Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group $G$, we prove that the smooth 2-group extensions of $G$ arising from our construction provide new models for the string group of $G$.


Introduction
This paper is motivated by the following problem from physics: In [BMS19] we showed how a bundle gerbe with connection on R d gives rise to a 3-cocycle on the translation group R d t of R d . Even though this 3-cocycle is trivial in group cohomology, it is very interesting from a physical as well as from a mathematical perspective: it gives a geometric explanation to the presence of nonassociativity in quantum mechanics with magnetic monopole backgrounds, and it implements the action of the parallel transport of a bundle gerbe on its 2-Hilbert space of sections. This appearence of nonassociativity in quantum mechanics goes back to [Jac85,GZ86], but as of yet the more natural extension to realistic scenarios involving periodically confined motion on configuration spaces such as tori T d has not been worked out. The discussion of [Jac85] was a response to the observed violation of the Jacobi identity for the algebra of field operators in quantum gauge theories with chiral fermions [Jo85], which is a manifestation of the chiral anomaly. Interest in these models has been recently revived through their conjectural relevance to non-geometric flux compactifications of string theory, which is based on backgrounds that are tori or more generally torus bundles [Lüs10,MSS12,BL14,MSS14]. However, the original finding [BP11] of nonassociativity in Wess-Zumino-Witten models based on other compact Lie groups has so far received considerably less attention, and in particular has not been understood from a geometric perspective.
In the present paper we work out the geometric framework and origin behind these results in complete generality. Subsequently, we present several applications of our results in both physics and mathematics, along the lines discussed above. We consider an action Φ : G × M −→ M of a connected Lie group G on a manifold M , where M is endowed with a bundle gerbe G. One can now ask whether G admits a G-equivariant structure. At the very least, such a structure should consist of a choice of 1-isomorphism G −→ Φ * g G for every g ∈ G. Instead of considering possible choices for such 1-isomorphisms individually, we assign to g the groupoid of all such 1-isomorphisms. This yields an object which can be understood as a bundle Sym G (G) −→ G of groupoids over G. Considering g = e, the identity element of G, we see that its typical fibre is the groupoid HLBdl(M ) of hermitean line bundles on M .
The definition of Sym G (G) so far does not capture the smooth structure of the gerbe G. We thus enhance the construction to take into account smooth families of elements of G. Then one can make sense of Sym G (G) as a category fibred in groupoids over a base category Cart that encodes smooth families of geometric objects. Categories fibred in groupoids over Cart assemble into a 2-category H, and there exists a fully faithful inclusion of the category of smooth manifolds into H. Motivated by [SP11] we define a smooth 2-group to be a group object in H. One of the central examples for us is the smooth 2-group HLBdl M of hermitean line bundles on M . We introduce a notion of smooth principal 2-bundle in H that lies between the definitions of higher principal bundles used in [SP11] and [NSS15] (see in particular Appendix A.2). We show that our principal 2-bundles are well behaved from a homotopical as well as from a geometric point of view (more precisely, they form effective epimorphisms while also admitting local sections). With the notion of smooth 2-group and principal 2-bundles, we can make precise what it means to be a (central) extension of smooth 2-groups in analogy to extensions of Lie groups. Then, our first main results can be summarised as Theorem 1.1. Let G be a connected Lie group acting on a manifold M , and let G be a bundle gerbe on M . Then: (1) There is a (non-central) extension of smooth 2-groups

2)
where G ∈ H denotes the category fibred in groupoids associated to G.
(2) The smooth 2-group Sym G (G) acts on G, and the action covers that of G on M .
(3) The gerbe G admits a G-equivariant structure if and only if there exists a morphism G −→ Sym G (G) of smooth 2-groups which splits the extension (1.2).
An extension similar to (1.2) was considered in [FRS16], where symmetries of a gerbe with connection were investigated in relation with higher geometric prequantisation. Infinitesimal versions of the extension (1.2) were considered in [Col11,FRS16], where it was shown that these give rise to the standard H-twisted Courant algebroid on M , where H is the 3-form curvature of the connection on G. These considerations have been expanded on and applied to higher versions of Kaluza-Klein reductions of string theory in [Alf20].
Our point here is that in many applications, such as nonassociativity in quantum mechanics and string theory, anomalies in quantum field theory, as well as interesting topological constructions, connections on G only play a secondary role: in this context, they can be seen as a tool to compute the extensions (1.2) and their associated cocycles. The key to this computability is an alternative presentation of Sym G (G) in terms of a categorified descent construction.
In order to work out this construction, we introduce a novel global approach to the parallel transport of a bundle gerbe. Parallel transport for gerbes has been constructed in [SW11,SW09,SW17], but for our purposes a global, rather than local, treatment is necessary. Our construction relies heavily on the transgression-regression machine for bundle gerbes [Wal16] together with the properties of the fusion product and the connection on the transgression line bundle that were studied in [Wal16,BW18]. Given a connection on G, we construct its parallel transport as a quadruple pt G = (pt G 1 , pt G 2 , pt G , ε G ), consisting of the following data: first, there is a 1-isomorphism pt G 1 : ev * 0 G −→ ev * 1 G over the path space P M of M , where ev t : P M −→ M is the evaluation of a path at t ∈ [0, 1]. Second, there is a 2-isomorphism pt G 2 : pt G 1|γ 0 −→ pt G 1|γ 1 for every smooth homotopy with fixed endpoints between paths γ 0 and γ 1 , which depends smoothly on the paths and the homotopy. The 2-isomorphisms pt G and ε G implement the compatibility of the parallel transport with concatenation of paths and with constant paths, respectively. Furthermore, the collection pt G is required to be invariant under thin homotopies in a precise way. We show Theorem 1.3. Every bundle gerbe with connection has a canonical parallel transport.
Using the parallel transport, we are able to write down a HLBdl M -valuedČech 1-cocycle on the covering of G by its space of based paths. These data are equivalently transition functions for an HLBdl M -principal 2-bundle Des L −→ G . We construct Des L explicitly by a homotopy-coherent version of the associated bundle construction. Then we prove Theorem 1.4. The principal 2-bundle Des L −→ G is a smooth 2-group extension of G by HLBdl M . There is a weakly commutative diagram of smooth 2-groups The morphism Ψ is an equivalence.
In the case M = R d , where G = R d t is the translation group of R d , and where G = I B is a trivial gerbe on R d with a connection B ∈ Ω 2 (R d ) corresponding to a magnetic field, we show that the extension Sym R d t (I) −→ R d t reproduces the 3-cocycles we obtained in [BMS19]. We achieve this by choosing a certain global section of the path fibration of R d t and implicitly pass through Des L in the computation. We show that the parallel transport we defined implements nonassociative magnetic translations on the sections of the gerbe, whereas the 2-group extension Sym R d t (I) −→ R d t allows us to understand the algebraic structure of nonassociative magnetic translations even without making any reference to sections. The latter is particularly useful in cases where there is no good notion of sections, such as when the Dixmier-Douady class of G is non-torsion. In particular, we study in detail the action of nonassociative magnetic translations on tori T d and give an explicit description of Sym R d t (G) for general choice of a gerbe G on T d . As a further application, we show that if Γ is a group of gauge transformations, the smooth 2group extensions Sym Γ (G) −→ Γ control the Faddeev-Mickelsson-Shatashvili anomalies in quantum field theory [Fad84,FS85,Mic85]. The relation between gerbes and these anomalies has been investigated in [CM95,CM96], but only as algebraic objects, disregarding the smooth structures. The relevant bundle gerbe G lives on the space A of gauge fields and describes the obstruction to a Fock bundle descending to the orbit space A/Γ . Here the extension Sym Γ (G) −→ Γ is split, so that G admits an equivariant structure. At the same time G is trivialisable as a bundle gerbe, but the anomaly is precisely the obstruction to choosing a Γ -equivariant trivialisation. This allows us to understand the anomaly in a conceptual way as a higher smooth 1-cocycle on Γ .
Finally, we consider the situation where M = G is a compact simply-connected Lie group, acting on itself by left multiplication, and where G is a bundle gerbe on G whose Dixmier-Douady class generates H 3 (G; Z) ∼ = Z. We motivate and propose a new smooth string 2-group model for the string group of G. For this, we first show that with our definition of principal 2-bundle, principal A-bundles on a manifold give rise to A-valuedČech 1-cocycles, for any smooth 2-group A. Then we call a smooth 2-group extension A −→ P −→ G a smooth 2-group model for the string group of G if A is equivalent to an Eilenberg-MacLane space K(Z; 2) in a certain sense and the class iň H 1 (G; BU(1)) ∼ = H 3 (G; Z) extracted from the 2-bundle P −→ G is a generator. Using this definition of smooth string 2-group models, we show Theorem 1.5. Let Sym G (G) and Des L be the smooth 2-group extensions of G by HLBdl G constructed from G with respect to the left action of G on itself via left multiplication. Then both Sym G (G) and Des L are smooth 2-group models for the string group of G.
The remainder of this paper is organised as follows. In Section 2 we briefly recall some background material on diffeological spaces, bundle gerbes, and transgression. Section 3 provides a motivation of the later constructions on the level of principal bundles; many concepts become clear already at this level. In Section 4 we provide our definition and construction of the parallel transport associated to a bundle gerbe with connection. The construction of Sym G (G) and Des L takes place in Section 5. Here we first motivate and then introduce the necessary language of Grothendieck fibrations, smooth 2-groups, and principal 2-bundles, before defining and studying the extensions Sym G (G) and Des L . We conclude this section by relating these extensions to equivariant structures on G. In the remaining three sections we apply our general results: in Section 6 we study nonassociative magnetic translations using our parallel transport, Section 7 contains the discussion of chiral anomalies and the Faddeev-Mickelsson-Shatashvili anomaly, and in Section 8 we show that Sym G (G) and Des L provide new models for the string group. We defer some technical results on categories fibred in groupoids and on principal 2-bundles to Appendix A.

Preliminaries on diffeological spaces and gerbes
In this section we review some of the relevant background material related to diffeological spaces and bundle gerbes that will be used throughout this paper.

Diffeological spaces
Throughout this paper we will use diffeological spaces (see [IZ13] for an extensive introduction) to describe the smooth structure on infinite-dimensional spaces such as path and mapping spaces. The idea behind diffeological spaces is to describe the smooth structure on a space X by specifying the set of smooth maps from Cartesian spaces to X. A Cartesian space c is a smooth manifold diffeomorphic to R n for some n ∈ N 0 . We denote by Cart the category with Cartesian spaces as objects and smooth maps as morphisms.
Example 2.6. Let X and Y be diffeological spaces. The set of smooth maps Y X from X to Y becomes a diffeological space by declaring a map f : c −→ Y X to be a plot if and only if the map is smooth. This is called the mapping space diffeology on Y X . Another source for subductions are quotient maps. Let X be a diffeological space and ∼ an equivalence relation on X. Then the space X/∼ becomes a diffeological space in a canonical way making the map π : X −→ X/∼ into a subduction: a map ϕ : c −→ X/∼ is a plot if and only if for all x ∈ c there exists an open neighbourhood U x ⊂ c of x and a plot ϕ x : U x −→ X such that ϕ |Ux = π • ϕ x . Clearly all subductions are of this type for appropriate equivalence relations. Diffeological quotients behave nicely with respect to quotients of manifolds when they exist.
Proposition 2.9. Let M be a manifold with a free and proper action of a Lie group G. Define an equivalence relation ∼ G on M by m 1 ∼ G m 2 if and only if there exists g ∈ G such that g · m 1 = m 2 . Then the manifold M/G and the diffeological space M/∼ G agree.
Proof. From [Lee13,Theorem 21.10] it follows that π : M −→ M/G is a surjective submersion. Since every surjective submersion is a subduction, the statement follows.
Most concepts from differential geometry generalise to diffeological spaces.
Definition 2.10. Let X be a diffeological space and k ≥ 0. A k-form ω on X consists of a family of differential forms ω ϕ ∈ Ω k (c) indexed by the plots ϕ : c −→ X of X such that ω ϕ 1 = f * ω ϕ 2 for all commuting triangles Definition 2.11 ([Wal12b, Section 3]). Let G be a Lie group and X a diffeological space. A principal G-bundle on X consists of a subduction π : P −→ X together with a fibre-preserving right action P × G −→ P such that the map is a diffeomorphism. A connection on a principal G-bundle P is a 1-form A ∈ Ω 1 (P ; g) satisfying on P × G, where ρ : P × G −→ P is the right G-action, θ is the left-invariant Maurer-Cartan 1-form on G, and pr P : P × G −→ P and pr G : P × G −→ G are the projections onto P and G, respectively.

Bundle gerbes and transgression
Bundle gerbes are higher categorical analogues of line bundles. They provide a geometric realisation for the third cohomology group with integer coefficients. Similarly to line bundles, bundle gerbes can be equipped with connections. We briefly recall the definition of the 2-groupoid of bundle gerbes and their transgression to loop space. For details we refer to [Wal07b,Wal16,Bun17,Mur96].
Let X be a diffeological space. We denote by HLBdl(X) (resp. HLBdl ∇ (X)) the category of hermitean line bundles (resp. with connection) on X. Before defining bundle gerbes we need to introduce some notation: for a subduction π : Y −→ X of diffeological spaces we denote by is a simplicial diffeological space corresponding to the subduction groupoid Y × X Y ⇒ X, and for k < n and 0 ≤ i 1 < · · · < i k < n we define the smooth face maps (y 0 , y 1 , . . . , y n−1 ) −→ (y i 1 , . . . , y i k ) .
The 2-form B is called a curving. The second condition implies that the closed 3-form dB = π * H descends to a unique closed 3-form H on X with integer periods, which is called the curvature of the bundle gerbe connection (∇ L , B) .
Schematically, the data corresponding to a bundle gerbe can be visualised by the diagram illustrating that hermitean bundle gerbes are equivalent to U(1)-central extensions of subduction groupoids.
Example 2.14. Let X be a diffeological space. The trivial hermitean bundle gerbe I on X consists of the identity subduction 1 X : X −→ X together with the trivial hermitean line bundle I := X × C over X [2] = X and bundle gerbe multiplication For every 2-form B ∈ Ω 2 (X) we can define a connection on I by setting ∇ I = d and taking B as the curving. We denote the resulting hermitean bundle gerbe with connection by I B . The curvature of I B is given by H = dB.
Remark 2.16. One can also define non-invertible 1-morphisms of bundle gerbes by using higher rank hermitean vector bundles E in Definition 2.15 [Wal07b]. In that case, a 1-morphism is weakly invertible if and only if the underlying hermitean vector bundle E is of rank 1 [Wal07a, Proposition 2.3.4]. However, with the exception of Section 6, we will only consider invertible 1-morphisms of bundle gerbes in the present paper.
be 1-isomorphisms G −→ G of hermitean bundle gerbes with connection. A 2-isomorphism of bundle gerbes is an equivalence class of a subduction ω : satisfying a natural compatibility condition, see e.g. [Wal07b] for details and the equivalence relation.
Bundle gerbes on a diffeological space X are classified by their Dixmier-Douady class in H 3 (X; Z), analogously to the Chern-Weil classification of line bundles by their Chern class in H 2 (X; Z). For a bundle gerbe with connection, the Dixmier-Douady class maps to the de Rham cohomology class of the curvature under the homomorphism H 3 (X; Z) −→ H 3 (X; R) induced by the inclusion of coefficient groups Z → R.
Let G be a hermitean bundle gerbe defined over a subduction π : Y −→ X, with underlying hermitean line bundle L −→ Y [2] . Let A : G −→ G be an endomorphism of G, with underlying hermitean vector bundle A over some subduction ξ : Z −→ Y [2] . Consider the hermitean vector bundle L ∨ ⊗ A on Z, where we denote the dual line bundle by L ∨ . This comes with a canonical descent isomorphism defined by the diagram [Wal07b,Bun17] In fact, this construction establishes an equivalence of categories R : BGrb(X)(G, G) −→ HLBdl(X).
From a hermitean bundle gerbe with connection G on a diffeological space X we can construct the transgression line bundle T G over the loop space LX of X. The fibre T G γ over a loop γ : S 1 −→ X consists of equivalence classes [[S], z] of a 2-isomorphism class of a trivialisation S : γ * G −→ I 0 in BGrb ∇ (S 1 ) over the unit circle S 1 and an element z ∈ C. Two pairs ([S], z) and ([S ], z ) are equivalent if and only if z = hol(S 1 , R(S • S −1 )) z. For the construction of a diffeological structure on T G := γ∈LX T G γ we refer to [Wal16]. A connection on a line bundle over the loop space LX is superficial if the holonomy around every thin loop 1 is equal to 1 and thin homotopic loops 2 have the same holonomy. In [Wal16] a superficial connection on T G is constructed from the connection on G. The bundle gerbe multiplication induces, for all triples of paths (γ 1 , γ 2 , γ 3 ) with sitting instants and the same start and end points, a fusion product where denotes the concatenation of paths and γ is the path t −→ γ(1 − t). The fusion product depends smoothly on the paths, is parallel with respect to the superficial connection, and is associative. The connection and fusion product satisfy one further compatibility condition, related to the rotation of all paths involved by 180 • (see [Wal16, Definition 2.1.5]). A line bundle over LX admitting all the structures discussed above is a fusion line bundle with superficial connection.
Transgression extends to a functor T from hBGrb ∇ (X), the 1-category obtained from BGrb ∇ (X) by identifying isomorphic 1-morphisms, to the category of fusion line bundles with superficial connection over LX. The central result of [Wal16] is that T defines an equivalence of categories. An explicit inverse functor R is constructed in [Wal16] and is called regression.

Group extensions from principal bundles
In this section we construct group extensions from group actions on manifolds with principal bundles. We generalise this extension to higher geometry in Section 5. We present two perspectives on this group extension. The first one is global. The second one is local and can be formulated in terms of the parallel transport of an auxiliary connection on a principal bundle.

Global description
Let H be a Lie group and P −→ M a principal H-bundle on a manifold M ; principal H-bundles on M and isomorphisms form a groupoid which we denote by Bun H (M ). We consider a Lie group action on the base manifold M , and ask whether and how this action lifts to P . An action of a Lie group G on M can equivalently be written as a smooth homomorphism of groups Φ : G −→ Diff(M ), where Diff(M ) is the diffeological group of diffeomorphisms M −→ M . In general, the action of G does not lift to P . Instead, we will construct a group extension of G by the gauge group Gau(P ) of P . The group Sym G (P ) acts on the total space P in a way compatible with the action of G on M . We show that it is the universal extension of G having this property.
We can pull back the bundle P along the source and target maps of the action groupoid We define a bundle where Bun H (M )(P, Φ * g P ) is the collection of gauge transformations from P to Φ * g P . In order for Sym G (P ) to be a bundle over G, we must ensure that the fibres of Sym G (P ) are actually pairwise diffeomorphic. It might happen that a pullback bundle Φ * g P is no longer isomorphic to P and hence the fibre over g is empty. As an example, consider the action of the group G = Z on the 2-torus M = T 2 generated by an orientation-reversing diffeomorphism f , and let P −→ T 2 be a U(1)-bundle with non-trivial Chern class. Then [f * P ] = −[P ], and thus Sym Z (P ) |1 = Bun U(1) (P, f * P ) = ∅. Hence in (3.1) we have to ensure that the fibres of Sym G (P ) are actually all non-trivial.
We restrict our attention to connected Lie groups G; otherwise, if G is not connected, we consider only the connected component of the identity e ∈ G. We show that in this case the fibres are always non-trivial: we need to show that for any g ∈ G the fibre of Sym G (P ) −→ G over g is non-empty. That is, we need to show that there exists an isomorphism P −→ Φ * g P of H-bundles over M . Let f P : M −→ BH be a map that classifies the bundle P −→ M . Then Φ * g P is classified by the map f P • Φ g : M −→ BH. Since G is connected, we can find a smooth path γ : [0, 1] −→ G with γ(0) = e and γ(1) = g. Consider the smooth map We can postcompose this map by f P to obtain a homotopy from f P to f P • Φ g . This shows that there exists a bundle isomorphism P −→ Φ * g P . We note for later use that this argument generalises to n-gerbes G, as these are classified by maps f G : M −→ B n+1 U(1).
In order to equip the set Sym G (P ) with a diffeology, we note that Sym G (P ) can be identified with the subspace of the Cartesian product of the space of H-equivariant diffeomorphisms P −→ P which cover the action of an arbitrary element g ∈ G on M with G, and equip Sym G (P ) with the subspace diffeology. Concretely, for c ∈ Cart, a map f : c −→ Sym G (P ) is a plot if and only if the composition π • f : c −→ G is smooth and the induced map pr * The automorphism group or group of gauge transformations Gau(P ) := Bun H (M )(P, P ) acts simply and transitively on each fibre Sym G (P ) |g from the right via precomposition.
Proof. We verify that the map π : Sym G (P ) −→ G is a subduction. Let f : c −→ G be a plot. We can pick an isomorphism ϕ f : pr * M P −→ Φ * f P (since c is contractible) and define the map The map f is a smooth lift of the plot f , showing that Sym G (P ) −→ G is a subduction.
The map provides a smooth inverse to the map Sym G (P ) × Gau(P ) −→ Sym G (P ) × G Sym G (P ) from (2.12), and the result follows.
Proposition 3.3. Sym G (P ) is a diffeological group. The principal bundle Sym G (P ) −→ G is part of an extension of diffeological groups Proof. To complete the proof we need to equip Sym G (P ) with a diffeological group structure such that the map Sym G (P ) −→ G becomes a morphism of diffeological groups. Consider isomorphisms ψ : P −→ Φ * g P and φ : P −→ Φ * g P for g, g ∈ G. We set This is associative by the associativity of pullbacks, the multiplication in G, and composition of morphisms. The inverse of an element ψ : P −→ Φ * g P with respect to µ is the isomorphism , and the result follows from the observation that these maps are smooth.
Proposition 3.4. The group Sym G (P ) acts smoothly on P , lifting the action of G on M . It is universal in the following sense: let G be a Lie group, ϕ : G −→ G a Lie group homomorphism and ψ : G × P −→ P an action of G on P making the diagram Then their exists a unique smooth group homomorphism G −→ Sym G (P ) such that the diagram Proof. The action is via the evaluation The unique smooth group homomorphism in the universality statement is g −→ ψ g : P −→ Φ * ϕ( g) P , and the result follows.
The construction of the group Sym G (P ) is functorial in P , i.e. an isomorphism of bundles ψ : P −→ P induces an isomorphism of group extensions

Equivariant bundles
Let G be a connected Lie group, M a manifold with G-action Φ : G × M −→ M , and P a principal H-bundle over M . A G-equivariant structure on P consists of an isomorphism χ : pr * M P −→ Φ * P of principal bundles over G × M such that the diagram P Φg(x) commutes for all g, g ∈ G and x ∈ M . We denote by E(P ) the set of equivariant structures on P . A splitting s of π : Sym G (P ) −→ G is a smooth group homomorphism s : G −→ Sym G (P ) such that π • s = 1 G . We denote the set of splittings of π : Sym G (P ) −→ G by S(G; Sym G (P )).
Proposition 3.6. There is a natural bijection of sets Ξ : E(P ) −→ S(G; Sym G (P )). In particular, the bundle P admits an equivariant structure if and only if the extension is trivial as an extension of diffeological groups.
Proof. Let (P, χ) be an equivariant bundle. We define Ξ(P, χ)(g) : P −→ Φ * g P to be χ |{g}×M . The inverse Ξ −1 : S(G; Sym G (P )) −→ E(P ) can be constructed by sending a splitting s : The equivariant structures on P and P can be described by smooth group homomorphisms s P : G −→ Sym G (P ) and s P : G −→ Sym G (P ). Since the isomorphism ψ defined in (3.5) intertwines the action of Sym G (P ) and Sym G (P ) on P and P , respectively, it follows that ψ is equivariant if and only if s P = ψ • s P . Hence the smooth group extension Sym G contains all information on equivariance.

Description via parallel transport
The extension Sym G (P ) can be described more explicitly using the parallel transport of a connection on P , as we will now explain. In Section 6 we apply this to the description of magnetic translations in quantum mechanics. We consider a principal H-bundle P −→ M . Let P 0 G denote the diffeological space of smooth paths in G with sitting instants based at e ∈ G, ev 1 : P 0 G −→ G the evaluation at the end point, (P 0 G) [2] the fibre product P 0 G × G P 0 G with respect to ev 1 , and LM the space of smooth loops in M . We denote by the concatenation of paths. For a path γ : [0, 1] −→ G we denote by γ the precomposition of γ with For a path γ ∈ P 0 G and a point x ∈ M , set The set Gau(P ) forms a diffeological group with respect to the composition of automorphisms and the smooth structure induced from the mapping space diffeology on P P . Endow P with an arbitrary connection A. The H-bundle P with connection then induces a principal Gau(P )-bundle on G as follows: we set where we define the equivalence relation for all (γ, α) ∈ (P 0 G) [2] and x ∈ M , and we interpret the holonomy of P along a loop starting and ending at x as an endomorphism of the fibre P x . We endow L G with the quotient diffeology.
Then the Gau(P )-bundle L G −→ G can be defined in terms of descent data as follows: the action Φ of G on M induces a smooth map for all t ∈ [0, 1] and x ∈ M . The descent data for the bundle L G consists of the subduction P 0 G −→ G, the trivial bundle P 0 G × Gau(P ) −→ P 0 G, and the smooth map Proposition 3.8. The total space L G is a smooth group extension which is controlled by a smooth Gau(P )-valued group cohomology class on G of degree 2.
Proof. Let γ and γ be two paths in G. The evaluation ev 1 : P 0 G −→ G is a group homomorphism with respect to the pointwise product of paths.
Let x ∈ M be an arbitrary point. To any triple (x, γ, γ ), we can associate a map Diagrammatically, this is a homotopy between the product path γ γ ∈ P 0 G and the concatenated path (γ γ (1)) γ ∈ P 0 G.
For γ, γ ∈ P 0 G and φ, φ ∈ Gau(P ), we define where we denote by pt γ the isomorphism P −→ Φ * γ(1) P defined at a point x ∈ M by the parallel transport along the path γ x . This is well-defined: let α, α ∈ P 0 G with γ(1) = α(1) and γ (1) = α (1). Then where we used Φ * γ (1) pt γ = pt γ (γ (1)) . Associativity then follows immediately from the associativity of the products in P 0 G and Gau(P ), together with associativity of taking pullbacks. Smoothness follows from the definition of the quotient diffeology and the smooth dependence of parallel transport on the path.
Remark 3.10. For abelian structure group H, we can use the fact that parallel transport commutes with gauge transformations to get the simplified expression for the multiplication (3.9).
Remark 3.11. If G is abelian, then the multiplicative structure yields isomorphisms L G|g × L G|g −→ L G|g g = L G|g g −→ L G|g × L G|g for all g, g ∈ G. That is, the group extension L G spoils the commutativity of G, since its fibres multiply commutatively only up to coherent isomorphism.
We summarise the connection to the construction from Section 3.1 in Proposition 3.12. Let G be a connected Lie group, and let P −→ M be a principal H-bundle on a manifold M with smooth G-action. The map is an isomorphism of diffeological group extensions of G.
Proof. The map is well-defined: consider two representatives (γ, φ) and (α, hol(P, α, γ) • φ) of the same equivalence class in L G , and calculate The map is bijective, because two gauge transformations P −→ Φ * g P differ by exactly one gauge transformation of P . It also follows directly from the definition that Γ is a morphism of extensions.
We check that Γ is a group homomorphism: for [(γ, φ)], [(γ , φ )] ∈ L G we compute Finally, we verify that Γ is smooth. Let f : c −→ L G be a plot admitting a lift f : c −→ P 0 G × Gau(P ). We denote the components of f by f γ and f Gau(P ) . It is enough to show that is a gauge transformation. This follows from the smoothness of parallel transport (recalled in Section 4.1 below).

A global approach to parallel transport for bundle gerbes
In Section 3 we have constructed two diffeological groups, Sym G (P ) and L G , which extend G and control the existence of G-equivariant structures on a principal bundle P over M . The key to constructing L G , as well as to comparing the groups Sym G (P ) and L G (see Section 3.3), was the parallel transport on the principal bundle P .
If one replaces the principal bundle P by a bundle gerbe G on M , there exist categorified versions of both these constructions which will be given in Section 5. However, in order to write down the categorification of L G we need a notion of parallel transport for G. In this section we give a definition of parallel transport for G suited for our purposes and explicitly construct such a parallel transport from any connection on G. Our construction relies heavily on Waldorf's transgression-regression machine [Wal16].
There is a different approach to the parallel transport on a bundle gerbe developed by Schreiber and Waldorf [SW09,SW11,SW17]. It relies on their technology of transport functors and is based on local constructions, which are then glued to global objects. Here, in contrast, we directly define and construct a global version of parallel transport suitable for our purposes. As our main goal in this paper is the construction of categorified smooth group extensions, we leave it for future work to prove in detail that our notion of parallel transport for G agrees with that of Schreiber and Waldorf, and instead focus on building the necessary input for the constructions in Section 5.

A path space approach to parallel transport on line bundles
Before we give our definition and construction of the parallel transport for bundle gerbes, we recast the parallel transport on line bundles from a global perspective. Our notion of parallel transport for bundle gerbes will then be a categorification of this picture. Let M be a connected smooth manifold, and fix a base point x ∈ M ; if M is not connected, we restrict to its connected components individually. We denote by P M the diffeological space of smooth paths with sitting instants in M and by P 0 M the subspace of paths starting at x. Let L be a line bundle on M with connection. The smoothness of the parallel transport on L can be encoded as follows: for t ∈ [0, 1], denote by ev t : P M −→ M , γ −→ γ(t), the evaluation at t. Parallel transport on L is in particular an isomorphism There is a different way to construct this isomorphism using descent. Via transgression and regression [Wal12b] we can construct a bundle RT (L), which is isomorphic to L, from the descent data U(1) with respect to the path fibration. Here f is constructed as in Section 3.3 from the holonomy of L.
The pullbacks ev * 0 RT (L) and ev * 1 RT (L) are thus described in terms of descent data with respect to the covers ev * 0 P 0 M ∼ = P 0 M × M P M −→ M and ev * 1 P 0 M ∼ = P M × M P 0 M −→ M , respectively. In order to construct the isomorphism pt RT (L) explicitly we use the space (see Figure 1) which fits into the diagram An isomorphism from ev * 0 RT (L) to ev * 1 RT (L) can be described by a function P ∂∆ 2 M −→ U(1) which is compatible with the descent data. There is a canonical choice for such a function given by Concretely, the induced map is where x is the fixed base point of M while y, z ∈ M are arbitrary points, and γ ab denotes a path from a to b for a, b ∈ {x, y, z}. The holonomy appearing here agrees with hol(L, (γ xz γ yz ) γ xy ). The construction is independent of all choices involved. Now a straightforward computation shows that the diagram commutes. This shows that we can construct the parallel transport on L completely in terms of the descent data with respect to the path fibration. For bundle gerbes, the analogue of pt L is difficult to define directly, but an analogous approach via descent data on P 0 M allows us to solve this problem.

Global definition of parallel transport on bundle gerbes
As before, let M be a manifold, and let G ∈ BGrb(M ) be a bundle gerbe on M . A parallel transport on G should in particular be a 1-isomorphism where c : M −→ P M is the embedding of M into P M as constant paths. Note that the parallel transport is, in general, an isomorphism of gerbes without connections. The same is true for bundles: the parallel transport on a vector bundle with connection respects the connection if and only if the connection is flat.
To proceed further, we need some definitions. Let i, n ∈ N with 1 ≤ i ≤ n. For each s = (s 1 , . . . , s n−1 ) ∈ [0, 1] n−1 , define a smooth map We also consider the subspaces P n * M of the diffeological spaces P n M consisting of maps Σ ∈ P n M satisfying the following property: for all s ∈ [0, 1] n−1 , and for each j = 1, . . . , n − 1 such that s j ∈ {0, 1}, the map Σ • ι n i;s is constant for all i > j. The space P n * M describes iterated smooth homotopies with fixed endpoints in M . For example, P * M = P M is the space of paths with sitting instants, P 2 * M consists of maps Σ ∈ P 2 M such that for all t ∈ [0, 1] and so is the space of homotopies of paths with fixed endpoints in M , and an element in P 3 * M is a family of fixed-ends homotopies between two fixed paths in M . We say that an element Σ in P n * M or in P n M is thin if its differential Σ * has non-maximal ranks rk(Σ * |s ) < n for all s ∈ [0, 1] n . Let s = (s 1 , . . . , s k ) ∈ [0, 1] k and n = k + l. For 0 ≤ i 1 < · · · < i l ≤ n, we define a map which inserts the coordinates of t = (t 1 , . . . , t l ) ∈ [0, 1] l into the k-tuple s such that For the parallel transport of a bundle gerbe, there should also be a 2-isomorphism in BGrb(P 2 * M ). In other words, any map Σ ∈ P 2 * M is in particular a smooth map [0, 1] 2 −→ M from the square to M . This map is constant on the vertical edges of the square. Pulling back the isomorphism pt G 1 to the horizontal edges of the square gives two 1-morphisms G Σ(0,0) −→ G Σ(1,0) , and the 2-morphism pt G 2 relates these. The data (pt G 1 , pt G 2 ) are required to satisfy the following two properties, which are motivated by [BW19,Wal16]: (1) For any two thin maps Σ, Σ ∈ P 2 * M with Σ • ι 2 1;s = Σ • ι 2 1;s for s = 0, 1, there is an equality That is, the 2-morphism pt G 2 evaluated on any pair of fixed-ends thin homotopies between any two given paths in M gives the same result.
(2) We further demand that for any thin map h ∈ P 3 * M , there is an equality As we will be using P n * M mostly for n = 0, 1, 2, we adopt the convention to write γ 2 γ 1 for the concatenation of smooth paths in M , and if Σ, Σ ∈ P 2 * M are homotopies Σ : γ −→ γ and Σ : γ −→ γ , we write Σ 2 Σ : γ −→ γ for their vertical concatenation. If Ξ : α −→ α is a further homotopy in P 2 * M such that the starting point of α is the endpoint of γ, then we write Ξ Σ : α γ −→ α γ for the horizontal concatenation of the homotopies. We will also often use the term 'composition' instead of 'concatenation'. These data are required to satisfy properties (4.1) and (4.2). Due to property (4.1), there is a for every (γ 1 , γ 2 , γ 3 ) ∈ P M × M P M × M P M , and we demand that pt G is coherently associative with respect to this isomorphism. The morphism pt G also needs to be compatible with the unitors in BGrb(P M ) and sit in a commutative diagram for all x, y, z ∈ M , all paths γ 1 , α 1 from x to y, all paths γ 2 , α 2 from y to z in M , and for all fixed-ends homotopies Σ i : γ i −→ α i . Furthermore, pt G 2 has to respect vertical composition and satisfy the interchange law , and for all fixed-ends homotopies Remark 4.6. The associativity condition in detail reads as follows: for every concatenable triple where the bottom arrow is the canonical 2-isomorphism obtained via (4.1) from any reparameterisation of [0, 1] that yields a homotopy γ 3 (γ 2 γ 1 ) ∼ (γ 3 γ 2 ) γ 1 .
Remark 4.7. By property (4.2), our definition factors through the path 2-groupoid of M as defined by Schreiber and Waldorf [SW11,SW17]. Given a manifold M , they construct a 2-groupoid internal to diffeological spaces, whose level sets are essentially M , P M and the quotient P 2 * M/∼ of P 2 * M by thin homotopies. Rather than taking this quotient, we demand that (4.2) holds. However, condition (4.1) is new as compared to [SW11,SW17]; it is motivated by [BW19].
Remark 4.8. There is a more slick and conceptual way of phrasing Definition 4.3: Given the path 2-groupoid P 2 M of Schreiber and Waldorf [SW11,SW17], one can check that the Duskin nerve of this 2-groupoid is a simplicial diffeological space N D (P 2 M ) : ∆ op −→ Dfg. We can then define the 2-category BGrb pt (M ) of bundle gerbes on M with parallel transport as the homotopy limit The diagram in the homotopy limit is a cosimplicial diagram in weak 2-categories, and the homotopy limit can be modelled explicitly as in [NS11, Definition 7.1]. While this formulation certainly has advantages when it comes to a conceptual treatment of parallel transport, for our present purposes the more hands-on model given in Definition 4.3 is more convenient.
In contrast to the case of parallel transport on vector bundles, we can define morphisms between parallel transports on a given bundle gerbe.
2 with pt 2 G , the 2-isomorphism pt G with pt G , and the 2-isomorphism ε G with ε G . This defines a groupoid PT(G) of parallel transports on G.
This notion of morphism of parallel transports is not an analogue of a gauge transformation, since it does not necessarily come from an automorphism of the bundle gerbe G.

Construction of the parallel transport
We now proceed to show that every bundle gerbe with connection on a manifold M has a canonical parallel transport. Let M be a connected manifold, and fix a base point x ∈ M ; otherwise, if M is not connected, we treat the connected components of M separately. By results of Waldorf [Wal16], any bundle gerbe G ∈ BGrb ∇ (M ) is isomorphic to a bundle gerbe G ∈ BGrb ∇ (M ) that is defined over the diffeological path fibration P 0 M −→ M . Given a choice of base point x ∈ M , Waldorf constructs a bundle gerbe G = RT (G) as the regression of the transgression line bundle of G, together with a 1-isomorphism A G : G −→ G in the homotopy category of BGrb ∇ (M ); that is, A G is determined only up to 2-isomorphism.
Consider the bundle gerbe G = RT (G) ∈ BGrb ∇ (M ) with connection on M , defined with respect to the path fibration π : P 0 M −→ M . Its line bundle L is the pullback of the transgression line bundle T G −→ LM along the map By a slight abuse of notation, we also denote this pullback by T G −→ (P 0 M ) [2] .
We would like to construct a 1-isomorphism in BGrb(P M ). For t = 0, 1, the bundle gerbe ev * t G is defined over the subduction ev * t P 0 M −→ P M . There are canonical isomorphisms of diffeological spaces Recall from Section 4.1 the space A point in the total space P ∂∆ 2 M is a triple (α 0 , γ, α 1 ) of a path γ ∈ P M and based paths α t ∈ P 0 M such that γ(t) = α t (1) for t = 0, 1. Any 1-morphism ev * 0 G −→ ev * 1 G is defined over (possibly a refinement of) the subduction ξ : There is a smooth map, i.e. a morphism of diffeological spaces There is also the smooth maps The maps s ands are smoothly homotopic via precomposition by a homotopy h of piecewise smooth homeomorphisms [0, 1] −→ [0, 1]; these fail to be smooth exactly at those points of the interval where the concatenations happen, but at these points all three paths have sitting instants, so that at each time the homotopy maps to LM , as desired. For each triple of paths (α 0 , γ, α 1 ), this results in a thin homotopy in LM from α 1 (γ α 0 ) to (α 1 γ) α 0 . By the superficiality of the parallel transport pt T G on the transgression line bundle [Wal16, Definition 2.2.1] (see also the end of Section 2.2), we thus obtain a canonical isomorphism r : s * T G −→s * T G in HLBdl ∇ (P ∂∆ 2 M ). The fact that this isomorphism preserves connections is a direct consequence of [Wal16, Lemma 2.3.3]. Since pt T G is thin-invariant, it follows that the morphism r is defined independently of the choice of homotopy h.
We define a morphism pt G 1 : ev * 0 G −→ ev * 1 G as follows: its underlying line bundle is the line bundle s * T G −→ P ∂∆ 2 M . To turn this into a morphism of bundle gerbes, we need to provide an isomorphism of line bundles . Let us unravel this: the fibre product (P ∂∆ 2 M ) [2] = P ∂∆ 2 M × P M P ∂∆ 2 M consists of pairs ((α 0 , γ, α 1 ), (α 0 , γ, α 1 )) where (α 0 , γ, α 1 ) and (α 0 , γ, α 1 ) are elements of P ∂∆ 2 M . For t = 0, 1, there are the projection maps , which provides the bundle gerbe multiplication on G . At a point (α 0 , α 1 , α 2 ) ∈ (P 0 M ) [3] the fusion product consists of unitary isomorphisms The diffeological space (P ∂∆ 2 M ) [2] comes with smooth maps This morphism is compatible with the bundle gerbe multiplication on G : consider an arbitrary point Then there is a commutative diagram The commutativity follows from the associativity of the fusion product λ and the fact that it respects the connection on T G [Wal16] so that, in particular, λ is compatible with the morphism r.

Construction of pt G 2
Next we construct the 2-isomorphism in BGrb(P 2 * M ) that is part of the parallel transport data for G . For this, we recall that the fibre of the hermitean line bundle T G at a loop γ is constructed from pairs ([S], z) of a 2-isomorphism class [S] of trivialisations S : γ * G −→ I 0 in BGrb ∇ (S 1 ) and a complex number z ∈ C. The complex line T G γ is the set of equivalence classes of such pairs under the equivalence relation Let M D 2 be the diffeological space of smooth maps from the unit disk D 2 to M . Let denote the smooth map induced by restriction to the boundary of the unit disk. The hermitean line bundle ∂ * T G on M D 2 has a canonical trivialisation which is defined as follows: for a smooth map f : D 2 −→ M , choose a trivialisation S : f * G −→ I B for some B ∈ Ω 2 (D 2 ). Define a unitary isomorphism of hermitean complex lines This isomorphism is defined independently of the choice of S: let S : f * G −→ I B be another trivialisation. Then the line bundle R(S • S −1 ) has curvature B − B, which implies that This construction works equally well if we replace the 'round' unit disk D 2 by the unit square [0, 1] 2 , as long as we consider maps f : [0, 1] 2 −→ M whose restrictions to ∂[0, 1] 2 have sitting instants at the corners. By the construction of the fusion product λ on T G, the section σ is compatible with fusion, .) Now consider the following setup: let Σ : [0, 1] 2 −→ M be an element in P 2 * M , presenting a fixed-end homotopy from a path γ to a path γ in M . We want to compare the 1-isomorphisms (ι 2 * 1;0 ) * pt G 1 and (ι 2 * 1;1 ) * pt G 1 of bundle gerbes over P 2 * M . The source bundle gerbes of both these morphisms have subductions while the target bundle gerbes live over The fibre product Y := Y 0 × P 2 * M Y 1 is the space of triples (α 0 , Σ, α 1 ) of based paths α 0 , α 1 ∈ P 0 M and fixed-ends homotopies Σ ∈ P 2 * M between arbitrary paths in M such that α t (1) = Σ(0, t) for t = 0, 1 (see Figure 2). The 1-isomorphism (ι 2 * 1;i ) * pt G 1 , for i = 0, 1, is defined over the subduction which is actually an isomorphism. Consequently, the 2-isomorphism pt G 2 should be defined with respect to the subduction which again is an isomorphism. Its elements are triples (α 0 , Σ, α 1 ) as above. Set γ t := Σ • ι 2 1;t for t = 0, 1, and let x = γ t (0) and y = γ s (1) for t, s = 0, 1.
At a point (α 0 , Σ, α 1 ), the morphism of hermitean line bundles over Z that defines pt G 2 is given by the morphism pt G 2|(α 0 ,Σ,α 1 ) : T G α 1 (γ 0 α 0 ) −→ T G α 1 (γ 1 α 0 ) of complex lines obtained as follows: (1) Using a smooth family of rotations of S 1 , apply parallel transport on T G to obtain an isomorphism This is achieved by parallel transport along a thin path in LM . Hence, since the parallel transport on T G is superficial, this isomorphism is independent of the choice of a smooth family of rotations.
(2) Use the canonical section σ Σ (1) ∈ T G ∂Σ from (4.13) to obtain an isomorphism (3) The boundary loop ∂Σ is smoothly and thinly homotopic (via reparameterisations) to ((id y γ 1 ) id x ) γ 0 , where id x is the constant path at the point x ∈ M . This loop is, in turn, thinly homotopic to γ 1 γ 0 . We thus obtain a canonical isomorphism (4) The fusion product on T G yields an isomorphism (5) Finally, we again use parallel transport along a path in LM that arises from a smooth family of rotations to obtain a canonical isomorphism This is compatible with vertical composition in P 2 * M : let Σ, Σ ∈ P 2 * M be two maps [0, 1] 2 −→ M that can be concatenated vertically. Since the connection on T G is superficial and compatible with the fusion product, we can replace the morphism ψ 1 bỹ Applying the fusion product with ∂(Σ 2 Σ) yields an isomorphism . Combining the fact that the fusion product λ is associative and compatible with the parallel transport on T G, that the parallel transport on T G is superficial (in particular, parallel transport along thin paths is independent of the choice of thin path), and that the section σ from (4.13) is compatible with λ, it follows that pt G 2 respects vertical concatenation. Since all morphisms involved in the construction of pt G 2 are smooth, it follows that pt G 2 is in fact a smooth morphism of bundle gerbes as desired.
is directly constructed from the fusion product λ on the transgression line bundle T G. Define q : Given a point (α 0 , γ, α 1 ), (α 1 , γ , α 2 ) ∈ Q 1 , the morphism pt G is given by the diagram where the horizontal morphisms are induced by smooth families of reparameterisations. The compatibility of this morphism with the morphism β from (4.12) follows again from the superficiality of the connection on T G and the associativity of the fusion product λ.
The compatibility of pt G with pt G 2 as in (4.4) is seen analogously to how we proved the compatibility of pt G 2 with vertical concatenation of homotopies. The interchange law (4.5) is satisfied by the associativity of λ, its compatibility with the parallel transport on T G and with the section σ from (4.13), as well as the superficiality of the connection on T G.

Construction of ε G
Finally, the 2-isomorphism ε G : c * pt G 1 −→ 1 G is obtained directly from the superficial connection on T G: it is defined over the space of triples (α, id x , α) ∈ P ∂∆ 2 M , and all paths of the form α α are canonically contractible by thin homotopies.
All necessary coherences in Definition 4.3 then follow from the superficiality of the parallel transport on T G, the associativity of the fusion product λ and its compatibility with the section σ, and the fact that the parallel transport on T G is compatible with the fusion product. Thus we have Theorem 4.14. Let G ∈ BGrb ∇ (M ) be a bundle gerbe with connection on M , and let G := RT (G) ∈ BGrb ∇ (M ) be the regression of the transgression of G. Then the quadruple pt G = (pt G 1 , pt G 2 , pt G , ε G ) defines a parallel transport on the bundle gerbe G .

Transfer to arbitrary bundle gerbes
In [Wal16], Waldorf shows that the functors T and R come with a canonical natural isomorphism as endofunctors of the homotopy 1-category hBGrb ∇ (M ). Given a bundle gerbe G ∈ BGrb ∇ (M ), we thus get a 2-isomorphism class of 1-isomorphisms G −→ G = RT (G). Let A G : G −→ G be a representative for this class.
That is, for any pair of parallel unitary 1-isomorphisms B, B : G −→ G for which there exists some

The transgression line bundle as a holonomy
Let G ∈ BGrb ∇ (M ) be a bundle gerbe with connection on M and write G = RT (G). We will now determine the holonomy of the parallel transport on G . For this, consider the diffeological space L * M of smooth maps S 1 −→ M that have a sitting instant at 1 ∈ S 1 . In other words, L * M is the pullback where ι denotes the inclusion map and ∆ is the diagonal embedding. The pullback ι * pt G 1 is an automorphism of ev * 1 G , which we understand as the holonomy of pt G . It is defined over the subduction ι * P ∂∆ 2 M −→ L * M . Recall from Section 2.2 that a 1-automorphism of a bundle gerbe defines a line bundle via descent. Thus the holonomy ι * pt G 1 gives rise to a descended line bundle hol(G) ∈ HLBdl(L * M ). Our goal is to understand this descended line bundle more explicitly.

Let ev
in Dfg. The hermitean line bundle (with connection) underlying the bundle gerbe ev * 1 G is the pullback bundle We now apply the construction from the diagram (2.18) that produces a line bundle R(A) from an automorphism A of a bundle gerbe: the tensor product bundle L ∨ ⊗ pt G 1 on ι * P ∂∆ 2 M has fibres Now the thin-invariant parallel transport and the fusion product on T G yield an isomorphism By the associativity of λ, this is an isomorphism of descent data for line bundles on L * M . This shows Thus the parallel transport pt G reproduces the transgression line bundle T G as its holonomy.
Remark 4.20. For a generic bundle gerbe G with parallel transport pt G , the morphism pt G 2 induces a parallel transport on hol(G). It should be possible to construct from this a fusion line bundle with connection on LM in the sense of [Wal16], which then regresses to a bundle gerbe with connection on M . Its underlying bundle gerbe should be canonically isomorphic to G (up to 2-isomorphism), and that should allow the reconstruction of the connection on G from its parallel transport in our sense. However, this would go beyond the scope of this paper, and since for our applications in Sections 5 and 6 having an explicit construction for pt G is sufficient, we leave this reconstruction of the connection on G for future work.

2-group extensions from bundle gerbes
Let G be a connected Lie group with a smooth group action on a manifold M . In Section 3 we saw how a principal bundle P −→ M gives rise to a group extension Sym G (P ) −→ G which encodes all information about equivariant structures on P . We were able to give two equivalent constructions for Sym G (P ), one as a subgroup of Diff(P ), and one as descent data associated to the path fibration P 0 G −→ G and a parallel transport on P .
In this section we study the analogous situation for a bundle gerbe G ∈ BGrb(M ) instead of a principal bundle P ∈ Bun H (M ). There are two main differences to the situation in Section 3: equivariant structures on G form a groupoid rather than a set, and they do not assemble into a topological or smooth space. We thus cannot expect a universal extension Sym G (G) −→ G as diffeological groups. A good framework to describe this extension is that of group objects in categories fibred in groupoids over Cart, where the fibering describes the smooth structure. After carefully setting up this framework, we give two constructions of Sym G (G), in analogy to the two constructions of Sym G (P ) in Section 3. We conclude this section by showing that, again, the extension Sym G (G) −→ G encodes all information about equivariant structures on G.

Smooth groupoids and symmetries of gerbes
Let G ∈ BGrb(M ) be a bundle gerbe on M . Let Φ : G × M −→ M be an action of a connected Lie (or diffeological) group G on M . Let Cart denote the category of smooth manifolds that are diffeomorphic to R n for some n ∈ N 0 . The morphisms in Cart are the smooth maps between these manifolds.
We can view M and G as presheaves on Cart by setting and By adding identity morphisms, we can canonically enhance the presheaf G to a (pre)stack on Cart, i.e. a (pre)sheaf of groupoids, which we still denote by G. Given a section f ∈ G(c), i.e. a smooth map f : c −→ G, we can define a map We can then assign to f the groupoid By a slight abuse of notation, we will denote the functor Sym PSh By construction of the 2-category of bundle gerbes, this defines a pseudofunctor where Grpd is the 2-category of groupoids, functors, and natural transformations.
is not a strict presheaf of groupoids on G , as it is only pseudofunctorial [Moe02]. There are several ways to technically treat such pseudo-presheaves of groupoids: (1) Encode the coherence morphisms by viewing pseudo-presheaves of groupoids as coherent simplicial presheaves, i.e. as simplicial functors C • N ∆ (G) op −→ Set ∆ in the notation of [Lur09].
We will follow the third approach here because the transition between the parameterising categories G and Cart becomes particularly easy in that framework.
We will frequently make use of the Grothendieck construction to pass from Grpd-valued pseudofunctors to categories fibred in groupoids; for background on the Grothendieck construction and fibred categories we refer to [Vis05,Lur09]. We will, however, describe the resulting fibred categories explicitly. For example, the canonical projection functor pr : G −→ Cart is the category fibred in groupoids obtained by applying the Grothendieck construction to the (pseudo)functor c −→ G(c), where G(c) is regarded as a groupoid with only identity arrows.
Definition 5.2. A functor π : D −→ C between categories is a Grothendieck fibration in groupoids, or makes D into a category fibred in groupoids over C, if it satisfies the properties: (1) For every object d ∈ D and for every morphism f : (2) For every pair of diagrams in D and C, respectively, there exists a unique lift f 01 of f 01 that makes the upper triangle commute.
The first requirement resembles a path-lifting condition. The second requirement can be viewed as a relative horn-filling property: given any Λ 2 2 -horn σ in D and a filling of π(σ) in C to a 2-simplex, there exists a unique filling of σ to a 2-simplex in D that lifts the 2-simplex in C. Alternatively, consider an arbitrary functor π : D −→ C between categories and a morphism ζ 12 : d 1 −→ d 2 in D.
If for every pair of solid arrow diagrams as in (5.3) the dashed arrow exists such that the upper triangle commutes and such that π( f 01 ) = f 01 , one says that ζ 12 is π-Cartesian. In particular, if π is a Grothendieck fibration in groupoids, then property (2) of Definition 5.2 is equivalent to saying that every morphism in D is π-Cartesian. If π : D −→ C is a Grothendieck fibration in groupoids and c ∈ C, we denote by D |c = π −1 (c) the fibre over c, which is the groupoid with objects d ∈ D such that π(d) = c and morphisms f : d −→ d such that π( f ) = 1 c .
Definition 5.4. A category fibred in groupoids over Cart is a smooth groupoid. Let H denote the strict 2-category of smooth groupoids. Its objects are smooth groupoids, its morphisms are (strictly) commutative diagrams of functors Definition 5.6. Let p : Sym G (G) −→ G denote the category fibred in groupoids obtained by applying the Grothendieck construction to the pseudofunctor Sym PSh G (G) : G op −→ Grpd. Explicitly, the category Sym G (G) consists of: The functor p : Sym G (G) −→ G is automatically a fibration in groupoids, since it arises as the Grothendieck construction of a pseudo-presheaf of groupoids. Since Grothendieck fibrations are stable under composition [Vis05], the composite functor makes Sym G (G) into a smooth groupoid.

Smooth 2-groups
We would now like to establish that Sym G (G) is not just a smooth groupoid, but can also be regarded as a higher group in some sense. That is, we would like to find on Sym G (G) the same type of structure as we found on the bundle Sym G (P ) −→ G in Section 3.1. Here, however, we are working inside the ambient 2-category H, and so we will need to make precise what we mean by a group in H. The notion of a group object in a 2-category goes back to [BL04]. The following definitions are taken from [SP11] which are strongly based on [BL04]. Let C be a 2-category with finite products; in particular, it has a terminal object * . Examples are the 2-categories Grpd and H. Definition 5.8. A 1-morphism of monoid objects (H, u, ⊗, a, l, r) −→ (H , u , ⊗ , a , l , r ) in C consists of a triple (F 1 , F ⊗ , F u ) of These are required to satisfy the coherence conditions in [SP11, Definition 42].
Morphisms of abelian monoid objects satisfy an additional compatibility condition for the symmetries β and β , which can be found in [SP11,Definition 48].
commute. 2-morphisms of abelian monoid objects are 2-morphisms of the underlying monoid objects.
Example 5.10. A monoid object in the 2-category Cat of categories is precisely a monoidal category. Similarly, 1-morphisms and 2-morphisms between monoid objects in Cat are precisely the monoidal functors and the monoidal natural transformations, respectively. The abelian monoids in Cat are precisely the symmetric monoidal categories.
Definition 5.11 ([SP11, Definition 41]). A group object in C is a monoid object (H, u, ⊗, a, l, r) in C such that the 1-morphism (⊗, pr 1 ) : invertible. An abelian group object in C is an abelian monoid object whose underlying monoid object is a group object.
For C a 2-category with finite products, we denote the 2-category of group objects in C by 2Grp(C).
Both these 2-categories are enriched in groupoids. Let us examine Definition 5.12 a little more closely. Consider two objects π C : C −→ Cart and π D : D −→ Cart in H. The product in H is given by the pullback in Cat: Explicitly, the category C × Cart D has • Objects : pairs (c ∈ C, d ∈ D) such that π C (c) = π D (d).
A monoid structure on C ∈ H thus allows us to multiply pairs of objects in the same fibre and pairs of morphisms that lie over the same morphism in Cart.
Example 5.13. The tensor product of line bundles turns the presheaf of groupoids of hermitean line bundles with connection HLBdl ∇ −→ Cart into an abelian group object in H. Similarly, for any manifold M it also turns the internal hom HLBdl ∇ M into an abelian group object in H.

Smooth principal 2-bundles
We shall now establish our precise notion of an extension of smooth 2-groups.
Definition 5.14. Let C be a 2-category with finite products, let (H, u, ⊗ H , a, l, r) be a monoid object in C, and let C ∈ C. A right action of H on C is a morphism ⊗ : C × H −→ C in C, together with 2-morphisms α and u in C that witness the commutativity of the diagrams and that are coherent with respect to the 2-isomorphism a, l and r. Left actions are defined analogously.
Example 5.15. The standard example for an action of a monoid object is that of a module category C over a monoidal category H in C = Cat.
Definition 5.16. Let C be a category. Suppose there are categories fibred in groupoids π i : D i −→ C, for i = 0, 1, and π E : E −→ C over C, and suppose there is a diagram This comes with a canonical functor which, as we show in Appendix A, is a Grothendieck fibration in groupoids.
Definition 5.17. Let H be a smooth 2-group, and let X ∈ H be any smooth groupoid. An Hprincipal 2-bundle on X is an object P ∈ H with a morphism π : P −→ X, a right action (⊗, α) of H on P and a 2-isomorphism (1) the functor π : P −→ X is an essentially surjective Grothendieck fibration, (2) the action (⊗, α) of H on P and the 2-isomorphism η are compatible in the sense that the diagram P × H P is coherent, where the front face carries the 2-isomorphism α that is part of the action of H on P, the back, right-hand, and bottom faces carry the 2-isomorphism η, and the left-hand face commutes strictly, (3) the composition P × H −→ P × X P × H −→ P × h X P is an equivalence, where the first functor is induced by the diagonal functor P −→ P × X P.
The first condition can be understood as demanding that P −→ X has local sections (see Lemma A.1 from Appendix A). The second condition implements the property that the H-action preserves the projection to X up to coherent homotopy. The third condition says that the Haction is principal. Note that upon choosing an inverse to the equivalence P × X P → P × h X P, one could equivalently formulate condition (3) using strict pullbacks alone (again by Lemma A.1 from Appendix A).
In order to understand the notion of an extension of smooth 2-groups, we first need to define the kernel of a morphism of smooth 2-groups. Naively, the kernel could easily be defined as a fibre over u, but the resulting category will not generally be fibred in groupoids over Cart. As it turns out, the homotopy pullback does satisfy this property.
We readily observe that the restrictions of the structure morphisms ⊗ H , a H , l H and r H , together with the morphism u H , turn ker h (p) into a smooth 2-group. It should also be possible to turn the strict kernel ker(p) into a smooth 2-group in this case, using an inverse to the equivalence ker(p) → ker h (p) (compare Lemma A.2 from Appendix A), but the homotopy-kernel ker h (p) carries a canonical 2group structure, and using ker(p) instead of ker h (p) would make Construction 5.21 below rather cumbersome.
Lemma 5.19. In the setting of Definition 5.18, the functor κ is a Grothendieck fibration in groupoids.

Proof. This follows directly by Lemma A.2 (1) from Appendix A.
Let ker(p) denote the strict pullback of the diagram Cart Definition 5.20. Let A and G be smooth 2-groups. An extension of G by A is a pair (F, p) of a morphism of smooth 2-groups p : H −→ G that turns H into a ker h (p)-principal 2-bundle over G, and an equivalence of smooth 2-groups F : A −→ ker h (p).
By Lemma A.2 from Appendix A, we could equivalently require p to turn H into a ker(p)principal 2-bundle, but then we would need to use the non-canonical 2-group structure on ker(p). This essentially amounts to choosing an inverse for the equivalence ker(p) → ker h (p).
Our goal now is to define when an extension of smooth 2-groups is central. Again, we follow the ideas of [SP11], where the criterion for an extension of G by A to be central is formulated using a functor G −→ Aut(A) from G into the automorphisms of A as a 2-group; the smooth structure does not matter here. In [SP11], this functor is obtained from abstract arguments.
Construction 5.21. In our formalism, we can understand this construction as follows: consider smooth 2-groups G and A, where A is abelian, and let (F, p) be a smooth 2-group extension of G by A, with morphism p : H −→ G. Then A is abelian if and only if ker h (p) is abelian, which is true if and only if ker(p) is abelian (since the 2-group structure induces Picard groupoid structures on the fibres of these smooth 2-groups, where F induces monoidal equivalences). Fix an arbitrary Cartesian space c ∈ Cart. Let Aut(ker h (p) |c ) denote the Picard groupoid of monoidal autoequivalences of the fibre ker h (p) |c of ker h (p) over c. Note that we do not claim that the Picard groupoids Aut(ker h (p) |c ) assemble into a smooth 2-group (though it might be possible to achieve this). We claim that there is a functor G |c −→ Aut(ker h (p) |c ) which is canonical up to unique natural isomorphism.
Let ( · ) ∨ : H |c −→ H |c denote a choice of functorial inverse in H |c . This can always be enhanced to a functorial choice of adjoint inverse, i.e. a functor k −→ (k ∨ , ev k , coev k ) that maps k to a triple of a dual object k ∨ , and duality morphisms (which are isomorphisms in this case) ev k : k⊗ H k ∨ −→ u H (c) and coev k : u H (c) −→ k ∨ ⊗ H k which satisfy the triangle identities. The functor ( · ) ∨ acts on morphisms ψ : k −→ k by taking the dual of ψ −1 with respect to the chosen duality data on k and k ∨ . This enhancement can be achieved by choosing an adjoint inverse for the equivalence of categories (⊗ H , pr 1 ) |c : H |c × H |c −→ H |c × H |c (which is always possible).
To an object k ∈ H |c we associate the functor where the morphism ϕ k is the composition Given another object k ∈ H |c such that p(k) = p(k ) in G, the principality condition implies that there exists an object (b, β) ∈ ker h (p) and an isomorphism ψ : k −→ k ⊗ H b. Since ker h (p) is abelian, this induces an isomorphism By the functoriality of ( · ) ∨ , any other choice of (b, β) and ψ yields the same isomorphism Ad k (h) −→ Ad k (h) in this way. Furthermore, this isomorphism is natural in k and h by the functoriality of ⊗ H and ( · ) ∨ . That is, the pair (Ad, α) defines an object |c −→ G |c denotes theČech nerve of the functor p |c . As we show in the proof of Proposition A.3 in Appendix A, any choice of preimages of the objects g ∈ G |c under p |c now induces a functor G |c −→ Aut(ker h (p) |c ) from these data. Moreover, any other choice of such preimages will induce a canonical natural isomorphism of functors. Hence we obtain a well-defined isomorphism class of functors, which we denote by [Ad, α] c ∈ π 0 2Grp G |c , Aut(ker h (p) |c ) .
This class allows us to state when a smooth 2-group extension is central.

Global description of the 2-group extension
We shall now apply the general considerations of Sections 5.2 and 5.3 to the smooth groupoid Sym G (G) constructed in Section 5.1.
Proof. First we show that Sym G (G) carries the structure of a monoid object in H. The terminal object * H ∈ H is the identity functor 1 Cart : Cart −→ Cart. We start by defining the 1-morphism u : * H −→ Sym G (G); it is the functor that assigns to every object c ∈ Cart the object (e c , pr * M 1 G ) ∈ Sym G (G), where e c : c −→ G is the constant map at the identity object e ∈ G.
Next we define the 1-morphism in the following way: consider two arbitrary objects (f 0 , A 0 ), (f 1 , A 1 ) ∈ Sym G (G) in the same fibre of π : Sym G (G) −→ Cart, i.e. f 0 , f 1 are defined over the same object c ∈ Cart. We define the map (1, Φ f 0 ) as the composition where the second and third identities use the fact that Φ is a group action. Thus we can form the 1-morphism The solid unlabelled arrows are canonical isomorphisms that stem from the fact that BGrb is a (pre)sheaf of 2-categories on the category of manifolds Mfd [Wal07b,NS11]. By a slight abuse of notation, we denote the composite morphism by (1, and analogously on 2-isomorphisms. The associator and unitors are readily obtained from those in the sheaf of 2-categories BGrb. The coherence conditions in BGrb imply that Sym G (G), endowed with the multiplication and coherence morphisms defined here, is a monoid object in H. and analogously on morphisms, where f −1 denotes the composition of the map f : c −→ G with the inversion map in the group G. It follows from the properties (5.24) of Φ ( · ) that this provides a functorial (two-sided) inverse object with respect to the 1-morphism ⊗, and hence shows that the morphism (⊗, pr 1 ) :

Now we show that Sym
is an equivalence in H (where the product is taken in H). Thus Sym G (G) −→ Cart is indeed a group object in H.
Theorem 5.27. There is a smooth 2-group extension where we abbreviate * H = Cart by 1.
Proof. The projection functor pr : It is evident from (5.25) that p : Sym G (G) −→ G is a morphism of smooth 2-groups. It is a Grothendieck fibration in groupoids by construction, and it is surjective on objects since G is connected (as we have argued at the beginning of this section).
Next we define the morphism ι : HLBdl M −→ Sym G (G) in H. Over a Cartesian space c ∈ Cart, it is simply the canonical inclusion . Here e c : c −→ G is the constant map at the unit element of G. Since the inclusion of line bundles into morphisms of bundle gerbes strictly maps the tensor product to the composition [Wal07b,Bun17], we readily find that ι is a morphism of smooth 2-groups.
To see that (5.28) is an extension of smooth 2-groups, we first show that the inclusion ι is an equivalence HLBdl M −→ ker h (p). By Lemma A.2 from Appendix A and the fact that p : Sym G (G) −→ G is a Grothendieck fibration in groupoids, it follows that the canonical inclusion ker(p) → ker h (p) is an equivalence. Consequently, it suffices to show that ι induces an equivalence HLBdl M −→ ker(p). Over an object c ∈ Cart, the fibre of ker(p) consists of the automorphism groupoid of pr * M G ∈ BGrb(c × M ). It is well-known [Wal07b] that the inclusion HLBdl(c × M ) −→ BGrb(c × M ) given by L −→ L ⊗ 1 pr * M G is an equivalence of groupoids. To see that the functor p : Sym G (G) −→ G is an HLBdl M -principal 2-bundle (see Definition 5.17), it now suffices to show that the functor is an equivalence in H, where we have written out the product in H as the fibre product over Cart. Observe that by the equivalence HLBdl M −→ ker h (p), it is enough to consider the action of HLBdl M , and since G has discrete fibres, i.e. the fibres have no non-identity morphisms, there is an identity Sym G (G) × h G Sym G (G) = Sym G (G) × G Sym G (G), and hence we can work with the strict pullback instead of the homotopy pullback.
Since both sides are fibred over Cart, it suffices to show that this functor is an equivalence on all fibres [Vis05, Proposition 3.36]. Thus we fix an object c ∈ Cart. We need to check that the functor is an equivalence. By construction, both sides decompose into coproducts and so the functor (1, α) |c decomposes into functors This functor acts as the identity on the first factor and as the standard action of line bundles on isomorphisms of bundle gerbes in the second factor. Thus (1, α) |f is an equivalence since on any manifold X, the category of 1-isomorphisms between any given bundle gerbes is a torsor category over HLBdl(X) with respect to this action [Wal07b].
Proposition 5.29. If G acts non-trivially on M , then the extension (5.28) is not central.
Proof. This follows readily from the explicit forms (5.25) and (5.26) of the product and the inverse in Sym G (G), together with the fact that composition of morphisms of bundle gerbes is compatible with tensoring by line bundles. Explicitly, given (f, A) ∈ Sym G (G) |c and L ∈ HLBdl(c × M ) we find Observe that since G has discrete fibres, we have ker h (p) = ker(p), and by the equivalence HLBdl M −→ ker(p) it is sufficient to consider the adjoint action on the smooth 2-group HLBdl M here.
Let c = * , so that the data f corresponds to an element g ∈ G. Assume that the extension (5.28) is central. Then, by Construction 5.21 and Definition 5.22, there is an isomorphism ϕ : (Ad, α) * −→ 1 HLBdl(M ) of morphisms of 2-groups G −→ Aut(HLBdl(M )). Let I ∈ HLBdl(M ) denote the trivial line bundle, and let ψ : I −→ I be any automorphism, i.e. any U(1)-valued function on M . The naturality of ϕ then implies, in particular, that the diagram g * I = I I = g * I I I g * ψ ϕ I,g ϕ I,g ψ commutes. But this is equivalent to the identity ψ = g * ψ for any g ∈ G and ψ : M −→ U(1), which is a contradiction if the G-action on M is non-trivial.
We now obtain an action of Sym G (G) on G in the following sense: let G ∈ H denote the category fibred in groupoids over Cart which is defined as follows. Consider the presheaf of groupoids on M that assigns to f : c −→ M the category BGrb(c)(I, f * G). Then q : G −→ M is obtained by applying the Grothendieck construction to this presheaf. The action of Sym G (G) on G is then the morphism of categories over G × M ∼ = G × M obtained through the diagram

Descent description of the 2-group extension
We can describe the smooth 2-group Sym G (G) in a way analogously to Section 3.3; that is, we can construct Sym G (G) in terms of descent data for the path fibration P 0 G −→ G and the parallel transport on G introduced in Section 4. However, for a bundle gerbe G this construction is more involved compared to the case of a principal bundle P . In particular, we need to replace the associated bundle construction (P 0 G × Gau(P ))/∼ of L G (cf. Section 3.3) by a homotopy-coherent version. Once established, the descent presentation of Sym G (G) allows us to study and compute this smooth 2-group very explicitly in certain situations, as we demonstrate in Section 6.
Recalling the notation of Section 3.3, let G ∈ BGrb ∇ (M ) be a bundle gerbe with connection on M . Using the smooth map (3.7) we obtain a diffeological hermitean line bundle This object is completely analogous to (3.7) when one views the holonomy of a line bundle L on M as the transgression of L to the loop space LM , and subsequently the transgression line bundle T G as the holonomy of the bundle gerbe G on M (cf. Section 4.4). In the adjoint picture, we can interpret L as a smooth assignment of a line bundle with connection L (γ,α) on M to each pair of based paths (γ, α) ∈ (P 0 G) [2] .
Consider the simplicial diffeological space (P 0 G) [•] × M with face maps for 0 ≤ i ≤ n − 1 defined by deleting the i-th entry of (P 0 G) [n] . The fusion product λ on the transgression line bundle T G over the loop space LM induces an isomorphism Definition 5.30. Let c ∈ Cart be a Cartesian space and f : c −→ G a smooth map. We define a category Des PSh L (f ) with • Objects : pairs (J, ), where J ∈ HLBdl(f * P 0 G × M ) and where  is an isomorphism of hermitean line bundles is the canonical map induced by the pullback of the subduction P 0 G −→ G along f . These data are required to satisfy the compatibility condition that is a commutative diagram in HLBdl((f * P 0 G) [3] × M ), where we use the simplicial relations.
• Morphisms : a morphism (J, ) −→ (J ,  ) is an isomorphism ψ : J −→ J such that Pullbacks of morphisms of bundle gerbes turns the assignment (f : c −→ G) −→ Des PSh L (f ) into a presheaf of groupoids on G . (This is actually even a sheaf of groupoids, but we will not need this fact here.) Applying the Grothendieck construction, we obtain a category fibred in groupoids over G , p L : Des L −→ G , and composing with the canonical projection functor pr : G −→ Cart we obtain a category fibred in groupoids over Cart which defines a smooth groupoid Des L .
Proposition 5.32. The functor π L : Des L −→ Cart is a smooth 2-group.
First, observe that it should lie in the fibre of Des L over the pointwise product map f 1 f 0 : c −→ G, u −→ f 1 (u) f 0 (u). Consider the smooth map where m : G × G −→ G is the multiplication of G, and pr 1 and pr 2 are the projections to the first and second factors of G × G. Let us denote by Φ ev 1 the composition where the first map evaluates a based path at its end point and the second map is the action Φ of G on M . The pair (f 1 , f 0 ) defines a map c −→ G × G. Let s : P ∂∆ 2 G −→ LG be the map defined in (4.11), and let (by a slight abuse of notation) LΦ : . Consider the hermitean line bundle We claim that the bundle K descends along the projection p 1 : The descended bundle is the hermitean line bundle underlying the product (f 1 , We thus endow the bundle K with an isomorphism κ : f 1 ,f 0 , which is required to satisfy a cocycle relation over Y [3] f 1 ,f 0 . An element of Y [2] f 1 ,f 0 can be identified with a pair of triples ((γ 10 , γ 1 , γ 0 ), (γ 10 , γ 1 , γ 0 )), where (γ i , γ i ) ∈ (P 0 G) [2] for i = 0, 1. We define the isomorphism κ as the composition of the fusion product λ on T G with (1 × Φ ev 1 ) *  1 ⊗  0 . Then the cocycle condition simply follows from the compatibility condition (5.31) and the associativity of the fusion product. (We also need to use thin reparameterisations, but these are implemented in a completely coherent way by the thin-homotopy invariant connection on T G.) Thus we obtain a descended hermitean line bundle Des(K, κ) on (f 1 f 0 ) * P 0 G × M (for descent properties of diffeological vector bundles, see [Bun20]). Applying the fusion product in the first tensor factor of K, we obtain an isomorphism which (by the associativity of λ) descends to an isomorphism Again by the associativity of λ and thin-homotopy invariance, the pair (Des(K, κ),  K ) satisfies the relation (5.31), and hence it makes sense to set The action of the product ⊗ in Des L on morphisms simply sends (ψ 1 , ψ 0 ) to the descent along p 1 of the isomorphism 1 T G ⊗ (1 × Φ ev 1 ) * ψ 1 ⊗ ψ 0 . The unitors of ⊗ are readily obtained from the construction, and the associator is defined from the fusion product; its coherence is yet another application of the associativity of λ and the superficiality of the parallel transport on T G. Inverses are constructed analogously to (5.26). Finally, all constructions are compatible with pullbacks along maps ϕ : c −→ c of Cartesian spaces, so that we obtain the structure of a smooth 2-group on Des L .
Theorem 5.33. There is a weakly commutative diagram of smooth 2-groups where the functor Ψ is an equivalence.
Proof. By the functoriality of G −→ Sym G (G) (see Section 5.4) we can assume that we are in the case where G = RT (G) is the regression of a transgression, so that we have direct access to our construction of a parallel transport on G from Section 4.4. We start by constructing the functor Ψ . For this, we construct a diagram in the 2-category H of the form and the functor Ψ is the composition from left to right. Each of the functors in (5.35) is an equivalence of categories fibred in groupoids over Cart, and hence so is Ψ .
For a smooth map f : c −→ G, let f : f * P 0 G × M −→ c × M denote the canonical projection. First we define the category Des(Sym G (G )). It is obtained via the Grothendieck construction applied to the presheaf Des PSh (Sym G (G )) of groupoids on G , which assigns to a smooth map The functor * simply pulls back 1-morphisms A : pr * M G −→ Φ * f G along the subductions f . This functor is an equivalence since morphisms of bundle gerbes satisfy descent 3 [Bun17, Theorem A.19]. Next we introduce some notation: we define the map Thus the pullback of the parallel transport 1-isomorphism (4.10) by the map P 0 Φ is a morphism . . , g n−2 , Φ g n−1 (x) for i = n (g 0 , g 1 , . . . , g i−1 g i , . . . , g n−1 , x) for 0 < i < n (g 1 , . . . , g n−1 , x) for i = 0 .
On G × M , the face maps d 0 = pr M and d 1 = Φ are the source and target maps of the action groupoid.
Definition 5.38. Let G be a connected Lie group, M a manifold with G-action Φ : G × M −→ M , and G a hermitean bundle gerbe over M . A G-equivariant structure on G consists of a 1-isomorphism

commutes. We denote by E(G) the groupoid of equivariant structures on G.
A splitting of p : Sym G (G) −→ G is a smooth 2-group homomorphism s : G −→ Sym G (G) such that p • s = 1 G . We assume here for simplicity and without loss of generality that unitors are strictly preserved. We denote by S(G ; Sym G (G)) the groupoid of splittings of p : Sym G (G) −→ G . Concretely, a splitting consists of and In what follows we construct an equivalence of categories. Let (A, χ) be an equivariant structure on G and (f : since Sym G (G) −→ Cart is a Grothendieck fibration. Hence the 1-isomorphisms s(c i ) glue together to a 1-isomorphism A s : pr * M G −→ Φ * G over G × M . Let pr 1 , pr 2 : G ×2 −→ G be the projections to the first and second factors, and m : G ×2 −→ G the multiplication in G. From A s we can construct three 1-morphisms pr * 1 A s , pr * 2 A s , and m * A s over G ×2 × M . We would like to show that these 1-morphisms are canonically isomorphic to the 1-morphisms constructed from the good open cover {c i × c j } i,j∈Λ by applying s to the morphisms pr 1|c i ×c j , pr 2|c i ×c j , and m |c i ×c j on c i × c j −→ G, respectively. For this, consider the commutative diagram where the 2-isomorphism is part of the data of the section s. The same argument shows the claim for pr 2 . To show the corresponding statement for m we need to pick a refinement { c a } a∈ Λ of the commutes. The cover { c a } a∈ Λ can be constructed by choosing a common refinement of the covers The multiplication of c i × c j The structure of a smooth 2-group homomorphism on s now provides natural 2-isomorphisms which glue together to a 2-isomorphism χ s : ) is a 2-isomorphism of smooth 1-isomorphisms. The coherence condition for χ s over G ×3 × M follows from the observation that the various pullbacks to G ×3 × M can be constructed by applying s to different functions from c i × c j × c k to G and the coherence condition for s. This shows that Ξ is essentially surjective.
Finally, we show that the functor Ξ is full: let (A, χ) and (A , χ ) be equivariant structures on G and ω : Ξ(A, χ) −→ Ξ(A , χ ) a morphism of splittings. Evaluating ω on the good open cover {c i } i∈Λ from above provides isomorphisms ω : A |c i ×M −→ A |c i ×M . Since ω is a morphism of splittings, these morphisms glue together to a 2-isomorphism ϑ ω : A −→ A . That this is an isomorphism of equivariant structures follows from the coherence conditions for ω and the observation that it suffices to check the conditions locally.
in BGrb(G × M ), such that for every g, g ∈ G there is an equality of diagrams Being an equivariant 1-isomorphism is a structure and not a property: given a 1-morphism E : G −→ G of bundle gerbes there is a set E(E) of equivariant structures on E. According to Theorem 5.39 we can describe the equivariant structures on G and G by splittings s : G −→ Sym G (G) and s : G −→ Sym G (G ). We shall now give a description of an equivariant structure on E using these homomorphisms of 2-groups. For this, recall from the end of Section 5.4 that any 1-isomorphism E : G −→ G in BGrb(M ) gives rise to a morphism of smooth 2-groups E : Sym G (G) −→ Sym G (G ): choose an adjoint inverse E ∨ for E and define E via Being an equivariant 2-morphism is a property.
The 2-group extension Sym G (G) −→ G can also be used to study the existence of equivariant structures on 1-morphisms. A condition for 2-isomorphisms of bundle gerbes to be equivariant is Proof. This follows from Definition 5.44 using the fact that the inverses E ∨ and E ∨ are adjoints to E and E .

Application I: Nonassociative magnetic translations
Nonassociativity in quantum mechanics has a long history dating back to foundational work on the theory in the 1930's. Its interest was revived in the 1980's with the realisation that the magnetic translation operators on the states of a charged particle moving in a magnetic monopole background generally form a nonassociative algebra [Jac85,GZ86]; see [Sza19a] for a mathematical introduction to the subject together with a survey and comparison of the various approaches to the quantisation of the pertinent twisted Poisson structures. The recent revived interest in these models has come about from their conjectural relevance to the low-energy dynamics of closed strings in non-geometric backgrounds, which are based on arguments invoking T-duality applied to target spaces that are tori or more generally total spaces of torus bundles [Lüs10,MSS12,BL14,MSS14], and other compact Lie groups [BP11]. See e.g. [Sza19b] for a contemporary introduction to the subject with further references.
As a first application of the general framework presented in this paper, we reformulate the wellknown theory of magnetic translations for source-free magnetic fields in the language of Section 3. We then use the results of Section 5 to describe nonassociative magnetic translations, which were first studied from a geometric perspective in [BMS19] on R d . They were subsequently studied from a quantum field theory perspective in [Mic19], where generalisations from R d to connected Lie groups are also considered. Here we will show that they are induced by a natural section of Sym G (G) −→ G constructed using the parallel transport of Section 4.

Magnetic translations on T d
Magnetic translations appear in the quantum mechanics of an electrically charged particle moving on a manifold M in the presence of a magnetic field, which is given by a 2-form B ∈ Ω 2 (M ). In the semi-classical Maxwell theory of electromagnetism, the 2-form B is closed and has integer periods. The first requirement H = dB = 0 is the statement that there are no magnetic monopoles. The second requirement is the Dirac charge quantisation condition which states that B is the curvature of a connection on a hermitean line bundle L over M . In Bloch theory (see e.g. [Gru00]), the line bundle L is used in geometric quantisation of the shift of the canonical symplectic structure on the cotangent bundle T * M by the 2-form B, so that the quantum Hilbert space of wavefunctions for the particle is H = L 2 (M ; L), the space of square-integrable sections of L. The (global) symmetry group G of the particle acts on M , and one would like to promote the G-action to an action on the Hilbert space by linear operators. In quantum mechanics, this action on H is only required to define a projective representation of G. If G acts via translations the resulting operators are called magnetic translations. The construction in Section 3 provides a universal mechanism to construct magnetic translations, which we will illustrate on the example of a d-dimensional torus M = T d .
Magnetic translations on T d have been studied in e.g. [Fio13,DGTS20] (for constant magnetic fields B), but our treatment is more general and also fits in with expectations from string theory.
Instead of working on T d directly, we work equivariantly on the universal cover R d by viewing T d = R d /Z d as the quotient of the natural free action τ on R d of the discrete subgroup Z d ⊂ R d by translations. The corresponding projection π : R d −→ T d is a surjective submersion. To describe line bundles on T d we consider the diagram of manifolds U(1) A smooth section of the short exact sequence t on the quantum Hilbert space H = L 2 (T d ; L f ); here we do not require this section to be a group homomorphism, and the 2-cocycle twisting this action takes values in C ∞ (T d ; U(1)). We can construct such a section from the choice of a connection on P f , which reproduces the usual expression for magnetic translations. A connection on P f can be described by a 1-form A ∈ Ω 1 (R d ) satisfying A . This condition implies that the closed 2-form dA = π * B descends to a well-defined magnetic field B on T d . The section corresponding to A is given by parallel transport: We check that this is indeed an element of Sym R d t (P f ) |v : One easily checks that P(v)ψ ∈ H for ψ ∈ H, i.e. P(v)ψ (x + i) = f i (x) P(v)ψ (x). However, they only provide a projective representation of the translation group R d t since s A is not a group homomorphism. Explicitly, using Stokes' Theorem we find that the magnetic translations satisfy the relations of the twisted group algebra and we used the relation in the simplicial complex in R d .
Remark 6.1. By dropping the (quasi-)periodicity conditions everywhere one gets back the description of magnetic translations corresponding to (necessarily trivialisable) line bundles over R d (cf. [BMS19]).

Nonassociative magnetic translations from parallel transport
Dirac's extension of the classical Maxwell theory assumes a singular magnetic field B whose 3-form curvature H = dB is distributional, with zero-dimensional support consisting of the locations of magnetic monopoles on the configuration manifold M . However, in applications to string theory the closed 3-form H corresponds to an NS-NS flux and is typically smooth, as we now assume. The framework described in Section 6.1 is not capable of encoding magnetic fields with non-vanishing magnetic charge H = dB, since in this case B can never be realised as the curvature of a line bundle. The quantisation problem now concerns an H-twisted Poisson structure on the cotangent bundle T * M [Sza19a], with twisting of the canonical Poisson structure which spoils the Jacobi identity for functions in C ∞ (T * M ; C) that vary along the vertical directions. The corresponding quantum operators do not associate; it is not possible to realise a nonassociative algebra by linear operators acting on a Hilbert space. Different approaches to describing the nonassociative quantum mechanics of charged particles moving in the background of a magnetic field with smooth monopole sources are described in [MSS14,BBBS15,KS18].
Fluxes in string theory obey a generalised version of Dirac charge quantisation (see e.g. [Sza13]); in particular, the closed 3-form H has integer periods and hence is the curvature of a connection on a hermitean bundle gerbe over M . Based on this observation, in [BMS19] we suggested the following approach: geometrically the magnetic field B can be interpreted as the curving on a trivial gerbe I B with curvature H. We proposed to use the 2-Hilbert space of sections of I B as a replacement for the usual Hilbert space of quantum mechanics. The category of sections of a gerbe G on a manifold M is the morphism category Γ(M ; G) := BGrb(I, G); for details on the 2-Hilbert space structure on this category we refer to [BSS18,BS17,Bun17]. As evidence for our approach we constructed a projective action of nonassociative magnetic translation operators on this 2-Hilbert space, which naturally encodes the relations of the H-twisted Poisson algebra on T * M . However, the drawback of this approach is that it does not work for topologically nontrivial fluxes H, or equivalently for gerbes G with non-torsion Dixmier-Douady class. Extending our geometric approach to nonassociative quantum mechanics along these lines was in fact one of our original motivations behind the present paper.
Let us first explain how the action of nonassociative magnetic translations for magnetic fields with sources on M = R d , which was described in [BMS19], can be constructed via the 2-group extensions from Section 5. Every gerbe on R d is isomorphic to a trivial gerbe I B represented by the diagram with trivial connection A = 0 and curving B ∈ Ω 2 (R d ). The connection on I B induces a section 7 7 To simplify the presentation, in the following we disregard smooth structures and work in the 2-category of 2-groups 2Grp(Grpd). The extensions to categories fibred in groupoids over Cart is straightfoward.
Combining s B with the action of Sym R d t (I) on I induces a higher projective action of R d t on Γ(R d ; I). We refer to [BMS19, Section 4] for precise definitions.
Since all line bundles over R d are isomorphic to a trivial line bundle, the category Sym R d t (I) |v at v ∈ R d t is equivalent to the category with one object and morphisms described by smooth functions R d −→ U(1). Thus However, under this equivalence the action of the magnetic translation operators becomes elusive. In [BMS19] we equipped the 1-morphisms with a connection to circumvent this problem. We did not require the 2-morphisms to preserve these connections leading to equivalent categories. The parallel transport 1-morphisms pt I B 1 can be equipped with such a connection in a canonical way. This reproduces the constructions from [BMS19].
In order to describe the parallel transport, we pull everything back to the path space P M along the two evaluation maps ev 0 , ev 1 : P M −→ M . The parallel transport defined in Section 4 is a 1-morphism given by a line bundle with connection over the space P ∂∆ 2 M . An element of P ∂∆ 2 M is a triple of paths (γ xy , γ yz , γ xz ), where γ xy is a path from the base point x to some other point y ∈ M , γ yz is a path from y to a third point z ∈ M , and γ xz is a path from x to z. Again, the line bundle for the parallel transport is trivial since we work with a trivial bundle gerbe. Its connection is given by where the notation means that the evaluation on a tangent vector V is given by We obtain a 1-morphism representing the colimit from Definition 4.17. Concretely, this is a trivial line bundle over P M with connection This is exactly the formula used for the magnetic translations in [BMS19] in the case M = R d , hence it provides a conceptual underpinning of the constructions in [BMS19], and moreover generalises them to trivial bundle gerbes on arbitrary manifolds M .

Nonassociative magnetic translations on T d
Now we generalise the description of nonassociative magnetic translations to the d-dimensional torus T d , see also [Mic19] for a discussion from a quantum field theory point of view. A problem in this context is that for topologically non-trivial gerbes on T d , there are no non-trivial sections. This makes the 2-Hilbert space of sections an uninteresting object to study. 8 However, our 2-group extension still exists and should encode the geometry of nonassociative magnetic translations in this context, regardless of whether or not sections exist. Non-trivial gerbes over T d are similarly treated as coming from Z d -equivariant (topologically trivial) gerbes over R d , as in e.g. [MW16, Section 7.1].
Bundle gerbes on T d can be described using the surjective submersion π : R d −→ T d and the corresponding diagram Here we used the identification ( [3] . Concretely, a bundle gerbe consists of a line bundle L over R d × Z d , which we can assume to be trivial without loss of generality, and an isomorphism f : π * 0,1 L ⊗ π * 1,2 L −→ π * 0,2 L of line bundles over R d × Z d × Z d satisfying a coherence condition over R d ×(Z d ) ×3 . We can describe this isomorphism by a collection of smooth maps f i,j : R d −→ U(1) for all i, j ∈ Z d , and the coherence condition translates to for all x ∈ R d and i, j, k ∈ Z d , which is the 2-cocycle condition for We denote the gerbe described by f as G f .
For v ∈ R d t , the pullback of G f along the translation τ v can be described by the 2-cocycle t up to equivalence in the following way: its objects are functions g : for all x ∈ R d and i, j ∈ Z d . It is straightforward to deduce the morphisms in Sym R d t (G f ) from [Bun17, Proposition A.31]; we find that a morphism from g to g is described by a function h : for all x ∈ R d and i ∈ Z d . Note that for the trivial 2-cocycle f i,j = 1 this describes the category of line bundles over T d with arbitrary gauge transformations as morphisms. The 2-group structure on from Theorem 5.23 takes the form fitting into the 2-group extension from Theorem 5.27: As in the case of line bundles from Section 6.1, a connection (A, B) on G f induces a section of the extension (6.2). A connection on G f is described by a 2-form B ∈ Ω 2 (R d ) and a 1-form for all x ∈ R d and i, j ∈ Z d . The second condition implies that the closed 3-form dB = π * H descends to a well-defined flux H on T d . Using the connection we can construct a section We check that this is indeed an element of Sym R d t (G f ) |v : For the multiplication we find This particular product is associative on the nose. However, the line bundle on T d described by the is non-trivial. We can use the decomposition where £ is the Lie derivative, to construct a 2-isomorphism which has the advantage that the line bundle on the right-hand side is trivial.
Remark 6.3. In this last representation the nonassociativity of the higher magnetic translations is realised by the composition property Concretely this means that there are two different ways to go from the triple product s A,B (u) ⊗ s A,B (v) ⊗ s A,B (w) to s A,B (u + v + w). Their difference is controlled by the 3-cocycle ω · , · , · on the translation group R d t with values in C ∞ (T d ; U(1)), as depicted in the commutative diagram This is the implementation of nonassociativity in the higher categorical framework.

Application II: Anomalies in quantum field theory
In this section we begin by using the group extensions Sym G (P ) from Section 3 to study the existence of equivariant trivialisations of line bundles. This has direct applications to the path integral description of the chiral anomaly in quantum field theory. Then, using the 2-group extension Sym G (G) from Section 5, we study the analogous questions for gerbes and apply our findings to the Hamiltonian description of the chiral anomaly.

Even dimensions: Chiral anomalies
Let G be a connected Lie group, M a manifold with smooth G-action Φ : G × M −→ M , and (P, χ) a G-equivariant U(1)-bundle on M . The equivariant structure on P can be described by a splitting s P : G −→ Sym G (P ). Assume furthermore that P is trivial as a line bundle, i.e. there exists a 1-isomorphism ψ : I −→ P . The trivial bundle carries a canonical equivariant structure with corresponding splitting s I : G −→ Sym G (I).
Rewriting the result of Section 3.2 slightly, we see that ψ is equivariant if and only if the smooth 1-cocycle is trivial. Every other isomorphism I −→ P differs from ψ by a uniquely determined element of C ∞ (M ; U(1)). Their corresponding 1-cocycles differ by the coboundary defined by this element. Hence the obstruction for an equivariant bundle which is trivial as a line bundle to be also trivial as an equivariant bundle is an element of the degree one group cohomology H 1 G; C ∞ (M ; U(1)) . This has also been observed in [CM95] from a different perspective.
The question of whether a bundle is equivariantly trivial is important in the path integral perspective on chiral anomalies in quantum field theory. Let M be a based even-dimensional compact Riemannian spin manifold, G a Lie group, Q a principal G-bundle on M , and ρ : G −→ End(V ) a unitary representation of G on a finite-dimensional vector space V which encodes the matter content of the field theory. Denote by S + and S − the positive and negative chirality spinor bundles on M , respectively, by Γ the group of based gauge transformations of Q and by A the affine space of connections on Q. The field content of the theory are chiral spinors, which are smooth sections of the vector bundle S + ⊗ V , where here V is the hermitean vector bundle associated to Q via the representation ρ. There is a family of (twisted) Dirac operators parameterised by gauge fields A ∈ A, which are first order elliptic differential operators acting on chiral spinors. These data together define the content of a chiral gauge theory. Being an affine space, A is contractible, so over A we can trivialise the determinant line bundle and hence identify the effective action functional Z with a complex function. However, this might not be possible over the orbit space A/Γ : if the descended line bundle is non-trivial then we cannot identify the effective action functional with a complex function in a gauge-invariant way, i.e. the gauge symmetry is anomalous. The line bundle over A/Γ is trivial if and only if we can choose a Γ -equivariant trivialisation of the line Z(A) = det(D / A ). By our general discussion above, the obstruction to this is an element of H 1 Γ ; U(1) A , where U(1) A is the diffeological space of maps from A to U(1). An explicit formula for this smooth 1-cocycle is obtained in [CM95].

Odd dimensions: The Faddeev-Mickelsson-Shatashvili anomaly
We shall now generalise the construction from the Section 7.1 to bundle gerbes. For this, we need to introduce a categorification of the first group cohomology which takes values in a smooth abelian 2-group. We use a definition along the lines of [BMS19], adjusted to the smooth setting.
Definition 7.1. Let G be a Lie group and A a smooth abelian 2-group equipped with a left action ρ of G . A smooth higher 1-cocycle on G with values in A consists of • a morphism λ : G −→ A, g −→ λ g , in H, and • a smooth natural isomorphism χ g,g : λ g ⊗ ρ g (λ g ) −→ λ g g of smooth functors G × Cart G −→ A, such that for every c ∈ Cart: • λ ec = I c where I c is the monoidal identity object of the fibre A |c of A over c ∈ Cart, and where e c : c −→ G is the constant map at the identity element of G, • χ ec, · and χ · ,ec agree with the left and right unitor morphisms in A |c , and • the diagram 1⊗ρg(χ g,g ) χ g,g g χ g,g ⊗1 χ g g ,g commutes for all g, g , g ∈ G(c).
We will also need the concept of a higher coboundary.
Definition 7.2. Let (λ, χ) and (λ , χ ) be higher 1-cocycles on a Lie group G valued in a smooth abelian 2-group A. A higher coboundary between (λ, χ) and (λ , χ ) consists of • a morphism θ : * −→ A, and such that ω ec agrees with the symmetry isomorphism β A , and the diagram commutes for all c ∈ Cart and g, g , g ∈ G(c).
Remark 7.3. There is a natural definition of morphisms between higher coboundaries, but these are not relevant for our purposes.
Let G be a connected Lie group, M a manifold with smooth G-action Φ : G × M −→ M , and (G, A, χ) a G-equivariant bundle gerbe on M . The equivariant structure on G can be described by a splitting s G : G −→ Sym G (G), as explained in Section 5.6. Assume furthermore that G is trivial as bundle gerbe, i.e. there exists a 1-isomorphism E : I −→ G. From the results in Section 5.6 we can deduce that the obstruction to the existence of an equivariant structure on E is the higher 1-cocycle Let us now explain the relation to the Hamiltonian description of chiral anomalies in terms of bundle gerbes, which was worked out in [Mic85,CM95,CM96,CMM00]. Let M be a based odd-dimensional compact Riemannian spin manifold, P a principal G-bundle on M , and ρ : G −→ End(V ) a representation of G describing the matter content of the gauge theory. Again we denote by A the affine space of connections on P and by Γ the pointed group of gauge transformations. For every A ∈ A we can construct a massless Dirac operator where S −→ M is the spinor bundle. The Dirac operator is a first order self-adjoint elliptic differential operator, which serves as the first quantised Hamiltonian acting on the one-particle Hilbert space H = Γ(M ; S ⊗ V ).
To define the fermionic Fock space F A (H) of the quantum field theory in the presence of a gauge field A ∈ A, one has to pick a polarisation H = H + (A) ⊕ H − (A). In general there are gauge fields A ∈ A for which the Dirac operator D / A has zero modes; for these fields there is no universal and natural way of choosing such a polarisation. Denote by A 0 ⊂ A × R the subset of pairs (A, r) such that the real number r is not contained in the spectrum of D / A . To equip this space with a diffeology we use the discrete diffeology on R. For every point (A, r) ∈ A 0 we get a decomposition of the one-particle Hilbert space into the positive and negative eigenstates of the operator D / A −r 1 H . The corresponding Fock bundle F(H) −→ A 0 has fibres It is shown in [CM96] that the corresponding projective Hilbert bundle descends to a bundle over A, and hence it induces a bundle gerbe G on A. Since A is contractible, over A the projective Hilbert bundle is trivial and hence is associated to a bundle of Hilbert spaces. Again the action of Γ on A lifts to an equivariant structure on G. Therefore G as well as the projective Hilbert bundle descends to the orbit space A/Γ . The Faddeev-Mickelsson-Shatashvili anomaly is the obstruction to the existence of a well-defined bundle of Hilbert spaces over A/Γ , i.e. to the existence of a trivialisation of the descended projective Hilbert bundle. Equivalently, the anomaly vanishes if and only if G descends to a trivial bundle gerbe on A/Γ . This in turn is the case if and only if G is trivial as a Γ -equivariant bundle gerbe on A.
From the general discussion above it follows that the obstruction to the equivariant triviality of G is a smooth higher 1-cocycle on Γ with values in HLBdl A . Because A is contractible, there is an equivalence with the smooth category with one object and the diffeological mapping space U(1) A as morphisms.
Since this is a smooth 2-group with one object, Definition 7.1 in this instance is equivalent to the definition of an ordinary group 2-cocycle on Γ with values in U(1) A . That the obstruction to the vanishing of the anomaly is a 2-cocycle of this form is well-known, see e.g. [CM96]; this cocycle reproduces the usual Schwinger terms in the commutator anomaly for the gauge group action. What is new here is the interpretation as a smooth higher 1-cocycle, which only reduces to an ordinary 2-cocycle because the space A is contractible, as well as a rigorous incorporation of the smooth structures.

Application III: The string group
Any compact simple Lie group G has homotopy groups π 3 (G) ∼ = H 3 (G; Z) ∼ = Z and π i (G) ∼ = H i (G; Z) ∼ = 0 for i = 0, 1, 2; that is, G is 2-connected. It is of interest in topology and geometry (see e.g. [DHH11,Sto96,ST04]), as well as in string theory (see e.g. [SS20]), to study 3-connected approximations to G that arise as group extensions of G. We denote such approximating objects by String(G) and call them models for the string group of G. There is a variety of interpretations of what this means, based on different choices of ambient higher categories in which one considers G to be a group object. The general theme, however, is that one needs a way to realise a generator of π 3 (G) ∼ = Z geometrically in the chosen framework, and a string group model for G will generally be a choice of such a generator.
In this final section we recall the definition and construction of a topological string group model, and show that our extensions Sym G (P ) from Section 3 provide a smooth enhancement thereof. We then propose the smooth 2-groups Sym G (G) and Des L from Section 5 as new string group models, for the specific choices of M = G and of a gerbe G on G whose Dixmier-Douady class generates H 3 (G; Z) ∼ = Z. A model for String(G) which is very similar in spirit to our model Sym G (G) was found in [FRS16]. However, that construction relies on the choice of connection on G, which may seem restrictive given that models for String(G) can be constructed by purely topological means. We defer further details and comments to Section 8.2.

A smooth string group model
The simplest and original framework for considering string group models is that of topological spaces.
Definition 8.1. Let G be a compact simple simply-connected Lie group. A topological model for the string group String(G) of G is a topological 3-connected group String t (G) along with a fibration String t (G) −→ G whose typical fibre is an Eilenberg-MacLane space K(Z; 2).
Using homotopy and cohomology groups one shows that String t (G) cannot be a finite-dimensional Lie group [NSW13]. If a topological string group model can be enhanced to consist of smooth spaces (such as Fréchet manifolds or diffeological spaces), we denote it by String(G) and refer to it as a smooth model for the string group of G.
We recall Stolz' model as a topological group [Sto96]: let PU denote the projective unitary group of an infinite-dimensional separable Hilbert space. As a consequence of Kuiper's Theorem, PU has homotopy type K(Z; 2). Hence the classifying space BPU has homotopy type K(Z; 3), while at the same time it classifies topological principal PU-bundles. In particular, isomorphism classes of PU-bundles on a space X are in one-to-one correspondence with elements of the set H 3 (X; Z).
Let P −→ G be a principal PU-bundle on G such that P corresponds to a generator of H 3 (G; Z) ∼ = Z; such PU-bundles on G are called basic. Let G denote the group of PU-equivariant homeomorphisms of P to itself which act on G as left multiplication by some element of G. We can topologise G as a subgroup of the topological group of homeomorphisms P −→ P . Thus G comes with a continuous surjective group homomorphism G −→ G. The gauge group Gau(P ) is the subgroup of G of those elements whose projection to G is the identity element e ∈ G. The crux of the proof of this theorem is showing that Gau(P ) is homotopy equivalent to PU, i.e. that it is an Eilenberg-MacLane space K(Z; 2). Part of the content in [NSW13] is to enhance this topological string group model to a smooth model in the sense that the groups appearing are Fréchet Lie groups.
The group extension G −→ G agrees with the extension Sym G (P ) −→ G constructed in Section 3 when we set M = G and H = PU, and let G act on itself by left multiplication. Thus we immediately obtain Corollary 8.3. Let P −→ G be a basic PU-bundle. The extension of diffeological groups exhibits Sym G (P ) as a smooth model for String(G).

Smooth string 2-group models
Let G be a compact simply-connected Lie group, and let G be a bundle gerbe on G whose Dixmier-Douady class generates H 3 (G; Z); such a bundle gerbe is said to be basic. Let Φ : G × G −→ G denote the left action of G on itself by left multiplication. In the spirit of Section 5, it is reasonable to expect that we should also be able to interpret the smooth 2-groups Sym G (G) and Des L as models for String(G). The idea of constructing String(G) as a smooth 2-group has also been considered in e.g. [BCSS07,SP11,Wal12a,NSW13,FRS16]. In the remainder of this section we will describe how Sym G (G) can be seen as a string 2-group model. By Theorem 5.33 it then follows that Des L is also a model for String(G).
Smooth string 2-group models usually consist of extensions of G by the smooth 2-group BU(1), the delooping of the smooth abelian group U(1). However, recall that in Theorem 5.27 we established Sym G (G) as an extension of G by the smooth abelian 2-group HLBdl G . Our point here is that what matters for string group models is only the homotopy type of the fibre and the total space of the map String(G) −→ G, so that there is a lot of flexibility in choosing the smooth 2-group A that extends G. Observe that this ambiguity is inherent already in Definition 8.1. This forces us to state which smooth 2-groups A are admissible in order to obtain smooth 2-group extensions of G that deserve to be called string group models. Our proposed definition for smooth string 2-group models emphasises the structure of A as a smooth analogue of an Eilenberg-MacLane space K(Z; 2). Note that for every smooth abelian 2-group A and any manifold M , we can defineČech cohomology of M with coefficients in A by evaluating (a delooping of A) on theČech nerve of good open coverings of M .
The definition of a smooth string 2-group model is thus a two-step process: we first fix the homotopy type of the extending 2-group A in a geometric way, and then we have to make precise when an A-extension of G has the correct homotopy type.
Definition 8.4. Let H be a diffeological group. The delooping BH ∈ H is the category fibred in groupoids over Cart whose objects are the Cartesian spaces c ∈ Cart, and whose morphisms If H is abelian, then BH naturally becomes a smooth abelian 2-group.
Definition 8.5. A smooth 2-group A is string-admissible if it is abelian and equivalent (as a smooth 2-group) to the delooping BH of a diffeological abelian group H whose underlying topological space is an Eilenberg-MacLane space K(Z; 2).
From the equivalence A ∼ = BH it follows thatČech cohomology with coefficients in A is equivalent toČech cohomology with coefficients in H, shifted by one degree. Then since H has homotopy type K(Z; 2), it follows that there are isomorphismš From any smooth principal 2-bundle P −→ M over a manifold M with structure 2-group A, we can distil aČech cohomology class as follows: let U = {U i } i∈I be a good open covering of M . Denote intersections by U i 1 ···in := U i 1 ∩ · · · ∩ U in . Viewing the (intersections of) open patches U i 1 ···in → M as objects in M , we denote by P |U i 1 ···in the fibres of P over these objects. By Definition 5.17, we can choose an object ψ i ∈ P |U i for every i ∈ I. We can further choose an object a ij ∈ A |U ij for every i, j ∈ I and an isomorphism g ij : ψ i|U ij ⊗ a ij −→ ψ j|U ij in P |U ij (where we have chosen pullbacks of ψ i and ψ j to P |U ij ). Over the triple overlaps U ijk we obtain isomorphisms which are uniquely determined by the properties of the Grothendieck fibration P −→ M (as previously, since M has discrete fibres, it follows that P × h M P = P × M P). Since the morphisms (1 ψ i|U ijk , β ijk ) lie in the image of the action functor P×A −→ P× M P, there are unique isomorphisms in A |U ijk , which satisfy the required coherence condition over quadruple overlaps by the fact that they are constructed as Cartesian lifts of identity morphisms. Hence these data assemble into an A-valuedČech 1-cocycle on M with respect to the open covering U. One can check that other choices of coverings and sections lead to 1-cocycles that become equivalent to (a ij , α ijk ) when passing to a common refinement of good open coverings.
Definition 8.6. Let G be a compact simply-connected Lie group, and let A be a string-admissible smooth 2-group. A smooth 2-group model for String(G) is a smooth 2-group extension such that the principal 2-bundle String(G) −→ G represents a generator of H 3 (G; Z) ∼ = Z under the isomorphismȞ 1 (G; A) ∼ = H 3 (G; Z).

With these definitions we have
Theorem 8.7. For any 2-connected manifold M , the smooth 2-group HLBdl M is string-admissible.
Theorem 8.8. Let G be a compact simply-connected Lie group, and let G ∈ BGrb(G) be a basic bundle gerbe. Let Sym G (G) and Des L be the smooth 2-group extensions of G by HLBdl G constructed from G with respect to the left action of G on itself by left multiplication. Then both Sym G (G) and Des L are smooth 2-group models for String(G) in the sense of Definition 8.6.
The rest of this section is devoted to the proofs of Theorems 8.7 and 8.8. We begin with a few results that will combine to prove that HLBdl M is string-admissible. Then we will prove Theorem 8.8 by observing that the HLBdl G -valuedČech 1-cocycles we obtain from the 2-bundle Sym G (G) agree with those obtained from local trivialisations of the bundle gerbe G. Proof. We readily see that the inclusion respects the group structures, and that it is fully faithful. If H 2 (M ; Z) = 0, then HLBdl M |c ∼ = HLBdl(c × M ) is connected, so that in this case the inclusion is also fully faithful on all fibres. Thus it is an equivalence on every fibre and hence an equivalence in the 2-category H by [Vis05, Proposition 3.36].
Combining Lemmas 8.9 and 8.10, we conclude that HLBdl M is string-admissible for any 2connected manifold M ; that is, we have proven Theorem 8.7.
For the diffeological group H = U(1) M and a 2-connected manifold M , there is an explicit isomorphismȞ k (X; U(1) M ) ∼ =Ȟ ( X; U(1)) for k > 0, for any manifold X, given by Proposition 8.11. Let M be a 2-connected manifold with a fixed base point x ∈ M . For any manifold X, evaluation at x ∈ M induces an isomorphismȞ k (X; U(1) M ) ∼ =Ȟ k (X; U(1)) for k > 0 ofČech cohomology groups with coefficients in the sheaves of smooth U(1) M -valued and U(1)-valued functions, respectively.
Proof. Consider the sequence of diffeological groups which is exact by the argument from the proof of Lemma 8.9. The sheaf R M admits a partition of unity by picking a partition of unity for the sheaf of smooth R-valued functions and a constant extension to R M -valued functions; henceȞ k (X; R M ) = 0 for any manifold X and for any k ≥ 1. Now the statement follows from applying the five lemma to the diagram H k+1 (X; R M )Ȟ k+1 (X; Z)Ȟ k (X; U(1) M )Ȟ k (X; R M )Ȟ k (X; Z) H k+1 (X; R)Ȟ k+1 (X; Z)Ȟ k (X; U(1))Ȟ k (X; R)Ȟ k (X; Z) induced by the long exact sequence in sheaf cohomology and the evaluation at x ∈ M .
It remains to determine theČech cohomology class inȞ 1 (G; HLBdl G ) ∼ = H 3 (G; Z) ∼ = Z determined by the extension Sym G (G) −→ G . The isomorphisms from Lemma 8.10 and Proposition 8.11 are useful in achieving this. From the smooth 2-group extension Sym G (G) −→ G we can extract ǎ Cech 2-cocycle on G with values in the sheaf of smooth U(1) G -valued functions. To construct it, we first follow the procedure of the paragraph preceding Definition 8.6 to extract HLBdl G -valued cocycle data and then choose local trivialisations of the line bundles which comprise it (which amounts to choosing an inverse for the equivalence from Lemma 8.10).
Let U = {U i } i∈I be a good open cover of G, let π i : U i × G −→ G denote the projection onto the second factor, and let m i : U i × G −→ G be the multiplication map restricted to U i × G. We choose and fix 1-isomorphisms ψ i : m * i G −→ π * i G along with adjoint inverses ψ −1 i , which induce equivalences Sym G (G) |U i ∼ = HLBdl(U i × G) of groupoids.
On double intersections U ij we can form the automorphism ψ ij := ψ −1 j|U ij • ψ i|U ij of m * ij G. The isomorphism ψ ij can be identified with a line bundle L ij on U ij × G. Since H 2 (G; Z) = 0, we can choose and fix a trivialisation of L ij for each i, j ∈ I.
On triple intersections U ijk we get a 2-isomorphism ψ ijk : ψ jk|U ijk • ψ ij|U ijk −→ ψ ik|U ijk inducing an isomorphism L jk ⊗ L ij −→ L ik of line bundles over U ijk × G. (In contrast to the construction in the paragraph above Definition 8.6, here the isomorphisms ψ ijk can be obtained directly from the choice of ψ ij .) Using the trivialisations of these line bundles we obtain a smooth map U ijk × G −→ U(1) or equivalently a map c ijk : U ijk −→ U(1) G . The collection c ijk form a smooth U(1) G -valueď Cech 2-cocycle.
The corresponding cohomology class is independent of all choices involved: let ψ i : m * i G −→ π * i G be a different set of 1-isomorphisms. The automorphism ψ −1 i • ψ i of m * i G can be identified with a line bundle Λ i over U i × G. The definition of L ij implies L ij ⊗ Λ i ∼ = Λ j ⊗ L ij . We can pick once and for all trivialisations of all bundles involved to identify this morphism with a function A ij : U ij × G −→ U(1). The diagram commutes over U ijk , which follows from the fact that all inverses were chosen to be adjoint so that the corresponding diagram involving ψ i and ψ i commutes. Applying the trivialisations we get This argument also shows that the cocycles define the same cohomology class if ψ i = ψ i and only the trivialisations of L ij differ.
The image of c ijk inȞ 2 (G; U(1) G ) under the isomorphismȞ 2 (G; U(1) G ) ∼ =Ȟ 2 (G; U(1)) of Proposition 8.11 can be computed by restricting each ψ i to U i × {e} ⊂ U i × G; the restriction ψ |U i ×{e} is a 1-isomorphism G |U i −→ G| e . After fixing once and for all a trivialisation of G |e , this is just a trivialisation of G |U i . This shows that the image of c ijk inȞ 2 (G; U(1)) ∼ = H 3 (G; Z) agrees with the cocycle c G classifying the bundle gerbe G, which proves Theorem 8.8.
Remark 8.12. The arguments involving cocycles can be adjusted to the simpler case of principal bundles over the Lie group G. In that case, starting from a principal U(1)-bundle P on G we get a principal U(1) G -bundle Sym G (P ) on G which is homotopy equivalent to P . The homotopy equivalence is induced by the maps ev x and c from Lemma 8.9. We can iterate the procedure to get larger and larger groups Sym G (· · · Sym G (Sym G (P )) · · · ). However, these groups are all topologically equivalent, so that iterating the procedure does not produce anything that is topologically novel.
A Properties of smooth principal 2-bundles A.1 Surjectivity on objects and homotopy pullbacks Here we provide some technical background on smooth groupoids, as introduced in Definition 5.4. Lemma A.1. Let π : X −→ Cart and π : P −→ Cart be objects in H.
(2) Let p : P −→ X be a morphism in H whose underlying functor is an essentially surjective Grothendieck fibration. Then p is surjective on objects.
Proof. To see (1), observe that Cart has a terminal object * ∈ Cart. Thus since π is a Grothendieck fibration, if X | * = π −1 ( * ) is non-empty then so is X |c for any c ∈ Cart. For any c ∈ Cart there exists a morphism x : * −→ c in Cart given by choosing any point x ∈ c. It follows that as soon as X = ∅, it has only non-empty fibres over Cart.
Claim (2) follows from the general observation that a Grothendieck fibration is essentially surjective if and only if it is surjective on objects.
We now consider the setup of Definition 5.16.
Lemma A.2. Let C be a category, let π i : D i −→ C, for i = 0, 1, and π E : E −→ C be Grothendieck fibrations in groupoids, and let F i : D i −→ E, for i = 0, 1, be morphisms of categories fibred in groupoids over C.
(2) Any morphism G = (G 0 , G E , G 1 ) of diagrams in H, where all vertical morphisms are equivalences, induces an equivalence D 0 × h E D 1 −→ D 0 × h E D 1 . (3) If F 1 (resp. F 0 ) is a Grothendieck fibration in groupoids, then the inclusion D 0 × E D 1 → D 0 × h E D 1 is an equivalence, and the projections pr h 0 : D 0 × h E D 1 −→ D 0 and pr 0 : D 0 × E D 1 −→ D 0 (resp. the projections pr h 1 and pr 1 to D 1 ) are Grothendieck fibrations in groupoids.
Next we need to show that for any morphism f : c −→ c in C and any object (d 0 , η , d 1 ) in D 0 × h E D 1 over c , there exists a lift f = (f 0 , f 1 ) of f to D 0 × h E D 1 with codomain (d 0 , η , d 1 ). Such a lift is obtained by lifting f to morphisms f i : d i −→ d i in D i using the fact that π i is a Grothendieck fibration in groupoids, for i = 0, 1. An isomorphism η : F 0 (d 0 ) −→ F 1 (d 1 ) compatible with f 0 , f 1 is obtained by filling the horn given by the morphisms η • F 0 (f 0 ) and F 1 (f 1 ) over the identity morphism 1 c in C. The filler is an isomorphism since the fibre E |c is a groupoid.
To prove (2), we note that by [Vis05,Proposition 3.36] the induced morphism G 0 × h G E G 1 is an equivalence in H if and only if it restricts to an equivalence of groupoids between all fibres of π h and π h . A direct inspection on any c ∈ C reveals that π −1 h (c) = π −1 0 (c) × h π −1 E (c) π −1 1 (c) as groupoids, and it is well-known that equivalences of spans of groupoids induce equivalences on homotopy pullbacks of groupoids.
Finally, consider the inclusion functor D 0 × E D 1 → D 0 × h E D 1 . Since pr h 0 is a Grothendieck fibration in groupoids, so is its restriction to each fibre over C. It is well-known that the inclusion of a pullback of groupoids into the homotopy pullback is an equivalence in case one of the functors in the diagram is a Grothendieck fibration. Thus our inclusion functor is an equivalence on each fibre over C, whence the result follows by [Vis05, Proposition 3.36].

A.2 Relation to principal ∞-bundles
Our notion of smooth principal 2-bundle does not have any notion of 'local triviality' built into it. This differs from the version of a principal 2-bundle defined in [SP11], but is very much in the spirit of the definition of a principal ∞-bundle from [NSS15]. The fact that we require essential surjectivity is our version of saying that the (homotopy) fibres of the bundle should be non-empty. In contrast to [NSS15] we have to require fibration properties because we do not work purely in an ∞-categorical framework. We shall now show that an H-principal 2-bundle in H in the sense of Definition 5.17 gives rise to a principal 2-bundle in the sense of [NSS15, Definition 3.4], adapted from a general ∞-topos (described e.g. by presheaves of ∞-groupoids) to our situation involving presheaves of groupoids. Let p : P −→ X be a morphism in H, and let P [•] be theČech nerve of p. We write hocolim C (resp. holim C ) for a homotopy colimit (resp. limit) taken in a simplicial model category C. Proposition A.3. Every morphism p : P −→ X in H whose underlying functor is an essentially surjective Grothendieck fibration in groupoids gives rise to an effective epimorphism: the morphism hocolim ∆ op H P [•] −→ X from itsČech nerve to X is an equivalence.
Because of Lemma A.2 and the assumption that p is a Grothendieck fibration in groupoids, it does not matter here if one uses the coherentČech nerve, formed using P × h X · · · × h X P, or the stricť Cech nerve, formed using P × X · · · × X P.
Proof. We work with Hollander's model structure on H [Hol08]. In this picture, H is a model category enriched, tensored and cotensored in the model category Grpd (seen as a strict category). In both H and Grpd all objects are fibrant, and the functor H : H op × H −→ Grpd is homotopical by [Vis05, Proposition 3.35], i.e. it preserves weak equivalences in each argument. The enrichment of H in Grpd even enhances to an enrichment over Set ∆ , the category of simplicial sets with the Kan-Quillen model structure. Thus homotopy (co)limits in H can be computed using (co)bar constructions [Rie14]. Let Q denote a cofibrant replacement functor in H, and let Z ∈ H be an arbitrary object. Then where the first equivalence stems from the fact that Z is fibrant and H is a Grpd-enriched model category, and the second equivalence stems from the fact that H is homotopical. It thus suffices to prove that the functor is an equivalence of groupoids.
An object in Des p (Z) is a pair (G, η) of a functor G : P −→ Z of categories fibred in groupoids over Cart, together with a natural isomorphism η |(p 0 ,p 1 ) : G(p 0 ) −→ G(p 1 ) from d * 1 G to d * 0 G of functors over Cart, where d i are the face maps in the simplicial object P [•] . This natural isomorphism is subject to the conditions d * 2 η • d * 0 η = d * 1 η over P [3] and ∆ * η = 1 G over P, where ∆ : P −→ P [2] is the diagonal map. A morphism (G, η) −→ (G , η ) in Des p (Z) is a natural isomorphism γ : G −→ G in H such that η • d * 1 γ = d * 0 γ • η. We first show that p * is essentially surjective: let (G, η) ∈ Des p (Z) be any object. We define a functor F : X −→ Z as follows: first, recalling that p is surjective on objects by Lemma A.1, we choose a section s : ob(X) −→ ob(P) of the map of objects defined by p. Then we set F (x) := G(s(x)) ∈ Z for x ∈ X. Now consider a morphism ψ : x −→ y in X. Since p is a Grothendieck fibration, ψ has a lift ψ : x −→ s(y) to a morphism in P with codomain s(y), where p( x) = x. Define F (ψ) : F (x) −→ F (y) via the diagram The naturality of η, together with the two conditions it satisfies and the fact that p is a Grothendieck fibration in groupoids, imply that F is a well-defined functor. Furthermore, η establishes an isomorphism p * F = (F, 1 F ) −→ (G, η) in Des p (Z). Thus p * is essentially surjective.
That p * is fully faithful follows from its explicit construction and the fact that p is essentially surjective.