ are connective structures, η^{αβ} + η^{βγ} = η^{αγ}, so η is a Čech 1-cocycle. Since Ȟ^{1}(U; \underline{A}_{X,ℂ}^{1}) = 0, this Čech 1-cocycle is a Čech 1-coboundary. ((ETC))

Proposition. To each connective structure ∇ on a smooth charted 2-line bundle L over X there is a naturally associated connective structure Co on the associated smooth ℂ^{∗}-gerbe G over X.

Proof. Consider a local homeomorphism f: Y → X and an object P ∈ G(Y), consisting of complex line bundles P^{α} over Y_{α} = f^{−1}(U_{α}) for each α ∈ I, and isomorphisms ξ^{αβ}: L^{αβ} ⊗ P^{β} ⟶^{≅} P^{α} over Y_{αβ} = f^{−1}(U_{αβ}) for each α < β. We need to define the \underline{A}_{Y,ℂ}^{1}-torsor Co(P) of connections on P, and its restrictions over further local homeomorphisms g: Z → Y.

Over Y the elements D = (D^{α})_{α∈I} of Co(P) consist of

(1) connections D^{α} on P^{α} over Y_{α} for all α ∈ I, such that

(2) the isomorphism ξ^{αβ}: L^{αβ} ⊗P^{β} ⟶^{≅} P^{α} takes the tensor product connection ∇^{αβ} + D^{β} to D^{α}, for all α < β, so that

(ξ^{αβ})_{∗}(∇^{αβ} + D^{β}) = D^{α}

over Y_{αβ}.

Given another such element, D̄ = (D̄^{α})_{α∈I}, we can write D̄^{α} = D^{α} + ϵ^{α} for a unique ϵ^{α} ∈ A^{1}(Y_{α}; ℂ), for all α ∈ I. By condition (2) for D and D̄, it follows that ϵ^{α} = ϵ^{β} over Y_{αβ}, so that the ϵ^{α} are all restrictions of a global complex 1-form ϵ ∈ A^{1}(Y; ℂ). This makes it clear that A^{1}(Y; ℂ) acts freely and transitively on the elements of Co(P) over Y. The extension to a sheaf is clear, as is the conclusion that Co(P) is a torsor under \underline{A}_{Y,ℂ}^{1}.

((Explain rest of requirements?)) □

Proposition. To each smooth charted 2-line bundle L over X, there is an associated Čech cohomology class [L] in Ȟ^{2}(X; \underline{ℂ}_{X}^{∗}) ≅ H^{2}(X; \underline{ℂ}_{X}^{∗}) ≅ H^{3}(X; ℤ).

Proof. After possibly refining the ordered cover (U, I), we may assume that the gluing line bundles L^{αβ} admit smooth sections s^{αβ}, giving trivializations U_{αβ}×ℂ ⟶^{≅} L^{αβ}. Comparing these trivializations via ϕ^{αβγ}, there is a unique smooth function h^{αβγ}: U_{αβγ} → ℂ^{∗} such that

ϕ^{αβγ}(s^{αβ} ⊗ s^{βγ}) = h^{αβγ} · s^{αγ} .

Letting α < β < γ vary, the h^{αβγ} combine to a Čech 2-cochain h ∈ Č^{2}(U; \underline{ℂ}_{X}^{∗}). By property (3) in the definition of a charted 2-line bundle, h is a Čech 2-cocycle, and [L] is its cohomology class in Ȟ^{2}(X; \underline{ℂ}_{X}^{∗}).

((Should check independence of choice of sections s^{αβ}.)) □

(For a suitable notion of equivalence of smooth charted 2-line bundles, this rule induces an isomorphism between the group of equivalence classes of smooth charted 2-line bundles over X and the cohomology group H^{3}(X; ℤ).)

Associated to the complex

0 → \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1} → 0

of abelian sheaves over X, and an ordered open cover (U, I) of X, we can associate the Čech bicomplex

0 → Č^{∗}(U; \underline{ℂ}_{X}^{∗}) ⟶^{d log} Č^{∗}(U; \underline{A}_{X,ℂ}^{1}) → 0

with Č^{p}(U; \underline{ℂ}_{X}^{∗}) in bidegree (p, 0) and Č^{p}(U; \underline{A}_{X,ℂ}^{1}) in bidegree (p, 1). The cohomology of the corresponding total complex is the hypercohomology

Ȟ^{∗}(U; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) .

Its colimit over all ordered open covers is the Čech hypercohomology

Ȟ^{∗}(X; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) ,

which for reasonable X agrees with sheaf hypercohomology H^{∗}. There is a long exact sequence

⋯ → H^{p}(X; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) → H^{p}(X; \underline{ℂ}_{X}^{∗}) ⟶^{d log} H^{p}(X; \underline{A}_{X,ℂ}^{1}) → ….

Here H^{p}(X; \underline{A}_{X,ℂ}^{1}) = 0 for p ≠ 0, so there are isomorphisms

H^{p}(X; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) ⟶^{≅} H^{p}(X; \underline{ℂ}_{X}^{∗}) ≅ H^{p+1}(X, ℤ)

for all p ≥ 2.

Proposition. To each smooth charted 2-line bundle L over X with connective structure ∇, there is an associated Čech cohomology class [L, ∇] in

Ȟ^{2}(X; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) .

(In view of the isomorphisms above, this says that the choice of connective structure is unique up to a suitable notion of equivalence.)

Proof. After possibly refining the ordered cover (U, I), we may choose sections s^{αβ} in the complex line bundles L^{αβ} over U_{αβ}. The connections ∇^{αβ} from the connective structure on L act on these sections, so that

∇^{αβ}(s^{αβ}) = η^{αβ} · s^{αβ}

for unique complex 1-forms η^{αβ} on U_{αβ}. These combine to a Čech 1-cochain η ∈ Č^{1}(U; \underline{A}_{X,ℂ}^{1}). It is not a Čech 1-cocycle, but its coboundary has a well-understood form, as we shall now see.

By assumption, the connections ∇^{αβ} + ∇^{βγ} on L^{αβ} ⊗ L^{βγ} and ∇^{αγ} on L^{αγ} are compatible under ϕ^{αβγ} over U_{αβγ}. Since

(∇^{αβ} + ∇^{βγ})(s^{αβ} ⊗ s^{βγ}) = (η^{αβ} + η^{βγ})(s^{αβ} ⊗ s^{βγ})

we get

∇^{αγ}(h^{αβγ} · s^{αγ}) = (η^{αβ} + η^{βγ})(h^{αβγ} · s^{αγ}) .

At the same time

∇^{αγ}(h^{αβγ} · s^{αγ}) = dh^{αβγ} · s^{αγ} + h^{αβγ} · ∇^{αγ}(s^{αγ})
= dh^{αβγ} · s^{αγ} + h^{αβγ}η^{αγ} · s^{αγ}

so that

(η^{αβ} + η^{βγ})h^{αβγ} = dh^{αβγ} + h^{αβγ}η^{αγ}

which we can rewrite as

η^{βγ} − η^{αγ} + η^{αβ} = (h^{αβγ})^{−1} dh^{αβγ} = d log h^{αβγ} .

Hence the pair (h, −η) forms a Čech 2-cocycle in the total complex

Č^{∗}(U; \underline{ℂ}_{X}^{∗} ⟶^{d log} \underline{A}_{X,ℂ}^{1}) .

We let [L, ∇] be its associated Čech cohomology class.

((Should check independence of choice of sections s^{αβ}.)) □

Connective structures on 2-vector bundles

Definition. Let X be a smooth manifold, with an ordered open cover (U, I). Let E → X be a smooth charted 2-vector bundle, with gluing bundle matrices E^{αβ} and coherence isomorphism matrices ϕ^{αβγ}. A connective structure ∇ on E is

(1) a matrix of connections ∇^{αβ} = (∇_{ij}^{αβ})_{i,j=1}^{n} on E^{αβ} over U_{αβ}, for each α < β in I, such that

(2) the matrix of isomorphisms ϕ^{αβγ}: E^{αβ} · E^{βγ} ⟶^{≅} E^{αγ} over U_{αβγ} takes the matrix product of connections ∇^{αβ} ·id+id·∇^{βγ} on E^{αβ} ·E^{βγ} to the matrix of connections ∇^{αγ} on E^{αγ}, so that

(ϕ^{αβγ})_{∗}(∇^{αβ} · id + id · ∇^{βγ}) = ∇^{αγ}

over U_{αβγ}. For matrices of sections s = (s_{ij})_{i,j=1}^{n} in E^{αβ} and t = (t_{jk})_{j,k=1}^{n} in E^{βγ}, with (s · t)_{ik} = ⨁_{j=1}^{n} s_{ij} ⊗ t_{jk} in ⨁_{j=1}^{n} E_{ij}^{αβ} ⊗ E_{jk}^{βγ}, we have

(∇^{αβ} · id + id · ∇^{βγ})(s · t) = ∇^{αβ}(s) · t + s · ∇^{βγ}(t)

as 1-forms of matrices of sections in E^{αβ} · E^{βγ} over U_{αβγ}. In the (i, k)-th entry, this asserts that

(∇^{αβ} · id + id · ∇^{βγ})_{ik}(s · t)_{ik} = ⨁_{j=1}^{n}(∇_{ij}^{αβ}(s_{ij}) ⊗ t_{jk} + s_{ij} ⊗ ∇_{jk}^{βγ}(t_{jk})) .

Do 2-vector bundles admit connective structures?

Proposition. Assume ((ETC)). Then E admits a connective structure ∇.

Proof. Give each E^{αβ} a matrix ˜∇^{αβ} of connections. For each α < β < γ we get a unique End(E^{αγ})-valued 1-form ω^{αβγ} on U_{αβγ}, such that

(ϕ^{αβγ})_{∗}( ˜∇^{αβ} · id + id · ˜∇^{βγ}) = ˜∇^{αγ} + ω^{αβγ} .

Here ω_{ik}^{αβγ} is a 1-form with values in the endomorphism bundle of E_{ik}^{αγ}, for each i, k = 1, …, n.

We can change the connections on E^{αβ} by End(E^{αβ})-valued 1-forms η^{αβ} on U_{αβ}, to

∇^{αβ} = ˜∇^{αβ} − η^{αβ} .

Here η_{ij}^{αβ} is a 1-form with values in the endomorphism bundle of E_{ij}^{αβ}, for each i, j = 1, …, n. Then

(∇^{αβ} · id + id · ∇^{βγ})_{ik} =

( ˜∇^{αβ} · id + id · ˜∇^{βγ})_{ik} − ⨁_{j=1}^{n}(η_{ij}^{αβ} ⊗ id_{E_{jk}^{βγ}} + id_{E_{ij}^{αβ}} ⊗ η_{jk}^{βγ})

and

∇_{ik}^{αγ} = ˜∇_{ik}^{αγ} − η_{ik}^{αγ} .

For the ∇^{αβ} to specify a connective structure on E, we need to have

−η_{ik}^{αγ} + (ϕ_{ik}^{αβγ})_{∗} ⨁_{j=1}^{n}(η_{ij}^{αβ} ⊗ id_{E_{jk}^{βγ}} + id_{E_{ij}^{αβ}} ⊗ η_{jk}^{βγ}) = ω_{ik}^{αβγ} ,

which we can rewrite as

(ϕ^{αβγ})_{∗}(id_{E^{αβ}} · η^{βγ}) − η^{αγ} + (ϕ^{αβγ})_{∗}(η^{αβ} · id_{E^{βγ}}) = ω^{αβγ} .

The left hand side gives a generalized Čech-style coboundary operator

∏_{α<β} Ω^{1}(U_{αβ}; End(E^{αβ})) ⟶^{d} ∏_{α<β<γ} Ω^{1}(U_{αβγ}; End(E^{αγ}))

taking the family of 1-forms η = (η^{αβ}) to dη. There is a similar operator

∏_{α<β<γ} Ω^{1}(U_{αβγ}; End(E^{αγ})) ⟶^{d} ∏_{α<β<γ<δ} Ω^{1}(U_{αβγδ}; End(E^{αδ}))

taking a general family of 1-forms ω = (ω^{αβγ}) to the family with value

(ϕ^{αβδ})_{∗}(id_{E^{αβ}} · ω^{βγδ}) − ω^{αγδ} + ω^{αβδ} − (ϕ^{αγδ})_{∗}(ω^{αβγ} · id_{E^{γδ}})

at α < β < γ < δ.

Then dd = 0 and dω = 0, when ω is defined as at the beginning of the proof. ((Spell out the proofs.)) We would like to have ω = dη. ((Is the complex exact?))

((ETC)) □

((Easier to give a Proof II like in the 2-line bundle case?))

((Can we specify a “connective structure” Co on the stack P with right GL_{n}(V)- action, associated to a 2-vector bundle E?))