Transitive Double Lie Algebroids Via Core Diagrams

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6 Integration of transitive double Lie algebroids 44

A The Lie bialgebroid condition for R and D˜ 46

B Double Lie algebroid morphisms – proofs 48 B.1 ProofofTheorem4.10 ...... 48 B.2 ProofofTheorem5.23 ...... 49

C Notation for linear and core sections. 50

1 Introduction

Double structures in geometry where first studied by the school of Ehresmann and later extensively by the second author, see [19]. In particular, [16, 18] define double Lie algebroids and prove that they are the infinitesimal objects associated to double Lie groupoids. Differentiating one side of a double Lie groupoid yields an LA-groupoid [16], and a second differentiation process applied to this LA-groupoid yields the tangent double Lie algebroid of the double Lie groupoid [18]. The converse integration of double Lie algebroids to double Lie groupoids has not been completely solved yet. Stefanini proposes in [25] an integration of LA-groupoids with integrable top Lie algebroid and strongly transitive source and target maps. Burzstyn, Cabrera and del Hoyo integrate in [4] double Lie algebroids with one integrable (top) side to LA-groupoids. This paper integrates transitive double Lie algebroids in one step to transitive double Lie groupoids, provided the side and core Lie algebroids are all integrable. This integration is the motivation behind the results in this work. It is indeed very simple using the equivalence of transitive double Lie groupoids with their core diagrams in [3], and the equivalence of transitive double Lie algebroids and their core diagrams proved here. A double groupoid is a groupoid object in the category of groupoids. That is, a double groupoid consists of a set Γ that has two groupoid structures over two bases G and H, which are themselves groupoids over a base M, such that the structure maps of each groupoid structure on S are morphisms with respect to the other. Γ G

HM The groupoids G and H are the side groupoids and the set M is the double base. The core K of a double groupoid pΓ,G,H,Mq is the set of elements which project under the two sources of Γ to units of G and H. It inherits from the double structure of Γ a groupoid structure over M, and the two targets of Γ induce two groupoid morphisms tG|K : K Ñ G and tH |K : K Ñ H over the identity on M. Elements of the kernel of the morphism K Ñ G commutes with elements of the kernel of K Ñ H. These two groupoid morphisms build together the core diagram of the double groupoid [3]. t | KG K G

tH |K HM

2 A double Lie groupoid is a double groupoid pΓ,G,H,Mq such that all four groupoids Γ Ñ G, Γ Ñ H, G Ñ M and H Ñ M are Lie groupoids and such that the double source map psG, sH q: Γ Ñ G ˆs H “ tpg,hq P G ˆ H | spgq “ sphqu is a smooth surjective submersion. In that case, the core diagram is a core diagram of Lie groupoids. A double Lie groupoid is locally trivial if the Lie groupoid K Ñ M is locally trivial and BG :“ tG|K and BH :“ tH |K are both surjective submersions. It is transitive if only BG and BH are both surjective submersions. Brown and Mackenzie proved in [3] that locally trivial double Lie groupoids are completely determined by their core diagrams. Section 1.2 reviews their construction and shows that K Ñ M does not need to be locally trivial for the equivalence to work. Hence, transitive double Lie groupoids are completely determined by their core diagrams. Mackenzie proved then in [16] that locally trivial LA-groupoids are completely determined by their core diagrams. This paper completes this series of results by proving that transitive double Lie algebroids are completely deﬁned by their core diagrams. Double Lie groupoids are described inﬁnitesimally by double Lie algebroids [16, 18]. A double Lie algebroid with core C is a double vector bundle pD; A, B; Mq

π D A / A

πB qA B / M qB with core C and four Lie algebroid structures on A Ñ M, B Ñ M, D Ñ A and D Ñ B such that B ˚ pπB, qAq and pπA, qBq are Lie algebroid morphisms, and the induced Lie algebroids D A Ñ C B ˚ and D B Ñ C form a VB-Lie bialgebroid [18]. The anchor ΘA : D Ñ T A is a morphism of double vector bundles and its core morphism is denoted by BA : C Ñ A. Likewise the core morphism of the linear anchor ΘB : D Ñ TB is written BB : C Ñ B. The core C of the double Lie algebroid inherits a Lie algebroid structure over M, such that the two core-anchors BA and BB are Lie algebroid morphisms over M [20, 7]. The compatiblity of the Lie algebroid structures on D Ñ A and D Ñ B implies that rc1,c2s “ 0 for all c1 P Γpker BAq and c2 P Γpker BBq. In other words, ker BA and ker BB commute in C. Hence, the core of a double Lie algebroid deﬁnes a diagram of Lie algebroids morphisms as in Figure 1, with commuting kernels. It is called the core diagram of the double Lie algebroid.

C / B BB

BA A

Figure 1. The core diagram of a double Lie algebroid.

If both BA and BB are surjective, then the double Lie algebroid and its core diagram are both called transitive. It is easy to see that the map sending a double Lie algebroid to its core diagram deﬁnes a functor from the category of double Lie algebroids with double base M, to the category of core diagrams over M. It turns out that the restriction of this functor to transitive double Lie algebroids versus transitive core diagrams has an inverse, which is constructed in this paper. In particular, it proves a more precise version of the following theorem (see Theorem 5.22). Theorem 1.1. A transitive double Lie algebroid pD,A,B,Mq is uniquely determined (up to isomorphism) by its core diagram. This work relies heavily on Gracia-Saz and Mehta’s equivalence of 2-representations with decomposed VB-algebroids [8] and shows how powerful the tools developped in [8, 6, 7] are in the

3 study of VB-algebroids and double Lie algebroids. Along the way, the authors define the comma- double Lie algebroid defined by a Lie algebroid morphism, which is an interesting structure in its own right. The dg-Lie algebroid defined [27] by the comma double Lie algebroid associated to a Lie pair A ãÑ L is used by Stiénon, Vitagliano and Xu [26] in their extension to arbitrary Lie pairs of the Kontsevich-Duflo type theorem for matched pairs in [14]. The following two sections explain comma double Lie groupoids, as well as the generalised version of Brown’s and Mackenzie’s equivalence of locally trivial double Lie algebroids with locally trivial core diagrams [3]. The construction steps of the inverse functor in that setting are basically the same as for the inverse functor in this paper. The equivalence of the category of transitive double Lie algebroids with the one of transitive core diagrams, and the equivalence of categories of locally trivial double Lie groupoids with the one of locally trivial core diagrams of Lie groupoids established in [3] yield finally a very simple integration method for transitive double Lie algebroids. Recall however that the notion of transitivity discussed here corresponds to a weaker notion of local triviality than the one used in [3], see Section 1.2. Section 6 explains this integration of transitive double Lie algebroids to transitive double Lie groupoids, which is now a straightforward exercise using many of Mackenzie’s prior results with coauthors [3, 16, 21]. It proves the following theorem (see Section 6 and Theorem 6.1 for the proof and more details).

Theorem 1.2. Let pD,A,B,Mq be a transitive double Lie algebroid with a transitive core diagram A Ð[ C ÞÑ B where A, B and C are all integrable Lie algebroids over M. Then pD,A,B,Mq integrates to a transitive double Lie groupoid, the core diagram of which integrates A Ð[ C ÞÑ B.

1.1 Comma double Lie groupoid and comma double Lie algebroid Given categories and functors D ÝÑF C ÐÝG E, the comma category pF, Gq has as objects the triples pd, e, ϕq with d P ObjpDq, e P ObjpEq and 1 1 1 1 ϕ P MorCpF pdq, Gpeqq, see [15]. Its arrows pd, e, ϕq Ñ pd ,e , ϕ q are the pairs ph,lqP MorDpd, d qˆ 1 MorE pe,e q such that F phq F pdq / F pd1q

ϕ ϕ1 Gpeq / Gpe1q Gplq commutes. If defined, the composition ph1,l1q ˝ ph,lq is ph1h,l1lq. If K Ñ M and G Ñ M are two (Lie) groupoids over a common base M and Φ: K Ñ G is a (Lie) groupoid morphism fixing the base, then the diagram K ÝÑΦ G ÐÝΦ K, defines as above the comma category pΦ, Φq, which has a double (Lie) groupoid structure with sides K and G and with core K. Denote the space of arrows of pΦ, Φq by Γ. The objects of pΦ, Φq are hence triples pm1,g,m2q P M ˆ G ˆ M with spgq “ m1 and tpgq “ m2. Hence, the objects of pΦ, Φq are just the arrows of G Ñ M, and the elements of Γ (that is, the arrows of pΦ, Φq) are

4 written as triples pk2,g,k1qP K ˆ G ˆ K such that spk2q“ tpgq and spk1q“ spgq.

Φpk2q n o m O2 O 2

´1 g Φpk2q¨g¨Φpk1 q pk2,g,k1q

n1 o m1 Φpk1q This diagram is pictured in the usual manner for elements of a double groupoid [3] and will just be written k2 n2 m2

´1 Φpk2q¨g¨Φpk1 q g

n1 m1 k1 in the following, and the left vertical arrow will not be labeled since it is clear from the rest of the diagram. The source and target maps Γ Ñ G are the maps pk2,g,k1q ÞÑ g and pk2,g,k1q ÞÑ ´1 Φpk2q ¨ g ¨ Φpk1 q, respectively. The source and target maps Γ Ñ K are the maps pk2,g,k1q ÞÑ k1 and pk2,g,k1q ÞÑ k2, respectively. The composition over G is 1 ´1 1 1 1 pk2, Φpk2q ¨ g ¨ Φpk1 q, k1q ¨G pk2,g,k1q “ pk2k2,g,k1k1q and the composition over K is 1 1 pk3,g , k2q ¨K pk2,g,k1q “ pk3,g g, k1q. 1 ´1 It is easy to check that given a composable square as below, hence with g “ Φpl3qg Φpl2 q and 1 ´1 h “ Φpl2qh Φpl1 q, the order of the horizontal and vertical multiplications is not relevant, as pictured below. Hence, pΦ, Φq is a double groupoid.

k3l3 ‚‚‚‚‚k3 l3 g g1 g1 k2l2 ‚‚‚‚‚k2 l2

h h1 h1

‚‚‚‚‚k1 l1 k1l1

k3l3 ‚‚‚‚‚k3 l3 gh g1h1 g1h1

‚‚‚‚‚k1 k3 k1l1 If Φ: K Ñ G was a morphism of Lie groupoids, it is not diﬃcult to check that pΦ, Φq is a double Lie groupoid [3]. The core of pΓ,K,G,Mq consists of elements pk, 1m, 1mq

nk m

Φpkq 1m m m 1m

5 with multiplication given by ﬁlling the top right and bottom left squares in the diagram below and multiplying all obtained squares together,

1 tpkq k mk1 m

1 1 Φpkq 1m m

1 1 m1m mk m

1 1 Φpk q Φpk q 1m1

1 1 1 m 1m1 m 1m1 m hence yielding 1 1 pk, 1m, 1mq ¨ pk , 1m1 , 1m1 q “ pkk , 1m1 , 1m1 q. As a consequence, the core (Lie) groupoid is isomorphic to K Ñ M as a (Lie) groupoid. Note that Γ Ñ G is the action groupoid of the (Lie) groupoid action of KˆK Ñ MˆM on pt,sq: G Ñ MˆM, ´1 pk2, k1q ¨ g “ Φpk2qgΦpk1 q. Section 4 constructs analogously a double Lie algebroid from a Lie algebroid morphism

C B / A ❆❆ ⑥ ❆❆ ⑥ ❆❆ ⑥⑥ ❆❆ ⑥⑥ ~⑥ M

The morphism B induces an action of the Lie algebroid T C Ñ TM on the anchor ρA : A Ñ TM of A. The total space R :“ T C ‘T M A of the action Lie algebroid T C ‘T M A Ñ A of this action carries as well automatically the pullback Lie algebroid structure from A Ñ M under the smooth map qC : C Ñ M. It is hence also a Lie algebroid over C, and these two Lie algebroid structures deﬁne a double Lie algebroid T C ‘T M A / A

C / M with sides A and C and with core C. This double Lie algebroid is the comma double Lie algebroid deﬁned by the morphism B : C Ñ A. The details of this construction are given in Section 4. The core diagram of this comma double Lie algebroid is pictured in the following diagram. C / C id B A

1.2 Transitive double Lie groupoids and their core diagrams A double Lie groupoid is called here transitive is the Lie groupoid morphisms in its core diagram are surjective submersions, hence ﬁbrations of Lie groupoids. The core diagram is then also called transitive. In particular, locally trivial double Lie groupoids in the sense of [3] are transitive double Lie groupoids. A careful study of the proof of the main result in [3], establishing an equivalence between locally trivial core diagrams and locally trivial double Lie groupoids, reveal that this equivalence is true in the more general setting of transitive double Lie groupoids versus transitive core diagrams. For completeness, and because the construction of the equivalence in [3] serves as a model for the construction of the equivalence of the categories of transitive double Lie algebroids with transitive core diagrams, it is sketched in this section.

6 KBG G

BH HM

Figure 2. Core diagram of Lie groupoids

Consider a transitive core diagram of Lie groupoids; i.e. a diagram as in Figure 2 of Lie groupoid morphisms, such that BG and BH are surjective submersions and the subgroupoids kerpBGq and kerpBH q commute in K. Build the comma double Lie groupoid pBG, BGq

Γ G

G with core K. Next, G Ñ M acts on K :“ kerpBH qĎ K via

G G ´1 ρ: G ˆs,t K Ñ K , ρpgqpκq“ kκk for any k P K such that BGpkq“ g. Consider the closed, embedded wide and normal sugbroupoid

G G N :“ tpκ2,g,κ1qP K ˆ G ˆ K | ρpgqpκ1q“ κ2u (1) of Γ Ñ G. The quotient Θ “ Γ{N has then a Lie groupoid structure over G. The elements of Θ are classes xk2,g,k1y“tpk2κ2,g,k1κ1q | pκ2,g,κ1qP Nu with pk2,g,k1qP Γ. Set sH , tH : Γ{N Ñ H

sH xk2,g,k1y“BH pk1q, tH xk2,g,k1y“BH pk2q,

and a partial multiplication ¨H :Θ ˆsH ,tH Θ Ñ Θ by

1 1 1 1 1 1 1 ´1 1 1 ´1 xk2,g , k1y ¨H xk2,g,k1y“xk2,g , k1y ¨H xk1,g,k1ρpg qpκqy“xk2,g g, k1ρpg qpκqy.

G 1 Here κ P K is such that k2κ “ k1. It is easy to check that Θ{N Ñ H becomes a groupoid with these structure maps and with the inversion

´1 xk2,g,k1y ÞÑ xk1,g , k2y and the unit inclusion h ÞÑ xk, 1sphq, ky for any k P K such that BH pkq“ h. Since those structure maps are deﬁned such that

ΓΘπ “ Γ{N

K H BH is a morphism of groupoids and the projections π : Γ Ñ Θ and BH : K Ñ H are surjective submersions, it is easy to check that Θ Ñ H is a Lie groupoid. The interchange law and the surjectivity of the double source map are also easily deduced from the one in pBG, BGq, and pΘ,G,H,Mq is a

7 ´1 double Lie groupoid. Its core consists in classes xk, 1spkq,κy“xkκ , 1spkq, 1spkqy, and is obviously isomorphic to K as a Lie groupoid. The core diagram of pΘ,G,H,Mq is hence again

KBG G

BH HM by construction. Now given a transitive double Lie groupoid

S G

S ´1,H with core K and core diagram as in Figure 2, set ΨS :Θ Ñ S, xk2,g,k1y ÞÑ k2 ¨H 1g ¨H k1 P S. Then ΨS is an isomorphism of double Lie groupoids, see [3] – where, again, only the surjectivity of BG and BH are needed. A morphism Φ: Γ1 Ñ Γ2 of double Lie groupoids with side morphisms ϕG : G1 Ñ G2, ϕH : H1 Ñ H2 and with core morphism ϕK : K1 Ñ K2 induces a morphism of the corresponding core diagrams as in the following diagram.

ϕK K1 K2

BG1 BG2

ϕG BH1 G1 G2 BH2 (2)

ϕH H1 H2

MMidM

Consider conversely a morphism of transitive core diagrams (of Lie groupoids) as in (2). Let Γi be the total space of the comma double Lie groupoid pBGi , BGi q for i “ 1, 2 and set Φ: Γ1 Ñ Γ2, Φpk1,g,k2q “ pϕK pk1q, ϕGpgq, ϕK pk2qq. The map Φ is obviously a morphism of double Lie groupoids, and a computation shows ΦpN1q “ N2 for the normal subgroupoids Ni of Γi Ñ Gi deﬁned as in (1) for i “ 1, 2. Therefore, it induces a morphism of the transitive double Lie groupoids Φ: Γ1{N1 Ñ Γ2{N2 with core morphism again the one in (2). Let M be a smooth manifold and consider the category trCDGpMq of transitive core diagrams of Lie groupoids with base M, and the category trDLGpMq of transitive double Lie groupoids with double base M. The functor C : trDLGpMq Ñ trCDGpMq sends transitive double Lie groupoids and their morphisms to the corresponding transitive core diagrams of Lie groupoids and morphisms of core diagrams. The above constructs a functor D : trCDGpMq Ñ trDLGpMq sending a transitive core diagram as in Figure 2 to the double Lie groupoid pΘ,G,H,Mq as above, and a morphism of core diagrams as in (2) to the morphism Φ:Θ1 Ñ Θ2 as in the previous paragraph. Then C ˝ D is obviously the identity functor. The assigment sending each transitive double Lie groupoid pS,G,H,Mq to the isomorphism ΨS : DCpSq Ñ S of double Lie groupoids deﬁnes a natural isomorphism Ψ¨ : D ˝ C Ñ idtrDLG. Therefore, Brown and Mackenzie’s equivalence of locally trivial core diagrams with locally trivial double Lie groupoids [3] is extended to an equivalence of transitive core diagrams with transitive double Lie groupoids.

8 Outline of the paper Section 2 collects necessary knowledge on ideals in Lie algebroids, on VB-algebroids and representations up to homotopy and on double Lie algebroids and their morphisms. Section 3 defines transitive core diagrams and studies the crossed modules associated to a transitive core diagram. It describes the (obvious) functor from transitive double Lie algebroids to transitive core diagrams. Section 4 defines the comma double Lie algebroid associated to a Lie algebroid morphism C Ñ A (over a fixed base). It describes the two VB-algebroid structures of this double Lie algebroid via apropriate representations up to homotopy. Section 5 is the core of this paper: it factors the comma double Lie algebroid defined by one side of a transitive core diagram, by a sub-structure defined by the other side. This structure is an ideal in one of the VB-algebroids and an infinitesimal ideal system (with trivial fiber) in the other VB-algebroid. The equivalence between the categories of transitive double Lie algebroids and of transitive core diagrams is then established in Section 5.5. Section 6 uses this for proving an integration of transitive double Lie algebroids (with integrable sides and core). The appendix collects some longer proofs, like the verification of the double Lie algebroid conditions [7] for the comma double Lie algebroid and its quotient.

Notation and conventions All manifolds and vector bundles in this paper are smooth and real. Vector bundle projections are ˚ written qE : E Ñ M, and pM : TM Ñ M for tangent bundles. Given a section ε of E , the map ℓε : E Ñ R is the linear function associated to it, i.e. the function defined by em ÞÑ xεpmq,emy for all em P E. The set of global sections of a vector bundle E Ñ M is denoted by ΓpEq, XpMq“ ΓpTMq is the space of smooth vector fields on a smooth manifold M, and Ω‚pMq “ Γp ‚ T ˚Mq is the space of smooth forms on M. Ź Acknowledgement This paper is the outcome of a joint project of the authors in 2012 and 2013, which had been left unfinished until the departure of the second author in 2020. The studied problem was Kirill Mackenzie’s idea and this research builds on many of his beautiful mathematical achievements. The first author dedicates to his memory her work on finishing this paper.

2 Preliminaries

This section collects necessary background on inﬁnitesimal ideal systems in Lie algebroids [13], on double Lie algebroids and their morphisms [20], and on the correspondence of decomposed VB-algebroids with 2-term representations up to homotopy [8].

2.1 Ideals in Lie algebroids Infinitesimal ideal systems [13, 10] are considered the right notion of ideal in Lie algebroids. They are the infinitesimal version of ideal systems [12, 19]. These are exactly the kernels of fibrations of Lie algebroids [12]. Definition 2.1. Let pq : A Ñ M, ρ, r¨ , ¨sq be a Lie algebroid, F Ď TM an involutive subbundle, J Ď A a subbundle over M such that ρpJq Ď F , and ∇ a flat F -connection on A{J with the following properties: 1. If a P ΓpAq is ∇-parallel1, then ra, jsP ΓpJq for all j P ΓpJq.

1 A section a P ΓpAq is said to be ∇-parallel if ∇X a¯ “ 0 for all X P ΓpF q. Here,a ¯ is the class of a in ΓpA{Jq» ΓpAq{ΓpJq.

9 2. If a,b P ΓpAq are ∇-parallel, then ra,bs is also ∇-parallel. Then the triple pF, J, ∇q is an infinitesimal ideal system in A. The first axiom implies immediately that J Ď A is a subalgebroid of A, and the following property of pF, J, ∇q follows from (2) above. 3. If a P ΓpAq is ∇-parallel, then ρpaq is ∇F -parallel, where F F ¯ ∇ : ΓpF qˆ ΓpTM{F q Ñ ΓpTM{F q, ∇X Y “ rX, Y s is the Bott connection associated to F . Infinitesimal ideal systems already appear in [10] (not under this name) in the context of geometric quantization as the infinitesimal version of polarizations on groupoids. Example 2.2 (Flat connections on vector bundles). Let E be a vector bundle over M and let K Ď E be a subbundle. Then any flat connection of an involutive subbundle FM Ď TM on E{K defines an infinitesimal ideal system pFM ,K, ∇q in E, i.e. an infinitesimal ideal system in the trivial Lie algebroid pE,ρ “ 0, r¨ , ¨s “ 0q. An infinitesimal ideal system in a Lie algebroid is therefore, by forgetting the Lie algebroid structure, automatically an infinitesimal ideal system in the underlying vector bundle. Example 2.3. 1. An ideal in a Lie algebroid A Ñ M is a vector subbundle I Ď A such that rΓpIq, ΓpAqs Ď ΓpIq. Given such an ideal I in A, the triple p0,I, 0q is an infinitesimal ideal system in A.

2. Let A Ñ M be a Lie algebroid. Let FM Ď TM be an involutive subbundle and let ∇: ΓpFM qˆ ΓpAq Ñ ΓpAq be a flat connection such that ra,bs is ∇-parallel for a,b P ΓpAq ∇-parallel. Then pFM , 0, ∇q is an infinitesimal ideal system in A. Definition 2.4. A morphism A Φ / B

qE qB M ϕ / N of Lie algebroids A Ñ M and B Ñ N is a fibration of Lie algebroids if the underlying morphism pΦ, ϕq of vector bundles is a fibration of vector bundles, i.e. ϕ is a surjective submersion and Φ is fiberwise surjective.

Let E Ñ M be a vector bundle, and let J Ď E be a subbundle and FM Ď TM an involutive subbundle. Assume that there is a flat FM -connection on E{J. Then define an equivalence relation 1 1 1 „∇ on E{J as follows: for e,e P E and their classese, ¯ e¯ P E{J,e ¯ „∇ e¯ if and only if qEpeq and 1 1 qEpe q are in the same leaf L of FM in M ande ¯ is the image ofe ¯ under ∇-parallel transport 1 along a path in the leaf L joining qEpeq and qEpe q. The quotient of E by this equivalence relation is written pE{Jq{∇. The projections E Ñ pE{Jq{∇ and M Ñ M{FM are written π and πM , repectively. The vector bundle map qE : E Ñ M factors to a map rqEs: pE{Jq{∇ Ñ M{FM . The following theorem shows that infinitesimal ideal systems in Lie algebroids define in this manner quotients of Lie algebroids, up to some topological obstructions [13]. The paper [6] proves that an infinitesimal ideal system defines a sub-representation (up to homotopy) of the adjoint representation of the Lie algebroid, after the choice of an extension of the infinitesimal ideal system connection. These two results suggest that indeed, an infinitesimal ideal systems is the right notion of ideal in a Lie algebroid. The following result from [13] is slightly reformulated here in order to emphasize the fact the two topological obstructions ensure the existence of a quotient vector bundle. The ‘reduced’ Lie algebroid structure then follows immediately.

10 Theorem 2.5. 1. Let qE : E Ñ M be a smooth vector bundle and let pFM , J, ∇q be an inﬁnites- imal ideal system in E (see Example 2.2). Assume that M¯ “ M{FM is a smooth manifold and that ∇ has trivial holonomy. Then the quotient pE{Jq{∇ Ñ M{FM carries a vector bundle structure such that the projection pπ, πM q is a ﬁbration of vector bundles:

π E / pE{Jq{∇ (3)

qE rqE s M / M{FM πM

2. Let now A Ñ M be a Lie algebroid and let pFM , J, ∇q be an inﬁnitesimal ideal system in A. If the quotient vector bundle pA{Jq{∇ Ñ M{FM exists as in (3), then it carries a unique Lie algebroid structure such that (3) is a ﬁbration of Lie algebroids.

An infinitesimal ideal system pFM , J, ∇q is defined by the kernel of a fibration of Lie algebroids if and only if it integrates to an ideal system in the sense of Higgins and Mackenzie [12, 19], see [13]. The reduced Lie algebroid structure in (2) of Theorem 2.5 is defined as follows. Write ΓpAq∇ for the space of sections of A Ñ M the class in A{J of which is ∇-flat. Then ΓpAq∇ is an R- 8 FM ˚ 8 ¯ ∇ Lie algebra and a C pMq “ πM C pMq-module. The space ΓpJq is an ideal in ΓpAq . By ∇ definition, for each a P ΓpAq there is a sectiona ¯ P ΓpA¯q such that π ˝ a “ a¯ ˝ πM (this is written a „π a¯ for short). In particular, the sections j P ΓpJq project in this manner to the zero section of A¯ Ñ M¯ . Conversely for each sectiona ¯ P ΓpA¯q there is a ∇-flat a P ΓpAq such that a „π a¯. The Lie algebroid bracket and the anchor on A¯ Ñ M¯ are then defined by

∇ ΓpAq Q ai „π a¯i P ΓpA¯q, i “ 1, 2 ñ ra1,a2s„π ra¯1, a¯2s, and ρpπpaqq “ T πM ρpaq for all a P A.

2.2 VB-algebroids and representations up to homotopy This section starts by brieﬂy recalling the deﬁnitions of double vector bundles, of their linear and core sections, and of their linear splittings and lifts. The reader is referred to [23, 19, 8] for more detailed treatments. A double vector bundle is a commutative square

π D B / B

πA qB A / M qA of vector bundles such that

pd1 À d2q`B pd3 À d4q “ pd1 `B d3qÀ pd2 `B d4q (4) for d1, d2, d3, d4 P D with πApd1q “ πApd2q, πApd3q “ πApd4q and πB pd1q “ πBpd3q, πBpd2q “ πBpd4q. Here, À and `B are the additions in D Ñ A and D Ñ B, respectively. In particular, pπA, qBq and pπB , qAq must be vector bundle morphisms. The vector bundles A and B are called the side bundles. The core C of a double vector bundle is the intersection of the kernels of πA and of πB . From (4) follows easily the existence of a natural vector bundle structure on C over ´1 A ´1 B M. The inclusion C ãÑ D is denoted by Cm Q c ÞÝÑ c P πA p0mqX πB p0mq.

11 8 The space of sections ΓBpDq is generated as a C pBq-module by two special classes of sections (see [20]), the linear and the core sections which are now described. For a section c: M Ñ C, : : D the corresponding core section c : B Ñ D is defined as c pbmq“ 0 À cpmq, m P M, bm P Bm. bm The corresponding core section A Ñ D is written c: also, relying on the argument to distinguish c between them. The space of core sections of D over B is written ΓBpDq. A section ξ P ΓBpDq is called linear if ξ : B Ñ D is a bundle morphism from B Ñ M to ℓ D Ñ A over a section a P ΓpAq. The space of linear sections of D over B is denoted by ΓBpDq. Given ψ P ΓpB˚ b Cq, there is a linear section ψ : B Ñ D over the zero section 0A : M Ñ A given ˜ by ψpbmq“ 0bm À ψpbmq. Such a section is called a core-linear section. r Example 2.6. Let A, B, C be vector bundles over M and consider D “ A ˆM B ˆM C. With r ! ! the vector bundle structures D “ qApB ‘ Cq Ñ A and D “ qB pA ‘ Cq Ñ B, one finds that pD; A, B; Mq is a double vector bundle called the decomposed double vector bundle with core C. The core sections are given by

: A c : bm ÞÑ p0m,bm,cpmqq, where m P M, bm P Bm, c P ΓpCq,

: ℓ and similarly for c : A Ñ D. The space of linear sections ΓBpDq is naturally identiﬁed with ΓpAq‘ ΓpB˚ b Cq via

˚ pa, ψq : bm ÞÑ papmq,bm, ψpbmqq, where ψ P ΓpB b Cq, a P ΓpAq.

In particular, the ﬁbered product A ˆM B is a double vector bundle over the sides A and B and has core t0uˆ M Ñ M.

A linear splitting of pD; A, B; Mq is an injective morphism of double vector bundles Σ: AˆM B ãÑ D over the identity on the sides A and B. The existence of a linear splitting is proved e.g. in [22], see also [11] for historical remarks. A linear splitting Σ of D is also equivalent to a splitting 8 σA of the short exact sequence of C pMq-modules

˚ ℓ 0 ÝÑ ΓpB b Cq ãÑ ΓBpDq ÝÑ ΓpAq ÝÑ 0, (5) where the third map is the map that sends a linear section pξ,aq to its base section a P ΓpAq. The ℓ splitting σA is called a horizontal lift. Given Σ, the horizontal lift σA : ΓpAq Ñ ΓBpDq is given by σApaqpbmq“ Σpapmq,bmq for all a P ΓpAq and bm P B. By the symmetry of a linear splitting, ℓ ℓ a lift σA : ΓpAq Ñ ΓB pDq is equivalent to a lift σB : ΓpBq Ñ ΓApDq.

Let qE : E Ñ M be a vector bundle. Then the tangent bundle T E has two vector bundle structures; one as the tangent bundle of the manifold E, and the second as a vector bundle over TM. The structure maps of T E Ñ TM are the derivatives of the structure maps of E Ñ M.

pE T E / E

T qE qE TM / M pM

´1 E The space T E is a double vector bundle with core bundle E Ñ M. The map¯: E Ñ pE p0 qX ´1 T M d  pT qEq p0 q sends em P Em toe ¯m “ tem P T E E. Hence the core vector field corre- dt t“0 0m Ò sponding to e P ΓpEq is the vertical lift eÒ : E Ñ T E, i.e. the vector field with flow ϕe : E ˆR Ñ E, 1 1 ℓ Xℓ ϕtpemq“ em ` tepmq. An element of ΓEpT Eq“ pEq is called a linear vector field. It is well- known (see e.g. [19]) that a linear vector field ξ P XlpEq covering X P XpMq corresponds to a derivation D : ΓpEq Ñ ΓpEq over X P XpMq. The precise correspondence is given by the following equations ˚ ˚ ξpℓεq“ ℓD˚pεq and ξpqEfq“ qEpXpfqq (6)

12 for all ε P ΓpE˚q and f P C8pMq, where D˚ : ΓpE˚q Ñ ΓpE˚q is the dual derivation to D [19, §3.4]. The linear vector field in XlpEq corresponding in this manner to a derivation D of ΓpEq is written D. Given a derivation D over X P XpMq, the explicit formula for D is  p d  p Dpemq“ TmepXpmqq È  pem ´ tDpeqpmqq (7) dtt“0 p for em P E and any e P ΓpEq such that epmq “ em. The choice of a linear splitting Σ for pT E; TM,E; Mq is equivalent to the choice of a linear connection ∇: XpMqˆ ΓpEq Ñ ΓpEq: the ∇ X Xl corresponding lift σT M : pMq Ñ pEq is defined by X ÞÑ ∇X . It is easy to see, using the equalities in (6), that y Ò Ò Ò Ò ∇X , ∇Y “ ∇rX,Y s ´ R∇pX, Y q, ∇X ,e “ p∇X eq , e1,e2 “ 0 (8) ” ı ” ı ” ı for all X, Y PyXpMyq and e,e{1,e2 P ΓpČEq. y The splitting Σ∇ defined by a linear TM-connection on E is, as above, equivalent as well to a ∇ l linear horizontal lift σE : ΓpEq Ñ ΓT M pT Eq. It is given by

∇ σE peq“ Te ´ ∇¨e for all e P ΓpEq.

: c The core section of T E Ñ TM deﬁned by e PĄΓpEq is written e P ΓT M pT Eq and explicitly deﬁned e: v T Ev d  E te m v TM by p mq“ m0 m ´ dt t“0 0m ` p q for all m P .

The remainder of this section collects necessary background on VB-algebroids and representations up to homotopy. A double vector bundle pD; A, B; Mq is a VB-algebroid ([17]; see also [8]) if D Ñ B is equipped with a Lie algebroid structure such that the anchor ΘB : D Ñ TB is a morphism of double vector bundles over ρA : A Ñ TM on one side and the Lie bracket on D Ñ B is linear:

ℓ ℓ ℓ ℓ c c c c rΓBpDq, ΓBpDqs Ă ΓBpDq, rΓBpDq, ΓB pDqs Ă ΓBpDq, rΓBpDq, ΓB pDqs “ 0.

The vector bundle A Ñ M is then also a Lie algebroid, with anchor ρA and bracket deﬁned as ℓ follows: if ξ1, ξ2 P ΓBpDq are linear over a1,a2 P ΓpAq, then the bracket rξ1, ξ2s is linear over ra1,a2s.

Let E0, E1 be two vector bundles over the same base M as A. A2-term representation up to homotopy of A on E0 ‘ E1 [2, 8] is the collection of

(R1) a map B : E0 Ñ E1,

0 1 0 1 (R2) two A-connections, ∇ and ∇ on E0 and E1, respectively, such that B˝ ∇ “ ∇ ˝B,

2 (R3) an element R P Ω pA, HompE1, E0qq such that R∇0 “ R ˝B, R∇1 “B˝ R and d∇Hom R “ 0, Hom 0 1 where ∇ is the connection induced on HompE1, E0q by ∇ and ∇ . Note that Gracia-Saz and Mehta call this structure a “superrepresentation”, following [24]. In this paper, 2-term representation up to homotopy are called 2-representations for short. Consider again a VB-algebroid pD Ñ B, A Ñ Mq and choose a linear splitting Σ: AˆM B Ñ D. : Since the anchor ΘB is linear, it sends a core section c , c P ΓpCq to a vertical vector field on B. : Ò This defines the core-anchor BB : C Ñ B which is given by ΘBpc q“pBBcq for all c P ΓpCq and does not depend on the splitting (see [16]). Since the anchor ΘB of a linear section is linear, for l each a P ΓpAq the vector field ΘBpσApaqq P X pBq defines a derivation of ΓpBq with symbol ρpaq. This defines a linear connection ∇AB : ΓpAqˆ ΓpBq Ñ ΓpBq:

AB ΘpσApaqq “ ∇a

z 13 for all a P ΓpAq. Since the bracket of a linear section with a core section is again a core section, there is a linear connection ∇AC : ΓpAqˆ ΓpCq Ñ ΓpCq such that

: AC : rσApaq,c s “ p∇a cq for all c P ΓpCq and a P ΓpAq. The diﬀerence σAra1,a2s´rσApa1q, σApa2qs is a core-linear section 2 for all a1,a2 P ΓpAq. This deﬁnes a vector valued form R P Ω pA, HompB, Cqq such that

rσApa1q, σApa2qs “ σAra1,a2s´ Rpa1,a2q, for all a1,a2 P ΓpAq. For more details on these constructions,Č see [8], where the following result is proved.

Theorem 2.7. Let pD Ñ B; A Ñ Mq be a VB-algebroid and choose a linear splitting Σ: AˆM B Ñ AB AC D. The triple p∇ , ∇ , Rq deﬁned as above is a 2-representation of A on BB : C Ñ B. Conversely, let pD; A, B; Mq be a double vector bundle such that A has a Lie algebroid structure AB AC and choose a linear splitting Σ: A ˆM B Ñ D. Then if p∇ , ∇ , Rq is a 2-representation of A on BB : C Ñ B, then the equations above deﬁne a VB-algebroid structure on pD Ñ B; A Ñ Mq.

2.3 Double Lie algebroids A double Lie algebroid with core C is a double vector bundle pD; A, B; Mq

π D A / A

πB qA B / M qB with core C and four Lie algebroid structures on A Ñ M, B Ñ M, D Ñ A and D Ñ B such that pD Ñ A, B Ñ Mq and pD Ñ B, A Ñ Mq are VB-algebroids, and the dual Lie algebroids D BA Ñ C˚ and D BB Ñ C˚ form a VB-Lie bialgebroid [20]. As it is central in this paper, this section recalls the equivalent deﬁnition given by the correspondence of decompositions of double Lie algebroids with matched pairs of 2-representations [7]. Consider a double vector bundle pD; A, B; Mq with core C and a VB-Lie algebroid structure on each of its sides. Since pD Ñ B, A Ñ Mq is a VB-Lie algebroid, it is described in any linear AB AC splitting Σ: A ˆM B Ñ D by a representation up to homotopy pBB, ∇ , ∇ , RAq of A on Cr0s‘ Br1s. Similarly, via the linear splitting Σ, the VB-algebroid pD Ñ A, B Ñ Mq is described BA BC by a representation up to homotopy pBA, ∇ , ∇ , RBq on Cr0s‘ Ar1s. Let ρA, respectively ρB , be the anchor of the Lie algebroid A Ñ M, respectively B Ñ M. [7] proves that pD BA, D BBq is a Lie bialgebroid over C˚ if and only if, for any linear splitting of D, the two induced 2-representations as above form a matched pair as in the following deﬁnition [7]. In other words, pD,A,B,Mq is a double Lie algebroid if and only if for any linear splitting of D, the two induced 2-representations satisfy the equations (M1) to (M7) below.

Deﬁnition 2.8. Let pA Ñ M,ρA, r¨ , ¨sq and pB Ñ M,ρB, r¨ , ¨sq be two Lie algebroids and assume AB AC that A acts on Cr0s‘ Br1s up to homotopy via pBB, ∇ , ∇ , RAq and B acts on Cr0s‘ Ar1s BA BC 2 up to homotopy via pBA, ∇ , ∇ , RBq . Then the two representations up to homotopy form a matched pair if

(M1) ∇BAc1 c2 ´ ∇BB c2 c1 “ ´p∇BAc2 c1 ´ ∇BB c1 c2q,

2For the sake of simplicity, all the four connections are denoted by ∇. It is always clear from the indexes which connection is meant. The connection ∇A is the A-connection induced by ∇AB and ∇AC on ^2B˚ b C and ∇B is the B-connection induced on ^2A˚ b C.

14 (M2) ra, BAcs“BAp∇acq´ ∇BB ca,

(M3) rb, BBcs“BBp∇bcq´ ∇BAcb,

(M4) ∇b∇ac ´ ∇a∇bc ´ ∇∇bac ` ∇∇abc “ RBpb, BBcqa ´ RApa, BAcqb,

(M5) BApRApa1,a2qbq“´∇bra1,a2s`r∇ba1,a2s`ra1, ∇ba2s` ∇∇a2 ba1 ´ ∇∇a1 ba2,

(M6) BBpRBpb1,b2qaq“´∇arb1,b2s`r∇ab1,b2s`rb1, ∇ab2s` ∇∇b2 ab1 ´ ∇∇b1 ab2, for all a,a1,a2 P ΓpAq, b,b1,b2 P ΓpBq and c,c1,c2 P ΓpCq, and

2 2 ˚ 2 2 ˚ (M7) d∇A RB “ d∇B RA P Ω pA, ^ B b Cq“ Ω pB, ^ A b Cq, where RB is seen as an element 1 2 ˚ 1 2 ˚ of Ω pA, ^ B b Cq and RA as an element of Ω pB, ^ A b Cq. Recall that the Lie bialgebroid pD BA, D BBq induces a Poisson structure (natural up to sign) on its base C˚; this Poisson structure is linear [20, §4] and so induces a Lie algebroid structure on B ˚ B ˚ C. Writing e: D B Ñ T pC q and e˚ : D A Ñ T pC q for the two anchors, then the Poisson ˚ # ˚ ˚ ˚ ˚ ˚ structure on C is deﬁned by its sharp morphism πC˚ :“ e˚ ˝ e “ ´e ˝ e˚ : T C Ñ T C . The sign of this Poisson structure is determined by the requirement that BA and BB be morphisms of Lie algebroids (see the next proposition). The proof of [7, Proposition 4.5.] shows the following 8 identities for c,c1,c2 P ΓpCq and f,g P C pMq:

˚ pe ˝ e qpdℓ 1 qpℓ 2 q“ ℓ 2 1 “ ℓ 1 2 ˚ c c ∇BAc1 c ´∇BB c2 c ´∇BAc2 c `∇BB c1 c ˚ ˚ ˚ ˚ pe˚ ˝ e qpdℓcqpqC˚ fqq “ qC˚ p£ρB BB pcqfq“ qC˚ p£ρABApcqfq ˚ ˚ ˚ pe˚ ˝ e qpdqC˚ fqpqC˚ gq“ 0. Hence the Lie algebroid on C Ñ M is deﬁned explicitly using Deﬁnition 2.8:

1. The anchor is ρC :“ ρA ˝BA “ ρB ˝BB. This equality follows easily from (M2) and (M3), see [7, Remark 3.2]. 2. The ﬁrst equation (M1) gives a simple formula for the induced Lie algebroid bracket on section of C:

rc1,c2s“ ∇BAc1 c2 ´ ∇BB c2 c1. (9) The Jacobi identity follows from (M2), (M3) and (M4), and the bracket does not depend on the choice of splitting Σ: A ˆM B Ñ D, see [7, Remark 3.5].

Proposition 2.9. Let pD; A, B; Mq be a double Lie algebroid. Then the morphisms BA : C Ñ A and BB : C Ñ B of vector bundles are morphisms of Lie algebroids.

Proof. Since ρC “ ρA ˝BA “ ρB ˝BB, the compatibility of the anchors with the morphisms is immediate. AC AB Equation (M3) with BB ˝ ∇ “ ∇ ˝BB gives

AC BC BBrc1,c2s“BBp∇BAc1 c2q´BBp∇BB c2 c1q AB AB “ ∇BAc1 pBBc2q´rBBc2, BBc1s´ ∇BAc1 pBBc2q“rBBc1, BBc2s

BC BA for all c1,c2 P ΓpCq. Similarly, (M2) and BA ˝ ∇ “ ∇ ˝BA yield together the compatibility of BA with the brackets on C and A. Finally, the following proposition is an immediate consequence of (9).

Proposition 2.10. Let pD; A, B; Mq be a double Lie algebroid. Let c1,c2 be sections of the core C with BApc1q“ 0 and BBpc2q“ 0. Then rc1,c2s“ 0.

15 2.4 Morphisms of double Lie algebroids This section quickly discusses morphisms of matched pairs of 2-representations versus morphisms of double Lie algebroids [6]. Let pD1 Ñ B1, A1 Ñ Mq and pD2 Ñ B2, A2 Ñ Mq be two VB-algebroids over a common double base M. A morphism Φ: D1 Ñ D2 of double vector bundles with side morphisms ϕA : A1 Ñ A2 and ϕB : B1 Ñ B2 is a morphism of VB-algebroids if

Φ D1 / D2

πB1 πB2 B / B 1 ϕB 2 is a morphism of Lie algebroids. The vector bundle morphism ϕA : A1 Ñ A2 is consequently also a morphism of Lie algebroids, over the identity on M. Let now pD1, A1,B1,Mq and pD2, A2,B2,Mq be two double Lie algebroids over a common double base M. A morphism Φ: D1 Ñ D2 of double vector bundles with side morphisms ϕA : A1 Ñ A2 and ϕB : B1 Ñ B2 is a morphism of double Lie algebroids if it is a morphism of VB- algebroids pD1 Ñ B1, A1 Ñ Mq Ñ pD2 Ñ B2, A2 Ñ Mq and a morphism of VB-algebroids pD1 Ñ A1,B1 Ñ Mq Ñ pD2 Ñ A2,B2 Ñ Mq.

E0 E1 F0 F1 Let A Ñ M be a Lie algebroid and let DE :“ pBE, ∇ , ∇ , REq and DF :“ pBF , ∇ , ∇ , RF q be 2-representations of A on E0r0s‘ E1r1s and F0r0s‘ F1r1s, respectively. A morphism of representations up to homotopy pA, DEq Ñ pA, DF q is determined by a triple pϕ0, ϕ1, ϕq, 1 where ϕ0 : E0 ÝÑ F0, ϕ1 : E1 Ñ F1 are vector bundle morphisms and ϕ P Ω pA, HompF1, E0qq, satisfying ϕ1 ˝BE “BF ˝ ϕ0, (10) Hom ∇a pϕ0, ϕ1q “ pϕa ˝BE, BF ˝ ϕaq for all a P ΓpAq (11) and d∇Hom ϕ “ RF ˝ ϕ1 ´ ϕ0 ˝ RE, (12) Hom where ∇ is the A-connection induced on HompE0r0s‘ E1r1s, F0r0s‘ F1r1sq by the four connections (see [2] for more details). Let A Ñ M and A1 Ñ M be two Lie algebroids over a smooth manifold M and consider a Lie 1 1 algebroid morphism ϕ: A Ñ A over the identity on M. Assume that A acts on E0r0s‘E1r1s up to 0 1 ˚ i homotopy via D :“ pB, ∇ , ∇ , Rq. For i “ 0, 1 the pullback connection ϕ ∇ : ΓpAqˆ ΓpEiq Ñ ΓpEiq is the A-connection on Ei given by

˚ i i pϕ ∇ qae :“ ∇ϕpaqe (13)

˚ 2 for a P ΓpAq and e P ΓpEiq. The pullback ϕ R P Ω pA, HompE1, E0qq is similarly deﬁned by

˚ pϕ Rqpa1,a2q“ Rpϕpa1q, ϕpa2qq (14)

˚ ˚ 0 ˚ 1 ˚ for a1,a2 P ΓpAq. Then the triple ϕ D :“ pB, ϕ ∇ , ϕ ∇ , ϕ Rq is a representation up to homotopy of A on E0r0s‘ E1r1s, see [2]. 1 If DE is a 2-representation of A on E0r0s ‘ E1r1s and DF is a 2-representation of A on F0r0s‘ F1r1s, then a morphism of representations up to homotopy from DE to DF is by deﬁnition [6] a morphism ˚ DE Ñ ϕ DF of 2-representations of A.

16 Let pD Ñ B, A Ñ Mq and pD1 Ñ B1, A1 Ñ Mq be two VB-algebroids over a common double base M and consider a morphism Φ: D Ñ D1 of double vector bundles with side morphisms 1 1 ϕA : A Ñ A and ϕB : B Ñ B . Choose linear splittings

1 1 1 1 Σ: A ˆM B Ñ D, Σ : A ˆM B Ñ D , of D and D1, respectively. The morphism Φ and the linear splittings deﬁne a form ϕ P Ω1pA, HompB, C1qq » ΓpA˚ b B˚ b C1q by

1 D1 a,b ϕ a , ϕ b 1 ϕ a,b Φ pΣp qq “ Σ p Ap q B p qq `B 0ϕApaq `B p q (15) ´ ¯ for all a P ΓpAq. 1 Then by [6, Theorem 4.11], Φ is a VB-algebroid morphism if and only if ϕA : A Ñ A is a Lie algebroid morphism and for any pair of linear splittings

1 1 1 1 Σ: A ˆM B Ñ D, Σ : A ˆM B Ñ D , the triple pϕB , ϕC , ϕq is a morphism ˚ 1 D Ñ ϕAD of the 2-representations D of A and D1 of A1 deﬁned by Σ and the VB-algebroid pD Ñ B, A Ñ Mq, and by Σ1 and the VB-algebroid pD1 Ñ B1, A1 Ñ Mq. Assume ﬁnally that pD,A,B,Mq and pD1, A1,B1,Mq are two double Lie algebroids and consider a morphism Φ: D Ñ D1 as above of double vector bundles. Choose linear splittings

1 1 1 Σ: A ˆM B Ñ D, Σ: A ˆM B Ñ D , of D and D1, respectively, and consider the form ϕ P ΓpA˚ b B˚ b C1q defined as in (15) by Φ 1 1 and the linear splittings. Let DA and DA be the 2-representations of A and of A defined by Σ and the VB-algebroid pD Ñ B, A Ñ Mq, and by Σ1 and the VB-algebroid pD1 Ñ B1, A1 Ñ Mq, 1 1 respectively. Similarly, Let DB and DB be the 2-representations of B and of B defined by Σ and the VB-algebroid pD Ñ A, B Ñ Mq, and by Σ1 and the VB-algebroid pD1 Ñ A1,B1 Ñ Mq, respectively. Then Φ is a double Lie algebroid morphism if and only if pϕB , ϕC , ϕq defines a morphism of 2-representations ˚ 1 DA Ñ ϕADA, (16) and pϕA, ϕC , ϕq defines a morphism of 2-representations

˚ 1 DB Ñ ϕB DB. (17) The following proposition is standard, but can now be proved easily using (9) and (11). Proposition 2.11. Let pD,A,B,Mq and pD1, A1,B1,Mq be two double Lie algebroids and consider 1 1 a morphism Φ: D Ñ D of double Lie algebroids. Then the core morphism ϕC : C Ñ C is a morphism of Lie algebroids.

1 1 Proof. Let ϕA : A Ñ A and ϕB : B Ñ B be the side morphisms of Φ. By the considerations above, they are Lie algebroid morphisms. Choose linear splittings

1 1 1 1 Σ: A ˆM B Ñ D, Σ : A ˆM B Ñ D , of D and D1, respectively, and consider the form ϕ P ΓpA˚ b B˚ b C1q deﬁned as in (15) by Φ and the linear splittings. Consider as above the four 2-representations3 deﬁned by the VB-algebroid

3For the sake of simplicity, all the eight connections are denoted by ∇. It is clear from the indexes which connection is meant.

17 and the splittings. Then for c1,c2 P ΓpCq:

ϕC rc1,c2s“ ϕC p∇BAc1 c2 ´ ∇BB c2 c1q ˚ ˚ “ pϕA∇qBAc1 pϕC c2q` ωpBAc1, BBc2q ´ pϕB ∇qBB c2 pϕC c1q´ ωpBAc1, BBc2q

“ ∇ϕABAc1 pϕC c2q´ ∇ϕB BB c2 pϕC c1q

“ ∇BA1 ϕC c1 pϕC c2q´ ∇BB1 ϕC c2 pϕC c1q“rϕCc1, ϕC c2s. The anchor condition is checked as follows using (10)

ρC1 ˝ ϕC “ ρA1 ˝BA1 ˝ ϕC “ ρA1 ˝ ϕA ˝BA “ ρA ˝BA “ ρC .

3 Transitive double Lie algebroids and transitive core diagrams

This section discusses transitive double Lie algebroids, and establishes a functor from the category of transitive double Lie algebroids, to the category of transitive core diagrams. Recall from [19, 4.4.1] that a morphism of Lie algebroids ϕ: A1 Ñ A over a surjective submersion f : M 1 Ñ M is a fibration of Lie algebroids if and only if the associated map 1 1 1 ! 1 a ÞÑ pqA1 a , ϕpa qq into the vector bundle pullback f A Ñ M is a surjection. Proposition 3.1. Let pD; A, B; Mq be a double Lie algebroid with core C. Assume that the anchors ρA : A Ñ TM and ρB : B Ñ TM are both surjective. (i) The morphism BA : C Ñ A is surjective if and only if the anchor ΘA : D Ñ T A is a fibration of Lie algebroids. (ii) The morphism BB : C Ñ B is surjective if and only if the anchor ΘB : D Ñ TB is a fibration of Lie algebroids.

Proof: In order to prove (i), assume ﬁrst that BA is surjective. Take b P B and ξ P T A, projecting to the same point v P TM: T qAξ “ ρBpbq. Write a for the projection of ξ to A. Take 1 1 1 1 any d P D which projects to a and b and write ξ “ ΘApd q P T A. Then ξ and ξ project to the same elements of A and TM and therefore there is a core element BApcqP A such that

1 TA ξ “ ξ À p0a `T M BApcqq. 1 Now define d :“ d À p0a `B cq. Clearly ΘApdq “ ξ. This shows that ΘA is a fibration. The converse is straightforward since BA is the core morphism of ΘA. 2 Although the considerations above are the authors’ original intuition for transitive double Lie algebroids, the definition below drops the surjectivity of the anchors of A and B because they are not needed for the main results of this paper. Hence, a transitive double Lie algebroid pD; A, B; Mq as in the following definition has two surjective anchors D Ñ T A and D Ñ TB if and only if the two base anchors ρA : A Ñ TM and ρB : B Ñ TM are surjective.

Deﬁnition 3.2. A double Lie algebroid pD; A, B; Mq is transitive if its two core-anchors BA and BB are surjective. Deﬁnition 3.3 ([1]). A crossed module of Lie algebroids pA, C, B,ρq comprises a Lie algebroid A on base M, a totally intransitive Lie algebroid C on the same base, a representation ∇: A Ñ DerpCq, and a morphism B : C Ñ A of Lie algebroids over M, such that

∇arc1,c2s“r∇ac1,c2s`rc1, ∇ac2s, (18a)

∇Bpc1qc2 “rc1,c2s, (18b)

Bp∇acq“ra, Bpcqs. (18c) for all c,c1,c2 P ΓpCq and a P ΓpAq.

18 Here DerpCq denotes the Lie algebroid of derivations of the vector bundle C: sections of DerpCq are R-linear maps D : ΓpCq Ñ ΓpCq for which there is a vector field X P XpMq such that Dpfcq“ fDpcq` Xpfqc for all f P C8pMq and c P ΓpCq. For C a totally intransitive Lie algebroid, Derr¨,¨spCq is the Lie subalgebroid of DerpCq the sections of which are also derivations of the bracket, Drc1,c2s“rDpc1q,c2s`rc1,Dpc2qs for D P ΓpDerr¨,¨spCqq and c1,c2 P ΓpCq. That is, r in Definition 3.3 is a morphism of Lie algebroids A Ñ Derr¨,¨spCq. For clarity, sections of Derr¨,¨spCq are called Lie derivations of C. A Consider now a transitive double Lie algebroid pD; A, B; Mq with core C. Write C “ kerpBBq B A B and C “ kerpBAq. Then C is a smooth subbundle of B and C is a subbundle of A. Proposition 3.4. Let pD; A, B; Mq be a transitive double Lie algebroid with core C. For a P ΓpAq A A A and γ P ΓpC q set ∇a γ “rc,γs where c P ΓpCq is such that BApcq“ a. Then ∇a γ is well-defined A A A and, together with the restriction of BA to C , gives pA, C , BA, ∇ q the structure of a crossed module of Lie algebroids.

Proof. First, BBrc,γs “ rBBc, 0s “ 0 since BB is a Lie algebroid morphism by Proposition 2.9. A 1 1 Therefore, rc,γs is a section of C . If c,c are two sections of C satisfying BApcq“BApc q “ a, 1 B 1 A then c ´ c P kerpBAq“ C and so, by Proposition 2.10, rc ´ c ,γs“ 0 for all γ P C “ kerpBBq. A The Lie algebroid C is totally intransitive as the kernel of the morphism BB of Lie algebroids. A A Then ∇a pfγq“rc,fγs“ frc,γs`ρCpcqpfqγ “ f∇a pcq`ρApaqpfqγ, since ρC “ ρA ˝BA. Likewise, A A A A A A ∇fapγq“rfc,γs“ f∇a pγq because C is totally intransitive. Hence, ∇ : ΓpAqˆΓpC q Ñ ΓpC q is a linear connection. (18b) and (18c) are immediate by the deﬁnition of ∇A and the fact that BA : C Ñ A is a Lie algebroid morphism. A A For the ﬂatness of ∇ , take a1,a2 P ΓpAq and γ P ΓpC q, as well as c1,c2 P ΓpCq with BApc1q“ a1 and BApc2q“ a2. Then BArc1,c2s“ra1,a2s since BA is a morphism of Lie algebroids, and

A A A A A ∇a1 ∇a2 γ ´ ∇a2 ∇a1 γ ´ ∇ra1,a2sγ “rc1, rc2,γss´rc2, rc1,γss´rrc1,c2s,γs“ 0. Similarly,

A A A ∇a rγ1,γ2s´r∇a γ1,γ2s´rγ1, ∇a γ2s“rc, rγ1,γ2ss´rrc,γ1s,γ2s´rγ1, rc,γ2ss “ 0

A for a P ΓpAq, c P ΓpCq with BApcq“ a and γ1,γ2 P ΓpC q, shows the equality (18a). Let pD; A, B; Mq be a transitive double Lie algebroid with core C. The core C and the two A A B B crossed modules pA, C , BA, ∇ q and pB, C , BB, ∇ q constructed in Proposition 3.4 are shown in Figure 3, picturing the transitive core diagram of the transitive double Lie algebroid pD; A, B; Mq. The remainder of this paper shows that a transitive double Lie algebroid can be reconstructed from its (transitive) core diagram.

B C ❆ ❆❆ ❆❆ ❆❆ ❆ CA / C / B ❉❉ BB ❉❉ ❉❉ BA ❉❉ ❉! A

Figure 3. The transitive core diagram of a transitive double Lie algebroid.

19 Definition 3.5. Let A and B be Lie algebroids over a same base M. A core diagram for A and B is a Lie algebroid C on M together with morphisms BA : C Ñ A and BB : C Ñ B of A B Lie algebroids such that the kernels C :“ kerpBBq and C :“ kerpBAq commute in C. A core diagram for A and B is transitive if the morphisms BA and BB are surjective. In the transitive case the proof of Proposition 3.4 applies, and there are representations ∇A : A Ñ DerpCAq, ∇B : B Ñ DerpCB q, (19) defined as in 3.4, which together with the restrictions of BA and BB, respectively, form crossed modules as in Figure 3. This paper shows that every transitive core diagram is the core diagram of a transitive double Lie algebroid. The construction follows the general outline of the analogous results for double Lie groupoids and LA –groupoids [3, 16]: first, Section 4 defines a ‘large’ double Lie algebroid in which the two structures are an action Lie algebroid and a pullback, and then a quotient is taken in Section 5.

Let now pD,A,B,Mq and pD1, A1,B1,Mq be two transitive double Lie algebroids over a common double base M. Consider a morphism Φ: D Ñ D1 of double Lie algebroids, with side morphisms 1 1 1 ϕA : A Ñ A and ϕB : B Ñ B , and with core morphism ϕC : C Ñ C . Then these two side morphisms and the core morphism are morphisms of Lie algebroids over M and, using (10) and 1 Proposition 2.11, the triple pϕA, ϕB, ϕC q is a morphism of the core diagrams of D and D , as in the following deﬁnition. Deﬁnition 3.6. Let A, A1,B,B1 be Lie algebroids over a base M and consider core diagrams 1 1 1 pC, BA, BBq and pC , BA1 , BB1 q for A and B and for A and B , respectively. A morphism 1 Φ: pC, BA, BBq Ñ pC , BA1 , BB1 q 1 1 1 of core diagrams is a triple of Lie algebroid morphisms ϕA : A Ñ A , ϕB : B Ñ B and ϕC : C Ñ C such that BA1 ˝ ϕC “ ϕA ˝BA and BB1 ˝ ϕC “ ϕB ˝BB.

ϕC | B 1 CB C / CB

ϕC | A 1 CA C / CA ❈❈ ❊❊ ❈❈ ❊❊ ❈❈ ❊❊ ❈❈ ❊❊ ❈ ϕ ❊ ! C " 1 C ❇ / C ❉ ❇ ❉❉ ❇❇ BB ❉ BB1 ❇❇ ❉❉ ❇ ❉❉ ❇ ϕ B ! 1 BA B / B

ϕA A / A1

Figure 4. A morphism of core diagrams.

1 Given a morphism Φ: pC, BA, BBq Ñ pC , BA1 , BB1 q of transitive core diagrams as in Figure 4, the 1 A A1 B B1 morphism ϕC : C Ñ C restricts to morphisms C Ñ C and C Ñ C of Lie algebroids. There- fore, all squares in 4 are commutative squares of Lie algebroid morphisms over M. A computation A yields as well for a P ΓpAq, c P ΓpCq such that BAc “ a and γ P ΓpC q:

A A1 A1 ! A1 ϕC p∇ γq“ ϕC rc,γs“rϕC pcq, ϕC pγqs “ ∇ pϕC pγqq “ ∇ pϕC pγqq “ pϕ ∇ qapϕC pγqq. a BA1 ϕC pcq ϕApaq A

20 4 The comma double Lie algebroid

This section constructs the comma double Lie algebroid defined by a morphism of Lie algebroids. It constructs more precisely a functor from morphisms of Lie algebroids over a fixed base, to double Lie algebroids over the fixed double base.

4.1 Action of TC Ñ T M on A Ñ T M Let C and A be Lie algebroids over a manifold M and let B : C Ñ A be a morphism of Lie algebroids over M. This section and the next construct a double Lie algebroid R from this data, the comma double Lie algebroid defined by B : C Ñ A. The construction follows the one of the comma double groupoids in [16, 2.5] and [3]. First an action of the Lie algebroid T C Ñ TM is defined on the anchor ρA : A Ñ TM. Recall that the Lie algebroid structure on T C Ñ TM may be defined as follows, using tangent sections : Tc and core sections c , for c P ΓpCq. For sections c1,c2 P ΓpCq, the bracket on ΓT M pT Cq is given by : : : : rTc1,Tc2s“ T rc1,c2s, rTc1,c2s“rc1,c2s , rc1,c2s“ 0. (20) 2 2 The anchor of T C Ñ TM is Θ :“ σM ˝ TρC where σM : T M Ñ T M is the canonical involution (denoted JM in [19]). An action of T C Ñ TM on ρA : A Ñ TM is a map Φ: ΓT M pT Cq Ñ XpAq such that

Φrξ1, ξ2s“rΦpξ1q, Φpξ2qs, (21a)

Φpξ1 ` ξ2q“ Φpξ1q` Φpξ2q, (21b) ˚ ΦpF ξq “ pρAF qΦpξq, (21c) ˚ ˚ £ΦpξqpρAF q“ ρAp£ΘpξqF q, (21d)

8 for ξ, ξ1, ξ2 P ΓT M pT Cq and F P C pTMq [19, 4.1.1].

Proposition 4.1. Let C and A be Lie algebroids over a manifold M and let B : C Ñ A be a morphism of Lie algebroids over M. For c P ΓpCq, set

ΦpTcq :“ rBc, ¨s P XlpAq, and Φpc:q :“ pBcqÒ P XcpAq. (22)

Then (22) extends to a unique{ action Φ of T C Ñ TM on ρA : A Ñ TM.

l Note that rBc, ¨s P X pAq is linear over ρApBcq“ ρCpcqP XpMq.

Proof. An arbitrary{ section χ of T C Ñ TM can be written k n : χ “ Fi Tci ` Gj γj i“1 j“1 ÿ ÿ 8 with F1,...,Fk, G1,...,Gn P C pTMq and c1,...,ck,γ1,...,γn P ΓpCq. Then set

k n k n ˚ ˚ : ˚ ˚ Ò Φpχq“ ρAFi ΦpTciq` ρAGj Φpγj q“ ρAFi rBci, ¨s ` ρAGj pBγjq . (23) i“1 j“1 i“1 j“1 ÿ ÿ ÿ ÿ {

(23) deﬁnes a map Φ: ΓT M pT Cq Ñ XpAq: For c P ΓpCq and f P C8pMq

: ˚ : ˚ : T pfcq“ ℓdf ¨ c `T M pM f ¨ T c, pfcq “ pM f ¨ c , (24)

21 where `T M and ¨ are the addition and scalar multiplication in T C Ñ TM, and pM : TM Ñ M is the canonical projection. Hence the following identities need to hold:

: ˚ ΦpT pfcqq “ Φpℓdf ¨ c `T M pM f ¨ Tcq (25) and : ˚ : Φppfcq q“ ΦppM f ¨ c q (26) for all c P ΓpCq and f P C8pMq. For the ﬁrst identity, compute

: ˚ ˚ Ò ˚ ˚ Ò ˚ Φpℓdf ¨ c `T M pM f ¨ Tcq“ ρAℓdf ¨ pBcq ` ρApM f ¨ rBc, ¨s “ ℓdAf ¨ pBcq ` qAf ¨ rBc, ¨s “ rBpfcq, ¨s “ ΦpT pfcqq. { { ‚ ‚`1 t ˚ Here dA is the de Rham like diﬀerential{ Ω pAq Ñ Ω pAq deﬁned by A, and ρA : T M Ñ ˚ t A is the pointwise transpose of ρA. The notation ρA avoids confusions with the pullback ˚ ‚ ˚ ‚ ˚ ˚ ρ : Γ T M Ñ Γ T A . It is easy to check that ρ ℓdf “ ℓ t d “ ℓd f for all f P A A ρA f A 8 C pMq. The second equality is then immediate with qA “ pM ˝ ρA. The third equality can be `Ź ˘ `Ź ˘ ˚ ˚ 8 checked easily on linear functions ℓα for α P ΓpA q and qAf, for f P C pMq. For (26), compute

˚ : ˚ ˚ Ò ˚ Ò Ò Ò : ΦppM f ¨ c q“ ρApM f ¨ pBcq “ qAf ¨ pBcq “ pfBcq “ pBpfcqq “ Φ pfcq . ` ˘ Φ: ΓT M pT Cq Ñ XpAq is an action: The identities (21b) and (21c) hold by construction. By the construction in (23) as well, it is suﬃcient to check (21d) on tangent and core sections ξ of T C Ñ TM. Consider ﬁrst ξ “ Tc where c P ΓpCq. Then (21d) reads

£ ρ˚ F ρ˚ £ F rBc,¨sp A q“ Ap pσM ˝T pρC pcqqq q

8 { ˚ for all F P C pTMq. Here again, it is enough to check this equality on pullbacks pM f for 8 1 f P C pMq and linear functions ℓθ for θ P Ω pMq. For am P A and any a P ΓpAq such that apmq“ am, ρ˚ p˚ f q˚ f q˚ f q˚ f , £rBc,¨sp A M q“ £rBc,¨sp A q“ Ap£pρA˝Bqpcq q“ Ap£ρC pcq q ˚ t t £ {pρ ℓθqpamq“ rB{c, ¨spℓ t qpamq“ ℓ£ t pamq“ ρC pcmqxρ θ,ay´xρ θm, rBc,aspmqy rBc,¨s A ρAθ BcρAθ A A “ ρ pc qxθ,ρ paqy´xθ , rρ pcq,ρ paqspmqy “ ℓ pa q, { {C m A m C A α m t ˚ where α :“ ρA£ρC pcqpθqP ΓpA q. Now consider the right hand side of (21d). Choose am P A, let γ be a path through m with ρC pcq γ9 p0q“ ρApamq and let ϕ¨ be the ﬂow of ρC pcqP XpMq. Then d  d    ρC pcq σM pT pρCpcqqρApamqq “ σM   ϕ pγptqq dt ds s ˆ t“0 s“0 ˙ d  d  d    ρC pcq  ρC pcq “   ϕs pγptqq “  Tmϕs pρApamqq, dss“0 dt t“0 dss“0 and so d  ˚  ρC pcq ˚ σM pT pρCpcqqρApamqqppM fq“  pf ˝ ϕs qpmq “ p£ρC pcqfqpmq“ qAp£ρC pcqfqpamq, dss“0 d   ρC pcq ρC pcq σM pT pρCpcqqρApamqqpℓθq“  xθpϕs pmqq,Tmϕs pρApamqqy dss“0

“x£ρC pcqθ,ρApamqy “ ℓαpamq,

22 t where α “ ρA£ρC pcqpθq as before. This completes the proof of (21d) for ξ “ Tc. : : Ò Now consider c P ΓT M pT Cq. The equality Θpc q “ pρC pcqq P XpTMq can be veriﬁed using the same method as above. It only remains to prove that

˚ ˚ Ò Ò £pBcq pρAF q“ ρAp£ρC pcq F q which is immediate for F a pullback or a linear function on TM. Finally the compatibility of Φ with the Lie algebroid brackets (21a) needs to be checked:

rΦpV q, ΦpW qs “ ΦrV, W s for all V, W P ΓT M pT Cq. Because of (21d), it is suﬃcient to prove (21a) on tangent and core sections of T C Ñ TM. Computations yield for c,c1 P ΓpCq:

ΦpTcq, ΦpTc1q “ rBc, ¨s, rBc1, ¨s “ rBc, ¨s˝rBc1, ¨s´rBc1, ¨s˝rBc, ¨s

“ ‰ ” 1 ı 1 1 “ rBr{c,c s,{¨s “ Φ T rc,c s “ ΦprTc,Tc{ sq, ` ˘ by the Jacobi identity for the Lie algebroid{ bracket on A, and

Ò ΦpTcq, Φ c1: “ rBc, ¨s, pBc1qÒ “ rBc, ¨spBc1q

” ´ ¯ı ” 1 Ò ı 1 Ò 1 : 1: “ rB{c, Bc s “ pBrc,c`sq “ Φ rc,c˘ s “ Φ Tc,c . ` ˘ ´” ı¯ The Lie algebroid brackets of core sections of T C Ñ TM, and of vertical vector ﬁelds on A vanish, so the proof is complete.

4.2 The double vector bundle R Consider two Lie algebroids A and C over a common base M and construct the pullback manifold R :“ T C ‘T M A :“ tpV,aq P T C ˆ A | T qCpV q “ ρApaqu, as shown in Figure 5(a). This section shows that R has a double vector bundle structure with sides A and C and with core C.

pullback R / A R / C

ρA action

T qC T C / TM A / M (a) The pullback manifold (b) The double Lie algebroid

Figure 5

The space R is equipped with two projections πA : R Ñ A, pV,aq Ñ a and πC : R Ñ C, pV,aq Ñ pC pV q, where as always, pC : T C Ñ C is the canonical projection. Deﬁne additions

À : R Â R Ñ R, pV1,aqÀ pV2,aq “ pV1 `T M V2,aq, (27) and `C : R ˆC R Ñ R, pV1,a1q`C pV2,a2q “ pV1 `C V2,a1 ` a2q. (28)

23 Proposition 4.2. An A-connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq on C deﬁnes a bijection

Ò I∇ : A ˆM C ˆM C Ñ R, pa,c,γq ÞÑ ∇apcq`C γ pcq,a , ´ ¯ such that x πC pI∇pa,c,γqq “ c and πApI∇pa,c,γqq “ a for pa,c,γqP A ˆM C ˆM C, and

I∇pa1 ` a2,c,γ1 ` γ2q“ I∇pa1,c,γ1q`C I∇pa2,c,γ2q as well as I∇pa,c1 ` c2,γ1 ` γ2q“ I∇pa,c1,γ1q`A I∇pa,c2,γ2q for pa1,c,γ1q, pa2,c,γ2q and pa,c1,γ1q, pa,c2,γ2qP A ˆM C ˆM C.

Proof. Take pa1,c1,γ1q and pa2,c2,γ2qP AˆM CˆM C such that I∇pa1,c1,γ1q“ I∇pa2,c2,γ2qP R. Then Ò Ò ∇a1 pc1q`C γ1 pc1q,a1 “ ∇a2 pc2q`C γ2 pc2q,a2 ,

´ ¯ ´ Ò ¯ Ò and so a1 “ a2, as welly as c1 “ pCp∇a1 pc1q`C γy1 pc1qq “ pCp∇a2 pc2q`C γ2 pc2qq “ c2. Then Ò Ò Ò Ò ∇a1 pc1q`C γ1 pc1q“ ∇a1 pc1q`C γ2 pc1q implies γ1 pc1q“ γ2 pc1q, and so γ1 “ γ2. This shows that I∇ is injective. y y yNow consider pV,ayq P R and set pC pV q “ c. Then T qCp∇apcqq “ ρApaq “ T qCpV q, which ´1 Ò implies the existence of γ P qC pqC pcqq such that V “ ∇apcq`C γ pcq. Hence pV,aq“ I∇pa,c,γq, which shows that I∇ is surjective. x The compatibilities of I∇ with the projections andx additions are immediate and left to the reader.

This implies immediately that R is a double vector bundle with sides A and C, with core C and the projections and additions above.

Corollary 4.3. R is a double vector bundle with side projections πA : R Ñ A, pV,aq Ñ a and πC : R Ñ C, pV,aq Ñ pC pV q with additions in (27) and (28) and with core C.

In the situation of Proposition 4.2, the isomorphism I∇ of double vector bundles is a decompo- A l sition of R. Recall that ∇ defines as a consequence linear horizontal lifts σ∇ : ΓpAq Ñ ΓC pRq and C l A σ∇ : ΓpCq Ñ ΓApRq. More precisely, given a section a P ΓpAq, the section a :“ σ∇paq P ΓC pRq is defined by apcmq “ p∇apcmq,apmqq, cm P C. p (29) C Given a section c P ΓpCq, the section c :“ σ∇pcqP ΓApRq is defined by p x cpqamq “ p∇apcpmqq,amq, (30) for am P A and any a P ΓpAq such that apmq“xam. Given a section γ P ΓpCq, the corresponding ˆ q core section γ P ΓC pRq is given by

ˆ Ò A γ pcmq “ pγ pcmq, 0mq, cm P C. (31)

; The corresponding core section γ P ΓApRq is given by

; : C Ò C γ pamq “ pγ pρApamqq,amq “ pT 0 pρApamqq `C γ p0mq,amq a P A. (32)

24 4.3 The comma double Lie algebroid Consider again two Lie algebroids A and C over a common base M and assume a Lie algebroid morphism B :“BA : C Ñ A as in Section 4.1. This section equips R constructed from A and C as in Section 4.2 with two Lie algebroid structures, over C and over A, and shows that they constitute together a double Lie algebroid pR,A,C,Mq with core C. As a Lie algebroid over C, R is the pullback Lie algebroid structure of A over qC : C Ñ M, !! denoted qC A in [19]. As a Lie algebroid over A, R is the action Lie algebroid structure for the action Φ of T C Ñ TM on ρA as deﬁned in (22) and (23). The remainder of this section describes these structures in detail.

Proposition 4.4. Consider a Lie algebroid morphism B : C Ñ A over M and equip R “ T C‘T M A with the pullback Lie algebroid structure of A over qC . Consider an A-connection ∇: ΓpAqˆ l ΓpCq Ñ ΓpCq on C and write ΘC for the anchor R Ñ T C. Then ΘC ˝ a “ ∇a P X pCq and ˆ Ò ΘC ˝ γ “ γ P XpCq for a P ΓpAq and γ P ΓpCq. Furthermore, the Lie algebroid bracket on R Ñ C is given by: p x

ˆ ˆ ˆ ˆ ra1, a2s“ ra1,a2s´ R∇pa1,a2q, a,γ “ p∇aγq , γ1 ,γ2 “ 0 (33) for a,a ,a P ΓpAq and γ,γ ,γ P ΓpCq. “ ‰ “ ‰ 1 2 p p {1 2 Č p Corollary 4.5. In the situation of Proposition 4.4, the Lie algebroid structure on R Ñ C is linear over A Ñ M. In other words, pR Ñ C, A Ñ Mq is a VB-algebroid. Proof of Proposition 4.4. The proof follows from the basic properties of pullbacks of Lie algebroids [19, p.156-157]. Following the formula (19) there

l ΘC paq“ ΘCp∇a,aq“ ∇a P X pCq for all a P ΓpAq, and x x pˆ Ò A Ò c ΘC pγ q“ ΘCpγ , 0 q“ γ P X pCq for all γ P ΓpCq. Following (20) in [19, p.157],

ra1, a2s“ ∇a1 ,a1 , ∇a2 ,a2 “ ∇a1 , ∇a2 , ra1,a2s ”´ ¯ ´ ¯ı ´” ı ¯ “ ∇ ´ R pa ,a q, ra ,a s “ ra ,a s´ R pa ,a q, p p yra1,a2s ∇y1 2 1 2y y1 2 ∇ 1 2 ´ ¯ ˆ {Ò A Č Ò A { Ò ČA ˆ a,γ “ ∇a,a , γ , 0 “ ∇a,γ , ra, 0 s “ p∇aγq , 0 “ p∇aγq and “ ‰ ”´ ¯ ` ˘ı ´” ı ¯ ` ˘ p ˆ ˆ x Ò A Ò A x Ò Ò A A TC A γ1 ,γ2 “ γ1 , 0 , γ2 , 0 “ γ1 ,γ2 , r0 , 0 s “ 0C , 0 “ 0 ”´ ¯ ´ ¯ı ´” ı ¯ for all a,a1,a“2 P ΓpA‰q and γ,γ1,γ2 P ΓpCq. ` ˘ Deﬁne now the connections ∇B : ΓpCqˆ ΓpAq Ñ ΓpAq,

B ∇c a “ rBc,as`Bp∇acq and ∇B : ΓpCqˆ ΓpCq Ñ ΓpCq, B 1 1 ∇c c “rc,c s` ∇Bc1 c, bas 2 and the tensor R∇ P Ω pC, HompA, Cqq:

B 1 1 1 1 1 R∇pc,c qpaq“´∇arc,c s`r∇ac,c s`rc, ∇ac s´ ∇∇Bac ` ∇∇B ac. (34) c c1

25 Proposition 4.6. Consider a Lie algebroid morphism B : C Ñ A over M and equip R “ T C‘T M A with the action Lie algebroid structure over A. Consider an A-connection ∇: ΓpAqˆΓpCq Ñ ΓpCq B Xl ; Ò on C and write ΘA for the anchor R Ñ T A. Then ΘApcq “ ∇c P pAq and ΘApγ q “ pBγq P XpAq for c,γ P ΓpCq. Furthermore, the Lie algebroid bracket on R Ñ A is given by: q x B ; B ; ; ; rc1, c2s“ rc1,c2s´ R∇pc1,c2q, c,γ “ p∇c γq , γ1 γ2 “ 0 (35) “ ‰ ” ı for c,c1,c2,γ,γ1,γq2 PqΓpCq. Č q Corollary 4.7. In the situation of Proposition 4.6, the Lie algebroid structure on R Ñ A is linear over C Ñ M. In other words, pR Ñ A, C Ñ Mq is a VB-algebroid. Proof of Proposition 4.6. The action Lie algebroid defined by the action (22) of T C Ñ TM on A Ñ TM lives on the pullback vector bundle of T C Ñ TM across ρA [19, Prop. 4.1.2]. As a manifold this is R, and the bundle projection is the projection onto the second factor, pV,aq ÞÑ ! a. As with any pullback vector bundle, the sections of R Ñ A are finite sums i Fi χi where 8 ! Fi P C pAq and χi P ΓT M pT Cq. Here, for χ P ΓT M pT Cq, the section χ P ΓApRq is given by ! ř χ paq “ pχpρApaqq,aq for all a P A. The anchor ΘA : R Ñ T A is then defined by ΘAp i Fi χiq“ i Fi Φpχiq. The Lie bracket on ΓApRq is defined by ! ! ř ! ř χ1,χ2 “rχ1,χ2s (36) for all χ1,χ2 P ΓT M pT Cq. “ ‰ ; c : ! By Definition, the section γ P ΓApRq defined by γ P ΓpCq equals pγ q P ΓApRq. Consider c P ΓpCq, and set n ΓpHompA, Cqq Q ∇¨c “ αj b cj j“1 ÿ ˚ with α1,...,αn P ΓpA q and c1,...,cn P ΓpCq. It is then easy to check that

n l ! : ! ΓApRqQ c “ pTcq ´ ℓαj ¨ pcj q . j“1 ÿ ; q : Ò Simple computations yield ΘApγ q“ Φpγ q“pBγq and

n n : Ò B ΘApcq“ ΦpTcq´ ℓαj ¨ Φpcj q“ rBc, ¨s ´ ℓαj ¨ pBcjq “ ∇c j“1 j“1 ÿ ÿ q { x for γ,c P ΓpCq. The Lie bracket on ΓApRq is then given by

! ; ; : : γ1,γ2 “ γ1,γ2 “ 0, ” ı ” ı n n ; ! : ! : ! ; ; : ! c,γ “ pTcq ´ ℓαj ¨ pcj q , pγ q “rc,γs ` £ΘAppγ q qpℓαj q ¨ cj « j“1 ﬀ j“1 “ ‰ ÿn n ÿ q ; ; ; ˚ ; ; B ; “rc,γs ` £pBγqÒ pℓαj q ¨ cj “rc,γs ` qAxBγ, αjy ¨ cj “ prc,γs` ∇Bγ cq “ p∇c γq , j“1 j“1 ÿ ÿ

26 ! n : ! 1 ! n : ! and for c “ pTcq ´ ℓ ¨ pc q and c “ pTcq ´ ℓ 1 ¨ pc q , j“1 αj j k“1 αk k

ř n n ř n n q ! 1 ! q; ; ; ; c, c1 “ pTcq , pTc q ´ c, ℓ 1 ¨ c ´ ℓ ¨ c , c1 ´ ℓ ¨ c , ℓ 1 ¨ c αk k αj j αj j αk k « k“1 ff «j“1 ff «j“1 k“1 ff ” ı “ ‰ ÿ ÿ ÿ ÿ q n n q n n q 1 ! q ; B ; ; B ; ˚ 1 ˚ “ pT rc,c sq ´ ℓp∇Bq α1 ¨ ck ´ ℓα ¨ p∇c ckq ` ℓp∇Bq α ¨ cj ` ℓαj ¨ p∇c1 cj q c k k c1 j k“1 k“1 j“1 j“1 n ÿ ÿ n ÿ ÿ ˚ 1 ; ˚ ; ´ ℓ ¨ q xBc , α y ¨ c ` ℓ 1 ¨ q xBc , α y ¨ c . αj A j k k αk A k j j j,k“1 j,k“1 ÿ ÿ

1 1 B 1 A straightforward computation yields that this is c, c “ rc,c s´ R∇pc,c q. ” ı q q ~ Č Let A and C be two Lie algebroids over M and B : C Ñ A a Lie algebroid morphism over M. Choose an A-connection on C: ∇: ΓpAqˆ ΓpCq Ñ ΓpCq. (55) and (51) give the representations up to homotopy describing the VB-algebroid structures R Ñ C and respectively R Ñ A in the splitting deﬁned by ∇ as in Proposition 4.2. Via the splitting given by ∇, the Lie algebroid R Ñ C is described by

2 id: C Ñ C, ∇: ΓpAqˆ ΓpCq Ñ ΓpCq, R∇ P Ω pA, HompC, Cqq. (37)

The Lie algebroid R Ñ A is described by

B : C Ñ A, ∇B : ΓpCqˆ ΓpCq Ñ ΓpCq, ∇B : ΓpCqˆ ΓpAq Ñ ΓpAq, (38) and B 2 R∇ P Ω pC, HompA, Cqq. (39) Straightforward computations (carried out in Appendix A) yield that these two representations up to homotopy are matched in the sense of [7] and so that R “ T C ‘T M A, with its two VB-Lie algebroid structures, is a double Lie algebroid with sides A and C, and with core C. This completes the proof of the following theorem.

Theorem 4.8. Let A and C be Lie algebroids over M and let B : C Ñ A be a morphism of Lie algebroids. Then the double vector bundle pR; A, C; Mq constructed in Section 4.2 inherits a double Lie algebroid structure with the following two Lie algebroid structures on R Ñ A and R Ñ C:

1. As a Lie algebroid over C, R is the pullback Lie algebroid structure of A over qC : C Ñ M. 2. As a Lie algebroid over A, R is the action Lie algebroid structure for the action Φ of T C Ñ TM on ρA as deﬁned in (22) and (23). The core of pR; A, C; Mq is C and its core diagram is as in Figure 6. As a consequence, R is transitive if and only if B is surjective.

4.4 Morphisms of comma-double Lie algebroids Consider in this section a square of Lie algebroid morphisms over M as in Figure 7. This section proves that such a square induces a morphism of the corresponding comma double Lie algebroids 1 1 T C ‘T M A Ñ T C ‘T M 1 A , with side morphisms ϕA and ϕC and with core morphism ϕC .

27 kern B ❋❋ ❋❋ ❋❋ ❋❋ ❋" 0 / C / C ❊❊ id ❊❊ ❊❊ B ❊❊ ❊ " A

Figure 6. The core diagram of R.

B C A / A

ϕC ϕA C1 / A1 BA1

Figure 7. Square of Lie algebroid morphisms.

Proposition 4.9. Consider a square of Lie algebroid morphisms as in Figure 7. Then

1 1 Φ: T C ‘T M A Ñ T C ‘T M 1 A , pV,aq ÞÑ pT ϕC V, ϕApaqq

1 1 deﬁnes a morphism of double vector bundles with side morphisms ϕA : A Ñ A and ϕC : C Ñ C 1 and with core morphism ϕC : C Ñ C .

Proof. First of all, Φ is well-deﬁned since for pV,aqP T C‘T M A, the image ΦpV,aq “ pT ϕC V, ϕApaqq satisﬁes T qC1 T ϕCV “ T qCV “ ρApaq“ ρA1 ϕApaq 1 1 and is hence an element of T C ‘T M C . Choose an A-connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq on C, and an A1-connection ∇1 : ΓpA1qˆ ΓpC1q Ñ ΓpC1q on C1. Consider the induced linear splittings as in Proposition 4.2

Ò I∇ : A ˆM C ˆM C Ñ T C ‘T M A, pa,c,γq ÞÑ ∇apcq`C γ pcq,a ´ ¯ and x 1 1 1 1 1 1 1 1 1 1 1Ò 1 1 I∇1 : A ˆM C ˆM C Ñ T C ‘T M A , pa ,c ,γ q ÞÑ ∇a1 pc q`C1 γ pc q,a .

˚ ˚ 1 ´ ¯ Deﬁne ω∇,∇1 P ΓpA b C b C q by y 1 ω 1 a,c ϕ c ϕ c ∇,∇ p q“ ∇ϕApaqp C p qq ´ C p∇a q (40) for a P ΓpAq and c P ΓpCq. Then an easy computation proves that

Ò 1 1 Tcϕc∇apcq´ ∇ϕpaqpϕC pcqq “ ω∇,∇ pa,cq pϕC pcqq for all c P C and a P A. Therefore,x {

´1 ´1 Ò ´1 Ò pI∇1 ˝ Φ ˝ I∇qpa,c,γq“ I∇1 ˝ Φqp∇apcq`C γ pcq,a “ I∇1 T ϕC p∇apcqq `C T ϕCpγ pcqq, ϕApaq

´´1 1 ¯ ´ Ò ¯ “ I 1 ∇ pϕpcqq `C pϕC γ ` ω , 1 pa,cqq pϕC pcqq, ϕApaq ∇ ϕApaxq ∇ ∇ x

“ pϕ p´aq, ϕ pcq, ϕ pγq` ω 1 pa,cqq. ¯ A {C C ∇,∇ (41)

28 ´1 1 1 1 This shows that I∇1 ˝Φ˝I∇ : AˆM C ˆM C Ñ A ˆM C ˆM C is a double vector bundle morphism, and consequently that Φ is a double vector bundle morphism. The following theorem is then easily proved in decompositions, see Appendix B. Theorem 4.10. Consider a square of Lie algebroid morphisms as in Figure 7. Then the morphism 1 1 Φ: T C ‘T M A Ñ T C ‘T M 1 A , pV,aq ÞÑ pT ϕC V, ϕApaqq of double vector bundles as in Proposition 4.9 is a morphism of the double Lie algebroids constructed in Theorem 4.8.

5 The quotient

Let now A and B be Lie algebroids over a common base M and consider a transitive core diagram for A and B as in Figure 8 (see Deﬁnition 3.5). Recall the representation ∇A : A Ñ DerpCAq

B C ❆ ❆❆ ❆❆ ❆❆ ❆ CA / C / B ❉❉ BB ❉❉ ❉❉ B“BA ❉❉ ❉! A

Figure 8. Setting of Section 5, the transitive core diagram C. deﬁned in (19). For a P ΓpAq, consider the vector ﬁeld

A Xl A ∇a P pC q corresponding to the derivation ∇A of CA over ρ paq. a y A Lemma 5.1. In the situation above, choose c P ΓpCq and set a “BAc. Then the vector field Xl A A A rc, ¨s P pCq restricts to ∇a on points of C . In other words, if ι: C ãÑ C is the inclusion, then A ∇a „ι rc, ¨s. y y Proof. Check first that rc, ¨spγ q P T CA for all γ P CA. In order to do this, it is sufficient to y y m γm m 8 A show rc, ¨spF q“ 0 for all F P C pCq such that F |CA is constant. Since C Ď C is a subbundle, it 8 A ˝ ˚ suffices to check this ony the linear functions ℓµ P C pCq for µ P Γ pC q Ď ΓpC q. But this is immediatey since ` ˘ rc, ¨spℓµqpγq“ ρC pcqxµ,γy´xµ, rc,γsy “ 0 for all γ P ΓpCAq, since rc,γsP ΓpCAq. y ˚ ˚ ˚ 8 ˚ ˚ Then ι qC f “ qCA f for all f P C pMq and ι ℓµ “ ℓιtµ for all µ P ΓpC q. The definitions of A A ˚ t t rc, ¨s and ∇a yield easily p∇a q pι µq“ ι £cµ and so

˚ ˚ ˚ t £ A pι ℓµq“ £ A ℓι µ “ ℓp∇Aq˚ιtµ “ ι ℓ£cµ “ ι prc, ¨spℓµqq y y ∇a ∇a a

˚ y y A for µ P ΓpC q. Similarly, together with ρC pcq “ ρApBcq “ ρApaq, they deﬁnitions of rc, ¨s and ∇a yield ˚ ˚ ˚ ˚ ˚ ˚ ˚ y˚ y £ A pι qC fq“ £ A pqCA fq“ qCA p£ρApaqpfqq “ ι qC p£ρC pcqpfqq “ ι p£ pqC fqq ∇a ∇a rc,¨s for f P C8ypMq. y z

29 Deﬁne the subset Π Ď R by

A A A Π:“ ∇a pγmq,apmq P T C ‘T M A a P ΓpAq, γm P C . (42) !´ ¯ ˇ ) ˇ By Lemma 5.1, it equals y ˇ

A A rc, ¨spγmq, Bcpmq P T C ‘T M A c P ΓpCq, γm P C . !´ ¯ ˇ ) ˇ A Proposition 5.2. Choosey any connection ∇: ΓpAqˆ ΓˇpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all A a P ΓpAq and γ P ΓpC q. Then the isomorphism I∇ : A ˆM C ˆM C Ñ R of double vector bundles in Proposition 4.2 and Corollary 4.3 does

A C I∇pA ˆM C ˆM 0 q“ Π.

As a consequence, Π is a sub-double vector bundle of R with sides A and CA and with trivial core.

Xl A A Xl Proof. As in the proof of Lemma 5.1, prove that pC q Q ∇a „ι ∇a P pCq. Then given C A C pam,γm, 0mqP A ˆM C ˆM 0 and any a P ΓpAq such that apmq“ am: y x C A A I∇pam,γm, 0mq“ ∇apγmq,am “ ∇a pγmq,am P Π Ď T C ‘T M A. ´ ¯ ´ ¯ The following corollary is immediate,x using they notation (30).

A Corollary 5.3. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq and γ P ΓpCAq. Then the subbundle Π Ñ A of R Ñ A is spanned by the sections

C l σ∇pγq“ γ P ΓApRq for all γ P ΓpCAq. q 5.1 The quotient of R by Π. This section deﬁnes the quotient R{Π and shows that it inherits a double vector bundle structure. In the situation above deﬁne „ on R by

1 1 1 A 1 pV,aq „ pV ,a q :ô a “ a and D W P T C : pW, aqP Π and V `T M W “ V .

Then it is easy to check that „ deﬁnes an equivalence relation on R. Two pairs pV,aq and pV 1,aqP R 1 are equivalent if and only if there exists pW, aqP Π such that pV,aq`A pW, aq “ pV ,aq. Hence, it is easy to see that R{„ “: R{Π has a vector bundle structure over A: the quotient of the vector bundle R Ñ A by its subbundle Π Ñ A. The reader should think about R{Π as this quotient. The following theorem shows that the vector bundle structure on R Ñ C induces a vector bundle structure on R{Π Ñ C{CA, such that pR{Π, A, C{CA,Mq becomes a double vector bundle. Theorem 5.4. 1. The space R{Π inherits from R a double vector bundle structure with core C and with side C{CA » B.

A 2. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq and A γ P ΓpC q. Then the induced decomposition I∇ : AˆM C ˆM C Ñ R induces a decomposition

A I∇ : A ˆM pC{C qˆM C Ñ R{Π, I∇pa, c,γq :“ I∇pa,c,γq`A Π.

The remainder of this section proves this theorem. Deﬁne FC Ď T C as the subbundle over C A spanned by the vertical vector ﬁelds corresponding to sections of C , i.e. for all cm P C:

Ò A FC pcmq“ γ pcmq | γ P ΓpC q ( 30 and so Ò A ΓpFC q“ spanC8pCq γ | γ P ΓpC q . A The subbundle FC Ď T C is involutive and simple since its leaf space( is C{C » B. Choose a linear A-connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq that restricts to ∇A on ΓpCAq, and l consider the induced sections a P ΓC pRq for all a P ΓpAq, as in (29). Recall also from (31) that ˆ c each section γ P ΓpCq deﬁnes a core section γ P ΓCpRq. Proposition 5.5. In the situationp above, there is a unique ﬂat connection

i ∇ : ΓC pFC qˆ ΓC pRq Ñ ΓC pRq, such that ∇ia “ 0 “ ∇iγˆ for all a P ΓpAq and all γ P ΓpCq. The connection ∇i does not depend on the choice of ∇ extending ∇A. p

Proof. The space of sections of the vector bundle R Ñ C is spanned as a C8pCq-module by sections of the form a and γˆ for a P ΓpAq and γ P ΓpCq. Deﬁne

∇i : Γ pF qˆ Γ pRq Ñ Γ pRq p C C C C by s t s t i ˆ ˆ ∇χ Fj aj ` Glγl “ £χpFj q ¨ aj ` £χpGlq ¨ γl ˜j“1 l“1 ¸ j“1 l“1 ÿ ÿ ÿ ÿ 8 for F1,...,Fs, G1,...,Gt P C ppCq, a1,...,as P ΓpAq and γ1,...,γp t P ΓpCq. For f P C8pMq, a P ΓpAq and γ P ΓpCq

˚ ˆ ˚ ˆ fa “ qC f ¨ a and pfγq “ qC f ¨ γ .

i i i ˚ i ˆ i ˚ ˆ 8 Hence ∇ is well-deﬁned ifx∇¨ fa “ ∇p¨ pqC f ¨ aq and ∇¨ pfγq “ ∇¨ pqC f ¨ γ q for all f P C pMq, i i ˆ a P ΓpAq and γ P ΓpCq. On the one hand, ∇χfa “ 0 “ ∇χpfγq and on the other hand x p i ˚ ˚ ∇χ pqC f ¨ aq“x £χpqC fq ¨ a “ 0 and i ˚ ˆp ˚ p ˆ ∇χ qC f ¨ γ “ £χpqC fq ¨ γ “ 0

qC since χ P ΓpFC q and FC Ď T C. ` ˘ The connection ∇i is obviously ﬂat with the required ﬂat sections. Let ∇1 : ΓpAqˆΓpCq Ñ ΓpCq be a second extension of ∇A : ΓpAqˆ ΓpCAq Ñ ΓpCAq. Then ϕ :“ ∇ ´ ∇1 P Ω1pA, pCAq˝ b Cq ∇1 ∇ k does σA paq “ σA paq´ ϕpaq “ a ´ ϕpaq for all a P ΓpAq. Writing ϕpaq “ j“1 νj b cj with ν ,...,ν P ΓppCAq˝q and c ,...,c P ΓpCq yields 1 k 1 k ř Ć p Ć k k i ∇1 i i ˆ ˆ ∇χ σA paq “ ∇χ a ´ ϕpaq “ ∇χ ℓνj ¨ cj “ £χpℓνj q ¨ cj “ 0 ˜j“1 ¸ j“1 ´ ¯ ´ ¯ ÿ ÿ Ć 8 pFC since χ P ΓpFC q and ℓνj P C pCq . The last proposition is reformulated as follows.

i Corollary 5.6. The triple pFC , 0, ∇ q is an inﬁnitesimal ideal system in the vector bundle R Ñ C considered as a trivial Lie algebroid.

31 i 1 1 The quotient R{∇ Ñ C{FC is construted as follows. Two vectors pV,aq and pV ,a q P R are 1 1 equivalent if cm “ pCpV q and cm “ pCpV q lie in the same leaf of FC , i.e. they can be joined by an integral path of FC and there exists a parallel section along this path taking values pV,aq at 1 1 1 1 1 the point cm and pV ,a q at cm. Note that here cm “ pCpV q and cm “ pC pV q lie in the same leaf 1 A 1 1 of FC if an only if cm ´ cm P C . Recall as well that pV,aq and pV ,a q P R project to the same element in R{Π if

1 1 A 1 A 1 a “ a , pC pV q´ pCpV qP C and V ´T M V “ ∇a ppC pV q´ pC pV qq.

Proposition 5.7. In the situation above, the two equivalence relationsy „ and „∇i coincide. 1 1 Proof. Assume ﬁrst that pV,aq and pV ,a q are equivalent via „∇i . Let cm “ pCpV q and dm1 “ 1 pCpV q. Since cm and dm1 are in the same leaf of FC , they must be in the same ﬁber of qC : C Ñ M 1 A and hence m “ m . Furthermore cm ´ dm P C pmq and there exists without loss of generality one i 1 1 ∇ -parallel section χ of R Ñ C such that χpcmq “ pV,aq and χpdmq “ pV ,a q. Write

k l ˆ χ “ Fiai ` Gj cj i“1 j“1 ÿ ÿ

p 8 FC with a1,...,ak P ΓpAq, c1,...,cl P ΓpCq and F1,...,Fk, G1,...,Gl P C pCq . Since the functions Fi, Gj are all FC -invariant, Fipcmq “ Fipdmq “: αi and Gj pcmq “ Gj pdmq “ βj for i “ 1,...,k and j “ 1,...,l. Thus

k l k l k ˆ Ò pV,aq“ αiaipcmq` βj cj pcmq“ αi∇ai pcmq` βj cj pcmq, αiaipmq i“1 j“1 ˜i“1 j“1 i“1 ¸ ÿ ÿ ÿ ÿ ÿ y and p

k l k l k 1 1 ˆ Ò pV ,a q“ αiaipdmq` βj cj pdmq“ αi∇ai pdmq` βj cj pdmq, αiaipmq i“1 j“1 ˜i“1 j“1 i“1 ¸ ÿ ÿ ÿ ÿ ÿ y k p 1 1 This implies a “ i“1 αiaipmq“ a and the diﬀerence V ´T M V in the ﬁber of T C Ñ TM over ρApaq equals by the interchange law ř k k l l 1 Ò Ò V ´T M V “ αi∇ai pcmq´T M αi∇ai pdmq `C βj cj pcmq´T M βj cj pdmq ˜i“1 i“1 ¸ ˜j“1 j“1 ¸ ÿ ÿ ÿ ÿ k y y k TC A A “ αi∇ai pcm ´ dmq `C 0cm´dm “ αi∇ai pcm ´ dmq “ ∇a pcm ´ dmq. ˜i“1 ¸ ˜i“1 ¸ ÿ ÿ y y y This shows that pV,aq and pV 1,a1q are equivalent in R{Π. 1 1 Conversely, consider pV,amq and pV ,amqP R, set cm “ pCpV q and dm “ pC pV q and choose a 1 section a P ΓpAq with apmq“ am. If pV,amq and pV ,amq are equivalent in R{Π, i.e. if cm ´ dm P A 1 A C pmq and V ´ V “ ∇a pcm ´ dmq, then, since T qCV “ ρApamq “ T qC∇apcmq, there exists Ò 1 γ P ΓpCq such that V “ ∇apcmq` γ pcmq. In the same manner, there is a section γ P ΓpCq such 1 1Òy 1 A x that V “ ∇apdmq` γ pdmq. Furthermore, since V ´ V “ ∇a pcm ´ dmq“ ∇apcm ´ dmq by the 1 x 1 ˆ choice of ∇, V ´ V “ ∇apcmq´ ∇apdmq and hence γpmq“ γ pmq. Set χ “ a ` γ . Then χ is a ∇i-parallelx section of R Ñ C and y x x x 1 p pV,amq“ χpcmq and pV ,amq“ χpdmq.

32 The previous proposition shows that there is a well-deﬁned projection

πB : R{Π Ñ B, pV,aq` Π ÞÑBBppC pV qq and a well-defined addition `B : pR{ΠqˆB pR{Πq Ñ R{Π 1 1 in the fibers of πB which is defined as follows. If pV,amq and pV ,amqP R are such that πB ppV,amq` 1 1 1 1 A Πq“BBppC pV qq“BBppC pV qq “ πB ppV ,amq` Πq, then pC pV q´ pC pV qP Cm. Set cm “ pCpV q 1 1 1 1 1 and cm “ pC pV q. Choose a section a P ΓpAq such that a pmq “ am, and γ P ΓpCq such that 1 1 1 1 ˆ 1 1 ˆ i 1 A 1 1 pV ,amq“ a pcmq`C γ pcmq. Since a and γ are ∇ -flat and cm ´ cm P C , the vectors pV ,a q 1 ˆ i and pa pcmq` γ pcmq are ∇ -equivalent and so p p 1 1 1 ˆ 1 p pV ,a q„ a pcmq` γ pcmq,a ´ ¯ by Proposition 5.7. Then set p

1 1 1 ˆ ppV,aqÀ Πq`B ppV ,a qÀ Πq :“ pV,aq`C pa pcmq`C γ pcmq À Π. ´ ´ ¯¯ Proof of Theorem 5.4. Choose as before a connection ∇: ΓppAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ A A ∇a γ for all a P ΓpAq and γ P ΓpC q. Consider the induced decomposition I∇ : AˆM C ˆM C Ñ R. A Since I∇pa, c, 0qP Π for pa, c, 0qP A ˆM C ˆM 0, the morphism

I∇ A ˆM C ˆM C / R

πA πA A / A idA of vector bundles over A factors to a morphism

A I∇ A ˆM C{C ˆM C / R{Π

πA πA A / A idA which is given by

A I∇ : A ˆM pC{C qˆM C Ñ R{Π, I∇pa, c,γq :“ I∇pa,c,γq`A Π.

A Since I∇pAˆM C ˆM 0q“ Π, the reduced morphism I∇ is again an isomorphism of vector bundles over A. 1 A 1 1 Consider bm “BBcm “BBcm P C{C , γ,γ P ΓpCq and a,a P ΓpAq. Then

1 1 1 I∇papmq, cm,γmq`B I∇pa pmq, cm,γmq

ˆ 1 1 1ˆ 1 “ apcmq`C γ pcmq À Π `B a pcmq`C γ pcmq À Π ´´ ˆ ¯ ¯ “ `àpc q` γˆpc q˘ ` Π˘ ` a1pc q` γ1 pc q ` Π p m C m A B p m C m A ´´ ¯ ¯ “ `à ` a1pc q` pγ `˘ γ1qˆpc˘ q ` Π “ I papmq` a1pmq, c ,γpmq` γ1pmqq. p m C m Ap ∇ m ´ ¯ A A Hence I∇ is compatible{ with the addition over B » C{C in A ˆM C{C ˆM C and with the addition `B in R{Π. Since I∇ is bijective and linear over A as well, R{Π is a double vector bundle, of which I∇ is a decomposition.

33 From now on the quotient C{CA is identiﬁed with B via the factorisation of the morphism BB : C Ñ B of vector bundles. The linear splitting I∇ : A ˆM B ˆM C Ñ R{Π deﬁnes a linear ∇ l horizontal lift σB : ΓpBq Ñ ΓApR{Πq: for all b P ΓpBq and am P A

∇ C C bpamq :“ σB pbqpamq“ I∇pam,bpmq, 0mq“ I∇pam,cpmq, 0mqÀ Π “ ∇apcpmqq,apmq À Π “ cpa q` Π ´ ¯ q m x (43) q for any a P ΓpAq such that apmq “ am and c P ΓpCq such that BBc “ b. The core sections of ; c R{Π Ñ A are γ P ΓApR{Πq defined by sections γ P ΓpCq: for am P A

; B C : γ pamq“ I∇pam, 0m,γpmqq “ I∇pam, 0 ,γpmqq À Π “ γ pρApamqq,am À Π (44) ; “ γ pamqÀ Π. ` ˘

; c ; c Note that the same notation is used for γ P ΓApR{Πq and γ P ΓApRq. It is always clear from the context which section is meant by the notation. ∇ The linear splitting I∇ : AˆM BˆM C Ñ R{Π deﬁnes as well a linear horizontal lift σA : ΓpAq Ñ l ΓBpR{Πq: for all a P ΓpBq and bm P B

∇ C C apbmq :“ σA paqpbmq“ I∇papmq,bm, 0mq“ I∇papmq,cm, 0mq`A Π “ ∇apcmq,apmq `A Π “ apc q` Π ´ ¯ m A x p (45)

p ˆ c for any cm P C such that BBcm “ bm. The core sections of R{Π Ñ B are γ P ΓBpR{Πq deﬁned by sections γ P ΓpCq: for bm P B and any cm P C such that BBcm “ bm

ˆ A A Ò A γ pbmq“ I∇p0m,bm,γpmqq “ I∇p0m,cm,γpmqq À Π “ γ pcmq, 0m À Π ˆ (46) “ γ pcmqÀ Π. ` ˘ The following corollary of Theorem 5.4 is immediate. Corollary 5.8. In the situation of Theorem 5.4, the map π : R Ñ R{Π is a double vector bundle morphism with sides idA and BB, and with core idC. A More precisely, choose a connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all A a P ΓpAq and γ P ΓpC q. Then in the induced decompositions I∇ : A ˆM C ˆM C Ñ R of R and A I∇ : A ˆM pC{C qˆM C Ñ R{Π of R{Π, the map π reads

´1 I∇ ˝ π ˝ I∇ : A ˆM C ˆM C Ñ A ˆM B ˆM C, pa,c,γq ÞÑ pa, BBc,γq. (47)

Finally, consider a morphism of transitive core diagrams as in Figure 4 and consider the mor- 1 1 A phism Φ: T C ‘T M A Ñ T C ‘T M A as in Proposition 4.9. Then for all a P ΓpAq and γ P ΓpC q,  A d  A T ϕC∇a pγpmqq “ T ϕCTγρApamq´  ϕC pγpmqq ` tϕC pp∇a γqpmqq dtt“0  y d  A1 (48) T ϕ γ ρ 1 ϕ a ϕ γ m t ϕ γ m “ p Cp qq A p Ap mqq ´  C p qp q` ∇ϕApaqp C qp qq dtt“0 A1 “ ∇ pϕC pγpmqqq, ϕApaq since for a section c P Γp{Cq such that BAc “ a, the equality BA1 pϕC cq“ ϕApBAcq“ ϕApaq follows and so 1 ϕ Aγ ϕ c,γ ϕ c, ϕ γ A ϕ γ . C p∇a q“ C r s“r C C s“ ∇ϕApaqp C q (49)

34 1 1 1 1 Hence, ΦpΠqĎ Π and the vector bundle morphism Φ: T C ‘T M A Ñ T C ‘T M A over ϕA : A Ñ A quotients to a vector bundle morphism

T C ‘ A T C1 ‘ A1 Φ: T M Ñ T M Π Π1

1 1 TC‘T M A 1 1 1 over ϕA : A Ñ A ; such that π˝Φ “ Φ˝π where π : T C ‘T M A Ñ Π and π : T C ‘T M A Ñ 1 1 TC ‘T M A Π1 are the projections. As before, the most eﬀective way to prove that Φ is a double vector bundle morphism from 1 1 TC‘T M A 1 TC ‘T M A R :“ Π to R :“ Π1 with side morphisms ϕA and ϕB and with core morphism ϕC is to show that the induced map in decompositions is a double vector bundle morphism. Choose a linear A-connection ∇ on C extending ∇A : ΓpAqˆ ΓpCAq Ñ ΓpCAq and a linear A1-connection 1 1 1 ∇1 on C1 extending ∇A : ΓpA1qˆ ΓpCA q Ñ ΓpCA q. Then (49) shows that the induced form ˚ ˚ 1 A ˚ ˚ 1 ω∇,∇1 P ΓpA b C b C q as in (40) vanishes on C . Hence it induces ω∇,∇1 P ΓpA b B b C q:

ω∇,∇1 pa, BBcq“ ω∇,∇1 pa,cq for all pa,cqP A ˆM C. A computation yields then

1 ΦpI∇pa, BBc,γqq “ ΦpI∇pa,c,γq`A Πq“ ΦpI∇pa,c,γqq `A1 Π

(41) 1 “ I 1 pϕ paq, ϕ pcq, ϕ pγq` ω 1 pa,cqq ` 1 Π ∇ A C C ∇,∇ A (50) “ I∇1 pϕApaq, BB1 ϕC pcq, ϕC pγq` ω∇,∇1 pa,cqq

“ I∇1 pϕApaq, ϕB pBBcq, ϕC pγq` ω∇,∇1 pa, BBcqq for pa,c,γqP A ˆM C ˆM C. Hence

´1 1 1 1 I∇1 ˝Φ˝I∇ : AˆM B ˆM C Ñ A ˆM B ˆM C , pa,b,cq ÞÑ pϕApaq, ϕB pbq, ϕC pγq`ω∇,∇1 pa,bqq, which shows the following theorem. Theorem 5.9. Given a morphism of transitive core diagrams as in Figure 4, the induced morphism 1 1 Φ: T C ‘T M A Ñ T C ‘T M A of double vector bundles as in Proposition 4.9 quotients to a double vector bundle morphism T C ‘ A T C1 ‘ A1 Φ: T M Ñ T M , Π Π1 1 i.e. with Φ ˝ π “ π ˝ Φ. The morphism Φ has side morphisms ϕA and ϕB and core morphism ϕC .

5.2 Π as an ideal of R Ñ A This section shows that Π deﬁned in (42) is an ideal in R Ñ A: i.e. it is totally intransitive and

rΓApΠq, ΓApRqs Ď ΓApΠq.

A Proposition 5.10. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq and γ P ΓpCAq. Then

B B B ∇γ a “ 0, ∇γ c “ 0 and R∇pγ,cq“ 0 for all a P ΓpAq, c P ΓpCq and γ P ΓpCAq.

B Proof. Choose any section a P ΓpAq and any section c P ΓpCq with Bc “ a. Then ∇γ a “ rBγ, Bcs` A A A Bp∇aγq “ ´Brc,γs`Bp∇a γq“Bp∇a γq`Bp∇a γq“ 0. The second identity is immediate and the last one is easy and left to the reader.

35 Proposition 5.11. The anchor ΘA : R Ñ T A sends Π to the zero section of T A Ñ A.

A Proof. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq A B Xl A and γ P ΓpC q. By Propositions 4.6 and 5.10 ΘApγq “ ∇γ “ 0 P pAq for all γ P ΓpC q. This completes the proof with Corollary 5.3. x Proposition 5.12. The subundle Π of R over A isq an ideal in R Ñ A.

A Proof. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq and γ P ΓpCAq. By the Leibnitz identity and since Π Ñ A is totally intransitive by Proposition 5.11, it is suﬃcient to check that

; rγ, csP ΓApΠq and rγ,c sP ΓApΠq for all γ P ΓpCAq and c P ΓpCq. But this follows immediately from Proposition 5.10, Proposition q q q 4.6 and rγ,csP ΓpCAq for all γ P ΓpCAq and c P ΓpCq. By Proposition 5.10, the connections ∇B : ΓpCqˆ ΓpCq Ñ ΓpCq and ∇B : ΓpCqˆ ΓpAq Ñ ΓpAq deﬁne connections ∇red : ΓpBqˆ ΓpCq Ñ ΓpCq and ∇red : ΓpBqˆ ΓpAq Ñ ΓpAq:

red B red B ∇BB c1 c2 “ ∇c1 c2 ∇BB ca “ ∇c a

B for c1,c2 P ΓpCq and a P ΓpAq. By the same proposition, R∇ deﬁnes a new tensor

red 2 red 1 B 1 R∇ P Ω pB, HompA, Cqq, R∇ pBBc, BBc q“ R∇pc,c q, for c,c1 P ΓpCq. Corollary 5.13. The vector bundle R{Π Ñ A has a linear Lie algebroid structure over the Lie A algebroid B Ñ M. Choose any connection ∇: ΓpAqˆ ΓpCq Ñ ΓpCq such that ∇aγ “ ∇a γ for all a P ΓpAq and γ P ΓpCAq. Then the Lie algebroid structure on R{Π Ñ A can be described as follows: red Xl ; Ò X ΘApbq“ ∇b P pAq and ΘApγ q“pBγq P pAq,

red ; red ; ; ; rb1, b1s“ rqb1,b2zsB ´ R∇ pb1,b2q, b,γ “ p∇b γq , γ1 γ2 “ 0 (51) ” ı ” ı for b,b1,b2 P ΓpBq q qand γ,γ1,γ2 P ΓpCqČ. q In other words, via the splitting I∇ of R{Π given by ∇, the Lie algebroid R{Π Ñ A is described by the following representation up to homotopy of B:

B : C Ñ A, ∇red : ΓpBqˆ ΓpCq Ñ ΓpCq, ∇red : ΓpBqˆ ΓpAq Ñ ΓpAq, (52) and red 2 R∇ P Ω pB, HompA, Cqq. (53)

Proof. Since Π Ñ A is an ideal in R Ñ A, the anchor ΘA of R{Π Ñ A is given by

ΘApξ ` Πq“ ΘApξq for all ξ P ΓApRq, and the bracket is given by

rξ1 À Π, ξ2 À Πs“rξ1, ξ2sÀ Π for ξ1, ξ2 P ΓApRq. This yields

B red Xl ΘApbq“ ΘApc `A Πq“ ΘApcq“ ∇c “ ∇b P pAq

q q q x z 36 for b P ΓpBq and a section c P ΓpCq such that BBc “ b, and

; ; Ò ΘApγ q“ ΘApγ `A Πq“pBγq for all γ P ΓpCq. Further computations show for b,b1,b2 P ΓpBq and γ,γ1,γ2 P ΓpCq:

B B b1, b2 “rc1 À Π, c2 À Πs“ rc1,c2s´ R∇pc1,c2q À Π “ rb1,b2s´ R∇pb1,b2q, ” ı ´ ¯ where c1q,c2qP ΓpCq are suchq that BBc1 “b1 und BBcČ2 “ b2, Č

; B Ò b, γ “ p∇c γq ” ı q ; ; where c P ΓpCq is such that BBc “ b, and γ1,γ2 “ 0 is immediate. ” ı 5.3 The VB-algebroid R{Π Ñ B Recall that the setting of this section is the transitive core diagram C in Figure 8: A and B are Lie algebroids over a common base M and C is a Lie algebroid on M together with morphisms A B :“BA : C Ñ A and BB : C Ñ B, both of which are surjective and such that the kernels C :“ B A A kerpBBq and C :“ kerpBAq commute in C. Recall as well the representation ∇ : A Ñ DerpC q A deﬁned in (19): ∇a γ “rc,γs for any c P ΓpCq such that Bc “ a. By its choice, the connection ∇: ΓpAqˆΓpCq Ñ ΓpCq restricts as before to ∇A : ΓpAqˆΓpCAq Ñ ΓpCAq. That is, it induces a connection

B B ∇ : ΓpAqˆ ΓpBq Ñ ΓpBq, ∇a pBBcq“BBp∇acq

A A for c P ΓpCq and a P ΓpAq. Since ∇ is ﬂat, the curvature R∇ of ∇ vanishes on sections of C 2 and induces similarly a tensor R∇ P Ω pA, HompB, Cqq,

R∇pa1,a2qpBB cq“ R∇pa1,a2qc for a1,a2 P ΓpAq and c P ΓpCq. The following lemma is easy to prove using (7).

Xl B Xl Xc Lemma 5.14. In the situation above, pCqQ ∇a „BB ∇a P pBq for all a P ΓpAq and pCqQ Ò Ò Xc γ „BB pBBγq P pBq for all γ P ΓpCq. x y Recall that the ﬂat connection

i ∇ : ΓC pFC qˆ ΓC pRq Ñ ΓC pRq, is deﬁned by ∇ia “ 0 “ ∇icˆ for all a P ΓpAq and all c P ΓpCq.

i Proposition 5.15. The triple pFC , 0, ∇ q is an inﬁnitesimal ideal system in R Ñ C. That is, p i FC Ď T C is an involutive subbundle, and ∇ a ﬂat FC -connection on R Ñ C with the following properties:

i 1. If χ1,χ2 P ΓC pRq are ∇ -parallel, then rχ1,χ2sC is also parallel.

i 2. If χ P ΓC pRq is ∇ -parallel, then rΘCpχq,XsP ΓpFC q for all X P ΓpFC q. That is, ΘCpχq is FC FC ∇ -parallel, where ∇ is the Bott connection associated to FC Ď T C. Proof. By the properties of an inﬁnitesimal ideal system, (2) follows from (1). Since sections l ˆ i 8 a P ΓC pRq and γ for a P ΓpAq and γ P ΓpCq are ∇ -parallel and span ΓCpRq as a C pCq-module, ˆ 8 FC it suﬃces to show (1) for χ1 and χ2 of the type F ¨ a and F ¨ γ with F P C pCq and a P ΓpAq, A 8 FC ˚ 8 pγ P ΓpCq. Since C{FC “ C{C “ B, the space C pCq equals BBC pBq. p 37 8 Choose F1, F2 P C pBq, a,a1,a2 P ΓpAq and γ,γ1,γ2 P ΓpCq. Then using Lemma 5.14,

˚ ˚ ˚ ˚ ˚ ˚ ˚ rB F1 ¨ a1, B F2 ¨ a2s“B pF1F2q¨ra1, a2s`B F1 ¨ £ pB F2q ¨ a2 ´B F2 ¨ £ pB F1q ¨ a1 B B B B ∇a1 B B ∇a2 B

“B˚ pF F q ¨ ra ,a s´ R pa ,azq z p p B 1 2 p 1 p 2 ∇ 1 2 p p ´ ¯ ˚ ˚ `BB F1 ¨ £{B F2 ¨ aČ2 ´BB F2 ¨ £ B F1 ¨ a1 ∇a1 ∇a2 ˆ ˙ ˆ ˙ z z ˚ ˚ i p p i A shows that rBBF1 ¨ a1, BBF2 ¨ a2s is a ∇ -flat section of R Ñ C if R∇pa1,a2q is ∇ -flat. Since ∇ is A ˝ ˚ t flat, R∇pa1,a2qP ΓppC q b Cq“ ΓpB b Cq. Hence it equals without loss of generality pBBνqb c ˆ ˚ ˆ i with ν P ΓpBq and c P ΓpCq, and consequently R pa ,a q “ ℓ t Č¨ γ “B ℓ ¨ γ is a ∇ -flat p p ∇ 1 2 BB ν B ν section of R Ñ C. Similarly, Č

˚ ˚ ˆ ˚ ˆ ˚ ˚ ˆ ˚ ˚ B F ¨ a, B F ¨ γ “B pF F q ¨ a,γ `B F ¨ £ pB F q ¨ γ ´B F ¨ £ Ò pB F q ¨ a B 1 B 2 B 1 2 B 1 ∇a B 2 B 2 γ B 1

“ ‰ ˚ “ ‰ˆ ˚ ˆ ˚ y Ò “BBpF1F2q ¨ p∇aγq `BB F1 ¨ £ B F2 ¨ γ ´BB F2 ¨ £pBB γq F1 ¨ a1 p p ∇a1 p ˆ ˙ z ` ˘ and p

˚ ˆ ˚ ˆ ˚ ˆ ˆ ˚ ˚ ˆ ˚ ˚ ˆ B F1 ¨ γ , B F2 ¨ γ “B pF1F2q ¨ γ ,γ `B F1 ¨ £ Ò pB F2q ¨ γ ´B F2 ¨ £ Ò pB F1q ¨ γ B 1 B 2 B 1 2 B γ1 B 2 B γ2 B 1 ˚ ˆ ˚ ˆ Ò Ò “ ‰ “BB F1 ¨ £pB“B γ1q F2‰ ¨ γ2 ´BB F2 ¨ £pBB γ2q F1 ¨ γ1

˚ ˚ ˆ` ˚ ˆ˘ ˚ ˆ` i ˘ show that rBBF1 ¨ a, BBF2 ¨ γ s and BBF1 ¨ γ1 , BBF2 ¨ γ2 are ∇ -flat sections of R Ñ C. The proof above shows as well the“ following lemma. ‰ p A ˝ l i Lemma 5.16. For ϕ P Γ pC q b C , the corresponding core-linear section ϕ P ΓC pRq is ∇ -flat. It projects to ϕ¯ P Γl pR{Πq, where ϕ¯ P ΓpB˚ b Cq is defined by ϕ¯pB cq“ ϕpcq for all c P C. B ` ˘ B r Corollary 5.17. There is a linear Lie algebroid structure on R{Π » R{∇i Ñ C{F » B such r C that π∇i R / R{∇i (54)

πC q C / B BB is a ﬁbration of VB-Lie algebroids.

Proof. By Theorem 5.4, R{Π has a double vector bundle structure with sides A and B and with core i i C. Since R{Π » R{∇ Ñ C{FC » B, and pFC , 0, ∇ q is an inﬁnitesimal ideal system in R Ñ C, the quotient R{Π has a Lie algebroid structure such that (54) is a ﬁbration of Lie algebroids, see Theorem 2.5,(2). It remains to show that the Lie algebroid structure on R{Π Ñ B is linear. Consider a linear A-connection ∇ on C that preserves CA. Then the sections of the type a and γˆ for a P ΓpAq and γ P ΓpCq are ∇i invariant and project to the sections a and γˆ of R{Π Ñ B as in (45) and (46): p

ˆ ˆ p π∇i ˝ a “ a ˝BB and π∇i ˝ γ “ γ ˝BB

The anchor Θ : R{Π Ñ TB is deﬁned by B p p B Xl ˆ Ò Xc ΘBpaq“ ∇a P pBq and ΘBpγ q“pBBγq P pBq,

p y 38 X B X ˆ Ò X since by Lemma 5.14, ΘCpaq“ ∇a P pCq is BB related to ∇a P pBq and ΘCpγ q“ γ P pCq Ò is BB-related to pBBγq P XpBq. The Lie algebroid bracketp ofx core sections of R{Π Ñ B isy deﬁned by γˆ,γˆ 1 2 R{Π “ 0

ˆ “ˆ ‰ˆ ˆ for γ1,γ2 P ΓpCq, since the bracket γ1 ,γ2 of γ1 ,γ2 P ΓCpRq vanishes and projects hence to the zero section of R{Π Ñ B. Similarly, “ ‰ a,γˆ γ ˆ R{Π “ p∇a q

“ ‰ ˆ ˆ ˆ for a P ΓpAq and γ P ΓpCq, since thep bracket ra,γ s of a and γ P ΓC pRq equals p∇aγq and ˆ c projects hence to p∇aγq P ΓBpR{Πq. Finally, as in Lemma 5.16, for a1,a2 P ΓpAq the curvature i l R∇pa1,a2q gives a ∇ -ﬂat core-linear section R∇ppa1,a2q PpΓC pRq, which projects under π∇i and l BB to R∇pa1,a2qP ΓBpR{Πq. As a consequence, the Lie bracket ra1, a2s“ ra1,a2s´ R∇pa1,a2qP l l Č ΓCpRq of a1 and a2 P ΓC pRq projects to Č p p { Č l p p ra1, a2sR{Π :“ ra1,a2s´ R∇pa1,a2qP ΓBpR{Πq. The proof of Corollary 5.17 shows as well the following Proposition. p p { Č Proposition 5.18. Let R{Π be constructed as above and consider as before an A-connection A ∇: ΓpAqˆ ΓpCq Ñ ΓpCq on C that extends ∇ . Then the anchor R Ñ TB is deﬁned by ΘB ˝ a “ B Xl ˆ Ò X ∇a P pBq and ΘB ˝ γ “ pBBγq P pBq for a P ΓpAq and γ P ΓpCq. Furthermore, the Lie algebroid bracket on R{Π Ñ B is given by: p y ˆ ˆ ˆ ˆ ra1, a2s“ ra1,a2s´ R∇pa1,a2q, a,γ “ p∇aγq , γ1 ,γ2 “ 0 (55) for a,a ,a P ΓpAq and γ,γ ,γ P ΓpCq. “ ‰ “ ‰ 1 2 p p {1 2 Č p In other words, via the splitting I∇ of R{Π given by ∇, the Lie algebroid R{Π Ñ B is described by the following representation up to homotopy of A:

B BB : C Ñ B, ∇: ΓpAqˆ ΓpCq Ñ ΓpCq, ∇ : ΓpAqˆ ΓpBq Ñ ΓpBq, (56) and 2 R∇ P Ω pA, HompB, Cqq. (57) Remark 5.19. The sections above show that the quotient R{Π can be deﬁned as the “double quotient” of the double Lie algebroid R by the two IM-foliations

i p0, Π, 0q and pFC , 0, ∇ q in R Ñ A and R Ñ C, respectively.

5.4 The quotient R{Π as a double Lie algebroid. In this section, the quotient double vector bundle is written R{Π “ R{∇i “: D˜ for simplicity. The next theorem shows that the two ‘reduced’ VB-algebroid structures found in Corollary 5.13 and Corollary 5.17 on the sides of D˜ are still compatible, so that D˜ “ R{Π is again a double Lie algebroid. The setting of this section is the same as in the rest of Section 5. Recall from Corollary 5.13 that D˜ Ñ A inherits a reduced Lie algebroid as the quotient of R Ñ A by the ideal Π Ñ A. By Corollary 5.17, D˜ Ñ B inherits a Lie algebroid structure, as the quotient of R Ñ C by the i inﬁnitesimal ideal system pFC , 0, ∇ q.

39 Theorem 5.20. With these two VB-Lie algebroid structures, the quotient D˜ “ R{Π has the structure of a transitive double Lie algebroid over the sides A and B, with core C and (transitive) core diagram C in Figure 8.

Proof. Let as before ∇: ΓpAqˆ ΓpCq Ñ ΓpCq be a linear connection extending the representation A A ∇ of A on C . In the induced linear splitting I∇ : A ˆM B Ñ D˜ of D˜, the two VB-algebroids D˜ Ñ A and D˜ Ñ B are represented by the two representations up to homotopy in (52), (53), and in (56), (57), respectively. It is not diﬃcult to check that these two representations up to homotopy are matched (see Appendix A). Hence, D˜ is a double Lie algebroid with sides A and B and with core C. By (9), the Lie algebroid structure induced on the core C by the double Lie algebroid D˜ is given by the anchor ρA ˝B“ ρB ˝BB, which is ρC , and by the bracket

red B ∇BAc1 c2 ´ ∇BB c2 c1 “ ∇BAc1 c2 ´ ∇c2 c1 “ ´rc2,c1s“rc1,c2s for all c1,c2 P ΓpCq. Hence, the Lie algebroid C in the core diagram C is the Lie algebroid C as the core of the double Lie algebroid D˜. By (52), the core-anchor of D˜ Ñ A is B“BA : C Ñ A, and by (56), the core-anchor of D˜ Ñ B is BB : C Ñ B. Therefore, the core diagram C is the core-diagram of D˜.

Proposition 5.21. In the situation of Theorem 5.20, the morphism π : R Ñ D˜ of double vector bundles is a morphism of double Lie algebroids.

Proof. By Corollary 5.8, a choice of linear connection ∇: ΓpAq ˆ ΓpCq Ñ ΓpCq that extends A A A ∇ : ΓpAqˆ ΓpC q Ñ ΓpC q yields linear splittings Σ∇ : A ˆM C Ñ R and Σ∇ : A ˆM B Ñ D˜ such that πpΣ∇pa,cqq “ Σ∇pa, BBcq for all pa,bqP A ˆM B. Therefore (10), (11) and (12) need to be checked for

1. the triple pidC, BB, 0q from the 2-representation pidC : Cr0s Ñ Cr1s, ∇, ∇, R∇q of A as in (37) B to the 2-representation pBB : Cr0s Ñ Br1s, ∇, ∇ , R∇q of A as in (56) and (57).

B B B 2. the triple pidC , idA, 0q from the 2-representation pBA : Cr0s Ñ Ar1s, ∇ , ∇ , R∇q of C as in ! red red red (38) and (39) to the 2-representation BBpBA : Cr0s Ñ Ar1s, ∇ , ∇ , R∇ q of C as in (52) and (53). In the first case, (10) is trivial and (11) and (12) follow immediately from the definitions of ∇B and ! red red red B B B R∇. In the second case, BBpBA : Cr0s Ñ Ar1s, ∇ , ∇ , R∇ q “ pBA : Cr0s Ñ Ar1s, ∇ , ∇ , R∇q red red by the definitions of ∇ and R∇ , so the three conditions are immediate.

5.5 Equivalence of transitive double Lie algebroids with transitive core diagrams Let trCDpMq be the category of transitive core diagrams over M, and let trDLApMq be the category of transitive double Lie algebroids over M. As discussed in Section 3, considering the core diagram of a transitive double Lie algebroid with double base M deﬁnes a functor

C : trDLApMq Ñ trCDpMq.

This section shows that the construction of a transitive double Lie algebroid D˜ “ R{Π from a given transitive core diagram deﬁnes a functor

D : trCDpMq Ñ trDLApMq, and proves the following theorem.

40 Theorem 5.22. Let M be a smooth manifold. The two functors C : trDLApMq Ñ trCDpMq and D : trCDpMq Ñ trDLApMq establish an equivalence between the category of transitive core diagrams over M and the one of transitive double Lie algebroids with double base M.

First consider a morphism of transitive core diagrams as in Figure 4 and the induced vector bundle morphism T C ‘ A T C1 ‘ A1 Φ: T M Ñ T M , Π Π1 as in Theorem 5.9, with side morphisms ϕA and ϕB and core morphism ϕC . Choose a linear 1 A-connection ∇ on C extending ∇A, and a linear A1-connection on C1 extending ∇A . Then by (50), ΦpI∇pa,b,γqq “ I∇1 pϕApaq, ϕB pbq, ϕC pγq` ω∇,∇1 pa,bqq for pa,b,cq P A ˆM B ˆM C. In order to check that Φ is a double Lie algebroid morphim, (10), (11) and (12) need to be checked for

B 1. the triple pϕC , ϕB, ω∇,∇1 q from the 2-representation pBB : Cr0s Ñ Br1s, ∇, ∇ , R∇q of A as 1 ˚ 1 1 1 1B in as in (56) and (57) to the 2-representation ϕApBB1 : C r0s Ñ B r1s, ∇ , ∇ , R∇1 q of A. red red red 2. the triple pϕC , ϕA, ω∇,∇1 q from the 2-representation pBA : Cr0s Ñ Ar1s, ∇ , ∇ , R∇ q of ˚ 1 1 1red 1red red B as in (52) and (53) to the 2-representation ϕB pBA1 : C r0s Ñ A r1s, ∇ , ∇ , R∇1 q of B. This is done in Section B.2, yielding the following theorem. Theorem 5.23. Given a morphism of transitive core diagrams as in Figure 4, then the induced double vector bundle morphism

T C ‘ A T C1 ‘ A1 Φ: T M Ñ T M , Π Π1 as in Theorem 5.9 is a morphism of double Lie algebroids with side morphisms ϕA and ϕB and core morphism ϕC . That is, the induced morphism of core diagrams is again pϕA, ϕB, ϕC q. Hence this paper has constructed the functor D : trCDpMq Ñ trDLApMq, sending a transitive TC‘T M A core diagram C over M as in Figure 3 to the transitive double Lie algebroid Π , and a morphism of transitive core diagrams as in Figure 4 to the corresponding double Lie algebroid morphism Φ as in the last theorem. By Theorem 5.20, the composition C ˝ D : trCDpMq Ñ trCDpMq is the identity functor. The remainder of this section proves that the composition D ˝ C : trDLApMq Ñ trDLApMq is a natural isomorphism, hence completing the proof of Theorem 5.22. Note that by construction, D ˝ C sends a transitive double Lie algebroid to a transitive double Lie algebroid with the same core, see Theorem 5.20. The needed natural isomorphism follows from the following theorem.

Theorem 5.24. 1. Let pD1; A,B,Mq and pD2; A, B; Mq be two transitive double Lie algebroids with the same core diagram C, as in Figure 8. Then there is a canonical isomorphism

ΦD1,D2 : D1 Ñ D2 of double Lie algebroids, which is the identity on the sides and on the core.

2. Let pD1; A,B,Mq and pD2; A, B; Mq be two transitive double Lie algebroids with the same 1 1 1 1 1 1 core diagram C, and let pD1; A ,B ,Mq and pD2; A ,B ; Mq be two transitive double Lie algebroids with the same core diagram C1. Consider two double Lie algebroid morphisms 1 1 Φ1 : D1 Ñ D1 and Φ2 : D2 Ñ D2 that both induce the same morphism of core diagrams C Ñ C1. Then 1 1 Φ2 ˝ ΦD1,D2 “ ΦD1,D2 ˝ Φ1.

41 1 2 Proof. 1. Choose a linear splitting Σ : A ˆM B Ñ D1 of D1 and Σ : A ˆM B Ñ D2 of D2. These two linear splittings induce four representations up to homotopy:

(a) For i “ 1, 2, the Lie algebroid Di Ñ A is described by the representation up to homotopy

Bi Bi BA : C Ñ A, ∇ : ΓpBqˆ ΓpCq Ñ ΓpCq, ∇ : ΓpBqˆ ΓpAq Ñ ΓpAq,

and Bi 2 R∇ P Ω pB, HompA, Cqq of B Ñ M on Ar0s‘ Cr1s.

(b) For i “ 1, 2, the Lie algebroid Di Ñ B is described by the representation up to homotopy

i Ai BB : C Ñ B, ∇ : ΓpAqˆ ΓpCq Ñ ΓpCq, ∇ : ΓpAqˆ ΓpBq Ñ ΓpBq,

and Ai 2 R∇ P Ω pA, HompB, Cqq of A Ñ M on Br0s‘ Cr1s.

A Choose γ P ΓpC q“ Γpker BBq and a P ΓpAq. Choose further a section c P ΓpCq with Bc “ a. By (9), i i Bi ∇aγ “ ∇Bcγ “rc,γs` ∇BB γc “rc,γs for i “ 1, 2. Hence, ∇1 ´ ∇2 P Ω1pA, pCAq˝ b Cq “ ΓpA˚ b B˚ b Cq; i.e. there exists a ˚ ˚ 1 2 (unique) section ϕ P ΓpA b B b Cq so dass ∇ac ´ ∇ac “ ϕpa, BBcq for all a P ΓpAq and 2 c P ΓpCq. Set Σ˜ : A ˆM B Ñ D2,

2 D2 pa,bq ÞÑ Σ pa,bq`A p0a `B ϕpa,bqq.

Then, by standard computations (see e.g. [7, Remark 2.12]), the connection ∇˜ 2 : ΓpAqˆ 2 ΓpCq Ñ ΓpCq induced by the VB-algebroid D2 Ñ B and the splitting Σ˜ is given by

˜ 2 2 2 1 2 1 ∇ac “ ∇ac ` ϕpa, BB cq“ ∇ac ` ∇ac ´ ∇ac “ ∇ac.

This shows that Σ1 and Σ2 can be chosen such that ∇ :“ ∇1 “ ∇2 : ΓpAqˆ ΓpCq Ñ ΓpCq. Consider then the linear decompositions I1 : AˆM BˆM C Ñ D1 and I2 : AˆM BˆM C Ñ D2 1 2 ´1 that are equivalent to Σ and Σ , respectively, and the isomorphism ΦD1,D2 :“ I2˝I1 : D1 Ñ 1 D2 of double vector bundles. Choose two diﬀerent linear splittings Σ˜ : A ˆM B Ñ D1 and 2 1 2 Σ˜ : A ˆM B Ñ D2 of D1 and D2 such that the induced A-connections ∇˜ and ∇˜ on C 1 2 ˚ ˚ are equal: ∇˜ “ ∇˜ “: ∇˜ . Then there exist forms ω1,ω2 P ΓpA b B b Cq such that ˜ i i D2 Σ pa,bq“ Σ pa,bq`B p0b `A ωipa,bqq for i “ 1, 2 and all pa,bqP A ˆM B. By [7, Remark 2.12], ∇ac ` ω1pa, BBcq“ ∇˜ ac “ ∇ac ` ω2pa, BBcq

for all a P ΓpAq and c P ΓpCq. Since BB is surjective, this shows ω1 “ ω2. The equality ˜ ˜´1 ´1 ˜ ˜ I2 ˝ I1 “ I2 ˝ I1 follows easily for the decompositions I1 and I2 of D1 and D2 that are ˜ 1 ˜ 2 equivalent to Σ and Σ . Hence ΦD1,D2 does not depend on the choice of the compatible linear splittings of D1 and D2.

It remains to show that ΦD1,D2 is an isomorphism of double Lie algebroids, i.e. that the representations up to homotopy in (a) coincide and that the ones in (b) coincide, if ∇1 “ 2 ∇ “: ∇. Choose a P ΓpAq, b P ΓpBq and a section c P ΓpCq such that BBc “ b. Then

A1 A1 A2 A2 ∇a b “ ∇a BBc “BB∇ac “ ∇a BBc “ ∇a b

42 by (R2) in the deﬁnition of a 2-representation, and

B1 B1 B2 B2 ∇b a “ ∇BB ca “ rBAc,as`BAp∇acq“ ∇BB ca “ ∇b a by (M2). 1 Similarly, for b “BBc P ΓpBq as above and c P ΓpCq:

B1 1 B1 1 1 B2 1 B2 1 1 ∇b c “ ∇BB cc “rc,c s` ∇BAc c “ ∇BB cc “ ∇b c

by (9) and (M1). Furthermore, for a1,a2 P ΓpAq: A1 A1 R pa1,a2qb “ R pa1,a2qpBBcq“ R∇pa1,a2qc A2 A2 “ R pa1,a2qpBBcq“ R pa1,a2qb by (R3). It remains to prove that RB1 “ RB2. This is done as follows using (M4). Choose a P ΓpAq, b1,b2 P ΓpBq and c2 P ΓpCq such that BBc2 “ b2. Then B1 B1 R pb1,b2qpaq“ R pb1, BBc2qa A1 B B B B “ R pa, BAc2qb1 ` ∇ 1 ∇ac2 ´ ∇a∇ 1 c2 ´ ∇ c2 ` ∇ A c2 b b ∇b1 a ∇a b1 A2 B B B B “ R pa, BAc2qb1 ` ∇ 1 ∇ac2 ´ ∇a∇ 1 c2 ´ ∇ c2 ` ∇ A c2 b b ∇b1 a ∇a b1 B2 B2 “ R pb1, BBc2qa “ R pb1,b2qpaq.

2. Choose compatible linear splittings Σ1 and Σ2 for D1 and D2 as in (1), and compatible linear 1 1 1 1 splittings Σ1 and Σ2 for D1 and D2. Then the A-connections on C induced as in (1) by Σ1 1 1 1 1 and Σ2 are equal, and the A -connections on C induced as in (1) by Σ1 and Σ2 are equal. 1 1 1 Let pϕA : A Ñ A , ϕB : B Ñ B , ϕC : C Ñ C q be the morphism of core diagrams underlying ˚ ˚ 1 both Φ1 and Φ2. The linear splittings and Φ1, Φ2 deﬁne ω1,ω2 P ΓpA b B b C q such that

1 1 Di ΦipΣipa,bqq “ Σ pϕApaq, ϕB pbqq `B1 p0 `A1 ωipa,bqq (58) i ϕBpbq 1 for i “ 1, 2 and pa,bq P A ˆM B. Since Φi : Di Ñ Di is a VB-algebroid morphism over 1 ϕB : B Ñ B for i “ 1, 2, (11) yields

Hom p∇a ϕC qpcq“ ωipa, BBcq

for i “ 1, 2 and pa,cq P A ˆM C. Since BB is surjective, this shows ω1 “ ω2. The equality 1 1 Φ2 ˝ ΦD1,D2 “ ΦD1,D2 ˝ Φ1 follows then immediately from (58) and the constructions of 1 1 ΦD1,D2 and ΦD1,D2 .

To conclude the proof of Theorem 5.22, consider for each transitive double Lie algebroid D the isomorphism IpDq :“ ΦpD ˝ CqpDq,D : pD ˝ CqpDq Ñ D. By the last theorem it deﬁnes a natural isomorphism

I : D ˝ C ñ idtrDLA, since IpD1q pD ˝ CqpD1q / D1

pD ˝ CqpΦq Φ pD ˝ CqpD2q / D2 IpD2q commutes for any morphism Φ: D1 Ñ D2 of transitive double Lie algebroids.

43 6 Integration of transitive double Lie algebroids

Let pΓ,G,H,Mq be a double Lie groupoid with core K. Then taking the Lie algebroids AGpΓq Ñ G and ApHq Ñ M yields an LA-groupoid, see [16].

AGpΓq G

ApHq M

That is, the horizontal arrows in the diagram above carry Lie algebroid stuctures, and the vertical structures are Lie groupoids, with a compatibility condition. Taking the Lie algebroids of AGpΓq Ñ H and G Ñ M yields a double Lie algebroid

AApHqpAGpΓqq ApGq

ApHq M which is isomorphic to the double Lie algebroid

AApGqpAH pΓqq ApGq

ApHq M see [18]. Any of these two double Lie algebroids is understood as the double Lie algebroid of the double Lie groupoid pΓ,G,H,Mq [18]. LA-algebroids have been integrated by [4] to LA-groupoids, see also [5], and a certain class of LA-groupoids were integrated by [25] to double Lie groupoids, but as far as the authors know the “direct” integration of (integrable) double Lie algebroids to double Lie groupoids has not been discussed yet in the literature. Consider here a transitive double Lie algebroid

π D A / A

πB qA B / M qB with transitive core diagram as in Figure 1. Assume that K Ñ M, G Ñ M and H Ñ M are source- simply connected Lie groupoids with ApKq “ C Ñ M, ApGq “ A Ñ M and ApHq “ B Ñ M – that is, assume that the Lie algebroids A, B and C are all integrable, to G, H and K, respectively. Then according to Lie’s second theorem for Lie algebroid morphisms (see the appendix of [21]), the core diagram in Figure 1 integrates to a diagram of Lie groupoid morphisms as in Figure 2. A study of the proof in [21] reveals that the surjectivity of BA and BB imply that the corresponding Lie groupoid morphisms BG : K Ñ G and BH : K Ñ H are surjective submersions. ´1 For each m P M, let Gm be the source fiber s pmq Ď G, let Km Ď K be similarly the source fiber of K through 1m, and let Fm :“ kerpBGqm Ď Km be space of elements h P Km with BGphq “ 1m P G. Then it is easy to see that Fm is a Lie group, that acts smoothly, freely and properly on the right on Km via the multiplication of K. The orbit space of that action is Gm, and the quotient map the surjective submersion BG|Km : Km Ñ Gm. That is, BG|Km : Km Ñ Gm is a principal Fm-bundle. It satisfies therefore the homotopy lifting property and the long exact

44 4 sequence in homotopy yields that Fm is connected since Gm and Km are simply connected (see [9, Chapter 4]). That is, the Lie subgroupoid kerpBGqĎ K integrating kerpBAq is source-connected, and similarly, the Lie subgroupoid kerpBH q Ď K integrating kerpBBq is source-connected. Hence the condition rc1,c2s “ 0 for all c1 P Γpker BAq and all c2 P Γpker BBq integrates to the elements of ker BG commuting with those of ker BH . In other words, the core diagram of the double Lie algebroid pD,A,B,Mq integrates to a core diagram of Lie groupoids as in Figure 2. According to the discussion in Section 1.2, there exists then a (up to isomorphism) unique transitive double Lie groupoid Θ G

HM with the core diagram in Figure 2. Following Propositions 2.6 and 2.9 in [18], and Deﬁnition 5.1 in [16], the transitive core diagram of the transitive double Lie algebroid

pAApGqpAH pΘqq, ApGq, ApHq,Mq » pAApHqpAGpΘqq, ApGq, ApHq,Mq of the double Lie groupoid pΘ,G,H,Mq is the one in Figure 1. Hence by Theorem 5.24 the double Lie algebroid of pΘ,G,H,Mq is isomorphic to pD,A,B,Mq, and so pΘ,G,H,Mq integrates pD,A,B,Mq. Since by Lie’s second theorem morphisms of transitive core diagrams (of Lie algebroids) integrate to morphisms of transitive core diagrams (of Lie groupoids), it is easy to see in the same manner that a morphism of transitive double Lie algebroids over a ﬁxed double base integrates to a morphism of transitive double Lie groupoids over the same double base. This proves the following theorem. Theorem 6.1. Let π D A / A

πB qA B / M qB be a transitive double Lie algebroid with core diagram C / B BB

BA A Assume that A,B and C are integrable Lie algebroids, with A » ApGq, B » ApHq and C » ApKq for source simply-connected Lie groupoids G Ñ M, H Ñ M and K Ñ M. Then the double Lie algebroid pD,A,B,Mq is integrable to a transitive double Lie groupoid pΘ,G,H,Mq, which is uniquely deﬁned by its core diagram KBG G

BH HM such that ApBGq“BA and ApBH q“BB. A morphism of integrable transitive double Lie algebroids over a ﬁxed base M integrates further to a morphism of transitive double Lie groupoids over the double base M. 4 1 This can also be seen without using further general tools: take k,k P Fm and a path γ : r0, 1sÑ Fm joining k 1 and k . Then BG ˝ γ is a loop based at 1m in Gm. Since Gm is simply connected, this loop is homotopic (with ﬁxed endpoints) to the constant loop at 1m. Lifting the homotopy to Km yields a homotopy between γ and a path in 1 Fm, by paths all with endpoints in Fm. This shows that k and k can be joined by a path in Fm.

45 A The Lie bialgebroid condition for R and D˜

This section checks in detail the seven conditions in Definition 3.1 and Theorem 3.6 of [7] for the two VB-Lie algebroid structures on R, and verifies at the same time that the conditions still hold for the “quotient representations up to homotopy” describing D˜. For clarity, each verification is numbered as the equation is in [7]. Note that in the case where A “ TM and B“ ρC , this is an alternative proof for pT C; TM,C; Mq satisfying the Lie bialgebroid condition [20], which is already given in [7]. Recall that, after a choice of linear A-connection ∇ on C extending ∇A, the two representations up to homotopy encoding the sides of R are the one of A given by pidC : C Ñ C, ∇, ∇, R∇q and the one of C given by B B B pB : C Ñ A, ∇ , ∇ , R∇q. The two representations up to homotopy encoding the sides of the quo- ˜ B ¯ red red red tient D are then pBB : C Ñ B, ∇, ∇ , R∇q and pBA : C Ñ A, ∇ , ∇ , R∇ q – now B : C Ñ A is written BA for clarity. (M1) R: By definition of ∇B,

B B ∇BAc1 c2 ´ ∇c2 c1 “rc1,c2s“´∇BAc2 c1 ` ∇c1 c2

for all c1,c2 P ΓpCq. D˜: From this follows immediately

red red ∇BAc1 c2 ´ ∇BB c2 c1 “rc1,c2s“´∇BAc2 c1 ` ∇BB c1 c2.

R a A c C a, c c B a (M2) : Choose P Γp q and P Γp q. The equation r BA s“BAp∇a q´ ∇idC c is the definition of ∇B. D˜ red a, c c red a : By definition of ∇ , this yields: r BA s“BAp∇a q´ ∇BB c . bas B (M3) R: rc1, idCc2s“ idC p∇c1 c2q´ ∇BAc2 c1 is again the definition of ∇ . D˜ b, c redc B b : Applying BB to both sides of this equation yields r BB s“BBp∇b q´ ∇BAc for all b P ΓpBq and c P ΓpCq.

(M4) R: For a P ΓpAq and c1,c2 P ΓpCq:

B B B ∇ ∇ac ´ ∇a∇ c ´ ∇ B c ` ∇ c “rc , ∇ac s` ∇ c2 c ´ ∇arc ,c s´ ∇a∇ c2 c c1 2 c1 2 ∇c1 a 2 ∇ac1 2 1 2 BA∇a 1 1 2 BA 1

´ ∇ B c `r∇ac ,c s` ∇ c2 ∇ac . ∇c1 a 2 1 2 BA 1

Since ∇ c2 c “ ∇ 2 B c , this yields BA∇a 1 ´rBAc ,as`∇c2 a 1

B B B B ∇ ∇ac ´ ∇a∇ c ´ ∇ B c ` ∇ c “ R pc , idC c qa ´ R pa, BAc qc . c1 2 c1 2 ∇c1 a 2 ∇ac1 2 ∇ 1 2 ∇ 2 1

D˜: Set c “ c2 P ΓpCq and b “BBc1 P ΓpBq. Then red red red red ¯ ∇ ∇ac ´ ∇a∇ c ´ ∇ red c ` ∇ B c “ R pb, BBcqa ´ R∇pa, BAcqb b b ∇b a ∇a b ∇ by deﬁnition of the B-connections ∇red and the A-connection ∇ on B.

(M5) R: Choose c P ΓpCq and a1,a2 P ΓpAq. Then B a ,a Ba ,a a , Ba B a B a ´ ∇c r 1 2s`r∇c 1 2s`r 1 ∇c 2s` ∇∇a2 c 1 ´ ∇∇a1 c 2 ✘✘ ✘✘ “´rBAc, ra1,a2ss´BAp∇ra1,a2scq`rrBAc,a1s`✘BAp∇a1 cq,a2s ✘✘✘ `ra , rBAc,a s`✘BA✘p∇a2 cqs 1 2 ✭ ✭ ✭✭✭✭ ✭✭✭✭ `✭rBA✭p∇a2 cq,a1s`BAp∇a1 ∇a2 cq´✭rBA✭p∇a1 cq,a2s´BAp∇a2 ∇a1 cq“BApR∇pa1,a2qcq by the Jacobi identity.

46 D˜: The equation

red red red red red ¯ ´ ∇ ra1,a2s`r∇ a1,a2s`ra1, ∇ a2s` ∇ B a1 ´ ∇ B a2 “BApR∇pa1,a2qb b b b ∇a2 b ∇a1 b

follows again immediately for b “BBc. Since BB is surjective, this shows (M5) for D˜.

B (M6) R: By deﬁnition of R∇,

B idC pR pc ,c qpaqq“´∇arc ,c s`r∇ac ,c s`rc , ∇ac s´ ∇ B c ` ∇ B c . ∇ 1 2 1 2 1 2 1 2 ∇c1 a 2 ∇c2 a 1

D˜: Setting bi “BBci P ΓpBq for i “ 1, 2 and applying BB to both sides of the equation yields then also

red B B B B B BBpR∇ pb1,b2qpaqq“´∇a rb1,b2s`r∇a b1,b2s`rb1, ∇a b2s´ ∇ red b2 ` ∇ red b1. ∇b1 a ∇b2 a

(M7) R: Checking the last equation is long, but straightforward. Choose c1,c2 P ΓpCq and B a1,a2 P ΓpAq. Then pd∇R∇qpa1,a2q evaluated on c1,c2 P ΓpCq reads

B B B ´ R∇pc1,c2qra1,a2s´ ∇a2 pR∇pc1,c2qa1q` ∇a1 pR∇pc1,c2qa2q B B B B ` R∇p∇a2 c1,c2qa1 ` R∇pc1, ∇a2 c2qa1 ´ R∇p∇a1 c1,c2qa2 ´ R∇pc1, ∇a1 c2qa2,

which equals

∇ 1 2 rc ,c s´r∇ 1 2 c ,c s´rc , ∇ 1 2 c s` ∇ B c ´ ∇ B c ra ,a s 1 2 ra ,a s 1 2 1 ra ,a s 2 ∇c1 ra1,a2s 2 ∇c2 ra1,a2s 1 ✘✘✘ ✘✘✘ ´ ∇a2 ´∇a1 rc ,c s` r∇✘a1 c ,c s` rc✘, ∇a1 c s´ ∇ B c ` ∇ B c 1 2 ✘ 1 2 ✘1 2 ∇c1 a1 2 ∇c2 a1 1 ´ ✘✘✘ ✘✘✘ ¯ ` ∇a1 ´∇a2 rc ,c s` r∇✘a2 c ,c s` rc✘, ∇a2 c s´ ∇ B 2 c ` ∇ B 2 c 1 2 ✘ 1 2 ✘1 2 ∇c1 a 2 ∇c2 a 1 ✭✭✭✭ ✭✭✭✭ ✭´✭ ✭✭ B ¯ B ´✭∇a1 r∇a2 c1,c2s`r∇a1 ∇a2 c1,c2s`✭r∇a2 c1, ∇a1 c2s´ ∇∇ a1 c2 ` ∇∇ a1 ∇a2 c1 ∇a2 c1 c2 ✭✭✭✭ ✭✭✭✭ ✭✭ ✭✭ B B ´✭∇a1 rc1, ∇a2 c2s`✭r∇a1 c1, ∇a2 c2s`rc1, ∇a1 ∇a2 c2s´ ∇∇ a1 ∇a2 c2 ` ∇∇ a1 c1 c1 ∇a2 c2 ✭✭✭✭ ✭✭✭✭ ✭✭ ✭✭ B B `✭∇a2 r∇a1 c1,c2s´r∇a2 ∇a1 c1,c2s´✭r∇a1 c1, ∇a2 c2s` ∇∇ a2 c2 ´ ∇∇ a2 ∇a1 c1 ∇a c1 c2 ✭✭ ✭✭ 1 ✭✭✭ ✭✭✭ `✭∇a✭2 rc1, ∇a1 c2s´✭r∇✭a2 c1, ∇a1 c2s´rc1, ∇a2 ∇a1 c2s` ∇∇B a2 ∇a1 c2 ´ ∇∇B a2 c1. c1 ∇a1 c2

Twelve terms of this equation cancel pairwise, and a reordering of the remaining terms yields

´ R∇pa1,a2qrc1,c2s`rR∇pa1,a2qc1,c2s`rc1, R∇pa1,a2qc2s

` ∇ B c ´ ∇ B c ´ ∇a2 ´∇ B c ` ∇ B c ` ∇a1 ´∇ B c ` ∇ B c ∇c1 ra1,a2s 2 ∇c2 ra1,a2s 1 ∇c1 a1 2 ∇c2 a1 1 ∇c1 a2 2 ∇c2 a2 1 ´ ¯ ´ ¯ ´ ∇∇B a1 c2 ` ∇∇B a1 ∇a2 c1 ´ ∇∇B a1 ∇a2 c2 ` ∇∇B a1 c1 ` ∇∇B a2 c2 ´ ∇∇B a2 ∇a1 c1 ∇a2 c1 c2 c1 ∇a2 c2 ∇a1 c1 c2

` ∇∇B a2 ∇a1 c2 ´ ∇∇B a2 c1 c1 ∇a1 c2

“´ R∇pa1,a2qrc1,c2s`rR∇pa1,a2qc1,c2s`rc1, R∇pa1,a2qc2s B B ` R pa , ∇ a qc ` ∇ 2 B 1 c ´ R pa , ∇ a qc ´ ∇ 2 B 1 c ∇ 2 c1 1 2 ra ,∇c1 a s 2 ∇ 2 c2 1 1 ra ,∇c2 a s 1 B B ´ R pa , ∇ a qc ´ ∇ 1 B 2 c ` R pa , ∇ a qc ` ∇ 1 B 2 c ∇ 1 c1 2 2 ra ,∇c1 a s 2 ∇ 1 c2 2 1 ra ,∇c2 a s 1

` ∇∇B ra1,a2sc2 ´ ∇∇B ra1,a2sc1 ´ ∇∇B a1 c2 c1 c2 ∇a2 c1

` ∇∇B a1 c1 ` ∇∇B a2 c2 ´ ∇∇B a2 c1. ∇a2 c2 ∇a1 c1 ∇a1 c2

47 By (M5), this equals

´ R∇pa1,a2qrc1,c2s´rc2, R∇pa1,a2qc1s`rc1, R∇pa1,a2qc2s B B B ` R∇pa2, ∇c1 a1qc2 ´ R∇pa2, ∇c2 a1qc1 ´ R∇pa1, ∇c1 a2qc2 B ` R∇pa1, ∇c2 a2qc1 ` ∇BAR∇pa1,a2qc2 c1 ´ ∇BAR∇pa1,a2qc1 c2, which is

B B ´ R∇pa1,a2qrc1,c2s´ ∇c2 pR∇pa1,a2qc1q` ∇c1 pR∇pa1,a2qc2q B B B B ´ R∇p∇c1 a1,a2qc2 ` R∇p∇c2 a1,a2qc1 ´ R∇pa1, ∇c1 a2qc2 ` R∇pa1, ∇c2 a2qc1.

Since this is pd∇B R∇qpc1,c2q on a1,a2 P ΓpAq, Condition (M7) is checked for R.

D˜: Setting bi “BBci P ΓpBq for i “ 1, 2 and using the deﬁnitions of all the involved objects yields again immediately

red red red ´ R∇ pb1,b2qra1,a2s´ ∇a2 pR∇ pb1,b2qa1q` ∇a1 pR∇ pb1,b2qa2q red B red B red B red B ` R∇ p∇a2 b1,b2qa1 ` R∇ pb1, ∇a2 b2qa1 ´ R∇ p∇a1 b1,b2qa2 ´ R∇ pb1, ∇a1 b2qa2 ¯ red ¯ red ¯ “´ R∇pa1,a2qrb1,b2s´ ∇b2 R∇pa1,a2qb1 ` ∇b1 R∇pa1,a2qb2 ¯ red ¯ red ¯ red ¯ red ´ R∇p∇b1 a1,a2qb2 ` R∇p∇b2 a1,a2qb1 ´ R∇pa1, ∇b1 a2qb2 ` R∇pa1, ∇b2 a2qb1,

and the seventh condition is veriﬁed for D˜.

B Double Lie algebroid morphisms – proofs B.1 Proof of Theorem 4.10

The double vector bundle morphism Φ has sides ϕC and ϕA and core ϕC . By the proof of ∇ ∇1 1 Proposition 4.9, a pair of choices of linear splittings Σ : A ˆM C Ñ T C ‘T M A and Σ : A ˆM 1 1 1 ˚ ˚ 1 C Ñ T C ‘T M A deﬁnes the form ω∇,∇1 P ΓpA b C b C q as in (40). The two splittings deﬁne as well the following 2-representations.

1. The VB-algebroid pT C ‘T M A Ñ C, A Ñ Mq induces DA :“ p∇, ∇, R∇q of A on id: C Ñ C, as in (37).

BA BA BA 2. The VB-algebroid pT C‘T M A Ñ A, C Ñ Mq induces the 2-representation DC :“ p∇ , ∇ , R∇ q of C on BA : C Ñ A, as in (38) and (39).

1 1 1 1 1 1 1 3. The VB-algebroid pT C ‘T M A Ñ C , A Ñ Mq induces DA1 :“ p∇ , ∇ , R∇1 q of A on id: C1 Ñ C1, as in (37).

1 1 1 1 4. The VB-algebroid pT C ‘T M A Ñ A , C Ñ Mq induces the 2-representation DC1 :“ B 1 BA1 BA1 A 1 1 1 p∇ , ∇ , R∇1 q of C on BA1 : C Ñ A , as in (38) and (39).

By Section 2.4 it suﬃces to check that pϕC , ϕC ,ω∇,∇1 q deﬁnes a morphism

˚ DA Ñ ϕADA1 and that pϕC , ϕA,ω∇,∇1 q deﬁnes a morphism

˚ DC Ñ ϕC DC1 .

(10) for pϕC , ϕC ,ω∇,∇1 q is immediate, and (11) is the deﬁnition of ω∇,∇1 . (12) is an easy computation, which is left to the reader. (10) for pϕC , ϕA,ω∇,∇1 q is

ϕA ˝BA “BA1 ˝ ϕC

48 and (11) is checked as follows. For c P ΓpCq, a P ΓpAq and c1 P ΓpCq:

1 Hom ˚ BA1 BA BA BA p∇c ϕAqpaq “ pϕC ∇ qcpϕApaqq ´ ϕAp∇c aq“ ∇ pϕApaqq ´ ϕAp∇c aq ✭ ϕC pcq ✭✭✭ 1 ✘✘ 1 ϕ✭✭c , ϕ a 1 ϕ c ϕ ✘✘c,a c “ ✭rBA✭ C p q Ap qs`BA ∇ϕApaqp C p qq ´ AprBA s`BAp∇a qq ˚ 1 “BA1 pϕA∇ qapϕC pcqq ´ ϕC p∇acq “BA1 pω∇,∇1 pa,cqq, and ` ˘

1 Hom 1 ˚ BA1 1 BA 1 BA 1 BA 1 p∇c ϕC qpc q “ pϕC ∇ qcpϕC pc qq ´ ϕC p∇c c q“ ∇ϕ pcqpϕC pc qq ´ ϕC p∇c c q ✭✭ C ✭✭✭ 1 1 ✟✟1 “ ✭rϕ✭Cpcq, ϕC pc qs ` ∇ 1 pϕC pcqq ´ ϕC prc,c s` ∇B c1 cq BA1 ϕC pc q ✟ A 1 1 1 ϕ c ϕ 1 c ω 1 c ,c . “ ∇ϕABApc qp C p qq ´ C p∇BAc q“ ∇,∇ pBA q

Finally, (12) is

˚ BA1 BA pd∇Hom ω∇,∇1 qpc1,c2qpaq ´ pϕC R 1 qpc1,c2qpϕAaq` ϕC pR pc1,c2qaq ∇ ✭✭∇ Hom Hom ✭✭ 1 1 ✭1 ✭ “ ∇c1 pω∇,∇ c2qpaq´ ∇c2 pω∇,∇ c1qpaq´✭ω∇✭,∇ pa, rc1,c2sq ✭✭ 1 ✭✭✭ 1 1 ` ∇✭✭rϕ✭C c1, ϕC c2s´r∇ ϕC c1, ϕC c2s´rϕC c1, ∇ ϕC c2s ✭ϕAa ϕApaq ϕApaq (59) 1 1 B ϕ c B ϕ c ` ∇ A1 C 2 ´ ∇ A1 C 1 ∇ϕC c1 ϕAa ∇ϕC c2 ϕAa ✘✘✘ ` ϕC ´✘∇a✘rc1,c2s`r∇ac1,c2s`rc1, ∇ac2s´ ∇ BA c2 ` ∇ BA c1 . ∇c1 a ∇c2 a ´ ¯ for c1,c2 P ΓpCq and a P ΓpAq. Since

Hom 1 1 1 ∇c1 pω∇,∇ c2qpaq´rϕC c1, ∇ϕ a ϕC c2s` ϕC rc1, ∇ac2s` ∇ B 1 ϕC c2 ´ ϕC ∇ BA c2 Ap q A ∇c1 a ∇ϕC c1 ϕAa B 1 BA 1 A ω 1 a,c ω 1 a,c ϕ c , ϕ c ϕ c , ϕ c “ ∇ϕC c1 p ∇,∇ p 2qq ´ ∇,∇ p∇c1 2q´r C 1 ∇ϕApaq C 2s`r C 1 C ∇a 2s 1 ` ∇ B 1 ϕC c2 ´ ϕC ∇ BA c2 A ∇c1 a ∇ϕC c1 ϕAa

BA1 1 1 ω 1 a,c B ϕ c ϕ c ,ω 1 a,c B ϕ c “ ∇ϕC c1 p ∇,∇ p 2qq ´ ∇ A C 2 ´r C 1 ∇,∇ p 2qs ` ∇ A1 C 2 ϕA∇c a 1 ∇ϕC c1 ϕAa 1 1 1 ϕ c B ϕ c B ϕ c , “ ∇B 1 ω 1 pa,c2q C 1 ´ ∇ A C 2 ` ∇ A1 C 2 A ∇,∇ ϕA∇c a 1 ∇ϕC c1 ϕAa (59) becomes

1 1 1 B B ∇B 1 ω 1 pa,c2qϕC c1 ´ ∇ A ϕC c2 ` ∇ A1 ϕC c2 A ∇,∇ ϕA∇c a 1 ∇ϕC c1 ϕAa 1 1 1 ϕ c B ϕ c B ϕ c ´ ∇B 1 ω 1 pa,c1q C 2 ` ∇ A C 1 ´ ∇ A1 C 1 A ∇,∇ ϕA∇c a 2 ∇ϕC c2 ϕAa 1 1 (11) Hom Hom “ ∇ 2 ϕC c1 ´ ∇ 1 ϕC c2 “ 0. BA1 ω∇,∇1 pa,c q´p∇c2 ϕAqpaq BA1 ω∇,∇1 pa,c q´p∇c1 ϕAqpaq

B.2 Proof of Theorem 5.23 Here (10), (11) and (12) need to be checked for

49 (10) is given by the morphism of core diagrams since in the ﬁrst case it is ϕB ˝BB “BB1 ˝ ϕC , and in the second case it is ϕA ˝BA “BA1 ˝ ϕC . (11) is in both cases an easy computation, which is left to the reader. (10) is checked in the ﬁrst case as follows. Choose a,a1 P ΓpAq, b P ΓpBq and B c P ΓpCq such that BBc “ b. Then with ∇a b “BB∇ac and BB1 ˝ ϕC “ ϕB ˝BB:

1 Hom 1 Hom 1 pd∇Hom ω∇,∇1 qpa,a qpbq“ ∇a pω∇,∇1 pa qqpbq´ ∇a1 pω∇,∇1 paqqpbq´ ω∇,∇1 pra,a s,bq 1 1 1 ω 1 a ,c ω 1 a , c “ ∇ϕApaqp ∇,∇ p qq ´ ∇,∇ p ∇a q 1 1 1 ω 1 a,c ω 1 a, 1 c ω 1 a,a ,c ´ ∇ϕApa qp ∇,∇ p qq ` ∇,∇ p ∇a q´ ∇,∇ pr s q ˚ 1 1 “ pϕAR∇1 qpa,a qpϕC pcqq ´ ϕC pR∇pa,a qcq ˚ 1 1 “ pϕAR∇1 qpa,a qpϕB pbqq ´ ϕC pR∇pa,a qbq.

Finally, in order to check (12) in the second case, note that for b,b1 P ΓpBq, a P ΓpAq and 1 1 1 c,c P ΓpCq such that BBc “ b and BBc “ b ,

1 1 pd∇Hom ω∇,∇1 qpb,b qpaq “ pd∇Hom ω∇,∇1 qpc,c qpaq.

By the proof of Theorem 4.10, the right-hand side is

˚ BA1 1 BA 1 pϕC R∇1 qpc,c qpϕApaqq ´ ϕC pR∇ pc,c qaq, which equals ˚ red 1 red 1 pϕBR∇1 qpb,b qpϕApaqq ´ ϕC pR∇ pb,b qaq, red red by the deﬁnition of R∇1 and R∇ .

C Notation for linear and core sections.

The main notations of this paper are summarised here for the convenience of the reader.

1. Let pD,A,B,Mq be any double vector bundle with sides A and B and with core C. A l morphism Φ P ΓpHompA, Cqq defines a core-linear section Φ P ΓApDq, which is linear over 0B P ΓpBq: D Φpaq“ 0a `B ϕpaq. r Similarly, a morphism Φ P ΓpHompB, Cqq defines a core-linear section Φ P Γl pDq, which is r B linear over 0A P ΓpAq: D Φpbq“ 0b À ϕpbq. r Core-linear sections are always written with this notation, no matter over which side or in r which double vector bundle. It is always clear from the context where they are defined.

Ò c 2. Let qE : E Ñ M be a vector bundle. Given e P ΓpEq, then e P X pEq is the vertical vector field on E defined by e; i.e. the core section of T E Ñ E defined by e. The core section of : c T E Ñ TM defined by e is written e P ΓT M pT Eq. Given a derivation D of E with symbol X P XpMq, then D P XlpEq is the linear vector field on E that is equivalent to D. It is a linear section of T E Ñ E over X. A section e P ΓpEq l defines further the linear section Te P ΓT M pT Eq. It is linearp over e.

Let ∇: ΓpAqˆ ΓpCq Ñ ΓpCq be a linear A-connection on C. Then ∇ deﬁnes a linear splitting ∇ ! Σ : A ˆM C Ñ R of R :“ ρAT C “ T C ‘T M A.

50 3. Core sections of R Ñ C are written γˆ for γ P ΓpCq:

ˆ Ò A γ pcmq “ pγ pcmq, 0mq

∇ ∇ l for all cm P C. The splitting Σ is equivalent to a linear horizontal lift σA : ΓpAq Ñ ΓC pRq: ∇ l for all a P ΓpAq the section a :“ σA paqP ΓC pRq is deﬁned by

p apcmq“ ∇apcmq,apmq ´ ¯ for all cm P C. By definition it isp linear overxa P ΓpAq. ; ; : ! 4. Core sections of R Ñ A are written γ for γ P ΓpCq, and defined by γ “ pγ q P ΓApRq “ ! ΓApρAT Cq: ; : γ paq“ γ pρApaqq,a for all a P ΓpAq. A section c P ΓpCq defines` a linear section˘ pT Cq! of R Ñ A:

! pTcq paq “ pTcpρApaqq,aq

for all a P ΓpAq. It is linear over c P C. The splitting Σ∇ is equivalent to a linear horizontal ∇ l ∇ l lift σC : ΓpCq Ñ ΓApRq: for all c P ΓpCq the section c :“ σC pcqP ΓApRq is deﬁned by

cpapmqq “ ∇apcpmqqq,apmq ´ ¯ for all a P ΓpAq. By deﬁnition itq is linear overx c P ΓpCq.

References

[1] I. Androulidakis. Crossed modules and the integrability of Lie brackets, 2005. [2] C. Arias Abad and M. Crainic. Representations up to homotopy of Lie algebroids. J. Reine Angew. Math., 663:91–126, 2012. [3] R. Brown and K. C. H. Mackenzie. Determination of a double Lie groupoid by its core diagram. J. Pure Appl. Algebra, 80(3):237–272, 1992. [4] H. Bursztyn, A. Cabrera, and M. del Hoyo. Vector bundles over Lie groupoids and algebroids. Adv. Math., 290:163–207, 2016. [5] A. Cabrera, O. Brahic, and C. Ortiz. Obstructions to the integrability of VB-algebroids. J. Symplectic Geom., 16(2):439–483, 2018. [6] T. Drummond, M. Jotz Lean, and C. Ortiz. VB-algebroid morphisms and representations up to homotopy. Diﬀerential Geometry and its Applications, 40:332–357, 2015. [7] A. Gracia-Saz, M. Jotz Lean, K. C. H. Mackenzie, and R. A. Mehta. Double Lie algebroids and representations up to homotopy. J. Homotopy Relat. Struct., 13(2):287–319, 2018. [8] A. Gracia-Saz and R. A. Mehta. Lie algebroid structures on double vector bundles and representation theory of Lie algebroids. Adv. Math., 223(4):1236–1275, 2010. [9] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. [10] E. Hawkins. A groupoid approach to quantization. J. Symplectic Geom., 6(1):61–125, 2008. [11] M. Heuer and M. Jotz Lean. Multiple vector bundles: cores, splittings and decompositions. Theory and Applications of Categories, 35(19):665–699, 2020.

51 [12] P. J. Higgins and K. Mackenzie. Algebraic constructions in the category of Lie algebroids. J. Algebra, 129(1):194–230, 1990. [13] M. Jotz Lean and C. Ortiz. Foliated groupoids and infinitesimal ideal systems. Indag. Math. (N.S.), 25(5):1019–1053, 2014. [14] H.-Y. Liao, M. Stiénon, and P. Xu. Formality and Kontsevich-Duflo type theorems for Lie pairs. Adv. Math., 352:406–482, 2019. [15] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. [16] K. C. H. Mackenzie. Double Lie algebroids and second-order geometry. I. Adv. Math., 94(2):180–239, 1992. [17] K. C. H. Mackenzie. Double Lie algebroids and iterated tangent bundles, 1998. [18] K. C. H. Mackenzie. Double Lie algebroids and second-order geometry. II. Adv. Math., 154(1):46–75, 2000. [19] K. C. H. Mackenzie. General Theory of Lie Groupoids and Lie Algebroids, volume 213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005. [20] K. C. H. Mackenzie. Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids. J. Reine Angew. Math., 658:193–245, 2011. [21] K. C. H. Mackenzie and P. Xu. Integration of Lie bialgebroids. Topology, 39(3):445–467, 2000. [22] E. Meinrenken and J. Pike. The Weil algebra of a double Lie algebroid. International Math- ematics Research Notices, 01 2020. rnz361. [23] J. Pradines. Fibrés vectoriels doubles et calcul des jets non holonomes, volume 29 of Esquisses Mathématiques [Mathematical Sketches]. Universitéd’Amiens U.E.R. de Mathématiques, Amiens, 1977. [24] D. Quillen. Superconnections and the Chern character. Topology, 24(1):89–95, 1985. [25] L. Stefanini. On Morphic Actions and Integrability of LA-Groupoids. arXiv e-prints, page arXiv:0902.2228, February 2009.

[26] M. Sti´enon, L. Vitagliano, and P. Xu. A8-algebras from Lie pairs. Work in preparation, 2021. [27] Th. Th. Voronov. Q-Manifolds and Mackenzie Theory. Comm. Math. Phys., 315(2):279–310, 2012.