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Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.


Introduction
Gerbes provide a (more or less) geometric representation of integral cohomology 3-classes on a space [8,5]. Bundle gerbes, introduced by Murray in [16], are particularly geometric and have a well-known application in the form of the 'lifting bundle gerbe', representing the obstruction to the extension of a principal G-bundle to a principal bundle with structure group a U(1) central extension of G. Here we present a direct extension of the notion of a bundle gerbe to obtain a similar representation of integral 4classes. These bigerbes are special cases, in a sense more rigid, of the bundle 2-gerbes as defined by Stevenson [22], which in turn are a more geometric version of 2-gerbes as defined by Breen [3]. In particular our bigerbes induce bundle 2-gerbes in two ways. One application of this notion is to Brylinski-McLaughlin (bi)gerbes, corresponding to the existence of an extension of the principal bundle over the loop space induced by a principal G-bundle over the original space, to a bundle with structure group a central extension of the loop group [4].
A gerbe may be defined as a simplicial object [17,22]. We work in the context of locally split maps, which is to say continuous maps π : Y −→ X, with local right inverses over an open cover of the topological space X. Such a map determines an associated simplicial space, Y [•] , over X, formed from the fiber products Y [2] Y [3] .
This constitutes a contravariant functor ∆ −→ Top/X, where ∆ denotes the simplex category with objects the sets n = {1, . . . , n} for n ∈ N 0 with morphisms the order preserving maps between these, and Top/X denotes the category of spaces with commuting maps to X. Functions on Y [•] admit a simplicial differential, denoted by d, by taking the alternating sum of the pull-backs, and this operation extends to line (or circle) bundles and sections thereof by taking the alternating tensor product of the pull-backs. A bundle gerbe on X is specified in terms of the simplicial space (1) by the prescription of a complex line bundle L over Y [2] such that dL over Y [3] has a section s which pulls back to be the canonical section of d 2 L over Y [4] . The important special case of the lifting bundle gerbe is obtained when π : E −→ X is a principal G-bundle; then there is a natural map E [2] −→ G, and the line bundle is the pull-back of the line bundle over G associated to a given central extension of G by C * or U(1).
(4) The result is a commutative diagram in which all rows and columns are simplicial spaces. There are then two commuting simplicial differentials, d 1 and d 2 , corresponding to the horizontal and vertical maps, respectively.

Definition.
A bigerbe on the bisimplicial space (3) corresponding to a locally split square (2) is specified by a (locally trivial) complex line bundle L over W [2,2] with d 1 L and d 2 L, over W [3,2] and W [2,3] respectively, having trivializing sections s i , for i = 1, 2, such that ds i is the canonical trivialization of d 2 i L and d 2 s 1 = d 1 s 2 . As for bundle gerbes, there are straightforward notions of products, inverses, pullbacks, and morphisms of bigerbes, for which the characteristic 4-class defined below behaves naturally.
As noted above, among the natural examples is the Brylinski-McLaughlin bigerbe. Suppose that E −→ X is a principal G-bundle over a manifold with structure group a compact, connected, simply connected, simple Lie group. Then is a split square where P X and P E are the respective (based) path spaces, the vertical arrows are projections and the horizontal arrows are the endpoint maps. In the resulting bisimplicial space W [2,1] = ΩE is the based loop space of E which is a principal bundle with structure group the based loop group ΩG of G. The central extensions 1 U(1) ΩG ΩG 1 (6) are classified by H 3 (G; Z) = H 3 G (G; Z) = Z, and the associated line bundle for such a central extension pulls back over W [2,2] = ΩE [2] to a line bundle Q determining a bigerbe. Here the triviality of d 1 Q is the multiplicativity of the central extension, as for a lifting gerbe, whereas the (consistent) triviality of d 2 Q corresponds to the so-called 'fusion' property of the central extension with respect to certain configurations of loops [24,29,11], and which is equivalent to the gerbe property with respect to the path fibration P E [2] −→ E [2] . Incorporation of an additional 'figure-of-eight' condition as in [11,12] -a condition related to the simplicial space of products of X discussed in §3 -promotes this to a bigerbe involving free loops and paths, representing the obstruction of the lift of LE −→ LX to a 'loop-fusion ' LG-bundle which is discussed further below.
There are various 2-gerbe versions of this in the literature. In [4], Brylinski and McLaughlin define a 2-gerbe in the sense of Breen by pulling back the canonical gerbe on G (corresponding to the given class in H 3 (G; Z)) to E [2] by the difference map, particularly in the universal case where X = BG. In [6], and later [28], a similar construction was used to produce a bundle 2-gerbe in the sense of Stevenson. Furthermore, in [4], the authors discuss a correspondence between the 2-gerbe and the problem of extending the structure group of the free loop space LX from LG to LG. The bigerbe above demonstrates this correspondence explicitly.
Returning to the simplicial space, (1), arising from any locally split map, the simplicial differentials extend to theČech cochain spaces over the Y [k] with respect to a system of 'admissible' covers, as defined in §1.1 and §1.2, which can be taken to be arbitrarily fine. Then the simplicial complex 0Č ℓ (X, A)Č ℓ (Y, A)Č ℓ (Y [2] , A) . . .
is exact (see Proposition 2.3), with a homotopy inverse arising from local sections over an open cover. The simplicial differential commutes with thě Cech differential resulting in a double complex. For a bundle gerbe, the representative c(L) of the Chern class of L can be chosen to be a pure cocycle: δc(L) = dc(L) = 0. From the exactness of the simplicial differential this class descends: c(L) = −dβ, δβ = dα for some β ∈Č 1 (Y ; C * ), α ∈Č 2 (X; C * ) (8) and then DD(L) ∈Ȟ 3 (X; Z), the image of [α] ∈Ȟ 2 (X; C * ) under the Bockstein isomorphism, is the Dixmier-Douady class of the gerbe. This is not the original definition of the Dixmier-Douady class of a bundle gerbe as in [16,17]; we show that it is equivalent below in Proposition 2.8 and use the simplicial characterization to prove that a locally split map π : Y −→ X supports a bundle gerbe with a given 3-class on X if and only if the class vanishes when pulled back to Y (Theorem 2.10), and also to classify trivializations of bundle gerbes (in Proposition 2.11). For a general bigerbe there is a similarČech analysis in terms of the triple complex, formed by the three commuting differentials δ, d 1 and d 2 , on theČech spaces over the bisimplicial space (3). Now the Chern class c(L) ∈Č 1 (W [2,2] ; C * ) can again be chosen to be a pure cocycle: δc(L) = d 1 c(L) = d 2 c(L) = 0. As a consequence it descends to a cocycle on Y [2] : c(L) = d 2 β 1 , d 1 β 1 = 0, β 1 ∈Č 1 (W [2,1] ; C * ) δβ 1 = −d 2 λ 1 , d 1 λ 1 = 0, λ 1 ∈Č 2 (Y [2] 1 ; C * ), (9) essentially as for the gerbe. Thus the image of [λ 1 ] inȞ 3 (Y [2] ; Z) (under the Bockstein isomorphism) is the Dixmier-Douady class of L as a gerbe over Y [2] 1 . Significantly however, λ 1 is naturally a simplicial cocycle -a pure cocycle in the δ, d 1 complex -and so dλ 1 = 0.
The symmetry of the bigerbe allows Y 1 and Y 2 to be interchanged, but this also reverses the sign of G(L).
(i) The characteristic 4-class of a bigerbe is natural with respect to pullbacks, morphisms, products, and inverses. It vanishes if and only if the bigerbe admits a trivialization, and two bigerbes have the same 4-class if and only if they are stably isomorphic. (ii) The bisimplicial space generated by a split square, as in (2), supports a bigerbe for a given class inȞ 4 (X; Z) if and only if this class lifts to the Y i to be trivial, with primitives which when pulled back to W have exact difference.
For the Brylinski-McLaughlin bigerbe Q associated to a principal bundle E −→ X with structure group a compact, connected, simply connected and simple Lie group G, the 4-class G(Q) = [γ] is the transgression to X of the 3-class on G corresponding to a central extension LG of LG: X E E [2] G, q α ∈ H 3 (G; Z), q * α = dµ, dγ = δµ.
In particular this applies to the spin frame bundle of a spin manifold. There it represents the obstruction to a lift of the principal loop spin bundle over the loop space to a loop-fusion principal bundle for the basic central extension of the loop spin group. The obstruction is then the Pontryagin class, usually denoted 1 2 p 1 because of its relation to the Pontryagin class of the oriented orthogonal frame bundle, of the spin bundle [29,6,11]. Such loop-fusion lifts are, by the above theorem, in bijection with so-called 'string structures' on the manifold (see Corollary 5.13).
In addition to the Brylinski-McLaughlin bigerbes, we provide other natural examples of bigerbes representing 'decomposable' 4-classes which are the cup product of either 2-classes or a 1-class and a 3-class (see §5.1). Moreover we show that for a simply connected and locally contractible space X, every 4-class is represented by a bigerbe with respect to the locally split square in which the Y i are the based path spaces P X and W = P P X is the based mapping space of the square into X (see §5.5). In particular K(Z, 4) supports a universal bigerbe. Likewise, for X not necessarily simply connected, we show that every 4-class is represented by a bigerbe using the free path spaces; in particular W [2,2] = LLX is the double free loop space in this case.
There is a direct extension of bigerbes to 'multigerbes', higher versions of (bi)gerbes in which the locally split squares are replaced by n-cubes, the line bundles satisfy simplicial conditions with respect to n commuting differentials; these represent cohomology classes of degree n + 2. The symmetry of the (multi)simplicial conditions replaces the ever higher and more complex 'associativity' conditions associated to higher gerbes. This extension is quite straightforward, and for this reason, and since we are unaware of examples apart from decomposable and path multigerbes, we only outline the theory briefly at the end of this paper.
In order to restrict attention to a simple category of topological spaces, and to avoid expanding the paper further, we do not develop the theory of connections on bigerbes here, though this will be done in a future work. We also do not discuss here the bigerbe analogue of bundle gerbe modules or the related theory of generalized morphisms due to Waldorf [26].
Section 1 below contains a discussion of covers and locally split maps, which is the context for the rest of the paper, and our notation forČech theory is introduced in §1.2.
Section 2 is a review of Murray's theory of bundle gerbes (without connections), as developed from theČech-simplicial point of view, with the basic properties of bundle gerbes over split maps recalled in §2.1. The extension of theČech cohomology complex to a bicomplex over the simplicial space of a split map in §2.2 leads to an alternative definition of the Dixmier-Douady class for a bundle gerbe in §2.3, the classification of gerbes over a given split map in §2.4, and the classification of trivializations in §2.5. Examples of bundle gerbes are recalled in §2.6.
A 'product-simplicial' condition, which we refer to as 'doubling', on bundle gerbes is defined in §3.1 with particular application to the free loop space, and the connection with results from [12] is described in §3.2.
In §4.1 the basic properties of locally split squares of maps are given, leading to the definition of bigerbes in §4.2. The characteristic 4-class of a bigerbe is obtained in §4.3 and conversely §4.4 contains a necessary and sufficient condition for representability of a 4-class over a given locally split square.
Section §5 is devoted to examples, with explicit bigerbes corresponding to decomposable classes constructed in §5.1, extending some of the results of [14]. After a brief discussion of doubling for bigerbes in §5.2, our main application of bigerbes -the Brylinski-McLaughlin lifting gerbe -is discussed in §5.3 and its relation to string structures is discussed in §5. 4. Further examples of path bigerbes can be found in §5.5, and finally, we end with a brief account of multigerbes in §6.

1.Čech theory
1.1. Covers and locally split maps. Since we will make substantial use ofČech theory, we start with some conventions on spaces, open covers, and maps. We work throughout in the standard topologist's category of compactly generated Hausdorff spaces and continuous maps, without additional conditions unless explicitly noted.
An open cover of a topological space X is a collection of open sets, U , for which X = U ∈U U . Such a cover defines anétale space by taking the disjoint union with the map to X consisting of the inclusion map on each U .
Note that since the individual sets may not be connected, it is not generally possible to recover the collection U from Et(U ) without specifying additional indexing information. We regard X as its own minimal cover.
If U and V are covers of X and Y then a map of covers is a continuous map g : Et(U ) −→ Et(V) where each element U ∈ U is mapped to a specific element V ∈ V. In particular, there is an underlying map of index sets. Definition 1.1. A continuous map π : Y −→ X of topological spaces is locally split if it admits continuous local sections; thus π is surjective and there exists a cover U of X with respect to which the local sections constitute a continuous map of covers s : Et(U ) −→ Y such that π • s : Et(U ) −→ X is inclusion of the cover in X. In particular the inclusion map of covers Et(U ) −→ X is itself locally split.
If U and V are covers of the same space X then a map of covers such that commutes makes V a refinement of U ; often in the literature the underlying map of index sets is omitted but we always retain it, even if implicitly. In this way the covers of X define a category with refinements as morphisms.
Observe also that if V is a cover of Et(U ) considered as a space, then V is a cover of X as well, the composite map Et(V) −→ X is an inclusion map of covers, and V constitutes a refinement of U . If f : Y −→ X is a continuous map then the pullback is a cover of Y and f lifts to a well defined map of covers f : . If U and V are both covers of X, then is a cover of X mutually refining U and V; indeed, this is the same thing as pulling back V to a cover of Et(U ) by the inclusion map to X or vice versa, and Et(U ∩ V) ∼ = Et(U ) × X Et(V). Note that gives the cover by ℓ-fold intersections of sets in U (we refer to Et(U (ℓ) ) as ǎ Cech space) and that there is a canonical identification of covers over Y whenever U is a cover of X and f : Y −→ X is continuous.
where A is a sheaf of abelian groups on X, Γ(U (•+1) ; A) denotes the group of local sections of A on the intersection cover U (•+1) and δ : Γ(U (•) ; A) −→ Γ(U (•+1) ; A) is given by the alternating sum of the pullbacks by the various inclusion maps U (•+1) −→ U (•) . For our purposes, A will always be a fixed topological abelian group such as C, C * , Z, or U(1) and we will work on the sheaf of continuous maps to A, so thať C • U (X; A) = C(Et(U (•+1) ); A) is a space of continuous maps to A from theétale space. The fullČech cohomology is defined as the direct limiť under refinement. Here the limit is taken with respect to the directed set of covers of X, where U < V if there exists a refinement Et(U ) −→ Et(V) inducing a pullbackČ • V (X; A) −→Č • U (X; A); while there may be many such distinct refinements, the associated pullbacks are chain homotopic and the induced maps on cohomology agree.
We use the standard terminology of cochains, cocycles and coboundaries for elements ofČ • (X; A), and also borrow the adjectives closed and exact from de Rham theory for cocycles and coboundaries, respectively. Proposition 1.2. Fix a topological abelian group A, and suppose a continuous map f : Y −→ X of topological spaces lifts to a map of covers f : Et(V) −→ Et(U ) for covers V and U of Y and X, respectively. Then this induces a chain map Proof. The map of covers induces a map of covers Et(V (ℓ) ) −→ Et(U (ℓ) ) for each ℓ, giving a chain mapČ ℓ V (Y ; A) −→Č ℓ U (X; A), and the natural maps Et(V (ℓ) ) −→ Et(U (ℓ) ) commute with the inclusions Et(U (ℓ) ) ֒−→ Et(U (ℓ−1) ) in each factor of the ℓ-fold intersections into the original covers, so f * commutes with theČech differential δ and defines a chain morphism. Furthermore, this is well-defined with respect to refinement; specifically, if Et(U ′ ) −→ Et(U ) and Et(V ′ ) −→ Et(V) are refinements such that f lifts to a map of covers f : commutes, descending in the direct limit in cohomology to the map f * : We proceed to define a more limited form of pullback on chains with respect to the sections of a locally split map. This is essential to the exactness of the simplicial complex (7) ofČech chains on the simplicial space induced from a locally split map mentioned in the introduction and discussed in detail in §2.2. Definition 1.3. For a locally split map π : Y −→ X, an admissible pair of covers consists of covers U of X and V of Y along with maps of covers π : Et(V) −→ Et(U ) and s : Et(U ) −→ Et(V) such that (i) π is a lift of π, (ii) s is a lift of a local section s : Et(U ) −→ Y , (iii) s is a section (right inverse) of π, i.e. π s = Id U : Et(U ) −→ Et(U ), and (iv) for each U 1 , i.e., the section defined on U 1 restricts over U 2 to a map into some V 2 over U 2 . The key consequence of (iv) is that for each ℓ ≥ 1 there exist canonical sections s ℓ : so that U (ℓ) and V (ℓ) also constitute an admissible pair.
Lemma 1.4. If π : Y −→ X is a locally split map with arbitrary covers U of V of X and Y , respectively, admitting a section s : Et(U ) −→ Y , then there exists an admissible pair U ′ and V ′ refining U and V such that s lifts s.
In fact, if π i : Y i −→ X, i = 1, . . . , N , is a finite number of locally split maps over X, then any covers U of X and V i of Y i can together be refined to U ′ and V ′ i , respectively, constituting an admissible pair for each i simultaneously, with s i lifting any given sections s i : Remark. Since admissible covers may be taken to refine arbitrary covers of X and Y , it follows that covers participating in admissible pairs are final in the directed set of all covers; in particular the direct limitsȞ ℓ (X; A) are equivalent to the direct limit taken over admissible covers alone [13,Thm. 1,p. 213]. Equivalently, any class in the cohomology of X or Y may be represented as aČech cochain with respect to a cover which is part of an admissible pair.
Proof. We construct the refinement in two steps, first 'pulling down' V to get U ′ = s −1 V and then 'pulling up' U ′ to get V ′ = π −1 U ′ ∩ V. More precisely, if s U : U −→ Y is the section of π over U ∈ U then the sets s −1 U V ⊂ U for V ∈ V, define a cover of U , and hence together a refinement of U as a cover of X. This is the cover U ′ = s −1 V defined above. Then s lifts to a naturally defined map of covers s : Et(U ′ ) −→ Et(V) given by s U : s −1 U V −→ V . The pull-back π −1 U ′ of the new cover of X defines a refinement V ′ = π −1 U ′ ∩ V of V to which π lifts as a map of covers π : Et(V ′ ) −→ Et(U ′ ). Moreover, as a consequence of the fact that πs = Id on each element U ∈ U ′ , it follows that s lifts to a map of covers s : Et(U ′ ) −→ Et(V ′ ), and π s = Id.
To see that this satisfies the final condition of Definition 1.3, note that by definition each Given a finite number of locally split maps π i : Y i −→ X, we set U ′ = i s −1 i V i , which refines each s −1 i V i and hence lifts the map of covers s i defined above, followed by V ′ i = π −1 i U ′ ∩ V for each i. In preparation for §2.2 we make the following observations. Lemma 1.5. Denote by Y [k] = Y × X · · · × X Y the k-fold fiber product of π : Y −→ X and fix an admissible pair of covers (U , V) for (X, Y ). Then for each k ≥ 2, The sections s ℓ of Definition 1.3.(iv) determine a map of covers s ℓ : Proof. It is clear that V [k] is an open cover of Y [k] . Note that there is a canonical identification ( , the latter fiber product taken over U (ℓ) ; indeed, for k = ℓ = 2 this amounts to observing that with similar identification holding in general.
Taking the set of all such O i over O ∈ O gives a cover V ′ of Y such that V ′ × X · · · × X V ′ is a refinement of O, and then invoking Lemma 1.4 produces an admissible pair such that V [k] = V × U · · · × U V refines V ′ × X · · · × X V ′ and therefore O.

Bundle gerbes
2.1. Simplicial line bundles. We recall the notion of a bundle gerbe [16], which for our purposes is most efficiently defined in terms of simplicial line bundles.
Denote by Y [k] the k-fold fiber product Y × π · · · × π Y , with projection maps π j : Y [k] −→ Y [k−1] , j = 0, 1, . . . , k − 1, where π j (y 0 , . . . , y k−1 ) = (y 0 , . . . , y j , . . . , y k−1 ) omits the jth factor enumerated from 0. Then is a simplicial space with face maps π j and degeneracy maps the fiber diagonal maps Y [k−1] −→ Y [k] (of which we will not make use). More precisely, Y [•] is a simplicial space over X, meaning that all maps commute with the projections π : Y [k] −→ X, and X itself may be regarded as an augmentation in (2.1). For notational convenience, we set Y [1] Remark. Our enumeration (which is geometrically natural here) differs unfortunately from the standard convention for simplicial spaces, under which one would typically write Y 0 = Y (as the image of the 0 simplex), Y 1 = Y [2] (as the image of the 1-simplex), etc., augmented by Y −1 = X. For consistency we use this alternative convention throughout, and beg the pardon of readers who would prefer to use the standard one.
Given a complex line bundle L −→ Y [k] , its differential is defined to be the line bundle Using the commutation relations between the π j , it follows that Remark. While we will mostly work with complex line bundles, we could equivalently take L to be a principal C * or U(1) bundle. At times we will use these objects interchangeably without further elaboration.
A bundle gerbe (L, Y, X) as defined by Murray is equivalent to a simplicial line bundle on the simplicial space Y [•] in the sense of Brylinski and McLaughlin [4]; this consists of a complex line bundle L −→ Y [2] along with a trivialization of the bundle dL = π * 0 L ⊗ π * 1 L −1 ⊗ π * 2 L over Y [3] , which in turn induces the canonical trivialization of d 2 L when pulled back over Y [4] . The trivialization of dL −→ Y [3] is equivalent to the 'gerbe (or groupoid) product' isomorphism φ : π * 2 L ⊗ π * 0 L ∼ = −→ π * 1 L, which multiplies (composes) pairs of respective elements in the fibers L y 0 ,y 1 and L y 1 ,y 2 to get elements in L y 0 ,y 2 ; the condition that the trivialization coincide with the canonical one on d 2 L over Y [4] is equivalent to associativity of this product.
A bundle gerbe (L, Y, X) is trivial if there exists a bundle L ′ −→ Y and an isomorphism L ∼ = dL ′ on Y [2] ; such an isomorphism is called a trivialization of L.
If (L, Y, X) is a bundle gerbe on X, then its pullback by a continuous map f : X ′ −→ X is the bundle gerbe ( f * L, f * Y, X ′ ); here we use the naturality [2] by f . Likewise, the product of two bundle gerbes (L i , Y i , X), i = 1, 2 on X is given by where L 1 ⊗ L 2 is shorthand for pr * 1 L 1 ⊗ pr * 2 L 2 with pr i : Y 1 × X Y 2 −→ Y i denoting the projections from the fiber product. It is straightforward to verify that this is a bundle gerbe, which we denote for simplicity as L 1 ⊗ L 2 . The definitions of product and pullback implicitly use the following standard result which we record for later use.
Lemma 2.1. Pullbacks and fiber products of locally split maps are locally split. More precisely, if π : Y −→ X is locally split and f : [2] which intertwines the sections of dL and dL ′ ; a (strong) isomorphism is a morphism for which X = X ′ and f = Id. In particular, a morphism f : Finally, two gerbes (L i , Y i , X), i = 1, 2 are said to be stably isomorphic if L 1 ⊗ L −1 2 is trivial, or equivalently, there exist trivial gerbes (T i , Z i , X) such that in the sense of a strong isomorphism over a space Z [2] where Z −→ X admits maps to Y 1 , Y 2 , Z 1 , and Z 2 . This is strictly weaker than an isomorphism as defined above, and was introduced in [17] in order to obtain a classification of bundle gerbes up to stable isomorphism by their Dixmier-Douady class.
Remark. There is a weaker notion of gerbe morphism due to Waldorf [26], which naturally incorporates the theory of gerbe modules, and has the property that the invertible morphisms are precisely the stable isomorphisms; moreover a trivialization becomes the same thing as an isomorphism to a canonical trivial gerbe over X. Because we leave the generalization to bigerbes of Waldorf's morphisms to a future work, we will not pursue this further here.

SimplicialČech theory.
A primary motivation for the definition of gerbes is that they represent a class inȞ 3 (X; Z) ∼ =Ȟ 2 (X; C * ) -the Dixmier-Douady class; for a bundle gerbe (L, Y, X) this will be denoted by DD(L). This class is natural with respect to products, pullbacks, and inverses, and it determines (L, Y, X) up to stable isomorphism [17]. We give an alternative (though not necessarily simpler) derivation of these facts based on a closer study of simplicial spaces; this approach is used in the generalization to bigerbes below. In doing so we identify the 3-classes on X which can be represented by a bundle gerbe with respect to a given locally split map Y −→ X, and recover the classification of the trivializations of a trivial bundle gerbe.
Consider the simplicial space, (2.1), consisting of the fiber products of a locally split map π : Y −→ X, and fix an admissible pair of covers (U , V). Proposition 1.2 and Lemma 1.5 show that the induced maps ; A) are chain maps and the simplicial differential onČech cochains is defined by where we have omitted the coefficient group and covers from the notation for simplicity. We take the rowČ • (Y ) to have vertical degree 0 in (2.3) (corresponding to its true simplicial degree), so the rowČ • (X) has degree −1.
Related versions of thisČech-simplicial double complex appear more generally in algebraic geometry [7], and are also discussed in [4] in the context of simplicial gerbes.
Convention 2.2. Our convention for double (and higher) complexes is that the two differentials commute, as above. This necessitates the introduction of a sign (depending on an ordering of the differentials) in the total differential, which we take to be Another possible sign convention is given by changing the formal order of d and δ, namely In general, whenever we have a multicomplex C p 1 ,...,p k with k commuting differentials d 1 , d 2 , . . . , d k , the total differential will be defined inductively by Switching the order of the indices requires composing with an automorphism given in each degree by an appropriate power of −1 as above.
is exact. In particular, a collection of local sections s : Et(U ) −→ Y determines a chain homotopy contraction via the maps in Lemma 1.5.
Compare the exactness of the de Rham complex in §8 of [16]. This may be understood as a manifestation of the fact that the geometric realization of the simplicial set Y [•] is known to be homotopy equivalent to X.
Proof. The chain contraction is defined by . That s * ℓ forms a chain homotopy retraction follows directly from (1.7). Remark. It is important that the maps s * ℓ do not generally commute with theČech differential; in other words, we do not obtain a chain map, and in particular we do not claim that s * ℓ descends to cohomology. Indeed the lift, s ℓ , of s in (1.6) corresponds to an (arbitrary) preference for the map corresponding to the first factor, U 1 , in the ℓ-fold intersection.
2.3. Dixmier-Douady class of a gerbe. The setting for our analysis of theČech cohomology of a bundle gerbe is the truncated complex where we have omitted the bottom row of (2.3), and wherě theČech cohomology of the simplicially trivial classes on Y [2] , or the horizontal cohomology of the top row in (2.5), taken in the direct limit over covers of the form V [2] for admissible pairs. For later use we note the following result.
Theorem 2.5. The total cohomology of the double complex (2.5) is isomorphic toȞ • U (X; A). Proof. Owing to exactness of the columns, the (d, δ) spectral sequence of (2.5) degenerates at the E 1 page to and therefore stabilizes at E 2 to the cohomologyȞ • U (X; A). Next we will show that a bundle gerbe is represented by a pure cocycle in the double complex (2.5) concentrated atŽ 1 (Y [2] ; C * ), so with δc(L) = 0 and dc(L) = 0 and this descends to the Dixmier-Douady class.
Proof. As a complex line bundle, L gives rise to a Chern cocycle c(L) ∈ C 1 O (Y [2] , C * ) for some cover O of Y [2] , which by Lemma 1.5 may be assumed to be of the form O = V [2] for an admissible pair, and dc(L) ∈Č 1 V [3] (Y [3] , C * ) represents the bundle dL on Y [3] . The trivialization of dL is encoded by [3] , C * ) such that δγ = dc(L), and the fact that this induces the canonical trivialization of d 2 L on Y [4] means that dγ = 0. Thus by exactness we can alter c(L) by δ applied to a d-preimage of γ to arrange that dc(L) = 0.
Conversely, given α ∈Ž 1 V [2] (Y [2] ; C * ) representing a cocycle with respect to some fixed open cover V [2] of Y [2] associated to an admissible pair, the usual construction uses α on (V [2] ) (2) to assemble a line bundle L −→ Y [2] out of trivial bundles on V ∈ V [2] . Then since dα = 0 it follows that dL is assembled trivially out of trivial bundles on the open cover V [3] of Y [3] , and hence is globally trivial (with the trivialization agreeing with the canonical one on d 2 L).
Finally, L admits a trivialization [2] ; C * ) in the double complex (2.5) and is obtained explicitly by a zig-zag construction The sign in −dβ = c(L) arises from the fact that the total differential D involves the term −d on that column according to Convention 2.2. Note that if π i : Y i −→ X i , i = 1, 2 are locally split maps which are intertwined by f : 1 ; A) of double complexes, in that the ( f [k] ) * commute with δ and d. Indeed, we may fix an admissible pair (U 2 , V 2 ) for (Y 2 , X 2 ) and then invoke Lemma 1.4 to obtain a refinement of (f −1 U 2 , f −1 V 2 ) by an admissible pair (U 1 , V 1 ), and then it follows that there are maps of covers 2 ) the pullbacks of which intertwine the two double complexes.
Proposition 2.8. The Dixmier-Douady class as defined above coincides with the definition given by Murray and is natural with respect to inverse, product, and pullback; it vanishes if and only if the gerbe is trivial.
Proof. The (well-known) naturality properties follow directly from the preceding remark. To see the coincidence of our definition with that of Murray given in [16], we first recall the latter.
Suppose s : U −→ Y is a set of local sections of the locally split map, and consider the pullback L ′ = (s 2 ) * L to U (2) of L via the map s 2 : U (2) −→ Y [2] . Since L is locally trivial, this cover can be refined so that L ′ is trivial over each component, and so has a nonvanishing section σ : [3] pulls back to give a trivialization of δL ′ = (s 3 ) * dL −→ U (3) which allows g := δσ to be regarded as a cochain g : U (3) −→ C * and the associativity condition over Y [4] implies that g is closed, To see that this is equivalent to Definition 2.7, it suffices to show that [g] represents the image of c(L) in the total cohomology of the double complex (Č • (Et(U (•) ); C * ), δ, d), where we use Et(U ) −→ X itself as the locally split map. For convenience we suppose that U is a 'good cover', meaning that each element of U (ℓ) is contractible for each ℓ. Note that by this contractibility, with the simplicial differential now identified with theČech differential oň C • U (X), and then stabilizes at

Representability of 3-classes.
We proceed to give a characterization of the classes in H 3 (X; Z) which are represented by bundle gerbes (L, Y, X) for a given locally split map Y −→ X. Note that the augmented double complex 0 0 0 for any admissible pair (U , V) has exact columns and therefore trivial total cohomology. Since the (δ, d) spectral sequence of this complex (beginning with the horizontal differential) must necessarily stabilize at the E 3 page (as there are only three rows), it follows that the E 2 differentials are necessarily isomorphisms, which we record in the following form after passing to the direct limit over covers in cohomology.
Theorem 2.9. There are isomorphisms [2] ; A)/Ȟ k (Y ; A) (2.10) for each k ∈ N and coefficient group A; these isomorphisms are natural with respect to pullback by maps of locally split spaces, and also with respect to the Bockstein isomorphismš It is reasonable to call the isomorphism (2.10) transgression from classes on X to (equivalence classes of) classes on Y [2] ; the map is realized at the chain level by the zig-zag (2.8).
In particular there is a natural isomorphism under which the Chern class c(L) ∈Ȟ 1 Z (Y [2] ; C * )/dȞ 1 (Y ; C * ) is the image of DD(L) inȞ 3 (X; Z). We see again that DD(L) = 0 if and only if c(L) ∈ dȞ 1 (Y ; C * ), which by Proposition 2.6 holds precisely when L is trivial. In combination with Proposition 2.6 this proves the following result.
Then on Y [2] the shift map composed with s [2] determines a map χ : Y [2] −→ PU(H), along which the universal line bundle can be pulled back to give a simplicial bundle L = χ * U(H) −→ Y [2] with DD(L) = α.
There is an action on the set of trivializations of L by H 2 (X; Z) (in the form of equivalence classes of line bundles) as follows. Given a line bundle Proposition 2.11. Let (L, Y, X) be a trivial gerbe. Then the set of trivializations of L is a torsor for the group Im π * : Proof. Clearly the action of H 2 (X; Z) factors through its image in H 2 (Y ; Z); to see that this image acts transitively suppose P −→ Y and P ′ −→ Y are two trivializations of L, represented by classes [2] ; Z), and from the (δ, d) spectral sequence for (2.9), the E 2 term associated toČ 2 (Y ; Z) of which must vanish identically, it follows that 2.6. Decomposable and universal gerbes. One consequence of Theorem 2.10 is the existence of the decomposable gerbes of [14]. Given a 3-class on X which is the cup product α ∪ β of α ∈ H 2 (X; Z) and β ∈ H 1 (X; Z), we may take π : Y −→ X to be the circle bundle with Chern class c(Y ) = α, and then since Y is canonically trivial when pulled back to itself, it follows that π * (α ∪ β) = 0 ∪ π * β = 0 ∈ H 3 (Y ; Z), so by Theorem 2.10 the following is immediate.
Remark. Note that [14] goes further for X a smooth manifold by constructing a connection on the gerbe from a connection on Y and a function u ∈ C ∞ (X; U(1)) representing β.
In fact, the image of α ∈ H 2 (X; Z) in H 1 Z (Y [2] ; Z)/H 1 (Y ; Z) with respect to the isomorphism (2.11) has a geometric interpretation that will be of use in the construction of decomposable bigerbes in §5.1.
It also follows from Theorem 2.10 that for a connected, locally contractible, space X, a gerbe can be constructed representing any integral three class using the (based) path fibration P X −→ X. Indeed, the hypotheses on X imply that the end-point map P X −→ X is locally split, and since P X is contractible, any 3-class on X vanishes when lifted to P X. The fiber product P [2] X may be identified with the based loop space ΩX, and the isomorphism (2.11) takes the form , from which we recover the following well-known result.
Theorem 2.14. For a connected, locally contractible, space X each α ∈ H 3 (X; Z), corresponds to a unique bundle gerbe L −→ ΩX (up to simplicial isomorphisms of the line bundle) with DD(L) = α.
This 'canonical gerbe' on the loop space goes back at least to Brylinski [5]. Murray defines a bundle gerbe version in [16] under the assumption that X is 2-connected; a hypothesis which is removed in [6].
In particular, since K(Z, 3) may be realized as a CW complex, its path space carries a universal gerbe.
The simplicial structure on ΩX coming from P [k] X is related to what has been called the fusion product in the literature [24,27,29,12]. A point γ = (γ 0 , γ 1 , γ 2 ) ∈ P [3] X consists of three paths with common endpoints and so defines three loops, ℓ i = π i γ ∈ ΩX = P [2] X, i = 0, 1, 2, by the simplicial maps, and we say ℓ 1 = (γ 0 , γ 2 ) is the fusion product of ℓ 2 = (γ 0 , γ 1 ) and A fusion structure on a line bundle L −→ ΩX is a collection of associative isomorphisms for all such triples, which is equivalent to a simplicial line bundle structure on L with respect to P [•] X. In this language then, Theorem 2.14 shows that fusion line bundles on ΩX, which are equivalent to bundle gerbes (L, P X, X), are classified by H 3 (X; Z) (see also Waldorf's related results in [27]).

Doubling and the free loop space
3.1. Simplicial bundle gerbes and figure-of-eight. Replacing the simplicial line bundle in the definition of a bundle gerbe with a bundle gerbe over X 2 of a simplicial space X • leads to the notion of a simplicial bundle gerbe, which has been defined by Stevenson [22] and is the setting for his definition of bundle 2-gerbes. Here we consider a more limited 'product-simplicial' version, which we call simply doubled, of this theory, not yet to obtain a version of 2-gerbes as we shall do in §4 below, but rather to promote the examples of bundle gerbes involving the based loop space ΩX to those involving the free (unbased) loop space LX by satisfying an additional condition with respect to the simplicial space X k : k ∈ N of products, with face maps the projections; this space is often denoted by EX in the literature. While we specialize to this simplicial space of products below, we proceed for the moment in some generality for an arbitrary simplicial space X • , where we continue to use our unusual enumeration convention. Suppose then that (L, Y 2 , X 2 ) is a bundle gerbe over X 2 . Using products, inverses and pullbacks, we may define the gerbe over X 3 , where π j : X 3 −→ X 2 for j = 0, 1, 2 are the face maps of the simplicial space.
Definition 3.1. A simplicial trivialization of a bundle gerbe L over X 2 is a trivialization of the bundle gerbe ∂L over X 3 . It follows by naturality that for such a gerbe the Dixmier-Douady class DD(L) ∈ H 3 (X 2 ; Z) satisfies Note that the gerbe ∂L is defined a priori with respect to the locally split map (3.2) However, using the notion of gerbe morphism, we may specialize to the setting in which there exists a locally split map Indeed, it then follows that Y 3 maps through the product space (3.2), and we may require that is trivial as a bundle gerbe over X 3 , where we continue to denote the extensions of π j as maps from Y 2 by the same notation. When such data is available, we will abuse notation by referring to (3.3) itself as ∂L (as these are (strongly) isomorphic as bundle gerbes over X 3 ) and a trivialization of (3.3) is a simplicial trivialization of L. Explicitly, this then is the data of a By naturality of the Dixmier-Douady class, the conclusion (3.1) remains valid.
Remark. We do not require that the split maps Y • −→ X • be compatible by the π j in the sense of §1.2. We also do not require that Y • extend to form (part of) a simplicial space over X • , as indeed our example of interest will not. By contrast, in the setting of the bigerbes defined in §4 below, we will employ a bisimplicial space of compatible locally split maps.
We now specialize to the case in which X • = X • consists of products of a fixed space X. As a special case of the fiber product construction over the unique map π : X −→ * to a 1-point space, this map is globally split, with section s : * −→ x * ∈ X for any choice of x * ∈ X. TheČech theory constructions of §1.1 and §2.2 give a map s * : A) which in this case does commute with theČech differential (since s is global), and hence descends to a chain homotopy contraction for each ℓ of the cohomology complex which is therefore exact. (This is a reflection of the well-known fact that the geometric realization |EX| of the simplicial set EX is contractible.) Indeed, denoting by s = s × 1 × · · · × 1 : , it follows that s * ∂ + ∂s * = 1 onȞ • (X k ; A). Note that throughout this section and below, we denote this product simplicial differential by ∂ = k−1 j=0 (−1) j π * j rather than d to avoid confusion whenever both appear together.
As a consequence of (3.5) and (3.1), we have the following result.
Proposition 3.2. For a gerbe (L, Y 2 , X 2 ) with simplicial trivialization over X 3 , the Dixmier-Douady class of L descends from X 2 to X itself, so is well-defined.
We refer to such a gerbe as a doubled gerbe.
Remark. When X • = X [•] is a more general simplicial space of fiber products of a locally split map X = X 1 −→ X 0 , Stevenson in [22] defines additional conditions for a simplicial gerbe, including higher associativity conditions over X 4 and X 5 under which the class of a bundle gerbe further descends to a degree four cohomology class on X 0 and such an object is defined to be a bundle 2-gerbe on X 0 . Here we only use the simplicial condition to descend the 3-class from X 2 = X 2 to X 1 = X, and will not make use of these additional conditions.
The locally split map of present interest consists of the free (unbased) path space mapping to X 2 by the evaluation map on both endpoints. Instead of the fiber products of the pullbacks of IX to X 3 we will take Y 3 = IX also, with evaluation map (1) , mapping to the midpoint as well as the endpoints. For disambiguation, we will often distinguish these two incarnations of the free path space by writing them as I 2 X and I 3 X, respectively. The three liftings π i of the projection maps π j : X 3 −→ X 2 taking γ ∈ I 3 X to I 2 X are obtained by reparameterizing to obtain the three paths π 1 γ(t) = γ(t), π 2 γ(t) = γ( 1 2 t), and π 0 γ(t) = γ 1 2 (1 + t) . Remark. While it is possible to continue to the right, with Y n = I n X mapping to X n by evaluating along n points, the need to reparameterize paths to define the lifts π i means that these do not satisfy the simplicial relations, so we do not in fact obtain simplicial spaces Y

[k]
• over X • . In particular, the associated maps ∂ on line bundles do not form a complex, i.e., ∂ 2 L is not canonically trivial, except at the bottom level.
Observe that Y [2] 2 = I [2] 2 X may be naturally identified with the free loop space, LX = C(R/2πZ; X), while the space Y 3 X consists of pairs of paths which coincide at their midpoint in addition to their endpoints. The latter may be identified with those loops ℓ in LX for which ℓ(π/2) = ℓ(3π/2), which we call figure-of-eight loops, and we accordingly denote the figure-ofeight loop (sub)space by 3 X. In fact, in this case the product doubling condition for a gerbe over X 2 can be strengthened. Proof. Retraction of paths onto their initial points determines a deformation retract of I k X onto X, with respect to which the two simplicial maps both become identified with the evaluation map I [2] k X −→ X at a single parameter value. Thus every line bundle P −→ I k X is isomorphic to a bundle Q pulled back from X, and then dP ∼ = Q ⊗ Q −1 −→ I [2] k X is isomorphic to a trivial bundle. This result is independent of k.
Alternatively, we may use the equality (2.12) proved in Proposition 2.11, which here takes the form since with respect to the retraction IX ≃ X the map π : I k X −→ X k is identified with the diagonal map ∆ : X −→ X k . Since ∆ * is surjective on cohomology, it follows that d ≡ 0 : H 2 (I k X; Z) −→ H 2 Z (I [2] k ; Z), so every trivial gerbe is in fact trivial as a line bundle.
Remark. In other words, there are no 'nontrivial trivial gerbes' with respect to the path spaces. This seems at first confusing in light of Proposition 2.11, since the classifying set Im π * : H 2 (X 2 ; Z) −→ H 2 (IX; Z) ∼ = H 2 (X; Z) of gerbe trivializtions may well be nontrivial, yet these facts are not inconsistent. Indeed, while the only trivial gerbe with respect to IX −→ X 2 is the equivalence class of the trivial line bundle on LX, the set of gerbe trivializations of this trivial gerbe may itself be nontrivial.
On the other hand, we could restrict consideration to doubled trivializations, meaning line bundles P −→ I 2 X with dP = L such that ∂P −→ I 3 X is a trivial bundle. The set of these doubled trivializations is indeed trivial, since under the retractions I k X ≃ X, the reparameterization maps π j : I 3 X −→ I 2 X become the identity, and the operator ∂ ∼ = Id * − Id * + Id * likewise becomes the identity, so triviality of ∂P implies triviality of P itself.
The extension of this notion of doubling will be important in the setting of the Brylinski-McLaughlin bigerbe in §5.3.
From the point of view of fusion line bundles on loop space, the doubling property corresponds to the 'figure-of-eight' condition, as defined in [12,11]. The following definition is therefore just a repackaging of the above in a different language.  Remark. The figure-of-eight structure is weaker than other conditions that have been considered in the literature, such as thin homotopy equivariance in [27], or reparameterization equivariance in [11], which likewise identify categories of fusion line bundles on LX with gerbes on X.

3.2.
Loop-fusion cohomology. In fact, applying the above considerations toČech theory in place of line bundles leads to a general result, which recovers the main theorem in our previous paper [12]. There we defined loop-fusion cohomology on LX, which in the present language is equivalent to the group In particular, the setȞ 2 lf (LX; Z) classifies loop-fusion line bundles up to isomorphism. It is additionally shown in [12] that the isomorphism descends via the forgetful mapȞ • lf (LX; A) −→Ȟ • (LX; A) to the transgression homomorphism H ℓ (X; A) −→Ȟ ℓ−1 (LX; A); recall that the latter is defined by composing the pullback along the evaluation map S 1 × LX −→ X with the pushforward along the projection S 1 × LX −→ LX (given by cap product with the fundamental class of S 1 ).
Proof. The result follows from naturality of the isomorphism (2.10), and exactness of (3.5). Applied to the three maps from I 2 X this yields an isomorphism (omitting the coefficient group for brevity) 3 X)/dȞ ℓ−1 (I 3 X) (3.6) However, as noted in the proof of Lemma 3.3, the deformation retraction of the free path spaces I k X onto X implies that d : k X; A) is trivial, so the quotients in (3.6) disappear. Moreover, by exactness of (3.5), the kernel of ∂ inȞ ℓ (X 2 ; A) is the image ofȞ ℓ (X; A) and this is automatically in the kernel of π * :Ȟ ℓ (X 2 ; A) −→Ȟ ℓ (I 2 X; A) under the retraction I 2 X ≃ X, so (3.6) simplifies tǒ as claimed.

Bundle bigerbes
4.1. Locally split squares. Bigerbes as introduced below are based on the following notion.
There is manifest symmetry in the definition. Note that in fact all four maps in (4.1) are locally split; indeed, if s 1 : Et(U ) −→ Y 1 are local sections of π 1 for some cover U of X, then 1 × ( s 1 • π 2 ) : Et(π −1 2 U ) −→ Y 1 × X Y 2 form local sections of the projection Y 1 × X Y 2 −→ Y 2 ; moreover, refining this cover π −1 2 U of Y 2 if necessary, these may then be composed with local sections of W −→ Y 1 × X Y 2 to obtain local sections of W −→ Y 2 , and similarly for Y 1 .

Definition 4.2.
Given a locally split square as above, an admissible set of covers is a set of covers U X , U Y 1 , U Y 2 , and U W for X, Y 1 , Y 2 , and W , respectively for which each pair (U X , is an admissible pair and such that the following four diagrams commute: Given any set of covers for the spaces in a locally split square (X, Y 1 , Y 2 , W ), there exist refinements constituting an admissible set.
In fact, given any finite set of locally split squares over X which may share some of the same maps π i : Y i −→ X and any initial set of covers of these, there exists a set of covers for all spaces which forms an admissible set for each square seperately, and for which the section maps lift any initially prescribed local sections of the maps W } be the initial arbitrary set of covers for a given square. Without loss of generality by the stronger statement of Lemma 1.4, we may assume that (U ′ X , U ′ Y i ), i = 1, 2 form an admissible pair lifting prescribed local sections s 1 and s 2 of π 1 and π 2 . Under this condition, it follows that U ′ Y i support sections σ i of the fiber product projections Y 1 × X Y 2 −→ Y i defined by σ 1 := 1 × (s 2 • π 1 ) and σ 2 := (s 1 • π 2 ) × 1, respectively (here 1 denotes the inclusion of the cover 1 : for which the following three squares commute: We also fix an initial cover U ′ . This is the starting point, and wet set U 0 where the superscript denotes the step in the process. As in the proof of Lemma 1.4, we proceed to refine the covers in two steps, first "pulling down" by the sections, and then "pulling up" by the projections.
Step 1 : Starting at the top right, we subsequently set U 1 after which we have maps of covers constituting the commutative diagram and which each constitute sections the appropriate map The only nontrivial assertion is the horizontal identifications in (4.2), but this follows easily from viewing the refinements as pullbacks in light of naturality of pullback. Alternatively it can be checked directly: given U ∈ U 0 X mapping by s 1 into V 1 ∈ U 0 Y 1 , say, the latter becomes refined into sets of the form . For the stronger version involving multiple split squares over X, we note that U 1 Y i and then subsequently U 1 X may be further refined while retaining the diagram (4.3); in particular, we can replace these by the mutual refinements of the Step 1 covers for each locally split square in which the spaces are involved.
Step 2 : Starting at the bottom left, we subsequently set U 2 , after which we have maps of covers such that each subsequent pair is admissible. Indeed, the only assertion to check admissibility of the pair (U 2 is defined in a different way here compared to Lemma 1.4.
To check this, first note that U 2 Y 1 × X Y 2 is in general finer than either i ) for i = 1, 2, and in fact it factors through both of these since pr −1 It follows from this that pr i lift to U 2 Y 1 × X Y 2 and π 1 • pr 1 = π 2 • pr 2 . To see that σ i lift, consider for example a V ∈ U 1 Y 2 mapping by σ 2 to W ∈ U 1 Y 1 × X Y 2 after step 1. In Step 2, V is refined into sets of the form V ∩ π −1 2 (U ) for U ∈ U 2 X , and we have a diagram Finally, to see that the final (pairwise intersection) property for admissibility holds, take V 1 , V 2 ∈ U 2 Y 2 with non-empty intersection. By construction the V i are associated to U i ∈ U 2 X for i = 1, 2, and by the intersection property for s 1 it follows that ( s 1 ) U 1 : . In case multiple split squares over X are being considered, note that the refinements in Step 2 are mutually consistent, so there are no additional considerations to this step in this case.
The admissible set is given by U • = U 2 • for • ∈ {X, Y 1 , Y 2 , W }, with maps of covers between the U Y i and U W given by compositions pr i • p and t • σ i . The required commutativity of the four squares follows from the like commutativity of the squares in step 2 for the spaces X, Y 1 , Y 2 , and Y 1 × X Y 2 .
2 each form simplicial spaces over X, giving the bounding column and row in (4.4) below.
Setting W [1,1] = W and gives simplicial spaces over Y 1 and Y 2 extending above and to the right of W [1,1] in (4.4). That the rest of the quadrant can then be filled out unambiguously by fiber products is a consequence of the following result.
Proof. Both sides may be identified with the set of tuples (w i,j : 1 ≤ i ≤ m, 1 ≤ j ≤ n) ∈ W mn such that for each i, (w i,1 , . . . , w i,n ) all map to a fixed y 1,i ∈ Y 1 and for each j, (w 1,j , . . . , w m,j ) all map to a fixed y 2,j ∈ Y 2 , and where every y 1,i and y 2,j sit over a fixed x ∈ X. 2 , respectively, and the squares commute for consistent choices of maps. For notational convenience, we also set For reasons that will become clear below, the order of the spaces Y 1 and Y 2 , or equivalently the orientation of the square (4.1), is part of the data of the bigerbe.
Definition 4.6. If Q 1 −→ W [1,2] and Q 2 −→ W [2,1] are line bundles which are simplicial with respect to d 2 and d 1 , respectively -so d 2 Q 1 over W [1,3] is equipped with a trivialization inducing the canonical trivialization of d 2 2 Q 1 and similarly for Q 2 -then d 1 Q 1 ⊗d 2 Q −1 2 has a canonical bigerbe structure. A bigerbe L is said to be trivial if 2 , for Q 1 and Q 2 as above, with the isomorphism identifying the bigerbe structure on L with the canonical one on d 1 Q 1 ⊗ d 2 Q −1 2 ; such an isomorphism is referred to as a trivialization of L.
In particular, L is trivial if either (i) L ∼ = d 1 P where P −→ W [1,2] is a line bundle with trivialization d 2 P ∼ = C inducing the canonical trivialization of d 2 2 P , or (ii) L ∼ = d 2 Q where Q −→ W [2,1] is a line bundle with trivialization d 1 Q ∼ = C inducing the canonical trivialization of d 2 1 Q, as in either case we can take the trivial bundle on the other factor.
As for ordinary bundle gerbes, we proceed to define pullbacks and products of bigerbes.
is a locally split square over X. Proof. By hypothesis there are covers U i −→ X admitting sections s i : Pullback of these by f gives covers f −1 U i −→ X ′ and sections f * s i : For the fiber product, the cover ) may be equipped with the cover V × X V ′ , which admits the section t × t ′ to W × X W ′ , proving (ii). [2,2] is a bigerbe with respect to the locally split square (W, Y 2 , Y 1 , X), and f : X ′ −→ X is a continuous map, then the pullback of L is the bigerbe f * L −→ f * (W [2,2] ) with respect to the locally split square f * (W ), f * (Y 2 ), f * (Y 1 ), X ′ .
If L = (L, W, Y 2 , Y 1 , X) and L ′ = (L ′ , W ′ , Y ′ 2 , Y ′ 1 , X) are bigerbes on X, then the product of L and L ′ is the bigerbe

Next we define (strong) morphisms and stable isomorphisms for bigerbes. A morphism of locally split squares (W
is a collection of maps from each space in the first square to the corresponding space in the second, with each of the relevant squares commuting. As for bundle gerbes we do not require compatibility in the sense of §1.2 of the locally split maps of the first square with those of the second. By naturality of fiber products, these maps extend to maps W ′ [m,n] −→ W [m,n] for each (m, n) ∈ N 2 0 commuting with the various fiber projections in both directions. By abuse of notation we will denote all such maps by a single letter, say are bigerbes over X and X ′ respectively, then a (strong) morphism from L ′ to L consists of a morphism f : 2,2] which intertwines the sections of d i L and d i f * L for i = 1, 2. A (strong) isomorphism is a morphism for which X = X ′ and f = Id : X −→ X ′ .
Finally, a stable isomorphism of bigerbes L and L ′ over X is a (strong) isomorphism L ⊗ T ∼ = L ′ ⊗ T where T and T ′ are trivial bigerbes.   ) (ℓ) ) for each ℓ such that s 1 ℓ commutes with the projections π 2 j and satisfies π 1 j s 1 ℓ = 1 j = 0 s 1 ℓ π 1 j−1 1 ≤ j ≤ m and similarly for s 2 ℓ with respect to π 1 j and π 2 j .

Proof. The same argument as in Proposition 4.4 shows that U
[m,n] W is welldefined, namely both sides are identified with tuples such that for each i, (w i,1 , . . . , w i,n ) all map to a fixed y 1,i ∈ V 1,i for some V 1,i ∈ U Y 1 , and for each j, (w 1,j , . . . , w m,j ) all map to a fixed y 2,j ∈ V 2,j for some V 2,j ∈ U Y 2 , with (y 1,1 , . . . , 2 . Then the arguments in Lemma 1.5 apply, viewing each row or column of (4.4) as a simplicial space, with commutativity of the vertical and horizontal maps of covers deriving from their commutativity at the bottom level for the locally split square.

The abelian groups ofČech cochainsČ
are defined for each ℓ, j, and k ∈ N 0 .Using Lemma 4.10 we may define  11. An admissible set of covers determines commuting homotopy contractions for the d i which commute with the other simplicial differential, and for each fixed ℓ and k, the subcomplex is exact and similarly with indices reversed.
Just as a bundle gerbe has a Dixmier Douady class in H 3 (X; Z), a bigerbe determines a characteristic class in H 4 (X; Z). To see this, consider the truncation of the triple complex (Č In particular, the leftmost column and bottom row of (4.5) are taken to have d i degree 0. Then following Convention 2.2, the total differential on (4.5) is  1 ; A), which in turn is isomorphic toȞ • U (X; A) as in Theorem 2.5. Lemma 4.13. The line bundle L −→ W [2,2] of a bigerbe determines a pure cocycle c(L) ∈Ž 1 U (W [2,2] ; C * ) in the triple complex (4.5) for some admissible set of covers, and conversely any line bundle with c(L) ∈Ž 1 U (W [2,2] ; C * ) determines a bigerbe. Moreover, the pure cocycle c(L) ∈Ž 1 U (W [2,2] ; C * ) is a coboundary if and only if L admits a trivialization.
Because of the need to introduce signs in the (δ, d 1 , d 2 ) total complex following Convention 2.2, the sign of the class G(L) ∈ H 4 (X; Z) depends on the order of Y 1 and Y 2 , which is to say the orientation of the locally split square. In particular, reversing the roles of Y 1 and Y 2 while keeping the bundle L −→ W [2,2] fixed determines a bundle gerbe L ′ with class G(L ′ ) = −G(L). Indeed, this follows from the fact that the total differential , where we have interchanged the roles of d 1 and d 2 , is intertwined with D = δ + (−1) ℓ d 2 + (−1) ℓ+m+1 d 2 by the automorphism (−1) (m+1)(n+1) of the triple complex. In particular this amounts to multiplication by −1 onŽ • U (W [2,2] ), exchanging c(L) and −c(L). Alternatively, from the explicit zig-zag (4.10) and (4.11) it follows that c(L) is the double transgression of G(L) ∈ H 4 (X; Z) (in the sense of the isomorphism (2.10)) first to H 3 Z (Y [2] 1 ; Z) and then to H 2 Z (W [2,2] ; Z). The transgression the other way, first to Y [2] 2 and then to W [2,2] has the opposite image −c(L).
Theorem 4.15. The characteristic 4-class G(L) vanishes if and only if L is trivial as a bigerbe, and is natural with respect to pullback, product and inverses in that • is an admissible set of covers refining f −1 (U • ). The naturality of G with respect to pullbacks and morphisms is then a consequence of the naturality of the spectral sequences which identify the total cohomology of the triple complex with the cohomology of X and X ′ , respectively. Naturality with respect to products and inverses is a direct consequence of the fact that we can take [L −1 ] = −c(L) and [L 1 ⊗ L 2 ] = [pr * 1 L 1 ] + [pr * 2 L 2 ] as representatives. Finally, if L and L ′ are stably isomorphic, then G(L) = G(L ′ ) by triviality and products, and conversely if G(L) = G(L ′ ), then L −1 ⊗L ′ = T is trivial, from which a stable isomorphism L⊗T ∼ = L ′ may be constructed.

Representability of 4-classes.
To characterize those 4-classes which are represented by bigerbes over a given locally split square we follow a similar argument to that in §2.4, though it is necessary in this case to go further in a spectral sequence for the triple complex. Consider the augmented triple complex 0 0 0 2 )Ž • (W [1,2] )Ž • (W [2,2] ) 0 with the leftmost column and bottom row considered as degree −1 for d 1 and d 2 , respectively.
where β i ∈Č ℓ−1 U (Y i ; A) are any elements satisfying δβ i = π * i α ∈Č ℓ U (Y i ; A) for a representative α ∈Č ℓ U (X; A) for an appropriate admissible set of covers U • .
There is an analogue of Proposition 2.11 classifying trivializations of bundle gerbes.
Proposition 4.18. The trivializations of a bundle bigerbe (L, W, Y 2 , Y 1 , X) form a torsor for the group whereȞ • Z denotes the δ cohomology of the associated space in the diagram (4.8).
In specific cases, as for the Brylinski-McLaughlin bigerbe in §5.3, this may be simplified further.

Examples of bigerbes
5.1. Decomposable bigerbes. As for the decomposable bundle gerbes discussed in §5.1, we consider the special classes of bigerbes corresponding to decomposable classes in H 4 (X; Z). These are either of the form α 1 ∪ α 2 with the α i ∈ H 2 (X; Z) or of the form ρ∪α with ρ ∈ H 1 (X; Z) and α ∈ H 3 (X; Z). Stuart Johnson in his PhD thesis, [9], makes related constructions in the setting of 2-gerbes.
From Theorem 4.17 it follows that if, for i = 1, 2, α i ∈ H 2 (X; Z) and π i : Y i −→ X, are locally split maps such that π * i α i = 0 ∈ H 2 (Y i ; Z) then the cup product α 1 ∪ α 2 is represented by a bigerbe over the locally split square ( are primitives for the π * i α i then ρ 1 ∪ α 2 and α 1 ∪ ρ 2 are primitives for α 1 ∪ α 2 on Y 1 and Y 2 , respectively, and pulled back to If the spaces Y i are the total spaces of circle bundles representing the 2-classes the bigerbe is given quite explicitly in terms of the classifying line bundle, for decomposed 2-forms, over the torus. Lemma 5.1. The fundamental line bundle on T 2 (with Chern class generating H 2 (T 2 ; Z) = Z) has a 'bimultiplicative' representative S −→ T 2 , meaning there are natural isomorphisms between fibers S x 1 +x 2 ,y ∼ = S x 1 ,y ⊗ S x 2 ,y and S x,y 1 +y 2 ∼ = S x,y 1 ⊗ S x,y 1 such that S x 1 +x 2 ,y 1 +y 2 S x 1 ,y 1 +y 2 ⊗ S x 2 ,y 1 +y 2 commutes.
Proof. Line bundles over T 2 ∼ = R 2 /Z 2 are naturally identified with Z 2 equivariant line bundles over the universal cover, R 2 . We equip the trivial bundle R 2 × C with the Z 2 action covering translation via (n, m) · (x, y, z) = (x + n, y + m, e 2πimx z), (n, m) ∈ Z 2 , (x, y, z) ∈ R 2 × C and let S −→ T 2 be the quotient bundle. The vertical and horizontal identifications in (5.1) correspond to the invariance of the bilinear maps (x, y 1 , z 1 ) × (x, y 2 , z 2 ) −→ (x, y 1 + y 2 , z 1 z 2 ), (x 1 , y, z 1 ) × (x 2 , y, z 2 ) −→ (x 1 + x 2 , y, z 1 z 2 ) which commute under the Z 2 actions. Moreover, the Z 2 -action corresponds to parallel transport along the circles with respect to the invariant connection d + 2πiydx, the curvature of which is the fundamental class in H 2 (T 2 ; Z).
Proposition 5.2. For a decomposed 4-class α 1 ∪ α 2 ∈ H 4 (X; Z), with the α i ∈ H 2 (X; Z) represented by circle bundles Y i −→ X, the pullback under the product of the difference maps χ i : Y Proof. The bimultiplicative relations of Lemma 5.1 correspond under pullback by χ 1 ×χ 2 to the bisimplicial conditions for L, and that G(L) = α 1 ∪α 2 is a consequence of Lemma 2.13.
Similarly if ρ ∈ H 1 (X; Z) and α ∈ H 3 (X; Z) the representability condition is satisfied by the fiber product square given by any locally split maps π i : Y i −→ X, i = 1, 2 such that π * 1 ρ = 0 ∈ H 1 (Y 1 ; Z) and π * 2 α = 0 ∈ H 3 (Y 2 ; Z). Taking the 'logarithmic' covering X −→ X corresponding to ρ, meaning the pullback of the universal cover of U(1) by a homotopy class of maps X −→ U(1) representing ρ and a bundle gerbe (L, Y, X) with DD(L) = α ∈ H 3 (X; Z), there is again a direct construction of a bigerbe for the fiber product square. Proposition 5.3. If (L, Y, X) is a bundle gerbe with Dixmier-Douady class α ∈ H 3 (X; Z) and X −→ X is the logarithmic cover corresponding to a class [ρ] ∈ H 1 (X; Z) represented by ρ : X −→ U(1) then the line bundle where χ : X × X X −→ Z is the fiber-shift map, defines a bigerbe (L χ , X × X Y, Y, X, X) with G(L χ ) = ρ ∪ α ∈ H 4 (X; Z).
Proof. We view the covering space X −→ X as a principal Z bundle, and then the shift map χ : X [2] −→ Z defines the collective bundle L χ on X [2] × X Y [2] given by the tensor product L n over χ −1 (n). The bisimplicial space is and the line bundle L χ is simplicial in the d 2 direction, with trivializing section of d 2 (L χ ) = (d 2 L) χ over W [2,3] given by s χ , and the d 1 differential of L χ is given by so is canonically trivial. Thus this is indeed a bigerbe. To see that G(L χ ) = ρ ∪ α, observe that representative cocycles c(L) ∈ C 2 (Y [2] ; Z) and χ ∈Č 0 ( X [2] ; Z) pull back toČ • ( X [2] × X Y [2] ; Z) by the fiber product projections, and their cup product χ ∪ c(L) ∈Č 2 ( X [2] × X Y [2] ; Z) represents the transgression image of ρ∪α from X. Then χ∪c(L) = nc(L) = [L n ] locally on components χ −1 (n), so the result follows.
is trivial on X 3 with respect to the three projection maps π i : X 3 −→ X 2 .
In the absence of additional data, ∂L is defined with respect to the bisimplicial space over X 3 obtained by the fiber products of the three pullbacks of the bisimplicial space W and taking the alternating product, admits a bigerbe trivialization.
As a special case relevant in our primary example, a bigerbe is a double if ∂L is itself trivial as a line bundle over W [2,2] 3 . Naturality of the bigerbe characteristic class and exactness of the sequence (3.5) together lead to the following analogue of Proposition 3.2.
In §5.5 we will also consider a similar condition with respect to the bisimplicial space of products X m,n = X mn , with the two sets of projections π 1 j : X m,n −→ X m−1,n and π 2 j : X m,n −→ X m,n−1 . A bigerbe L over X 4 = X 2,2 is quadruple if ∂ 1 L and ∂ 2 L are respectively trivial over X 1,2 = X 2 and X 2,1 = X 2 . The natural differentials ∂ 1 and ∂ 2 , defined on cohomologyȞ ℓ (X •,• , A), commute, and from exactness of these we obtain the following result.

Brylinski-McLaughlin bigerbes.
Next we turn to our main application. The loop space of a principal G bundle over a manifold is a principal bundle over the loop space with structure group the loop group of G. The Brylinski-McLaughlin bigerbe captures the obstruction to lifting this bundle to a (loop-fusion) principal bundle for a central extension of the loop group. While the version involving based path and loop spaces is simpler, we focus from the beginning on the doubled version involving free path and loop spaces, as this gives the results of primary interest. Note that this theory most naturally involves U(1) principal bundles in place of line bundles, which we shall use for the remainder of the section without further comment.
Let G be a compact, simple, connected and simply connected group. As is well-known (see for instance [19]), there is a classification of U(1) central These extensions descend to the quotient LG/G ∼ = ΩG and so the classification of central extensions of the based loop group ΩG is equivalent.
Forgetting the group structure for the moment, such a central extension may be viewed as a circle bundle over LG ∼ = I [2] G, the Chern class c( LG) ∈ H 2 (LG; Z) of which is the transgression of the defining class in H 3 (G; Z). As such, it follows from Theorem 3.6 that c( LG) has a loop-fusion refinement. In the equivalent language of the loop-fusion structures of Definition 3.4, we may restate this as follows.
Theorem 5.7. As a U(1)-bundle, LG −→ LG has a canonical loop-fusion structure, meaning a trivialization of d LG −→ I [3] G inducing the canonical trivialization of d 2 LG −→ I [4] G and a trivialization of ∂ LG −→ L 8 G.
Remark. In fact, the additional structure that promotes a general U(1)principal bundle over LG to a central extension is also a simplicial one. Indeed, as noted by Brylinski and McLaughlin in [4] and attributed to Grothendieck, a U(1) central extension of any group H is equivalent to a simplicial circle bundle with respect to the simplicial space BH • defined by BH k = H k−1 with the face maps H k+1 −→ H k given by Thus, given a circle bundle Q −→ H = BH 2 , a trivialization of ∂Q = π * 0 Q ⊗ π * 1 Q −1 ⊗ π * 2 Q inducing the canonical trivialization of d 2 Q −→ H 3 equips Q with the (associative) multiplicative structure of a U(1) central extension of H and vice versa.
For the groups under consideration, we believe it can be shown that the classes in H 3 G (G; Z) = H 4 (|BG|; Z) are represented by cohomology classes α ∈ H 3 (G = BG 2 ; Z) satisfying ∂α = 0 ∈ H 3 (G 2 = BG 3 ; Z), and that the corresponding gerbe (W, P G, G), with circle bundle W −→ ΩG, admits a simplicial structure with respect to BΩG • , and thus a central extension of ΩG. Further considering a doubled structure with respect to G • = G • gives rise to the central extensions of LG. For reasons of space, and since the theory of central extensions of LG is already well-known, we will not elaborate further on this point.
To define the Brylinski-McLaughlin bigerbe, let X be a connected manifold with principal G-bundle E −→ X. One case of particular interest is the spin frame bundle over a spin manifold of dimension ≥ 5.
Lemma 5.8. With vertical maps projections and evaluation at end-and mid-points in the horizontal directions, the diagrams for k ≥ 2 are locally split squares.
Proof. The maps are locally trivial fiber bundles, so all maps are locally split, and I k E is likewise a fiber bundle over the fiber product I k X × X k E k , which is the space of paths in X along with prescribed points in E over the endpoints (for k = 2) and midpoint (for k = 3) of the path. A connection on E gives a horizontal lift of each path segment in X given an initial point in E, and from the connectedness of G this can be concatenated with a path in the fiber from the endpoint of the lifted path segment to any other prescribed point in the same fiber; this can be done for each segment defined between the k marked points of the path on X. This construction can be carried out locally continuously, so giving a local section of I k E over E k × X k I k X.
Filling out the bisimplicial space for k = 2 by fiber products leads to the diagram (E [3] ) 2 IE [3] LE [3] I [3] E [3] (E [2] ) 2 IE [2] LE [2] The third column of (5.4) is the simplicial space generated by the fibration LE −→ LX, itself a principal bundle with structure group LG, and thus supports a lifting bundle gerbe is the shift map of the principal bundle, and we consider LG −→ LG as a U(1)-bundle. The other columns are likewise the simplicial spaces of principal bundles, with structure groups G and I [k] G for k ≥ 1, and we denote their associated shift maps by the same letter.
Theorem 5.9. Given a central extension (5.2) of level ℓ ∈ Z = H 3 G (G; Z), the lifting bundle gerbe Q −→ LE [2] is the double bigerbe (Q, IE, E 2 , IX, X 2 ) with characteristic class where p 1 (E) is the first Pontryagin class of E.
Proof. It follows immediately from the lifting gerbe construction that Q is vertically simplicial, and the simplicial condition in the horizontal direction follows from naturality of the shift map and Theorem 5.7. Indeed, unwinding the definitions reveals that d 1 χ * LG = χ * d LG, which admits the trivialization noted in Theorem 5.7 inducing the canonical trivialization of χ * d 2 LG = d 2 1 χ * LG.
For doubling, we define ∂Q with respect to the locally split square (5.3) for k = 3, the induced bisimplicial space of which sits in the diagram (E [3] ) 3 IE [3] L 8 E [3] I [3] E [3] (E [2] with the obvious maps to (5.4), and where we have omitted the subscript 3 on the path spaces and made the identification L 8 X ∼ = I [2] 3 X, etc. Once again, it follows that ∂Q = ∂χ * LG = χ * ∂ LG with respect to the shift map for the principal L 8 G-bundle L 8 E −→ L 8 X, which admits a trivialization in light of Theorem 5.7. Thus Q is product simplicial in the sense of Definition 5.4 and its characteristic class descends to H 4 (X; Z).
Observe that this characteristic class can be obtained in two steps, first by regressing the Chern class c(Q) from H 2 Z (LE [2] ; Z) to H 3 Z (LX; Z) and then to H 4 (X; Z). The image of c(Q) in H 3 Z (LX; Z) is, essentially by definition, the Dixmier-Douady class of the lifting bundle gerbe (or more precisely, its loopfusion refinement), and it is well-known that this class is the transgression of the Pontryagin class of E on X (multiplied by ℓ in the case of higher levels), so from Theorem 3.6 we obtain (5.5).
Without the vertical simplicial condition, the gerbe (Q, LE [2] , LX) represents the obstruction to lifting the LG bundle LE −→ LX to an LG bundle LE −→ LX. The enhancement of this data to a bigerbe carries additional information, which is formalized in the following definition. Without the additional figure-of-eight structure, such fusion lifts have been considered by Waldorf in [29], and with stronger conditions (high regularity and equivariance with respect to diffeomorphisms of S 1 ) by the authors in [11]. Proof. Here by a doubled trivialization we mean a trivialization (P 1 , P 2 ) in the sense of Definition 4.6 with the additional property that ∂P i is trivial on the bisimplicial space (5.6). As argued in the proof of Lemma 3.3, the retractions for any space Y of I k Y onto Y itself for each k give a homotopy with respect to which ∂ = Id as an operator from line bundles on I 2 Y to those I 3 Y ; in particular the condition that ∂P 1 is trivial for P 1 −→ I 3 E [2] means that P 1 itself is trivial. Thus doubled trivializations for the bigerbe in question are reduced to loop-fusion line bundles P −→ LE satisfying d 2 P ∼ = Q. On the one hand, these are clearly equivalent to loop-fusion LG lifts of LE −→ LX, and on the other, they are classified by those classes iň H 2 lf (LE; Z) with image c(Q) under d 2 . By Lemma 5.12 below, the difference of any two such classes descends to a class inȞ 2 lf (LX; Z), and so doubled trivializations form a torsor for the image ofȞ 2 lf (LX; Z) inȞ 2 lf (LE; Z), which by Theorem 3.6 is equivalent to the image of H 3 (X; Z) in H 3 (E; Z). Finally, given the conditions on G, it follows by the Serre spectral sequence for E −→ X that H 3 (X; Z) −→ H 3 (E; Z) is an isomorphism in this case, so the trivializations are classified simply by H 3 (X; Z).
It remains to show that the exactness of d 2 in theČech-simplicial double complex is consistent with ∂, which is a consequence of the following.
Lemma 5.12. The homotopy chain contraction for d 2 in the triple complex (Ž • (W [•,•] ), δ, d 1 , d 2 ) for the locally split squares (I k E, E k , I k X, X k ) commutes with the product simplicial operator ∂.
Proof. This follows ultimately from the existence of local sections of E k −→ X k (resp. IE −→ IX) which are intertwined by the three projection maps E 3 −→ E 2 and X 3 −→ X 2 (resp. I 3 E −→ I 2 E and I 3 X −→ I 2 X), which we proceed to demonstrate. In the first case, we may fix an admissible pair of covers (V, U ) for (E, X) and then equip E k and X k with the covers V k and U k along with the induced sections Et(U k ) −→ Et(V k ), which are then automatically intertwined by the projection maps.
For the path spaces, we begin with the fact that (IE, IX, E 3 , X 3 ) is a locally split square, so I 3 E and I 3 X admit covers W and Y and sections Et(Y) −→ Et(W) lying over the sections Et(U 3 ) −→ Et(V 3 ). In general, the reparameterization maps I 3 Y I 2 Y are open, so we may equip I 2 E and I 2 X with the union of the three image covers W ′ = π 0 (W) ∪ π 1 (W) ∪ π 2 (W) and Y ′ = π 0 (Y) ∪ π 1 (Y) ∪ π 2 (Y), along with the induced section giving a set of local sections of I 2 E −→ I 2 X which is compatible with the three reparameterization maps π i , and which covers the local sections of There is a simpler version of this bigerbe using based path and loop spaces, starting with the locally split square (P E, E, P X, X), pulling back a central extension ΩG −→ ΩG to ΩE [2] , and omitting the doubling conditions. We leave the details as an exercise to the reader.

Loop spin structures.
There is a well-known relationship between string structures on a spin manifold X of dimension 2n > 4, and (loop) spin structures on its loop space LX.
Here, a string structure is a lift of the principal Spin(2n) bundle E −→ X to a principal bundle with structure group String(2n), a 3-connected topological group covering Spin(2n) in the sequence of ever more connected groups that form the Whitehead tower for O(2n), see for instance [23]. The string group cannot be a finite dimensional Lie group (having a subgroup with the topology of K(Z, 2)), though there are various realizations as a 2-group [2,21]. The obstruction to lifting the structure group is 1 2 p 1 (X) ∈ H 4 (X; Z) (the Pontryagin class of the Spin bundle being a refinement of the Pontryagin class of the oriented frame bundle), and if unobstructed, string structures are classified by H 3 (X; Z) [25].
As originally defined by Killingback in [10] and further developed by McLaughlin in [15], a spin structure on LX is a lift of the LSpin bundle LE −→ LX to the structure group LSpin, the fundamental U(1) central extension of LSpin. (By analogy, as originally suggested by Atiyah in [1], an orientation on LX is a refinement of the LSO(2n) bundle LE SO −→ LX to have structure group the connected component of the identity, L + SO(2n) ∼ = LSpin(2n), and therefore is typically related to a spin structure on X.) The obstruction to this lift is the 3-class on LX obtained by transgression of 1 2 p 1 (X) ∈ H 4 (X; Z).
As defined, string structures on X and spin structures on LX are not necessarily in bijection [18]. In fact, it was Stolz and Teichner in [24] who first noted the importance of the fusion structure on LX and showed that string structures on X were in correspondence with what they called 'stringor bundles' on (the piecewise smooth loop space) LX, essentially bundles associated to a lift LE along with a fusion condition. It was further proved by the authors in [11] that string structures in the sense of Redden [20] correspond with spin structures on the smooth loop space LX which are both fusion and equivariant for the group Diff + (S 1 ) of oriented diffeomorphisms of the loop parameter, and it was independently proved by Waldorf in [29] that string structures on X exist if and only if fusion spin structures on (piecewise smooth) LX exist, using a transgression theory relating the 2-gerbe obstructing string structures of [6] and fusion gerbes on loop space. Waldorf did not obtain a complete correspondence between string structures and fusion loop spin structures, noting that this would necessitate additional conditions such as equivariance with respect to thin homotopy; in the version considered here, it is the figure-of-eight (i.e., doubling) condition that provides the remedy.
In any case, the bigerbe formulation here leads to the following result. The sets are empty unless 1 2 p 1 (E) = 0 ∈ H 4 (X; Z) and otherwise are torsors for H 3 (X; Z).

Path bigerbes.
If X is a path-connected and simply connected space with basepoint b, from the based double path space there are two surjective restriction maps and any class γ ∈ H 4 (X, Z) arises from a bigerbe corresponding to (5.9).
Proof. The fiber product of the two copies of P X is the based loop space of X. The simple connectedness of X implies the fiber product of the two f i is surjective and from local contractibility it is locally split. Since P X and QX are both contractible, Theorem 4.17 applies to any 4-class on X.
Since the Eilenberg-Mac Lane spaces can be represented by CW complexes Theorem 5.14 applies in particular to K(Z, 4).
Theorem 5.15. There exists a universal bigerbe over K(Z, 4) with respect to the locally split square (5.9) with X = K(Z, 4).
Note the structure of the bisimplicial space in this case, in which * represents a contractible space: Finally, incorporation of a product-bisimplicial condition allows any 4class to be represented as a bigerbe on any connected, locally contractible space X, whether simply connected or not. Indeed, consider the locally split square (IIX, IX 2 , IX 2 , X 4 ), where IIX = u : [0, 1] 2 −→ X is the free double path space and the projection maps are given by evaluation at both endpoints of a given path factor. The induced bisimplicial space becomes and in particular W [2,2] = LLX is the double free loop space of X. We may view X 4 at the bottom as the factor X 2,2 in the bisimplicial space X m,n = X mn of products as discussed in §5.2. Over X 3,2 consider the locally split square and induced bisimplicial space and likewise for X 2,3 with factors reversed. The bisimplicial spaces (5.11) map to (5.10) over the product maps X 3,2 X 2,2 and X 2,3 X 2,2 , and there are associated operators ∂ 1 and ∂ 2 on line bundles. Theorem 5.16. For a connected, locally contractible space X, every class in H 4 (X; Z) is represented by a product-bisimplicial bigerbe with respect to (5.10), that is, a bigerbe (L, IIX, IX 2 , IX 2 , X 4 ) having in addition trivializations of the line bundles ∂ 1 L −→ L 8 LX and ∂ 2 L −→ LL 8 X.

Multigerbes
We end by sketching out the theory of multigerbes, the higher degree generalization of bigerbes. By contrast to bundle gerbes, this generalization to higher degree is straightforward, with symmetry of the simplicial conditions replacing the need for higher and ever more complicated associativity conditions. Fix a degree n ∈ N, where n = 1 and n = 2 correspond to bundle gerbes and bigerbes, respectively. To establish notation, let e j = (0, . . . , 1, . . . , 0) denote the jth standard basis vector, and for each integer k let k = (k, . . . , k) denote the vector with constant entries. For a multiindex α ∈ N n we let |α| = α 1 + · · · + α n ∈ N, and we use the partial order on multiindices where β < α if β i < α i for each 1 ≤ i ≤ n. We distinguish the sets of natural numbers starting at 1 and 0 respectively by N 0 and N 1 . Definition 6.1. Fix a set of spaces X α indexed by α = (α 1 , . . . , α n ) ∈ {0, 1} n along with continuous surjective maps X α −→ X α−e j whenever α j = 1. (6.1) For each α ∈ N n define X <α = lim β<α X β to be the limit space of the diagram formed by all spaces to which X α maps along with the maps between them. Then the set {X α : α ∈ {0, 1} n } is a locally split n-cube provided each of the induced maps X α −→ X <α is locally split.
In particular, a locally split 1-cube is just a locally split map X 1 −→ X 0 and a locally split 2-cube is a locally split square. It follows, in particular by the proof of Lemma 6.6, that all of the maps (6.1) are locally split, and that each subcube determined by freezing some of the coefficients of α is likewise locally split. Lemma 6.2. A locally split n-cube extends naturally by taking fiber products to a set of spaces {X α : α ∈ N n 0 } such that X •≥1 = {X α : α ∈ N n 1 } is an nfold multisimplicial space over X := X 0 ; in particular for each fixed α = (α 1 , . . . , 0, . . . , α n ) with vanishing jth coordinate the sequence X α X α+e j X α+2e j X α+3e j is the simplicial space of fiber products of the map X α+e j −→ X α .
Proof. The proof is by induction on n, the case n = 2 having been proved as Proposition 4.4. Assuming therefore that the result holds for n − 1, the 'hypersurfaces' {X α : α j ≡ 0} are well-defined for 1 ≤ j ≤ n, and for general α ∈ N n 1 define X α as a subspace of X α 1 ···αn 1 as follows. For each j, there are α j projection maps X α 1 ···αn , where the caret denotes omission, and these may be composed with the structure map Introducing the notation α(j) to mean the multiindex obtained from α by setting α j = 0, we may view X α(j) as a subspace of X α 1 ··· α k ···αn 1−e j , and then X α is well-defined as the subspace of X α 1 ···αn 1 in the mutual preimage of X α(j) under the α j maps X α 1 ···αn 1 −→ X α 1 ··· α j ···αn 1−e j for each j.
We denote the face maps of the multisimplicial space by π j • , for 1 ≤ j ≤ n. Then for each α ∈ N n 0 , there are n simplicial differentials defined on a line bundle L −→ X α by with the property that d 2 j L −→ X α+2e j is canonically trivial. Definition 6.3. A bundle n-multigerbe, or simply multigerbe, defined with respect to a locally split n-cube X α is a multisimplicial line bundle L −→ X 2 , meaning d j L is given a trivializing section s j inducing the canonical trivialization of d 2 j L for each j, and the induced trivializations of d j d k L ∼ = d k d j L are consistent for each pair j = k.
A set (P 1 , . . . , P n ) of line bundles P j −→ X 2−e j such that each P j is multisimplicial with respect to d i for i = j determines a multisimplicial line bundle n j=1 d j P (−1) j j −→ X 2 in the obvious way, and a trivialization of an n-multigerbe L consists of a set (P 1 , . . . , P n ) as above along with an isomorphism L ∼ = n j=1 d j P (−1) j j intertwining the multisimplicial structures of both sides.
Pullbacks, products and morphisms of multigerbes are defined by generalizing in the obvious way those same operations for bigerbes, and making use of the following result, which follows immediately from Lemma 4.7.
Lemma 6.4. The pullback by a continuous map X ′ −→ X 0 of a locally split n-cube X α is a locally split n-cube over X ′ . Likewise, if X α and X ′ α are locally split n-cubes over the same base X := X 0 = X ′ 0 , then the fiber products X α × X X ′ α form a locally split n-cube. The characteristic class is defined as before in terms of the total cohomology of aČech-simplicial multicomplex. Definition 6.5. An admissible set of covers for a locally split n-cube is a set of covers U Xα −→ X α such that each pair (U X α−e j , U Xα ) is admissible and each set (U X α−e i −ej , U X α−e i , U X α−e j , U Xα ) is an admissible set of covers for the associated split square. Lemma 6.6. Given any covers for the spaces in a locally split n-cube, there exists an admissible set refining them.
Proof. The proof is by induction on n of the the stronger statement that covers of adjacent n-cubes rooted at X = X 0 can be refined to a set which is simultaneously admissible for each cube. Here by adjacent we mean a finite set of n-cubes rooted at X which may share subcubes. The cases n = 1 and n = 2 are furnished by Lemmas 1.4 and 4.3.
For the inductive case, we consider, for each n-cube, X 1 at the top mapping to the limit space X <1 := lim β<1 X β , and we fix a cover U 0 X <1 supporting sections, denoted t, of the split map p : X 1 −→ X <1 . The topmost spaces in the diagram over which the limit is taken, namely X 1−e j , are the topmost spaces of a set of adjacent (n − 1)-cubes rooted at X, so by the inductive hypothesis there are covers U 0 X• , • < 1 refining the given covers, which constitute simultaneous admissible sets for the (n − 1)-cubes. In addition, at this point the covers U 0 X 1−e j of the penultimate spaces support sections, σ j , of the natural maps pr j : X <1 −→ X 1−e j from the limit space, which participate in commutative squares along with the maps of covers involving the U 0 X 1−e i −e j . Finally, we set U 0 X 1 to be the given cover of X 1 and this setup constitutes step 0.
Step 1 : We define U 1 X 1 to be the given cover of X 1 and then subsequently set U 1 X <1 = U 0 X <1 ∩ t −1 (U 1 X 1 ), and U 1 ; if there are multiple n-cubes in which X 1−e j participates, then we take U 1 X 1−e j to be the mutual refinement by the pullback of the step 1 covers of the associated limit spaces for each such n-cube.
Step 2 : Appealing to the inductive step again gives refinements, denoted U 2 • , of all the spaces up to the X 1−e j , and then we refine the cover of the symmetric group Σ n acting by permutation of the indices of the locally split n-cube X α .
Proof sketch. As for (bi)gerbes, the naturality of the characteristic class is a consequence of the naturality of the multicomplex Č • U (X • ; A), δ, d 1 , . . . , d n and the naturality of the Chern class of the line bundle L −→ X 2 , and the equivalence between vanishing of G(L) and multigerbe triviality of L follows from Lemma 6.7.(iii). Finally, that G(L) is odd with respect to permutations of the n-cube is a consequence of the sign convention, Convention 2.2, since changing the order of the differentials in the multicomplex by a permutation σ involves multiplying the complex by powers of −1, and in particular the sign (−1) sgn(σ) on the termČ 1 U (X 2 ; C * ). The question of representability of a given (n + 2)-class by a multigerbe supported by a given locally split n-cube can be addressed along similar lines as for bigerbes in §4.4. Consider the multicomplex Č • U (X • ; Z), δ, d 1 , . . . , d n truncated to involve only the spaces in the n-cube, so the X α with α ∈ {0, 1} n . The (δ, D 1...,n ) spectral sequence of this complex (with the d i rolled up into a single differential) has E 1 page consisting of the cohomology complexes H k (X; Z) |α|=1 H k (X α ; Z) |α|=2 H k (X α ; Z) · · · D 1...,n D 1...,n D 1...,n for each k ∈ N. At the bottom level, the D 1...,n differential of a class in H n+2 (X; Z) is just the sum of the pullbacks along the n-cube face maps to |α|=1 H n+2 (X α ; Z), and if this vanishes, then we say the class survives to the E 2 page. In this case the E 2 differential maps the class into the quotient |α|=2 H n+1 (X α ; Z)/ |α|=1 H n+1 (X α ; Z) (the cohomology of the E 1 page), and we say the class survives to the E 3 page if this E 2 differential vanishes and so on. Provided the class survives to the E n page, the associated differential maps it into the quotient of |α|=n H 2 (X α ; Z) = H 2 (X 1 ; Z) by some complicated subgroup, and this is the last nontrivial differential of the spectral sequence, which therefore stabilizes at E n+1 = E ∞ . We say the class survives to E ∞ , or simply stabilizes, if it survives to E n and has vanishing E n differential. Proposition 6.10. A given locally split n-cube X α supports an n-multigerbe representing a given class α ∈ H n+2 (X; Z) if and only if α stabilizes in the above sense.
We leave the details of the proof, which we claim is a relatively straightforward generalization of the proof of Theorem 4.17, as an exercise.
Then in the induced multisimplicial simplicial space, X 2 = X [2] × X X 2 and we define the line bundle by L χ = (pr 2 * L) ⊗pr * 1 χ −→ X [2] × X X 2 where χ : X [2] −→ Z is the fiber shift map with X −→ X thought of as a principal Z-bundle.
For the generalization of the path bigerbes of §5.5, let X be a locally contractible space with a chosen basepoint. Lemma 6.12. With notation P 1 Y = P Y and P 0 Y = Y , the iterated (based) path spaces X α = P α 1 · · · P αn X = P |α| X with evaluation maps X α −→ X α−e j form a locally split n-cube over X, provided X is (n − 1)-connected.
Proof. The obstruction to the locally split condition is the surjectivity of the maps X α −→ X <α . However, it can be checked that for each |α| ∈ N, the limit X <α ∼ = C * (S |α|−1 ; X) can be identified with the set of basepointed maps from the (|α|−1)-sphere into X. Indeed, the X α−e j ∼ = C * ([0, 1] |α|−1 ; X) are cube mapping spaces of the (|α| − 1)-cube, and the limit can be realized as the subspace of the product of these in which the cubes are identified along their boundaries in a way that assembles into an (|α| − 1)-sphere, with the image of an element in X α = C * ([0, 1] |α| , X) in X <α identified with the restriction map to the boundary of the disk [0, 1] |α| ∼ = D |α| . Since X is (n − 1) connected, each of these maps is surjective.
Since in this case all the X α in the n-cube for α = 0 are contractible spaces, every class in H n+2 (X; Z) survives to the E ∞ page in the (δ, D 1···n ) spectral sequence of the n-cube, so in light of Proposition 6.10 we conclude the following. Proposition 6.13. For X (n − 1)-connected and locally contractible, every class in H n+2 (X; Z) is represented by an n-multigerbe supported on the iterated path n-cube X α = P |α| X; in particular, the multisimplicial line bundle of the multigerbe lives on the iterated loop space X 2 = Ω n X.