# Transitive double Lie algebroids via core diagrams

M. Jotz Lean111Mathematisches Institut, Georg-August Universität Göttingen.
<EMAIL_ADDRESS>and K. C. H. Mackenzie222In
memoriam – School of Mathematics and Statistics, The University of Sheffield.

###### Abstract

The core diagram of a double Lie algebroid consists in the core of the double
Lie algebroid, together with the two core-anchor maps to the sides of the
double Lie algebroid. If these two core anchors are surjective, then the
double Lie algebroid and its core diagram are called _transitive_. This paper
establishes an equivalence between transitive double Lie algebroids, and
transitive core diagrams over a fixed base manifold. In other words, it proves
that a transitive double Lie algebroid is completely determined by its core
diagram.

The comma double Lie algebroid associated to a morphism of Lie algebroids is
defined. If the latter morphism is one of the core-anchors of a transitive
core diagram, then the comma double algebroid can be quotiented out by the
second core-anchor, yielding a transitive double Lie algebroid, which is the
one that is equivalent to the transitive core diagram.

Brown’s and Mackenzie’s equivalence of transitive core diagrams (of Lie
groupoids) with transitive double Lie groupoids is then used in order to show
that a transitive double Lie algebroid with integrable sides and core is
automatically integrable to a transitive double Lie groupoid.

_This research was a joint project with the sadly deceased second author. This
paper is dedicated to his memory._

###### Contents

  1. 1 Introduction
    1. 1.1 Comma double Lie groupoid and comma double Lie algebroid
    2. 1.2 Transitive double Lie groupoids and their core diagrams
  2. 2 Preliminaries
    1. 2.1 Ideals in Lie algebroids
    2. 2.2 VB-algebroids and representations up to homotopy
    3. 2.3 Double Lie algebroids
    4. 2.4 Morphisms of double Lie algebroids
  3. 3 Transitive double Lie algebroids and transitive core diagrams
  4. 4 The comma double Lie algebroid
    1. 4.1 Action of $TC\to TM$ on $A\to TM$
    2. 4.2 The double vector bundle $R$
    3. 4.3 The comma double Lie algebroid
    4. 4.4 Morphisms of comma-double Lie algebroids
  5. 5 The quotient
    1. 5.1 The quotient of $R$ by $\Pi$.
    2. 5.2 $\Pi$ as an ideal of $R\to A$
    3. 5.3 The VB-algebroid $R/\Pi\to B$
    4. 5.4 The quotient $R/\Pi$ as a double Lie algebroid.
    5. 5.5 Equivalence of transitive double Lie algebroids with transitive core diagrams
  6. 6 Integration of transitive double Lie algebroids
  7. A The Lie bialgebroid condition for $R$ and $\tilde{D}$
  8. B Double Lie algebroid morphisms – proofs
    1. B.1 Proof of Theorem 4.10
    2. B.2 Proof of Theorem 5.23
  9. C Notation for linear and core sections.

##  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 $\Gamma$ 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.

${\Gamma}$${G}$${H}$${M}$

The groupoids $G$ and $H$ are the side groupoids and the set $M$ is the double
base. The core $K$ of a double groupoid $(\Gamma,G,H,M)$ is the set of
elements which project under the two sources of $\Gamma$ to units of $G$ and
$H$. It inherits from the double structure of $\Gamma$ a groupoid structure
over $M$, and the two targets of $\Gamma$ induce two groupoid morphisms ${\rm
t}_{G}\arrowvert_{K}\colon K\to G$ and ${\rm t}_{H}\arrowvert_{K}\colon K\to
H$ over the identity on $M$. Elements of the kernel of the morphism $K\to G$
commutes with elements of the kernel of $K\to H$. These two groupoid morphisms
build together the _core diagram_ of the double groupoid [3].

${K}$${G}$${H}$${M}$$\scriptstyle{{\rm
t}_{G}\arrowvert_{K}}$$\scriptstyle{{\rm t}_{H}\arrowvert_{K}}$

A _double Lie groupoid_ is a double groupoid $(\Gamma,G,H,M)$ such that all
four groupoids $\Gamma\rightrightarrows G$, $\Gamma\rightrightarrows H$,
$G\rightrightarrows M$ and $H\rightrightarrows M$ are Lie groupoids and such
that the double source map $({\rm s}_{G},{\rm s}_{H})\colon\Gamma\to
G\times_{\rm s}H=\\{(g,h)\in G\times H\mid{\rm s}(g)={\rm s}(h)\\}$ 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 $K\rightrightarrows M$ is locally trivial and $\partial_{G}:={\rm
t}_{G}\arrowvert_{K}$ and $\partial_{H}:={\rm t}_{H}\arrowvert_{K}$ are both
surjective submersions. It is _transitive_ if only $\partial_{G}$ and
$\partial_{H}$ 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
$K\rightrightarrows M$ 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
$(D;A,B;M)$

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\scriptstyle{\pi_{B}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{A}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{B}}$$\textstyle{M}$

with core $C$ and four Lie algebroid structures on $A\to M$, $B\to M$, $D\to
A$ and $D\to B$ such that $(\pi_{B},q_{A})$ and $(\pi_{A},q_{B})$ are Lie
algebroid morphisms, and the induced Lie algebroids
$D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,A\to
C^{*}$ and
$D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,B\to
C^{*}$ form a VB-Lie bialgebroid [18]. The anchor $\Theta_{A}\colon\thinspace
D\to TA$ is a morphism of double vector bundles and its core morphism is
denoted by $\partial_{A}\colon C\to A$. Likewise the core morphism of the
linear anchor $\Theta_{B}\colon\thinspace D\to TB$ is written
$\partial_{B}\colon C\to B$. The core $C$ of the double Lie algebroid inherits
a Lie algebroid structure over $M$, such that the two _core-anchors_
$\partial_{A}$ and $\partial_{B}$ are Lie algebroid morphisms over $M$ [20,
7]. The compatiblity of the Lie algebroid structures on $D\to A$ and $D\to B$
implies that $[c_{1},c_{2}]=0$ for all $c_{1}\in\Gamma(\ker\partial_{A})$ and
$c_{2}\in\Gamma(\ker\partial_{B})$. In other words, _$\ker\partial_{A}$ and
$\ker\partial_{B}$ 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_.

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{B}}$$\scriptstyle{\partial_{A}}$$\textstyle{B}$$\textstyle{A}$
Figure 1: The core diagram of a double Lie algebroid.

If both $\partial_{A}$ and $\partial_{B}$ 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 $(D,A,B,M)$ 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 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 $A\hookrightarrow L$ 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 $(D,A,B,M)$ be a transitive double Lie algebroid with a transitive core
diagram $A\mapsfrom C\mapsto B$ where $A$, $B$ and $C$ are all integrable Lie
algebroids over $M$. Then $(D,A,B,M)$ integrates to a transitive double Lie
groupoid, the core diagram of which integrates $A\mapsfrom C\mapsto B$.

###  1.1 Comma double Lie groupoid and comma double Lie algebroid

Given categories and functors

$\mathcal{D}\overset{F}{\longrightarrow}\mathcal{C}\overset{G}{\longleftarrow}\mathcal{E},$

the _comma category_ $(F,G)$ has as objects the triples $(d,e,\varphi)$ with
$d\in{\rm Obj}(\mathcal{D})$, $e\in{\rm Obj}(\mathcal{E})$ and $\varphi\in{\rm
Mor}_{\mathcal{C}}(F(d),G(e))$, see [15]. Its arrows
$(d,e,\varphi)\to(d^{\prime},e^{\prime},\varphi^{\prime})$ are the pairs
$(h,l)\in{\rm Mor}_{\mathcal{D}}(d,d^{\prime})\times{\rm
Mor}_{\mathcal{E}}(e,e^{\prime})$ such that

$\textstyle{F(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{F(h)}$$\textstyle{F(d^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{G(e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(l)}$$\textstyle{G(e^{\prime})}$

commutes. If defined, the composition $(h^{\prime},l^{\prime})\circ(h,l)$ is
$(h^{\prime}h,l^{\prime}l)$. If $K\rightrightarrows M$ and $G\rightrightarrows
M$ are two (Lie) groupoids over a common base $M$ and $\Phi\colon K\to G$ is a
(Lie) groupoid morphism fixing the base, then the diagram

$K\overset{\Phi}{\longrightarrow}G\overset{\Phi}{\longleftarrow}K,$

defines as above the comma category $(\Phi,\Phi)$, which has a _double_ (Lie)
groupoid structure with sides $K$ and $G$ and with core $K$. Denote the space
of arrows of $(\Phi,\Phi)$ by $\Gamma$. The objects of $(\Phi,\Phi)$ are hence
triples $(m_{1},g,m_{2})\in M\times G\times M$ with ${\rm s}(g)=m_{1}$ and
${\rm t}(g)=m_{2}$. Hence, the objects of $(\Phi,\Phi)$ are just the arrows of
$G\rightrightarrows M$, and the elements of $\Gamma$ (that is, the arrows of
$(\Phi,\Phi)$) are written as triples $(k_{2},g,k_{1})\in K\times G\times K$
such that ${\rm s}(k_{2})={\rm t}(g)$ and ${\rm s}(k_{1})={\rm s}(g)$.

$\textstyle{n_{2}}$$\textstyle{m_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(k_{2})}$$\textstyle{(k_{2},g,k_{1})}$$\textstyle{n_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(k_{2})\cdot
g\cdot\Phi(k_{1}^{-1})}$$\textstyle{m_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(k_{1})}$$\scriptstyle{g}$

This diagram is pictured in the usual manner for elements of a double groupoid
[3] and will just be written

$\textstyle{n_{2}}$$\textstyle{m_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{2}}$$\textstyle{n_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(k_{2})\cdot
g\cdot\Phi(k_{1}^{-1})}$$\textstyle{m_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}}$$\scriptstyle{g}$

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 $\Gamma\to G$
are the maps $(k_{2},g,k_{1})\mapsto g$ and
$(k_{2},g,k_{1})\mapsto\Phi(k_{2})\cdot g\cdot\Phi(k_{1}^{-1})$, respectively.
The source and target maps $\Gamma\to K$ are the maps $(k_{2},g,k_{1})\mapsto
k_{1}$ and $(k_{2},g,k_{1})\mapsto k_{2}$, respectively. The composition over
$G$ is

$(k_{2}^{\prime},\Phi(k_{2})\cdot
g\cdot\Phi(k_{1}^{-1}),k_{1}^{\prime})\cdot_{G}(k_{2},g,k_{1})=(k_{2}^{\prime}k_{2},g,k_{1}^{\prime}k_{1})$

and the composition over $K$ is

$(k_{3},g^{\prime},k_{2})\cdot_{K}(k_{2},g,k_{1})=(k_{3},g^{\prime}g,k_{1}).$

It is easy to check that given a composable square as below, hence with
$g=\Phi(l_{3})g^{\prime}\Phi(l_{2}^{-1})$ and
$h=\Phi(l_{2})h^{\prime}\Phi(l_{1}^{-1})$, the order of the horizontal and
vertical multiplications is not relevant, as pictured below. Hence,
$(\Phi,\Phi)$ is a double groupoid.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{k_{3}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{l_{2}}$$\scriptstyle{g}$$\scriptstyle{k_{2}}$$\scriptstyle{k_{1}}$$\scriptstyle{l_{1}}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\scriptstyle{l_{3}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{h^{\prime}}$$\scriptstyle{l_{3}}$$\scriptstyle{g^{\prime}h^{\prime}}$$\scriptstyle{k_{3}}$$\scriptstyle{gh}$$\scriptstyle{k_{3}}$$\scriptstyle{k_{1}}$$\scriptstyle{k_{3}l_{3}}$$\scriptstyle{k_{2}l_{2}}$$\scriptstyle{k_{1}l_{1}}$$\scriptstyle{k_{3}l_{3}}$$\scriptstyle{g^{\prime}h^{\prime}}$$\scriptstyle{k_{1}l_{1}}$

If $\Phi\colon K\to G$ was a morphism of Lie groupoids, it is not difficult to
check that $(\Phi,\Phi)$ is a double _Lie_ groupoid [3].

The core of $(\Gamma,K,G,M)$ consists of elements $(k,1_{m},1_{m})$

$\textstyle{n}$$\textstyle{m\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{m\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi(k)}$$\textstyle{m\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{m}}$$\scriptstyle{1_{m}}$

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

${{{\rm
t}(k)}}$${m}$${{m^{\prime}}}$${m}$${m}$${{m^{\prime}}}$${{m^{\prime}}}$${{m^{\prime}}}$${{m^{\prime}}}$$\scriptstyle{\Phi(k)}$$\scriptstyle{k}$$\scriptstyle{1_{m}}$$\scriptstyle{1_{m}}$$\scriptstyle{\Phi(k^{\prime})}$$\scriptstyle{1_{m^{\prime}}}$$\scriptstyle{1_{m^{\prime}}}$$\scriptstyle{k^{\prime}}$$\scriptstyle{\Phi(k^{\prime})}$$\scriptstyle{1_{m^{\prime}}}$$\scriptstyle{k^{\prime}}$$\scriptstyle{1_{m^{\prime}}}$

hence yielding

$(k,1_{m},1_{m})\cdot(k^{\prime},1_{m^{\prime}},1_{m^{\prime}})=(kk^{\prime},1_{m^{\prime}},1_{m^{\prime}}).$

As a consequence, the core (Lie) groupoid is isomorphic to $K\rightrightarrows
M$ as a (Lie) groupoid. Note that $\Gamma\rightrightarrows G$ is the action
groupoid of the (Lie) groupoid action of $K\times K\rightrightarrows M\times
M$ on $({\rm t},s)\colon G\to M\times M$, $(k_{2},k_{1})\cdot
g=\Phi(k_{2})g\Phi(k_{1}^{-1})$.

Section 4 constructs analogously a double Lie algebroid from a Lie algebroid
morphism

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

The morphism $\partial$ induces an action of the Lie algebroid $TC\to TM$ on
the anchor $\rho_{A}\colon A\to TM$ of $A$. The total space
$R:=TC\oplus_{TM}A$ of the action Lie algebroid $TC\oplus_{TM}A\to A$ of this
action carries as well automatically the pullback Lie algebroid structure from
$A\to M$ under the smooth map $q_{C}\colon C\to M$. It is hence also a Lie
algebroid over $C$, and these two Lie algebroid structures define a double Lie
algebroid

$\textstyle{TC\oplus_{TM}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

with sides $A$ and $C$ and with core $C$. This double Lie algebroid is the
_comma double Lie algebroid_ defined by the morphism $\partial\colon C\to A$.
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.

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\scriptstyle{\partial}$$\textstyle{C}$$\textstyle{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.

Consider a transitive core diagram

${K}$${G}$${H}$${M}$$\scriptstyle{\partial_{G}}$$\scriptstyle{\partial_{H}}$
Figure 2: Core diagram of Lie groupoids

of Lie groupoids; i.e. a diagram as in Figure 2 of Lie groupoid morphisms,
such that $\partial_{G}$ and $\partial_{H}$ are surjective submersions and the
subgroupoids $\ker(\partial_{G})$ and $\ker(\partial_{H})$ commute in $K$.
Build the comma double Lie groupoid $(\partial_{G},\partial_{G})$

${\Gamma}$${G}$${K}$${M}$

with core $K$. Next, $G\to M$ acts on $K^{G}:=\ker(\partial_{H})\subseteq K$
via

$\rho\colon G\times_{{\rm s},{\rm t}}K^{G}\to
K^{G},\qquad\rho(g)(\kappa)=k\kappa k^{-1}$

for any $k\in K$ such that $\partial_{G}(k)=g$. Consider the closed, embedded
wide and normal sugbroupoid

$N:=\\{(\kappa_{2},g,\kappa_{1})\in K^{G}\times G\times
K^{G}\mid\rho(g)(\kappa_{1})=\kappa_{2}\\}$ (1)

of $\Gamma\rightrightarrows G$. The quotient $\Theta=\Gamma/N$ has then a Lie
groupoid structure over $G$. The elements of $\Theta$ are classes

$\langle
k_{2},g,k_{1}\rangle=\\{(k_{2}\kappa_{2},g,k_{1}\kappa_{1})\mid(\kappa_{2},g,\kappa_{1})\in
N\\}$

with $(k_{2},g,k_{1})\in\Gamma$. Set ${\rm s}_{H},{\rm t}_{H}\colon\Gamma/N\to
H$

${\rm s}_{H}\langle k_{2},g,k_{1}\rangle=\partial_{H}(k_{1}),\qquad{\rm
t}_{H}\langle k_{2},g,k_{1}\rangle=\partial_{H}(k_{2}),$

and a partial multiplication $\cdot_{H}\colon\Theta\times_{{\rm s}_{H},{\rm
t}_{H}}\Theta\to\Theta$ by

$\langle k_{2}^{\prime},g^{\prime},k_{1}^{\prime}\rangle\cdot_{H}\langle
k_{2},g,k_{1}\rangle=\langle
k_{2}^{\prime},g^{\prime},k_{1}^{\prime}\rangle\cdot_{H}\langle
k_{1}^{\prime},g,k_{1}\rho(g^{-1})(\kappa)\rangle=\langle
k_{2}^{\prime},g^{\prime}g,k_{1}\rho(g^{-1})(\kappa)\rangle.$

Here $\kappa\in K^{G}$ is such that $k_{2}\kappa=k_{1}^{\prime}$. It is easy
to check that $\Theta/N\rightrightarrows H$ becomes a groupoid with these
structure maps and with the inversion

$\langle k_{2},g,k_{1}\rangle\mapsto\langle k_{1},g^{-1},k_{2}\rangle$

and the unit inclusion

$h\mapsto\langle k,1_{{\rm s}(h)},k\rangle$

for any $k\in K$ such that $\partial_{H}(k)=h$. Since those structure maps are
defined such that

${\Gamma}$${{\Theta=\Gamma/N}}$${K}$${H}$$\scriptstyle{\partial_{H}}$$\scriptstyle{\pi}$

is a morphism of groupoids and the projections $\pi\colon\Gamma\to\Theta$ and
$\partial_{H}\colon K\to H$ are surjective submersions, it is easy to check
that $\Theta\rightrightarrows H$ is a Lie groupoid. The interchange law and
the surjectivity of the double source map are also easily deduced from the one
in $(\partial_{G},\partial_{G})$, and $(\Theta,G,H,M)$ is a double Lie
groupoid. Its core consists in classes $\langle k,1_{{\rm
s}(k)},\kappa\rangle=\langle k\kappa^{-1},1_{{\rm s}(k)},1_{{\rm
s}(k)}\rangle$, and is obviously isomorphic to $K$ as a Lie groupoid. The core
diagram of $(\Theta,G,H,M)$ is hence again

${K}$${G}$${H}$${M}$$\scriptstyle{\partial_{G}}$$\scriptstyle{\partial_{H}}$

by construction.

Now given a transitive double Lie groupoid

${S}$${G}$${H}$${M}$

with core $K$ and core diagram as in Figure 2, set $\Psi_{S}\colon\Theta\to
S$, $\langle k_{2},g,k_{1}\rangle\mapsto
k_{2}\cdot_{H}1^{S}_{g}\cdot_{H}k_{1}^{-1,H}\in S$. Then $\Psi_{S}$ is an
isomorphism of double Lie groupoids, see [3] – where, again, only the
surjectivity of $\partial_{G}$ and $\partial_{H}$ are needed.

A morphism $\Phi\colon\Gamma_{1}\to\Gamma_{2}$ of double Lie groupoids with
side morphisms $\varphi_{G}\colon G_{1}\to G_{2}$, $\varphi_{H}\colon H_{1}\to
H_{2}$ and with core morphism $\varphi_{K}\colon K_{1}\to K_{2}$ induces a
morphism of the corresponding core diagrams as in the following diagram.

${{K_{1}}}$${{K_{2}}}$${{G_{1}}}$${{G_{2}}}$${{H_{1}}}$${{H_{2}}}$${M}$${M}$$\scriptstyle{\varphi_{K}}$$\scriptstyle{\varphi_{H}}$$\scriptstyle{\partial_{H_{1}}}$$\scriptstyle{\partial_{H_{2}}}$$\scriptstyle{\partial_{G_{1}}}$$\scriptstyle{\partial_{G_{2}}}$$\scriptstyle{\operatorname{id}_{M}}$$\scriptstyle{\varphi_{G}}$
(2)

Consider conversely a morphism of transitive core diagrams (of Lie groupoids)
as in (2). Let $\Gamma_{i}$ be the total space of the comma double Lie
groupoid $(\partial_{G_{i}},\partial_{G_{i}})$ for $i=1,2$ and set
$\Phi\colon\Gamma_{1}\to\Gamma_{2}$,
$\Phi(k_{1},g,k_{2})=(\varphi_{K}(k_{1}),\varphi_{G}(g),\varphi_{K}(k_{2}))$.
The map $\Phi$ is obviously a morphism of double Lie groupoids, and a
computation shows $\Phi(N_{1})=N_{2}$ for the normal subgroupoids $N_{i}$ of
$\Gamma_{i}\rightrightarrows G_{i}$ defined as in (1) for $i=1,2$. Therefore,
it induces a morphism of the transitive double Lie groupoids
$\overline{\Phi}\colon\Gamma_{1}/N_{1}\to\Gamma_{2}/N_{2}$ with core morphism
again the one in (2).

Let $M$ be a smooth manifold and consider the category
$\operatorname{trCDG}(M)$ of transitive core diagrams of Lie groupoids with
base $M$, and the category $\operatorname{trDLG}(M)$ of transitive double Lie
groupoids with double base $M$. The functor
$\mathcal{C}\colon\operatorname{trDLG}(M)\to\operatorname{trCDG}(M)$ 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
$\mathcal{D}\colon\operatorname{trCDG}(M)\to\operatorname{trDLG}(M)$ sending a
transitive core diagram as in Figure 2 to the double Lie groupoid
$(\Theta,G,H,M)$ as above, and a morphism of core diagrams as in (2) to the
morphism $\overline{\Phi}\colon\Theta_{1}\to\Theta_{2}$ as in the previous
paragraph. Then $\mathcal{C}\circ\mathcal{D}$ is obviously the identity
functor. The assigment sending each transitive double Lie groupoid $(S,G,H,M)$
to the isomorphism $\Psi_{S}\colon\mathcal{D}\mathcal{C}(S)\to S$ of double
Lie groupoids defines a natural isomorphism
$\Psi_{\cdot}\colon\mathcal{D}\circ\mathcal{C}\to\operatorname{id}_{\operatorname{trDLG}}$.
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.

### 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 $C\to A$ (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 $q_{E}\colon E\to M$, and $p_{M}\colon TM\to M$
for tangent bundles. Given a section $\varepsilon$ of $E^{*}$, the map
$\ell_{\varepsilon}\colon E\to\mathbb{R}$ is the linear function associated to
it, i.e. the function defined by
$e_{m}\mapsto\langle\varepsilon(m),e_{m}\rangle$ for all $e_{m}\in E$. The set
of global sections of a vector bundle $E\to M$ is denoted by $\Gamma(E)$,
$\mathfrak{X}(M)=\Gamma(TM)$ is the space of smooth vector fields on a smooth
manifold $M$, and $\Omega^{\bullet}(M)=\Gamma(\bigwedge^{\bullet}T^{*}M)$ is
the space of smooth forms on $M$.

### 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 $(q\colon A\to M,\rho,[\cdot\,,\cdot])$ be a Lie algebroid, $F\subseteq
TM$ an involutive subbundle, $J\subseteq A$ a subbundle over $M$ such that
$\rho(J)\subseteq F$, and $\nabla$ a flat $F$-connection on $A/J$ with the
following properties:

  1. 1.

If $a\in\Gamma(A)$ is $\nabla$-parallel333A section $a\in\Gamma(A)$ is said to
be $\nabla$-parallel if $\nabla_{X}\bar{a}=0$ for all $X\in\Gamma(F)$. Here,
$\bar{a}$ is the class of $a$ in $\Gamma(A/J)\simeq\Gamma(A)/\Gamma(J)$., then
$[a,j]\in\Gamma(J)$ for all $j\in\Gamma(J)$.

  2. 2.

If $a,b\in\Gamma(A)$ are $\nabla$-parallel, then $[a,b]$ is also
$\nabla$-parallel.

Then the triple $(F,J,\nabla)$ is an infinitesimal ideal system in $A$.

The first axiom implies immediately that $J\subseteq A$ is a subalgebroid of
$A$, and the following property of $(F,J,\nabla)$ follows from (2) above.

  1. 3.

If $a\in\Gamma(A)$ is $\nabla$-parallel, then $\rho(a)$ is
$\nabla^{F}$-parallel, where

$\nabla^{F}\colon\Gamma(F)\times\Gamma(TM/F)\to\Gamma(TM/F),\quad\nabla^{F}_{X}\bar{Y}=\overline{[X,Y]}$

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 $K\subseteq E$ be a subbundle.
Then any flat connection of an involutive subbundle $F_{M}\subseteq TM$ on
$E/K$ defines an _infinitesimal ideal system $(F_{M},K,\nabla)$ in $E$_, i.e.
an infinitesimal ideal system in the trivial Lie algebroid
$(E,\rho=0,[\cdot\,,\cdot]=0)$. 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. 1.

An ideal in a Lie algebroid $A\to M$ is a vector subbundle $I\subseteq A$ such
that $[\Gamma(I),\Gamma(A)]\subseteq\Gamma(I)$. Given such an ideal $I$ in
$A$, the triple $(0,I,0)$ is an infinitesimal ideal system in $A$.

  2. 2.

Let $A\to M$ be a Lie algebroid. Let $F_{M}\subseteq TM$ be an involutive
subbundle and let $\nabla\colon\Gamma(F_{M})\times\Gamma(A)\to\Gamma(A)$ be a
flat connection such that $[a,b]$ is $\nabla$-parallel for $a,b\in\Gamma(A)$
$\nabla$-parallel. Then $(F_{M},0,\nabla)$ is an infinitesimal ideal system in
$A$.

######  Definition 2.4.

A morphism

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{E}}$$\scriptstyle{\Phi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{B}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N}$

of Lie algebroids $A\to M$ and $B\to N$ is a fibration of Lie algebroids if
the underlying morphism $(\Phi,\varphi)$ of vector bundles is a fibration of
vector bundles, i.e. $\varphi$ is a surjective submersion and $\Phi$ is
fiberwise surjective.

Let $E\to M$ be a vector bundle, and let $J\subseteq E$ be a subbundle and
$F_{M}\subseteq TM$ an involutive subbundle. Assume that there is a flat
$F_{M}$-connection on $E/J$. Then define an equivalence relation
$\sim_{\nabla}$ on $E/J$ as follows: for $e,e^{\prime}\in E$ and their classes
$\bar{e},\bar{e}^{\prime}\in E/J$, $\bar{e}\sim_{\nabla}\bar{e}^{\prime}$ if
and only if $q_{E}(e)$ and $q_{E}(e^{\prime})$ are in the same leaf $L$ of
$F_{M}$ in $M$ and $\bar{e}^{\prime}$ is the image of $\bar{e}$ under
$\nabla$-parallel transport along a path in the leaf $L$ joining $q_{E}(e)$
and $q_{E}(e^{\prime})$. The quotient of $E$ by this equivalence relation is
written $(E/J)/\nabla$. The projections $E\to(E/J)/\nabla$ and $M\to M/F_{M}$
are written $\pi$ and $\pi_{M}$, repectively. The vector bundle map
$q_{E}\colon E\to M$ factors to a map $[q_{E}]\colon(E/J)/\nabla\to M/F_{M}$.

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.

######  Theorem 2.5.

  1. 1.

Let $q_{E}\colon E\to M$ be a smooth vector bundle and let $(F_{M},J,\nabla)$
be an infinitesimal ideal system in $E$ (see Example 2.2). Assume that
$\bar{M}=M/F_{M}$ is a smooth manifold and that $\nabla$ has trivial holonomy.
Then the quotient $(E/J)/\nabla\to M/F_{M}$ carries a vector bundle structure
such that the projection $(\pi,\pi_{M})$ is a fibration of vector bundles:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{E}}$$\scriptstyle{\pi}$$\textstyle{(E/J)/\nabla\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[q_{E}]}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{M}}$$\textstyle{M/F_{M}}$
(3)

  2. 2.

Let now $A\to M$ be a Lie algebroid and let $(F_{M},J,\nabla)$ be an
infinitesimal ideal system in $A$. If the quotient vector bundle
$(A/J)/\nabla\to M/F_{M}$ 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 $(F_{M},J,\nabla)$ 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 $\Gamma(A)^{\nabla}$ for the space of sections of $A\to M$ the
class in $A/J$ of which is $\nabla$-flat. Then $\Gamma(A)^{\nabla}$ is an
$\mathbb{R}$-Lie algebra and a
$C^{\infty}(M)^{F_{M}}=\pi_{M}^{*}C^{\infty}(\bar{M})$-module. The space
$\Gamma(J)$ is an ideal in $\Gamma(A)^{\nabla}$. By definition, for each
$a\in\Gamma(A)^{\nabla}$ there is a section $\bar{a}\in\Gamma(\bar{A})$ such
that $\pi\circ a=\bar{a}\circ\pi_{M}$ (this is written $a\sim_{\pi}\bar{a}$
for short). In particular, the sections $j\in\Gamma(J)$ project in this manner
to the zero section of $\bar{A}\to\bar{M}$. Conversely for each section
$\bar{a}\in\Gamma(\bar{A})$ there is a $\nabla$-flat $a\in\Gamma(A)$ such that
$a\sim_{\pi}\bar{a}$. The Lie algebroid bracket and the anchor on
$\bar{A}\to\bar{M}$ are then defined by

$\Gamma(A)^{\nabla}\ni
a_{i}\sim_{\pi}\bar{a}_{i}\in\Gamma(\bar{A}),\,i=1,2\qquad\Rightarrow\qquad[a_{1},a_{2}]\sim_{\pi}[\bar{a}_{1},\bar{a}_{2}],$

and

$\overline{\rho}(\pi(a))=T\pi_{M}\rho(a)$

for all $a\in A$.

###  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

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B}}$$\scriptstyle{\pi_{A}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{B}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{A}}$$\textstyle{M}$

of vector bundles such that

$(d_{1}+_{A}d_{2})+_{B}(d_{3}+_{A}d_{4})=(d_{1}+_{B}d_{3})+_{A}(d_{2}+_{B}d_{4})$
(4)

for $d_{1},d_{2},d_{3},d_{4}\in D$ with $\pi_{A}(d_{1})=\pi_{A}(d_{2})$,
$\pi_{A}(d_{3})=\pi_{A}(d_{4})$ and $\pi_{B}(d_{1})=\pi_{B}(d_{3})$,
$\pi_{B}(d_{2})=\pi_{B}(d_{4})$. Here, $+_{A}$ and $+_{B}$ are the additions
in $D\to A$ and $D\to B$, respectively. In particular, $(\pi_{A},q_{B})$ and
$(\pi_{B},q_{A})$ 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 $\pi_{A}$ and of $\pi_{B}$. From (4) follows
easily the existence of a natural vector bundle structure on $C$ over $M$. The
inclusion $C\hookrightarrow D$ is denoted by $C_{m}\ni
c\longmapsto\overline{c}\in\pi_{A}^{-1}(0^{A}_{m})\cap\pi_{B}^{-1}(0^{B}_{m}).$

The space of sections $\Gamma_{B}(D)$ is generated as a $C^{\infty}(B)$-module
by two special classes of sections (see [20]), the linear and the core
sections which are now described. For a section $c\colon M\rightarrow C$, the
corresponding core section $c^{\dagger}\colon B\rightarrow D$ is defined as
$c^{\dagger}(b_{m})=0^{D}_{\vphantom{1}{}_{b_{m}}}+_{A}\overline{c(m)}$, $m\in
M$, $b_{m}\in B_{m}$. The corresponding core section $A\to D$ is written
$c^{\dagger}$ also, relying on the argument to distinguish between them. The
space of core sections of $D$ over $B$ is written $\Gamma_{B}^{c}(D)$.

A section $\xi\in\Gamma_{B}(D)$ is called linear if $\xi\colon B\rightarrow D$
is a bundle morphism from $B\rightarrow M$ to $D\rightarrow A$ over a section
$a\in\Gamma(A)$. The space of linear sections of $D$ over $B$ is denoted by
$\Gamma^{\ell}_{B}(D)$. Given $\psi\in\Gamma(B^{*}\otimes C)$, there is a
linear section $\widetilde{\psi}\colon B\to D$ over the zero section
$0^{A}\colon M\to A$ given by
$\widetilde{\psi}(b_{m})=\tilde{0}_{b_{m}}+_{A}\overline{\psi(b_{m})}$. Such a
section is called a core-linear section.

######  Example 2.6.

Let $A,\,B,\,C$ be vector bundles over $M$ and consider
$D=A\times_{M}B\times_{M}C$. With the vector bundle structures
$D=q^{!}_{A}(B\oplus C)\to A$ and $D=q_{B}^{!}(A\oplus C)\to B$, one finds
that $(D;A,B;M)$ is a double vector bundle called the decomposed double vector
bundle with core $C$. The core sections are given by

$c^{\dagger}\colon b_{m}\mapsto(0^{A}_{m},b_{m},c(m)),\text{ where }m\in
M,\,b_{m}\in B_{m},\,c\in\Gamma(C),$

and similarly for $c^{\dagger}\colon A\to D$. The space of linear sections
$\Gamma^{\ell}_{B}(D)$ is naturally identified with
$\Gamma(A)\oplus\Gamma(B^{*}\otimes C)$ via

$(a,\psi):b_{m}\mapsto(a(m),b_{m},\psi(b_{m})),\text{ where
}\psi\in\Gamma(B^{*}\otimes C),\,a\in\Gamma(A).$

In particular, the fibered product $A\times_{M}B$ is a double vector bundle
over the sides $A$ and $B$ and has core $\\{0\\}\times M\to M$.

A linear splitting of $(D;A,B;M)$ is an injective morphism of double vector
bundles $\Sigma\colon A\times_{M}B\hookrightarrow D$ 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 $\Sigma$ of $D$ is
also equivalent to a splitting $\sigma_{A}$ of the short exact sequence of
$C^{\infty}(M)$-modules

$0\longrightarrow\Gamma(B^{*}\otimes
C)\hookrightarrow\Gamma^{\ell}_{B}(D)\longrightarrow\Gamma(A)\longrightarrow
0,$ (5)

where the third map is the map that sends a linear section $(\xi,a)$ to its
base section $a\in\Gamma(A)$. The splitting $\sigma_{A}$ is called a
horizontal lift. Given $\Sigma$, the horizontal lift
$\sigma_{A}\colon\Gamma(A)\to\Gamma_{B}^{\ell}(D)$ is given by
$\sigma_{A}(a)(b_{m})=\Sigma(a(m),b_{m})$ for all $a\in\Gamma(A)$ and
$b_{m}\in B$. By the symmetry of a linear splitting, a lift
$\sigma_{A}\colon\Gamma(A)\to\Gamma_{B}^{\ell}(D)$ is equivalent to a lift
$\sigma_{B}\colon\Gamma(B)\to\Gamma_{A}^{\ell}(D)$.

Let $q_{E}\colon E\to M$ be a vector bundle. Then the tangent bundle $TE$ 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 $TE\to TM$
are the derivatives of the structure maps of $E\to M$.

$\textstyle{TE\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tq_{E}}$$\scriptstyle{p_{E}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{E}}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{M}}$$\textstyle{M}$

The space $TE$ is a double vector bundle with core bundle $E\to M$. The map
$\bar{}\,\colon E\to p_{E}^{-1}(0^{E})\cap(Tq_{E})^{-1}(0^{TM})$ sends
$e_{m}\in E_{m}$ to
$\bar{e}_{m}=\left.\frac{d}{dt}\right\arrowvert_{t=0}te_{m}\in
T_{0^{E}_{m}}E$. Hence the core vector field corresponding to $e\in\Gamma(E)$
is the vertical lift $e^{\uparrow}\colon E\to TE$, i.e. the vector field with
flow $\varphi^{e^{\uparrow}}\colon E\times\mathbb{R}\to E$,
$\varphi_{t}(e^{\prime}_{m})=e^{\prime}_{m}+te(m)$. An element of
$\Gamma^{\ell}_{E}(TE)=\mathfrak{X}^{\ell}(E)$ is called a linear vector
field. It is well-known (see e.g. [19]) that a linear vector field
$\xi\in\mathfrak{X}^{l}(E)$ covering $X\in\mathfrak{X}(M)$ corresponds to a
derivation $D\colon\Gamma(E)\to\Gamma(E)$ over $X\in\mathfrak{X}(M)$. The
precise correspondence is given by the following equations

$\xi(\ell_{\varepsilon})=\ell_{D^{*}(\varepsilon)}\,\,\,\,\text{ and
}\,\,\,\xi(q_{E}^{*}f)=q_{E}^{*}(X(f))$ (6)

for all $\varepsilon\in\Gamma(E^{*})$ and $f\in C^{\infty}(M)$, where
$D^{*}\colon\Gamma(E^{*})\to\Gamma(E^{*})$ is the dual derivation to $D$ [19,
§3.4]. The linear vector field in $\mathfrak{X}^{l}(E)$ corresponding in this
manner to a derivation $D$ of $\Gamma(E)$ is written $\widehat{D}$. Given a
derivation $D$ over $X\in\mathfrak{X}(M)$, the explicit formula for
$\widehat{D}$ is

$\widehat{D}(e_{m})=T_{m}e(X(m))+_{E}\left.\frac{d}{dt}\right\arrowvert_{t=0}(e_{m}-tD(e)(m))$
(7)

for $e_{m}\in E$ and any $e\in\Gamma(E)$ such that $e(m)=e_{m}$. The choice of
a linear splitting $\Sigma$ for $(TE;TM,E;M)$ is equivalent to the choice of a
linear connection $\nabla\colon\mathfrak{X}(M)\times\Gamma(E)\to\Gamma(E)$:
the corresponding lift
$\sigma_{TM}^{\nabla}\colon\mathfrak{X}(M)\to\mathfrak{X}^{l}(E)$ is defined
by $X\mapsto\widehat{\nabla_{X}}$. It is easy to see, using the equalities in
(6), that

$\begin{split}\left[\widehat{\nabla_{X}},\widehat{\nabla_{Y}}\right]&=\widehat{\nabla_{[X,Y]}}-\widetilde{R_{\nabla}(X,Y)},\qquad\left[\widehat{\nabla_{X}},e^{\uparrow}\right]=(\nabla_{X}e)^{\uparrow},\qquad\left[e_{1}^{\uparrow},e_{2}^{\uparrow}\right]=0\end{split}$
(8)

for all $X,Y\in\mathfrak{X}(M)$ and $e,e_{1},e_{2}\in\Gamma(E)$.

The splitting $\Sigma^{\nabla}$ defined by a linear $TM$-connection on $E$ is,
as above, equivalent as well to a linear horizontal lift
$\sigma^{\nabla}_{E}\colon\Gamma(E)\to\Gamma^{l}_{TM}(TE)$. It is given by

$\sigma^{\nabla}_{E}(e)=Te-\widetilde{\nabla_{\cdot}e}\qquad\text{ for all
}e\in\Gamma(E).$

The core section of $TE\to TM$ defined by $e\in\Gamma(E)$ is written
$e^{\dagger}\in\Gamma^{c}_{TM}(TE)$ and explicitly defined by
$e^{\dagger}(v_{m})=T_{m}0^{E}v_{m}-\left.\frac{d}{dt}\right\arrowvert_{t=0}0^{E}_{m}+te(m)$
for all $v_{m}\in TM$.

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

$[\Gamma^{\ell}_{B}(D),\Gamma^{\ell}_{B}(D)]\subset\Gamma^{\ell}_{B}(D),\qquad[\Gamma^{\ell}_{B}(D),\Gamma^{c}_{B}(D)]\subset\Gamma^{c}_{B}(D),\qquad[\Gamma^{c}_{B}(D),\Gamma^{c}_{B}(D)]=0.$

The vector bundle $A\to M$ is then also a Lie algebroid, with anchor
$\rho_{A}$ and bracket defined as follows: if
$\xi_{1},\xi_{2}\in\Gamma^{\ell}_{B}(D)$ are linear over
$a_{1},a_{2}\in\Gamma(A)$, then the bracket $[\xi_{1},\xi_{2}]$ is linear over
$[a_{1},a_{2}]$.

Let $E_{0},E_{1}$ be two vector bundles over the same base $M$ as $A$. A
$2$-term representation up to homotopy of $A$ on $E_{0}\oplus E_{1}$ [2, 8] is
the collection of

  1. (R1)

a map $\partial\colon E_{0}\to E_{1}$,

  2. (R2)

two $A$-connections, $\nabla^{0}$ and $\nabla^{1}$ on $E_{0}$ and $E_{1}$,
respectively, such that $\partial\circ\nabla^{0}=\nabla^{1}\circ\partial$,

  3. (R3)

an element $R\in\Omega^{2}(A,\operatorname{Hom}(E_{1},E_{0}))$ such that
$R_{\nabla^{0}}=R\circ\partial$, $R_{\nabla^{1}}=\partial\circ R$ and
$\mathbf{d}_{\nabla^{\operatorname{Hom}}}R=0$, where
$\nabla^{\operatorname{Hom}}$ is the connection induced on
$\operatorname{Hom}(E_{1},E_{0})$ by $\nabla^{0}$ and $\nabla^{1}$.

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 $(D\to B,A\to M)$ and choose a linear splitting
$\Sigma\colon A\times_{M}B\to D$. Since the anchor $\Theta_{B}$ is linear, it
sends a core section $c^{\dagger}$, $c\in\Gamma(C)$ to a vertical vector field
on $B$. This defines the core-anchor $\partial_{B}\colon C\to B$ which is
given by $\Theta_{B}(c^{\dagger})=(\partial_{B}c)^{\uparrow}$ for all
$c\in\Gamma(C)$ and does not depend on the splitting (see [16]). Since the
anchor $\Theta_{B}$ of a linear section is linear, for each $a\in\Gamma(A)$
the vector field $\Theta_{B}(\sigma_{A}(a))\in\mathfrak{X}^{l}(B)$ defines a
derivation of $\Gamma(B)$ with symbol $\rho(a)$. This defines a linear
connection $\nabla^{AB}\colon\Gamma(A)\times\Gamma(B)\to\Gamma(B)$:

$\Theta(\sigma_{A}(a))=\widehat{\nabla_{a}^{AB}}$

for all $a\in\Gamma(A)$. Since the bracket of a linear section with a core
section is again a core section, there is a linear connection
$\nabla^{AC}\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ such that

$[\sigma_{A}(a),c^{\dagger}]=(\nabla_{a}^{AC}c)^{\dagger}$

for all $c\in\Gamma(C)$ and $a\in\Gamma(A)$. The difference
$\sigma_{A}[a_{1},a_{2}]-[\sigma_{A}(a_{1}),\sigma_{A}(a_{2})]$ is a core-
linear section for all $a_{1},a_{2}\in\Gamma(A)$. This defines a vector valued
form $R\in\Omega^{2}(A,\operatorname{Hom}(B,C))$ such that

$[\sigma_{A}(a_{1}),\sigma_{A}(a_{2})]=\sigma_{A}[a_{1},a_{2}]-\widetilde{R(a_{1},a_{2})},$

for all $a_{1},a_{2}\in\Gamma(A)$. For more details on these constructions,
see [8], where the following result is proved.

######  Theorem 2.7.

Let $(D\to B;A\to M)$ be a VB-algebroid and choose a linear splitting
$\Sigma\colon A\times_{M}B\to D$. The triple $(\nabla^{AB},\nabla^{AC},R)$
defined as above is a $2$-representation of $A$ on $\partial_{B}\colon C\to
B$.

Conversely, let $(D;A,B;M)$ be a double vector bundle such that $A$ has a Lie
algebroid structure and choose a linear splitting $\Sigma\colon
A\times_{M}B\to D$. Then if $(\nabla^{AB},\nabla^{AC},R)$ is a
$2$-representation of $A$ on $\partial_{B}\colon C\to B$, then the equations
above define a VB-algebroid structure on $(D\to B;A\to M)$.

###  2.3 Double Lie algebroids

A double Lie algebroid with core $C$ is a double vector bundle $(D;A,B;M)$

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\scriptstyle{\pi_{B}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{A}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{B}}$$\textstyle{M}$

with core $C$ and four Lie algebroid structures on $A\to M$, $B\to M$, $D\to
A$ and $D\to B$ such that $(D\to A,B\to M)$ and $(D\to B,A\to M)$ are VB-
algebroids, and the _dual_ Lie algebroids
$D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,A\to
C^{*}$ and
$D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,B\to
C^{*}$ 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 $(D;A,B;M)$ with core $C$ and a VB-Lie
algebroid structure on each of its sides. Since $(D\to B,A\to M)$ is a VB-Lie
algebroid, it is described in any linear splitting $\Sigma\colon
A\times_{M}B\to D$ by a representation up to homotopy
$(\partial_{B},\nabla^{AB},\nabla^{AC},R_{A})$ of $A$ on $C{[0]}\oplus
B{[1]}$. Similarly, via the linear splitting $\Sigma$, the VB-algebroid $(D\to
A,B\to M)$ is described by a representation up to homotopy
$(\partial_{A},\nabla^{BA},\nabla^{BC},R_{B})$ on $C{[0]}\oplus A{[1]}$.

Let $\rho_{A}$, respectively $\rho_{B}$, be the anchor of the Lie algebroid
$A\to M$, respectively $B\to M$. [7] proves that
$(D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,A,D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,B)$
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, $(D,A,B,M)$ 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 $(A\to M,\rho_{A},[\cdot\,,\cdot])$ and $(B\to
M,\rho_{B},[\cdot\,,\cdot])$ be two Lie algebroids and assume that $A$ acts on
$C[0]\oplus B[1]$ up to homotopy via
$(\partial_{B},\nabla^{AB},\nabla^{AC},R_{A})$ and $B$ acts on $C[0]\oplus
A[1]$ up to homotopy via $(\partial_{A},\nabla^{BA},\nabla^{BC},R_{B})$444For
the sake of simplicity, all the four connections are denoted by $\nabla$. It
is always clear from the indexes which connection is meant. The connection
$\nabla^{A}$ is the $A$-connection induced by $\nabla^{AB}$ and $\nabla^{AC}$
on $\wedge^{2}B^{*}\otimes C$ and $\nabla^{B}$ is the $B$-connection induced
on $\wedge^{2}A^{*}\otimes C$. . Then the two representations up to homotopy
form a matched pair if

  1. (M1)

$\nabla_{\partial_{A}c_{1}}c_{2}-\nabla_{\partial_{B}c_{2}}c_{1}=-(\nabla_{\partial_{A}c_{2}}c_{1}-\nabla_{\partial_{B}c_{1}}c_{2})$,

  2. (M2)

$[a,\partial_{A}c]=\partial_{A}(\nabla_{a}c)-\nabla_{\partial_{B}c}a$,

  3. (M3)

$[b,\partial_{B}c]=\partial_{B}(\nabla_{b}c)-\nabla_{\partial_{A}c}b$,

  4. (M4)

$\nabla_{b}\nabla_{a}c-\nabla_{a}\nabla_{b}c-\nabla_{\nabla_{b}a}c+\nabla_{\nabla_{a}b}c=R_{B}(b,\partial_{B}c)a-R_{A}(a,\partial_{A}c)b$,

  5. (M5)

$\partial_{A}(R_{A}(a_{1},a_{2})b)=-\nabla_{b}[a_{1},a_{2}]+[\nabla_{b}a_{1},a_{2}]+[a_{1},\nabla_{b}a_{2}]+\nabla_{\nabla_{a_{2}}b}a_{1}-\nabla_{\nabla_{a_{1}}b}a_{2}$,

  6. (M6)

$\partial_{B}(R_{B}(b_{1},b_{2})a)=-\nabla_{a}[b_{1},b_{2}]+[\nabla_{a}b_{1},b_{2}]+[b_{1},\nabla_{a}b_{2}]+\nabla_{\nabla_{b_{2}}a}b_{1}-\nabla_{\nabla_{b_{1}}a}b_{2}$,

for all $a,a_{1},a_{2}\in\Gamma(A)$, $b,b_{1},b_{2}\in\Gamma(B)$ and
$c,c_{1},c_{2}\in\Gamma(C)$, and

  1. (M7)

$\mathbf{d}_{\nabla^{A}}R_{B}=\mathbf{d}_{\nabla^{B}}R_{A}\in\Omega^{2}(A,\wedge^{2}B^{*}\otimes
C)=\Omega^{2}(B,\wedge^{2}A^{*}\otimes C)$, where $R_{B}$ is seen as an
element of $\Omega^{1}(A,\wedge^{2}B^{*}\otimes C)$ and $R_{A}$ as an element
of $\Omega^{1}(B,\wedge^{2}A^{*}\otimes C)$.

Recall that the Lie bialgebroid
$(D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,A,D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,B)$
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 $C$. Writing $e\colon\thinspace
D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,B\to
T(C^{*})$ and $e_{*}\colon\thinspace
D\mathbin{\raisebox{3.0pt}{\varhexstar}\kern-3.7pt{\rule{0.15pt}{4.0pt}}}\,A\to
T(C^{*})$ for the two anchors, then the Poisson structure on $C^{*}$ is
defined by its sharp morphism $\pi^{\\#}_{C^{*}}:=e_{*}\circ e^{*}=-e\circ
e_{*}^{*}\colon T^{*}C^{*}\to TC^{*}$. The sign of this Poisson structure is
determined by the requirement that $\partial_{A}$ and $\partial_{B}$ be
morphisms of Lie algebroids (see the next proposition). The proof of [7,
Proposition 4.5.] shows the following identities for
$c,c_{1},c_{2}\in\Gamma(C)$ and $f,g\in C^{\infty}(M)$:

$\begin{split}&(e_{*}\circ
e^{*})(\mathbf{d}\ell_{c_{1}})(\ell_{c_{2}})=\ell_{\nabla_{\partial_{A}c_{1}}c_{2}-\nabla_{\partial_{B}c_{2}}c_{1}}=\ell_{-\nabla_{\partial_{A}c_{2}}c_{1}+\nabla_{\partial_{B}c_{1}}c_{2}}\\\
&(e_{*}\circ
e^{*})(\mathbf{d}\ell_{c})(q_{C^{*}}^{*}f))=q_{C^{*}}^{*}({{\pounds}}_{\rho_{B}\partial_{B}(c)}f)=q_{C^{*}}^{*}({{\pounds}}_{\rho_{A}\partial_{A}(c)}f)\\\
&(e_{*}\circ e^{*})(\mathbf{d}q_{C^{*}}^{*}f)(q_{C^{*}}^{*}g)=0.\end{split}$

Hence the Lie algebroid on $C\to M$ is defined explicitly using Definition
2.8:

  1. 1.

The anchor is $\rho_{C}:=\rho_{A}\circ\partial_{A}=\rho_{B}\circ\partial_{B}$.
This equality follows easily from (M2) and (M3), see [7, Remark 3.2].

  2. 2.

The first equation (M1) gives a simple formula for the induced Lie algebroid
bracket on section of $C$:

$[c_{1},c_{2}]=\nabla_{\partial_{A}c_{1}}c_{2}-\nabla_{\partial_{B}c_{2}}c_{1}.$
(9)

The Jacobi identity follows from (M2), (M3) and (M4), and the bracket does not
depend on the choice of splitting $\Sigma\colon A\times_{M}B\to D$, see [7,
Remark 3.5].

######  Proposition 2.9.

Let $(D;A,B;M)$ be a double Lie algebroid. Then the morphisms
$\partial_{A}\colon C\to A$ and $\partial_{B}\colon C\to B$ of vector bundles
are morphisms of Lie algebroids.

###### Proof.

Since $\rho_{C}=\rho_{A}\circ\partial_{A}=\rho_{B}\circ\partial_{B}$, the
compatibility of the anchors with the morphisms is immediate.

Equation (M3) with $\partial_{B}\circ\nabla^{AC}=\nabla^{AB}\circ\partial_{B}$
gives

$\begin{split}\partial_{B}[c_{1},c_{2}]&=\partial_{B}(\nabla^{AC}_{\partial_{A}c_{1}}c_{2})-\partial_{B}(\nabla^{BC}_{\partial_{B}c_{2}}c_{1})\\\
&=\nabla^{AB}_{\partial_{A}c_{1}}(\partial_{B}c_{2})-[\partial_{B}c_{2},\partial_{B}c_{1}]-\nabla^{AB}_{\partial_{A}c_{1}}(\partial_{B}c_{2})=[\partial_{B}c_{1},\partial_{B}c_{2}]\end{split}$

for all $c_{1},c_{2}\in\Gamma(C)$. Similarly, (M2) and
$\partial_{A}\circ\nabla^{BC}=\nabla^{BA}\circ\partial_{A}$ yield together the
compatibility of $\partial_{A}$ with the brackets on $C$ and $A$. ∎

Finally, the following proposition is an immediate consequence of (9).

######  Proposition 2.10.

Let $(D;A,B;M)$ be a double Lie algebroid. Let $c_{1},c_{2}$ be sections of
the core $C$ with $\partial_{A}(c_{1})=0$ and $\partial_{B}(c_{2})=0$. Then
$[c_{1},c_{2}]=0$.

###  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 $(D_{1}\to B_{1},A_{1}\to M)$ and $(D_{2}\to B_{2},A_{2}\to M)$ be two VB-
algebroids over a common double base $M$. A morphism $\Phi\colon D_{1}\to
D_{2}$ of double vector bundles with side morphisms $\varphi_{A}\colon
A_{1}\to A_{2}$ and $\varphi_{B}\colon B_{1}\to B_{2}$ is a morphism of VB-
algebroids if

$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\pi_{B_{1}}}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B_{2}}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{B}}$$\textstyle{B_{2}}$

is a morphism of Lie algebroids. The vector bundle morphism $\varphi_{A}\colon
A_{1}\to A_{2}$ is consequently also a morphism of Lie algebroids, over the
identity on $M$.

Let now $(D_{1},A_{1},B_{1},M)$ and $(D_{2},A_{2},B_{2},M)$ be two double Lie
algebroids over a common double base $M$. A morphism $\Phi\colon D_{1}\to
D_{2}$ of double vector bundles with side morphisms $\varphi_{A}\colon
A_{1}\to A_{2}$ and $\varphi_{B}\colon B_{1}\to B_{2}$ is a morphism of double
Lie algebroids if it is a morphism of VB-algebroids $(D_{1}\to B_{1},A_{1}\to
M)\to(D_{2}\to B_{2},A_{2}\to M)$ and a morphism of VB-algebroids $(D_{1}\to
A_{1},B_{1}\to M)\to(D_{2}\to A_{2},B_{2}\to M)$.

Let $A\to M$ be a Lie algebroid and let
$\mathcal{D}_{E}:=(\partial_{E},\nabla^{E_{0}},\nabla^{E_{1}},R_{E})$ and
$\mathcal{D}_{F}:=(\partial_{F},\nabla^{F_{0}},\nabla^{F_{1}},R_{F})$ be
$2$-representations of $A$ on $E_{0}[0]\oplus E_{1}[1]$ and $F_{0}[0]\oplus
F_{1}[1]$, respectively. A morphism of representations up to homotopy
$(A,\mathcal{D}_{E})\to(A,\mathcal{D}_{F})$ is determined by a triple
$(\varphi_{0},\varphi_{1},\varphi)$, where $\varphi_{0}\colon
E_{0}\longrightarrow F_{0}$, $\varphi_{1}\colon E_{1}\to F_{1}$ are vector
bundle morphisms and
$\varphi\in\Omega^{1}(A,\operatorname{Hom}(F_{1},E_{0}))$, satisfying

$\varphi_{1}\circ\partial_{E}=\partial_{F}\circ\varphi_{0},\\\ $ (10)
$\nabla^{\rm
Hom}_{a}(\varphi_{0},\varphi_{1})=(\varphi_{a}\circ\partial_{E},\,\partial_{F}\circ\varphi_{a})\quad\text{
for all }a\in\Gamma(A)\\\ $ (11)

and

$\mathbf{d}_{\nabla^{\rm Hom}}\varphi=R_{F}\circ\varphi_{1}-\varphi_{0}\circ
R_{E},$ (12)

where $\nabla^{\rm Hom}$ is the $A$-connection induced on
$\operatorname{Hom}(E_{0}[0]\oplus E_{1}[1],F_{0}[0]\oplus F_{1}[1])$ by the
four connections (see [2] for more details).

Let $A\to M$ and $A^{\prime}\to M$ be two Lie algebroids over a smooth
manifold $M$ and consider a Lie algebroid morphism $\varphi\colon A\to
A^{\prime}$ over the identity on $M$. Assume that $A^{\prime}$ acts on
$E_{0}[0]\oplus E_{1}[1]$ up to homotopy via
$\mathcal{D}:=(\partial,\nabla^{0},\nabla^{1},R)$. For $i=0,1$ the pullback
connection
$\varphi^{*}\nabla^{i}\colon\Gamma(A)\times\Gamma(E_{i})\to\Gamma(E_{i})$ is
the $A$-connection on $E_{i}$ given by

$(\varphi^{*}\nabla^{i})_{a}e:=\nabla^{i}_{\varphi(a)}e$ (13)

for $a\in\Gamma(A)$ and $e\in\Gamma(E_{i})$. The pullback
$\varphi^{*}R\in\Omega^{2}(A,\operatorname{Hom}(E_{1},E_{0}))$ is similarly
defined by

$(\varphi^{*}R)(a_{1},a_{2})=R(\varphi(a_{1}),\varphi(a_{2}))$ (14)

for $a_{1},a_{2}\in\Gamma(A)$. Then the triple
$\varphi^{*}\mathcal{D}:=(\partial,\varphi^{*}\nabla^{0},\varphi^{*}\nabla^{1},\varphi^{*}R)$
is a representation up to homotopy of $A$ on $E_{0}[0]\oplus E_{1}[1]$, see
[2].

If $\mathcal{D}_{E}$ is a $2$-representation of $A$ on $E_{0}[0]\oplus
E_{1}[1]$ and $\mathcal{D}_{F}$ is a $2$-representation of $A^{\prime}$ on
$F_{0}[0]\oplus F_{1}[1]$, then a morphism of representations up to homotopy
from $\mathcal{D}_{E}$ to $\mathcal{D}_{F}$ is by definition [6] a morphism

$\mathcal{D}_{E}\to\varphi^{*}\mathcal{D}_{F}$

of $2$-representations of $A$.

Let $(D\to B,A\to M)$ and $(D^{\prime}\to B^{\prime},A^{\prime}\to M)$ be two
VB-algebroids over a common double base $M$ and consider a morphism
$\Phi\colon D\to D^{\prime}$ of double vector bundles with side morphisms
$\varphi_{A}\colon A\to A^{\prime}$ and $\varphi_{B}\colon B\to B^{\prime}$.
Choose linear splittings

$\Sigma\colon A\times_{M}B\to D,\qquad\Sigma^{\prime}\colon
A^{\prime}\times_{M}B^{\prime}\to D^{\prime},$

of $D$ and $D^{\prime}$, respectively. The morphism $\Phi$ and the linear
splittings define a form
$\varphi\in\Omega^{1}(A,\operatorname{Hom}(B,C^{\prime}))\simeq\Gamma(A^{*}\otimes
B^{*}\otimes C^{\prime})$ by

$\Phi\left(\Sigma(a,b)\right)=\Sigma^{\prime}(\varphi_{A}(a),\varphi_{B}(b))+_{B^{\prime}}\left(0^{D^{\prime}}_{\varphi_{A}(a)}+_{B}\overline{\varphi(a,b)}\right)$
(15)

for all $a\in\Gamma(A)$.

Then by [6, Theorem 4.11], $\Phi$ is a VB-algebroid morphism if and only if
$\varphi_{A}\colon A\to A^{\prime}$ is a Lie algebroid morphism and for any
pair of linear splittings

$\Sigma\colon A\times_{M}B\to D,\qquad\Sigma^{\prime}\colon
A^{\prime}\times_{M}B^{\prime}\to D^{\prime},$

the triple $(\varphi_{B},\varphi_{C},\varphi)$ is a morphism

$\mathcal{D}\to\varphi_{A}^{*}\mathcal{D}^{\prime}$

of the $2$-representations $\mathcal{D}$ of $A$ and $\mathcal{D}^{\prime}$ of
$A^{\prime}$ defined by $\Sigma$ and the VB-algebroid $(D\to B,A\to M)$, and
by $\Sigma^{\prime}$ and the VB-algebroid $(D^{\prime}\to
B^{\prime},A^{\prime}\to M)$.

Assume finally that $(D,A,B,M)$ and $(D^{\prime},A^{\prime},B^{\prime},M)$ are
two double Lie algebroids and consider a morphism $\Phi\colon D\to D^{\prime}$
as above of double vector bundles. Choose linear splittings

$\Sigma\colon A\times_{M}B\to D,\qquad\Sigma\colon
A^{\prime}\times_{M}B^{\prime}\to D^{\prime},$

of $D$ and $D^{\prime}$, respectively, and consider the form
$\varphi\in\Gamma(A^{*}\otimes B^{*}\otimes C^{\prime})$ defined as in (15) by
$\Phi$ and the linear splittings. Let $\mathcal{D}_{A}$ and
$\mathcal{D}_{A}^{\prime}$ be the $2$-representations of $A$ and of
$A^{\prime}$ defined by $\Sigma$ and the VB-algebroid $(D\to B,A\to M)$, and
by $\Sigma^{\prime}$ and the VB-algebroid $(D^{\prime}\to
B^{\prime},A^{\prime}\to M)$, respectively. Similarly, Let $\mathcal{D}_{B}$
and $\mathcal{D}_{B}^{\prime}$ be the $2$-representations of $B$ and of
$B^{\prime}$ defined by $\Sigma$ and the VB-algebroid $(D\to A,B\to M)$, and
by $\Sigma^{\prime}$ and the VB-algebroid $(D^{\prime}\to
A^{\prime},B^{\prime}\to M)$, respectively.

Then $\Phi$ is a double Lie algebroid morphism if and only if
$(\varphi_{B},\varphi_{C},\varphi)$ defines a morphism of $2$-representations

$\mathcal{D}_{A}\to\varphi_{A}^{*}\mathcal{D}^{\prime}_{A},$ (16)

and $(\varphi_{A},\varphi_{C},\varphi)$ defines a morphism of
$2$-representations

$\mathcal{D}_{B}\to\varphi_{B}^{*}\mathcal{D}^{\prime}_{B}.$ (17)

The following proposition is standard, but can now be proved easily using (9)
and (11).

######  Proposition 2.11.

Let $(D,A,B,M)$ and $(D^{\prime},A^{\prime},B^{\prime},M)$ be two double Lie
algebroids and consider a morphism $\Phi\colon D\to D^{\prime}$ of double Lie
algebroids. Then the core morphism $\varphi_{C}\colon C\to C^{\prime}$ is a
morphism of Lie algebroids.

###### Proof.

Let $\varphi_{A}\colon A\to A^{\prime}$ and $\varphi_{B}\colon B\to
B^{\prime}$ be the side morphisms of $\Phi$. By the considerations above, they
are Lie algebroid morphisms. Choose linear splittings

$\Sigma\colon A\times_{M}B\to D,\qquad\Sigma^{\prime}\colon
A^{\prime}\times_{M}B^{\prime}\to D^{\prime},$

of $D$ and $D^{\prime}$, respectively, and consider the form
$\varphi\in\Gamma(A^{*}\otimes B^{*}\otimes C^{\prime})$ defined as in (15) by
$\Phi$ and the linear splittings. Consider as above the four
$2$-representations555For the sake of simplicity, all the eight connections
are denoted by $\nabla$. It is clear from the indexes which connection is
meant. defined by the VB-algebroid and the splittings. Then for
$c_{1},c_{2}\in\Gamma(C)$:

$\begin{split}\varphi_{C}[c_{1},c_{2}]&=\varphi_{C}\left(\nabla_{\partial_{A}c_{1}}c_{2}-\nabla_{\partial_{B}c_{2}}c_{1}\right)\\\
&=(\varphi_{A}^{*}\nabla)_{\partial_{A}c_{1}}(\varphi_{C}c_{2})+\omega(\partial_{A}c_{1},\partial_{B}c_{2})-(\varphi_{B}^{*}\nabla)_{\partial_{B}c_{2}}(\varphi_{C}c_{1})-\omega(\partial_{A}c_{1},\partial_{B}c_{2})\\\
&=\nabla_{\varphi_{A}\partial_{A}c_{1}}(\varphi_{C}c_{2})-\nabla_{\varphi_{B}\partial_{B}c_{2}}(\varphi_{C}c_{1})\\\
&=\nabla_{\partial_{A^{\prime}}\varphi_{C}c_{1}}(\varphi_{C}c_{2})-\nabla_{\partial_{B^{\prime}}\varphi_{C}c_{2}}(\varphi_{C}c_{1})=[\varphi_{C}c_{1},\varphi_{C}c_{2}].\end{split}$

The anchor condition is checked as follows using (10)

$\rho_{C^{\prime}}\circ\varphi_{C}=\rho_{A^{\prime}}\circ\partial_{A^{\prime}}\circ\varphi_{C}=\rho_{A^{\prime}}\circ\varphi_{A}\circ\partial_{A}=\rho_{A}\circ\partial_{A}=\rho_{C}.\qed$

##  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
$\varphi\colon\thinspace A^{\prime}\to A$ over a surjective submersion
$f\colon\thinspace M^{\prime}\to M$ is a fibration of Lie algebroids if and
only if the associated map
$a^{\prime}\mapsto(q_{A^{\prime}}a^{\prime},\varphi(a^{\prime}))$ into the
vector bundle pullback $f^{!}A\to M^{\prime}$ is a surjection.

######  Proposition 3.1.

Let $(D;A,B;M)$ be a double Lie algebroid with core $C$. Assume that the
anchors $\rho_{A}\colon A\to TM$ and $\rho_{B}\colon B\to TM$ are both
surjective.

(i) The morphism $\partial_{A}\colon\thinspace C\to A$ is surjective if and
only if the anchor $\Theta_{A}\colon\thinspace D\to TA$ is a fibration of Lie
algebroids.

(ii) The morphism $\partial_{B}\colon\thinspace C\to B$ is surjective if and
only if the anchor $\Theta_{B}\colon\thinspace D\to TB$ is a fibration of Lie
algebroids.

Proof: In order to prove (i), assume first that $\partial_{A}$ is surjective.
Take $b\in B$ and $\xi\in TA$, projecting to the same point $v\in TM$:
$Tq_{A}\xi=\rho_{B}(b)$. Write $a$ for the projection of $\xi$ to $A$. Take
any $d^{\prime}\in D$ which projects to $a$ and $b$ and write
$\xi^{\prime}=\Theta_{A}(d^{\prime})\in TA$. Then $\xi^{\prime}$ and $\xi$
project to the same elements of $A$ and $TM$ and therefore there is a core
element $\partial_{A}(c)\in A$ such that

$\xi=\xi^{\prime}+_{A}(0^{TA}_{a}+_{TM}\overline{\partial_{A}(c)}).$

Now define $d:=d^{\prime}+_{A}(0_{a}+_{B}\overline{c})$. Clearly
$\Theta_{A}(d)=\xi$. This shows that $\Theta_{A}$ is a fibration. The converse
is straightforward since $\partial_{A}$ is the core morphism of $\Theta_{A}$.
$\Box$

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 $(D;A,B;M)$ as in the
following definition has two surjective anchors $D\to TA$ and $D\to TB$ if and
only if the two base anchors $\rho_{A}\colon A\to TM$ and $\rho_{B}\colon B\to
TM$ are surjective.

######  Definition 3.2.

A double Lie algebroid $(D;A,B;M)$ is transitive if its two core-anchors
$\partial_{A}$ and $\partial_{B}$ are surjective.

######  Definition 3.3 ([1]).

A crossed module of Lie algebroids $(A,C,\partial,\rho)$ comprises a Lie
algebroid $A$ on base $M$, a totally intransitive Lie algebroid $C$ on the
same base, a representation $\nabla\colon\thinspace
A\to\operatorname{Der}(C)$, and a morphism $\partial\colon\thinspace C\to A$
of Lie algebroids over $M$, such that

$\displaystyle\nabla_{a}[c_{1},c_{2}]=[\nabla_{a}c_{1},c_{2}]+[c_{1},\nabla_{a}c_{2}],$
(18a) $\displaystyle\nabla_{\partial(c_{1})}c_{2}=[c_{1},c_{2}],$ (18b)
$\displaystyle\partial(\nabla_{a}c)=[a,\partial(c)].$ (18c)

for all $c,c_{1},c_{2}\in\Gamma(C)$ and $a\in\Gamma(A)$.

Here $\operatorname{Der}(C)$ denotes the Lie algebroid of derivations of the
vector bundle $C$: sections of $\operatorname{Der}(C)$ are $\mathbb{R}$-linear
maps $D\colon\Gamma(C)\to\Gamma(C)$ for which there is a vector field
$X\in\mathfrak{X}(M)$ such that $D(fc)=fD(c)+X(f)c$ for all $f\in
C^{\infty}(M)$ and $c\in\Gamma(C)$. For $C$ a totally intransitive Lie
algebroid, $\operatorname{Der}_{[\cdot,\cdot]}(C)$ is the Lie subalgebroid of
$\operatorname{Der}(C)$ the sections of which are also derivations of the
bracket,

$D[c_{1},c_{2}]=[D(c_{1}),c_{2}]+[c_{1},D(c_{2})]$

for $D\in\Gamma(\operatorname{Der}_{[\cdot,\cdot]}(C))$ and
$c_{1},c_{2}\in\Gamma(C)$. That is, $r$ in Definition 3.3 is a morphism of Lie
algebroids $A\to\operatorname{Der}_{[\cdot,\cdot]}(C)$. For clarity, sections
of $\operatorname{Der}_{[\cdot,\cdot]}(C)$ are called Lie derivations of $C$.

Consider now a transitive double Lie algebroid $(D;A,B;M)$ with core $C$.
Write $C^{A}=\ker(\partial_{B})$ and $C^{B}=\ker(\partial_{A})$. Then $C^{A}$
is a smooth subbundle of $B$ and $C^{B}$ is a subbundle of $A$.

######  Proposition 3.4.

Let $(D;A,B;M)$ be a transitive double Lie algebroid with core $C$. For
$a\in\Gamma(A)$ and $\gamma\in\Gamma(C^{A})$ set
$\nabla^{A}_{a}\gamma=[c,\gamma]$ where $c\in\Gamma(C)$ is such that
$\partial_{A}(c)=a$. Then $\nabla^{A}_{a}\gamma$ is well-defined and, together
with the restriction of $\partial_{A}$ to $C^{A}$, gives
$(A,C^{A},\partial_{A},\nabla^{A})$ the structure of a crossed module of Lie
algebroids.

###### Proof.

First, $\partial_{B}[c,\gamma]=[\partial_{B}c,0]=0$ since $\partial_{B}$ is a
Lie algebroid morphism by Proposition 2.9. Therefore, $[c,\gamma]$ is a
section of $C^{A}$. If $c,c^{\prime}$ are two sections of $C$ satisfying
$\partial_{A}(c)=\partial_{A}(c^{\prime})=a$, then
$c-c^{\prime}\in\ker(\partial_{A})=C^{B}$ and so, by Proposition 2.10,
$[c-c^{\prime},\gamma]=0$ for all $\gamma\in C^{A}=\ker(\partial_{B})$.

The Lie algebroid $C^{A}$ is totally intransitive as the kernel of the
morphism $\partial_{B}$ of Lie algebroids. Then
$\nabla^{A}_{a}(f\gamma)=[c,f\gamma]=f[c,\gamma]+\rho_{C}(c)(f)\gamma=f\nabla^{A}_{a}(c)+\rho_{A}(a)(f)\gamma$,
since $\rho_{C}=\rho_{A}\circ\partial_{A}$. Likewise,
$\nabla^{A}_{fa}(\gamma)=[fc,\gamma]=f\nabla^{A}_{a}(\gamma)$ because $C^{A}$
is totally intransitive. Hence,
$\nabla^{A}\colon\Gamma(A)\times\Gamma(C^{A})\to\Gamma(C^{A})$ is a linear
connection. (18b) and (18c) are immediate by the definition of $\nabla^{A}$
and the fact that $\partial_{A}\colon C\to A$ is a Lie algebroid morphism.

For the flatness of $\nabla^{A}$, take $a_{1},a_{2}\in\Gamma(A)$ and
$\gamma\in\Gamma(C^{A})$, as well as $c_{1},c_{2}\in\Gamma(C)$ with
$\partial_{A}(c_{1})=a_{1}$ and $\partial_{A}(c_{2})=a_{2}$. Then
$\partial_{A}[c_{1},c_{2}]=[a_{1},a_{2}]$ since $\partial_{A}$ is a morphism
of Lie algebroids, and

$\displaystyle\begin{split}\nabla^{A}_{a_{1}}\nabla^{A}_{a_{2}}\gamma-\nabla^{A}_{a_{2}}\nabla^{A}_{a_{1}}\gamma-\nabla^{A}_{[a_{1},a_{2}]}\gamma=[c_{1},[c_{2},\gamma]]-[c_{2},[c_{1},\gamma]]-[[c_{1},c_{2}],\gamma]=0.\end{split}$

Similarly,

$\nabla_{a}^{A}[\gamma_{1},\gamma_{2}]-[\nabla^{A}_{a}\gamma_{1},\gamma_{2}]-[\gamma_{1},\nabla^{A}_{a}\gamma_{2}]=[c,[\gamma_{1},\gamma_{2}]]-[[c,\gamma_{1}],\gamma_{2}]-[\gamma_{1},[c,\gamma_{2}]]=0$

for $a\in\Gamma(A)$, $c\in\Gamma(C)$ with $\partial_{A}(c)=a$ and
$\gamma_{1},\gamma_{2}\in\Gamma(C^{A})$, shows the equality (18a). ∎

Let $(D;A,B;M)$ be a transitive double Lie algebroid with core $C$. The core
$C$ and the two crossed modules $(A,C^{A},\partial_{A},\nabla^{A})$ and
$(B,C^{B},\partial_{B},\nabla^{B})$ constructed in Proposition 3.4 are shown
in Figure 3, picturing the _transitive core diagram of the transitive double
Lie algebroid $(D;A,B;M)$_. The remainder of this paper shows that a
transitive double Lie algebroid can be reconstructed from its (transitive)
core diagram.

$\textstyle{C^{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{B}}$$\scriptstyle{\partial_{A}}$$\textstyle{B}$$\textstyle{A}$
Figure 3: The transitive core diagram of a transitive double Lie algebroid.

######  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
$\partial_{A}\colon\thinspace C\to A$ and $\partial_{B}\colon\thinspace C\to
B$ of Lie algebroids such that the kernels $C^{A}:=\ker(\partial_{B})$ and
$C^{B}:=\ker(\partial_{A})$ commute in $C$. A core diagram for $A$ and $B$ is
transitive if the morphisms $\partial_{A}$ and $\partial_{B}$ are surjective.

In the transitive case the proof of Proposition 3.4 applies, and there are
representations

$\nabla^{A}\colon\thinspace
A\to\operatorname{Der}(C^{A}),\quad\nabla^{B}\colon\thinspace
B\to\operatorname{Der}(C^{B}),$ (19)

defined as in 3.4, which together with the restrictions of $\partial_{A}$ and
$\partial_{B}$, 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
$\mathscr{L}\\!\mathscr{A}$–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 $(D,A,B,M)$ and $(D^{\prime},A^{\prime},B^{\prime},M)$ be two
transitive double Lie algebroids over a common double base $M$. Consider a
morphism $\Phi\colon D\to D^{\prime}$ of double Lie algebroids, with side
morphisms $\varphi_{A}\colon A\to A^{\prime}$ and $\varphi_{B}\colon B\to
B^{\prime}$, and with core morphism $\varphi_{C}\colon C\to C^{\prime}$. Then
these two side morphisms and the core morphism are morphisms of Lie algebroids
over $M$ and, using (10) and Proposition 2.11, the triple
$(\varphi_{A},\varphi_{B},\varphi_{C})$ is a morphism of the core diagrams of
$D$ and $D^{\prime}$, as in the following definition.

######  Definition 3.6.

Let $A,A^{\prime},B,B^{\prime}$ be Lie algebroids over a base $M$ and consider
core diagrams $(C,\partial_{A},\partial_{B})$ and
$(C^{\prime},\partial_{A^{\prime}},\partial_{B^{\prime}})$ for $A$ and $B$ and
for $A^{\prime}$ and $B^{\prime}$, respectively. A morphism

$\Phi\colon(C,\partial_{A},\partial_{B})\to(C^{\prime},\partial_{A^{\prime}},\partial_{B^{\prime}})$

of core diagrams is a triple of Lie algebroid morphisms $\varphi_{A}\colon
A\to A^{\prime}$, $\varphi_{B}\colon B\to B^{\prime}$ and $\varphi_{C}\colon
C\to C^{\prime}$ such that
$\partial_{A^{\prime}}\circ\varphi_{C}=\varphi_{A}\circ\partial_{A}$ and
$\partial_{B^{\prime}}\circ\varphi_{C}=\varphi_{B}\circ\partial_{B}$.

$\textstyle{C^{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}\arrowvert_{C^{B}}}$$\textstyle{C^{B^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}\arrowvert_{C^{A}}}$$\textstyle{C^{A^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{A}}$$\scriptstyle{\partial_{B}}$$\scriptstyle{\varphi_{C}}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{B^{\prime}}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{B}}$$\textstyle{B^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{A^{\prime}}$
Figure 4: A morphism of core diagrams.

Given a morphism
$\Phi\colon(C,\partial_{A},\partial_{B})\to(C^{\prime},\partial_{A^{\prime}},\partial_{B^{\prime}})$
of transitive core diagrams as in Figure 4, the morphism $\varphi_{C}\colon
C\to C^{\prime}$ restricts to morphisms $C^{A}\to C^{A^{\prime}}$ and
$C^{B}\to C^{B^{\prime}}$ of Lie algebroids. Therefore, all squares in 4 are
commutative squares of Lie algebroid morphisms over $M$. A computation yields
as well for $a\in\Gamma(A)$, $c\in\Gamma(C)$ such that $\partial_{A}c=a$ and
$\gamma\in\Gamma(C^{A})$:

$\begin{split}\varphi_{C}(\nabla^{A}_{a}\gamma)=\varphi_{C}[c,\gamma]=[\varphi_{C}(c),\varphi_{C}(\gamma)]=\nabla^{A^{\prime}}_{\partial_{A^{\prime}}\varphi_{C}(c)}(\varphi_{C}(\gamma))=\nabla^{A^{\prime}}_{\varphi_{A}(a)}(\varphi_{C}(\gamma))=(\varphi_{A}^{!}\nabla^{A^{\prime}})_{a}(\varphi_{C}(\gamma)).\end{split}$

##  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 $TC\to TM$ on $A\to TM$

Let $C$ and $A$ be Lie algebroids over a manifold $M$ and let
$\partial\colon\thinspace C\to A$ 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 $\partial\colon\thinspace C\to A$_.
The construction follows the one of the _comma double groupoids_ in [16, 2.5]
and [3].

First an action of the Lie algebroid $TC\to TM$ is defined on the anchor
$\rho_{A}\colon\thinspace A\to TM$. Recall that the Lie algebroid structure on
$TC\to TM$ may be defined as follows, using tangent sections $Tc$ and core
sections $c^{\dagger}$, for $c\in\Gamma(C)$. For sections
$c_{1},c_{2}\in\Gamma(C)$, the bracket on $\Gamma_{TM}(TC)$ is given by

$[Tc_{1},Tc_{2}]=T[c_{1},c_{2}],\quad[Tc_{1},c_{2}^{\dagger}]=[c_{1},c_{2}]^{\dagger},\quad[c_{1}^{\dagger},c_{2}^{\dagger}]=0.$
(20)

The anchor of $TC\to TM$ is $\Theta:=\sigma_{M}\circ T\rho_{C}$ where
$\sigma_{M}\colon\thinspace T^{2}M\to T^{2}M$ is the canonical involution
(denoted $J_{M}$ in [19]).

An action of $TC\to TM$ on $\rho_{A}\colon\thinspace A\to TM$ is a map
$\Phi\colon\thinspace\Gamma_{TM}(TC)\to\mathfrak{X}(A)$ such that

$\displaystyle\Phi[\xi_{1},\xi_{2}]=[\Phi(\xi_{1}),\Phi(\xi_{2})],$ (21a)
$\displaystyle\Phi(\xi_{1}+\xi_{2})=\Phi(\xi_{1})+\Phi(\xi_{2}),$ (21b)
$\displaystyle\Phi(F\xi)=(\rho_{A}^{*}F)\Phi(\xi),$ (21c)
$\displaystyle{{\pounds}}_{\Phi(\xi)}(\rho_{A}^{*}F)=\rho_{A}^{*}({{\pounds}}_{\Theta(\xi)}F),$
(21d)

for $\xi,\xi_{1},\xi_{2}\in\Gamma_{TM}(TC)$ and $F\in C^{\infty}(TM)$ [19,
4.1.1].

######  Proposition 4.1.

Let $C$ and $A$ be Lie algebroids over a manifold $M$ and let
$\partial\colon\thinspace C\to A$ be a morphism of Lie algebroids over $M$.
For $c\in\Gamma(C)$, set

$\Phi(Tc):=\widehat{[\partial c,\cdot]}\in\mathfrak{X}^{l}(A),\qquad\text{ and
}\qquad\Phi(c^{\dagger}):=(\partial c)^{\uparrow}\in\mathfrak{X}^{c}(A).$ (22)

Then (22) extends to a unique action $\Phi$ of $TC\to TM$ on $\rho_{A}\colon
A\to TM$.

Note that $\widehat{[\partial c,\cdot]}\in\mathfrak{X}^{l}(A)$ is linear over
$\rho_{A}(\partial c)=\rho_{C}(c)\in\mathfrak{X}(M)$.

###### Proof.

An arbitrary section $\chi$ of $TC\to TM$ can be written

$\chi=\sum_{i=1}^{k}F_{i}\,Tc_{i}+\sum_{j=1}^{n}G_{j}{\gamma_{j}}^{\dagger}$

with $F_{1},\ldots,F_{k},G_{1},\ldots,G_{n}\in C^{\infty}(TM)$ and
$c_{1},\ldots,c_{k},\gamma_{1},\ldots,\gamma_{n}\in\Gamma(C)$. Then set

$\begin{split}\Phi(\chi)&=\sum_{i=1}^{k}\rho_{A}^{*}F_{i}\,\Phi(Tc_{i})+\sum_{j=1}^{n}\rho_{A}^{*}G_{j}\,\Phi({\gamma_{j}}^{\dagger})=\sum_{i=1}^{k}\rho_{A}^{*}F_{i}\,\widehat{[\partial
c_{i},\cdot]}+\sum_{j=1}^{n}\rho_{A}^{*}G_{j}\,(\partial\gamma_{j})^{\uparrow}.\end{split}$
(23)

(23) defines a map $\Phi\colon\Gamma_{TM}(TC)\to\mathfrak{X}(A)$:

For $c\in\Gamma(C)$ and $f\in C^{\infty}(M)$

$T(fc)=\ell_{\mathbf{d}f}\cdot c^{\dagger}+_{TM}p_{M}^{*}f\cdot
Tc,\qquad(fc)^{\dagger}=p_{M}^{*}f\cdot c^{\dagger},$ (24)

where $+_{TM}$ and $\cdot$ are the addition and scalar multiplication in
$TC\to TM$, and $p_{M}\colon TM\to M$ is the canonical projection.

Hence the following identities need to hold:

$\Phi(T(fc))=\Phi(\ell_{\mathbf{d}f}\cdot c^{\dagger}+_{TM}p_{M}^{*}f\cdot
Tc)$ (25)

and

$\Phi((fc)^{\dagger})=\Phi(p_{M}^{*}f\cdot c^{\dagger})$ (26)

for all $c\in\Gamma(C)$ and $f\in C^{\infty}(M)$. For the first identity,
compute

$\displaystyle\Phi(\ell_{\mathbf{d}f}\cdot c^{\dagger}+_{TM}p_{M}^{*}f\cdot
Tc)$ $\displaystyle=\rho_{A}^{*}\ell_{\mathbf{d}f}\cdot(\partial
c)^{\uparrow}+\rho_{A}^{*}p_{M}^{*}f\cdot\widehat{[\partial
c,\cdot]}=\ell_{\mathbf{d}_{A}f}\cdot(\partial
c)^{\uparrow}+q_{A}^{*}f\cdot\widehat{[\partial c,\cdot]}$
$\displaystyle=\widehat{[\partial(fc),\cdot]}=\Phi(T(fc)).$

Here $\mathbf{d}_{A}$ is the de Rham like differential
$\Omega^{\bullet}(A)\to\Omega^{\bullet+1}(A)$ defined by $A$, and
$\rho_{A}^{t}\colon T^{*}M\to A^{*}$ is the pointwise transpose of $\rho_{A}$.
The notation $\rho_{A}^{t}$ avoids confusions with the pullback
$\rho_{A}^{*}\colon\Gamma\left(\bigwedge^{\bullet}T^{*}M\right)\to\Gamma\left(\bigwedge^{\bullet}T^{*}A\right)$.
It is easy to check that
$\rho_{A}^{*}\ell_{\mathbf{d}f}=\ell_{\rho_{A}^{t}\mathbf{d}f}=\ell_{\mathbf{d}_{A}f}$
for all $f\in C^{\infty}(M)$. The second equality is then immediate with
$q_{A}=p_{M}\circ\rho_{A}$. The third equality can be checked easily on linear
functions $\ell_{\alpha}$ for $\alpha\in\Gamma(A^{*})$ and $q_{A}^{*}f$, for
$f\in C^{\infty}(M)$.

For (26), compute

$\Phi(p_{M}^{*}f\cdot c^{\dagger})=\rho_{A}^{*}p_{M}^{*}f\cdot(\partial
c)^{\uparrow}=q_{A}^{*}f\cdot(\partial c)^{\uparrow}=(f\partial
c)^{\uparrow}=(\partial(fc))^{\uparrow}=\Phi\left((fc)^{\dagger}\right).$

$\Phi\colon\thinspace\Gamma_{TM}(TC)\to\mathfrak{X}(A)$ 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 $\xi$ of
$TC\to TM$. Consider first $\xi=Tc$ where $c\in\Gamma(C)$. Then (21d) reads

${{\pounds}}_{\widehat{[\partial
c,\cdot]}}(\rho_{A}^{*}F)=\rho_{A}^{*}({{\pounds}}_{(\sigma_{M}\circ
T(\rho_{C}(c)))}F)$

for all $F\in C^{\infty}(TM)$. Here again, it is enough to check this equality
on pullbacks $p_{M}^{*}f$ for $f\in C^{\infty}(M)$ and linear functions
$\ell_{\theta}$ for $\theta\in\Omega^{1}(M)$.

For $a_{m}\in A$ and any $a\in\Gamma(A)$ such that $a(m)=a_{m}$,

$\displaystyle{{\pounds}}_{\widehat{[\partial
c,\cdot]}}(\rho_{A}^{*}p_{M}^{*}f)$
$\displaystyle={{\pounds}}_{\widehat{[\partial
c,\cdot]}}(q_{A}^{*}f)=q_{A}^{*}({{\pounds}}_{(\rho_{A}\circ\partial)(c)}f)=q_{A}^{*}({{\pounds}}_{\rho_{C}(c)}f),$
$\displaystyle{{\pounds}}_{\widehat{[\partial
c,\cdot]}}(\rho_{A}^{*}\ell_{\theta})(a_{m})$
$\displaystyle=\widehat{[\partial
c,\cdot]}(\ell_{\rho_{A}^{t}\theta})(a_{m})=\ell_{{{\pounds}}_{\partial
c}\rho_{A}^{t}\theta}(a_{m})=\rho_{C}(c_{m})\langle\rho_{A}^{t}\theta,a\rangle-\langle\rho_{A}^{t}\theta_{m},[\partial
c,a](m)\rangle$
$\displaystyle=\rho_{C}(c_{m})\langle\theta,\rho_{A}(a)\rangle-\langle\theta_{m},[\rho_{C}(c),\rho_{A}(a)](m)\rangle=\ell_{\alpha}(a_{m}),$

where $\alpha:=\rho_{A}^{t}{{\pounds}}_{\rho_{C}(c)}(\theta)\in\Gamma(A^{*})$.

Now consider the right hand side of (21d). Choose $a_{m}\in A$, let $\gamma$
be a path through $m$ with $\dot{\gamma}(0)=\rho_{A}(a_{m})$ and let
$\varphi^{\rho_{C}(c)}_{\cdot}$ be the flow of
$\rho_{C}(c)\in\mathfrak{X}(M)$. Then

$\displaystyle\sigma_{M}(T(\rho_{C}(c))\rho_{A}(a_{m}))$
$\displaystyle=\sigma_{M}\left(\left.\frac{d}{dt}\right\arrowvert_{t=0}\left.\frac{d}{ds}\right\arrowvert_{s=0}\varphi^{\rho_{C}(c)}_{s}(\gamma(t))\right)$
$\displaystyle=\left.\frac{d}{ds}\right\arrowvert_{s=0}\left.\frac{d}{dt}\right\arrowvert_{t=0}\varphi^{\rho_{C}(c)}_{s}(\gamma(t))=\left.\frac{d}{ds}\right\arrowvert_{s=0}T_{m}\varphi^{\rho_{C}(c)}_{s}(\rho_{A}(a_{m})),$

and so

$\displaystyle\sigma_{M}(T(\rho_{C}(c))\rho_{A}(a_{m}))(p_{M}^{*}f)$
$\displaystyle=\left.\frac{d}{ds}\right\arrowvert_{s=0}(f\circ\varphi^{\rho_{C}(c)}_{s})(m)=({{\pounds}}_{\rho_{C}(c)}f)(m)=q_{A}^{*}({{\pounds}}_{\rho_{C}(c)}f)(a_{m}),$
$\displaystyle\sigma_{M}(T(\rho_{C}(c))\rho_{A}(a_{m}))(\ell_{\theta})$
$\displaystyle=\left.\frac{d}{ds}\right\arrowvert_{s=0}\langle\theta(\varphi^{\rho_{C}(c)}_{s}(m)),T_{m}\varphi^{\rho_{C}(c)}_{s}(\rho_{A}(a_{m}))\rangle$
$\displaystyle=\langle{{\pounds}}_{\rho_{C}(c)}\theta,\rho_{A}(a_{m})\rangle=\ell_{\alpha}(a_{m}),$

where $\alpha=\rho_{A}^{t}{{\pounds}}_{\rho_{C}(c)}(\theta)$ as before. This
completes the proof of (21d) for $\xi=Tc$.

Now consider $c^{\dagger}\in\Gamma_{TM}(TC)$. The equality
$\Theta(c^{\dagger})=(\rho_{C}(c))^{\uparrow}\in\mathfrak{X}(TM)$ can be
verified using the same method as above. It only remains to prove that

${{\pounds}}_{(\partial
c)^{\uparrow}}(\rho_{A}^{*}F)=\rho_{A}^{*}({{\pounds}}_{\rho_{C}(c)^{\uparrow}}F)$

which is immediate for $F$ a pullback or a linear function on $TM$.

Finally the compatibility of $\Phi$ with the Lie algebroid brackets (21a)
needs to be checked:

$[\Phi(V),\Phi(W)]=\Phi[V,W]$

for all $V,W\in\Gamma_{TM}(TC)$. Because of (21d), it is sufficient to prove
(21a) on tangent and core sections of $TC\to TM$. Computations yield for
$c,c^{\prime}\in\Gamma(C)$:

$\displaystyle\left[\Phi(Tc),\Phi(Tc^{\prime})\right]$
$\displaystyle=\left[\widehat{[\partial c,\cdot]},\widehat{[\partial
c^{\prime},\cdot]}\right]=\widehat{[\partial c,\cdot]\circ[\partial
c^{\prime},\cdot]-[\partial c^{\prime},\cdot]\circ[\partial c,\cdot]}$
$\displaystyle=\widehat{[\partial[c,c^{\prime}],\cdot]}=\Phi\left(T[c,c^{\prime}]\right)=\Phi([Tc,Tc^{\prime}]),$

by the Jacobi identity for the Lie algebroid bracket on $A$, and

$\displaystyle\left[\Phi(Tc),\Phi\left({c^{\prime}}^{\dagger}\right)\right]$
$\displaystyle=\left[\widehat{[\partial c,\cdot]},(\partial
c^{\prime})^{\uparrow}\right]=\left([\partial c,\cdot](\partial
c^{\prime})\right)^{\uparrow}$ $\displaystyle=[\partial c,\partial
c^{\prime}]^{\uparrow}=(\partial[c,c^{\prime}])^{\uparrow}=\Phi\left([c,c^{\prime}]^{\dagger}\right)=\Phi\left(\left[Tc,{c^{\prime}}^{\dagger}\right]\right).$

The Lie algebroid brackets of core sections of $TC\to TM$, 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 $R:=TC\oplus_{TM}A:=\\{(V,a)\in TC\times A\mid
Tq_{C}(V)=\rho_{A}(a)\\}$, 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
$C$.

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{A}\phantom{XXXXX}}$$\textstyle{TC\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tq_{C}}$$\textstyle{TM}$
(a) Thepullbackmanifold

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$pullbackaction$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$
(b) ThedoubleLiealgebroid

Figure 5:

The space $R$ is equipped with two projections $\pi_{A}\colon R\to A$,
$(V,a)\to a$ and $\pi_{C}\colon R\to C$, $(V,a)\to p_{C}(V)$, where as always,
$p_{C}\colon TC\to C$ is the canonical projection. Define additions

$+_{A}\colon R\times_{A}R\to
R,\qquad(V_{1},a)+_{A}(V_{2},a)=(V_{1}+_{TM}V_{2},a),$ (27)

and

$+_{C}\colon R\times_{C}R\to
R,\qquad(V_{1},a_{1})+_{C}(V_{2},a_{2})=(V_{1}+_{C}V_{2},a_{1}+a_{2}).$ (28)

######  Proposition 4.2.

An $A$-connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ on $C$
defines a bijection

$I_{\nabla}\colon A\times_{M}C\times_{M}C\to
R,\qquad(a,c,\gamma)\mapsto\left(\widehat{\nabla_{a}}(c)+_{C}\gamma^{\uparrow}(c),a\right),$

such that

$\pi_{C}(I_{\nabla}(a,c,\gamma))=c\quad\text{ and
}\quad\pi_{A}(I_{\nabla}(a,c,\gamma))=a$

for $(a,c,\gamma)\in A\times_{M}C\times_{M}C$, and

$I_{\nabla}(a_{1}+a_{2},c,\gamma_{1}+\gamma_{2})=I_{\nabla}(a_{1},c,\gamma_{1})+_{C}I_{\nabla}(a_{2},c,\gamma_{2})$

as well as

$I_{\nabla}(a,c_{1}+c_{2},\gamma_{1}+\gamma_{2})=I_{\nabla}(a,c_{1},\gamma_{1})+_{A}I_{\nabla}(a,c_{2},\gamma_{2})$

for $(a_{1},c,\gamma_{1}),(a_{2},c,\gamma_{2})$ and
$(a,c_{1},\gamma_{1}),(a,c_{2},\gamma_{2})\in A\times_{M}C\times_{M}C$.

###### Proof.

Take $(a_{1},c_{1},\gamma_{1})$ and $(a_{2},c_{2},\gamma_{2})\in
A\times_{M}C\times_{M}C$ such that
$I_{\nabla}(a_{1},c_{1},\gamma_{1})=I_{\nabla}(a_{2},c_{2},\gamma_{2})\in R$.
Then

$\left(\widehat{\nabla_{a_{1}}}(c_{1})+_{C}\gamma_{1}^{\uparrow}(c_{1}),a_{1}\right)=\left(\widehat{\nabla_{a_{2}}}(c_{2})+_{C}\gamma_{2}^{\uparrow}(c_{2}),a_{2}\right),$

and so $a_{1}=a_{2}$, as well as
$c_{1}=p_{C}(\widehat{\nabla_{a_{1}}}(c_{1})+_{C}\gamma_{1}^{\uparrow}(c_{1}))=p_{C}(\widehat{\nabla_{a_{2}}}(c_{2})+_{C}\gamma_{2}^{\uparrow}(c_{2}))=c_{2}$.
Then
$\widehat{\nabla_{a_{1}}}(c_{1})+_{C}\gamma_{1}^{\uparrow}(c_{1})=\widehat{\nabla_{a_{1}}}(c_{1})+_{C}\gamma_{2}^{\uparrow}(c_{1})$
implies $\gamma_{1}^{\uparrow}(c_{1})=\gamma_{2}^{\uparrow}(c_{1})$, and so
$\gamma_{1}=\gamma_{2}$. This shows that $I_{\nabla}$ is injective.

Now consider $(V,a)\in R$ and set $p_{C}(V)=c$. Then
$Tq_{C}(\widehat{\nabla_{a}}(c))=\rho_{A}(a)=Tq_{C}(V)$, which implies the
existence of $\gamma\in q_{C}^{-1}(q_{C}(c))$ such that
$V=\widehat{\nabla_{a}}(c)+_{C}\gamma^{\uparrow}(c)$. Hence
$(V,a)=I_{\nabla}(a,c,\gamma)$, which shows that $I_{\nabla}$ is surjective.

The compatibilities of $I_{\nabla}$ with the projections and 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 $\pi_{A}\colon R\to A$,
$(V,a)\to a$ and $\pi_{C}\colon R\to C$, $(V,a)\to p_{C}(V)$ with additions in
(27) and (28) and with core $C$.

In the situation of Proposition 4.2, the isomorphism $I_{\nabla}$ of double
vector bundles is a decomposition of $R$. Recall that $\nabla$ defines as a
consequence linear horizontal lifts
$\sigma^{A}_{\nabla}\colon\Gamma(A)\to\Gamma_{C}^{l}(R)$ and
$\sigma^{C}_{\nabla}\colon\Gamma(C)\to\Gamma_{A}^{l}(R)$. More precisely,
given a section $a\in\Gamma(A)$, the section
$\widehat{a}:=\sigma^{A}_{\nabla}(a)\in\Gamma_{C}(R)$ is defined by

$\widehat{a}(c_{m})=(\widehat{\nabla_{a}}(c_{m}),a(m)),\quad c_{m}\in C.$ (29)

Given a section $c\in\Gamma(C)$, the section
$\widecheck{c}:=\sigma^{C}_{\nabla}(c)\in\Gamma_{A}(R)$ is defined by

$\widecheck{c}(a_{m})=(\widehat{\nabla_{a}}(c(m)),a_{m}),$ (30)

for $a_{m}\in A$ and any $a\in\Gamma(A)$ such that $a(m)=a_{m}$. Given a
section $\gamma\in\Gamma(C)$, the corresponding core section
$\gamma^{\times}\in\Gamma_{C}(R)$ is given by

$\gamma^{\times}(c_{m})=(\gamma^{\uparrow}(c_{m}),0^{A}_{m}),\quad c_{m}\in
C.$ (31)

The corresponding core section $\gamma^{\ddagger}\in\Gamma_{A}(R)$ is given by

$\gamma^{\ddagger}(a_{m})=(\gamma^{\dagger}(\rho_{A}(a_{m})),a_{m})=(T0^{C}(\rho_{A}(a_{m}))+_{C}\gamma^{\uparrow}(0^{C}_{m}),a_{m})\quad
a\in A.$ (32)

###  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 $\partial:=\partial_{A}\colon C\to A$ 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 $(R,A,C,M)$ with
core $C$.

As a Lie algebroid over $C$, $R$ is the pullback Lie algebroid structure of
$A$ over $q_{C}\colon\thinspace C\to M$, denoted $q_{C}^{!!}A$ in [19]. As a
Lie algebroid over $A$, $R$ is the action Lie algebroid structure for the
action $\Phi$ of $TC\to TM$ on $\rho_{A}$ as defined in (22) and (23). The
remainder of this section describes these structures in detail.

######  Proposition 4.4.

Consider a Lie algebroid morphism $\partial\colon C\to A$ over $M$ and equip
$R=TC\oplus_{TM}A$ with the pullback Lie algebroid structure of $A$ over
$q_{C}$. Consider an $A$-connection
$\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ on $C$ and write
$\Theta_{C}$ for the anchor $R\to TC$. Then
$\Theta_{C}\circ\widehat{a}=\widehat{\nabla_{a}}\in\mathfrak{X}^{l}(C)$ and
$\Theta_{C}\circ\gamma^{\times}=\gamma^{\uparrow}\in\mathfrak{X}(C)$ for
$a\in\Gamma(A)$ and $\gamma\in\Gamma(C)$. Furthermore, the Lie algebroid
bracket on $R\to C$ is given by:

$\left[\widehat{a_{1}},\widehat{a_{2}}\right]=\widehat{[a_{1},a_{2}]}-\widetilde{R_{\nabla}(a_{1},a_{2})},\quad\left[\widehat{a},\gamma^{\times}\right]=(\nabla_{a}\gamma)^{\times},\quad\left[\gamma_{1}^{\times}\,,\gamma_{2}^{\times}\right]=0$
(33)

for $a,a_{1},a_{2}\in\Gamma(A)$ and
$\gamma,\gamma_{1},\gamma_{2}\in\Gamma(C)$.

######  Corollary 4.5.

In the situation of Proposition 4.4, the Lie algebroid structure on $R\to C$
is linear over $A\to M$. In other words, $(R\to C,A\to M)$ 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

$\Theta_{C}(\widehat{a})=\Theta_{C}(\widehat{\nabla_{a}},a)=\widehat{\nabla_{a}}\in\mathfrak{X}^{l}(C)$

for all $a\in\Gamma(A)$, and

$\Theta_{C}(\gamma^{\times})=\Theta_{C}(\gamma^{\uparrow},0^{A})=\gamma^{\uparrow}\in\mathfrak{X}^{c}(C)$

for all $\gamma\in\Gamma(C)$.

Following (20) in [19, p.157],

$\begin{split}\left[\widehat{a_{1}},\widehat{a_{2}}\right]&=\left[\left(\widehat{\nabla_{a_{1}}},a_{1}\right),\left(\widehat{\nabla_{a_{2}}},a_{2}\right)\right]=\left(\left[\widehat{\nabla_{a_{1}}},\widehat{\nabla_{a_{2}}}\right],[a_{1},a_{2}]\right)\\\
&=\left(\widehat{\nabla_{[a_{1},a_{2}]}}-\widetilde{R_{\nabla}(a_{1},a_{2})},[a_{1},a_{2}]\right)=\widehat{[a_{1},a_{2}]}-\widetilde{R_{\nabla}(a_{1},a_{2})},\end{split}$
$\left[\widehat{a},\gamma^{\times}\right]=\left[\left(\widehat{\nabla_{a}},a\right),\left(\gamma^{\uparrow},0^{A}\right)\right]=\left(\left[\widehat{\nabla_{a}},\gamma^{\uparrow}\right],[a,0^{A}]\right)=\left((\nabla_{a}\gamma)^{\uparrow},0^{A}\right)=(\nabla_{a}\gamma)^{\times}$

and

$\left[\gamma_{1}^{\times},\gamma_{2}^{\times}\right]=\left[\left(\gamma_{1}^{\uparrow},0^{A}\right),\left(\gamma_{2}^{\uparrow},0^{A}\right)\right]=\left(\left[\gamma_{1}^{\uparrow},\gamma_{2}^{\uparrow}\right],[0^{A},0^{A}]\right)=\left(0^{TC}_{C},0^{A}\right)=0$

for all $a,a_{1},a_{2}\in\Gamma(A)$ and
$\gamma,\gamma_{1},\gamma_{2}\in\Gamma(C)$. ∎

Define now the connections
$\nabla^{\partial}\colon\Gamma(C)\times\Gamma(A)\to\Gamma(A)$,

$\nabla^{\partial}_{c}a=[\partial c,a]+\partial(\nabla_{a}c)$

and $\nabla^{\partial}\colon\Gamma(C)\times\Gamma(C)\to\Gamma(C)$,

$\nabla^{\partial}_{c}c^{\prime}=[c,c^{\prime}]+\nabla_{\partial
c^{\prime}}c,$

and the tensor $R_{\nabla}^{\rm bas}\in\Omega^{2}(C,\operatorname{Hom}(A,C))$:

$R_{\nabla}^{\partial}(c,c^{\prime})(a)=-\nabla_{a}[c,c^{\prime}]+[\nabla_{a}c,c^{\prime}]+[c,\nabla_{a}c^{\prime}]-\nabla_{\nabla^{\partial}_{c}a}c^{\prime}+\nabla_{\nabla^{\partial}_{c^{\prime}}a}c.$
(34)

######  Proposition 4.6.

Consider a Lie algebroid morphism $\partial\colon C\to A$ over $M$ and equip
$R=TC\oplus_{TM}A$ with the action Lie algebroid structure over $A$. Consider
an $A$-connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ on $C$
and write $\Theta_{A}$ for the anchor $R\to TA$. Then
$\Theta_{A}(\widecheck{c})=\widehat{\nabla^{\partial}_{c}}\in\mathfrak{X}^{l}(A)$
and
$\Theta_{A}(\gamma^{\ddagger})=(\partial\gamma)^{\uparrow}\in\mathfrak{X}(A)$
for $c,\gamma\in\Gamma(C)$. Furthermore, the Lie algebroid bracket on $R\to A$
is given by:

$\left[\widecheck{c_{1}},\widecheck{c_{2}}\right]=\widecheck{[c_{1},c_{2}]}-\widetilde{R_{\nabla}^{\partial}(c_{1},c_{2})},\quad\left[\widecheck{c},\gamma^{\ddagger}\right]=(\nabla^{\partial}_{c}\gamma)^{\ddagger},\quad\left[\gamma_{1}^{\ddagger}\,\gamma_{2}^{\ddagger}\right]=0$
(35)

for $c,c_{1},c_{2},\gamma,\gamma_{1},\gamma_{2}\in\Gamma(C)$.

######  Corollary 4.7.

In the situation of Proposition 4.6, the Lie algebroid structure on $R\to A$
is linear over $C\to M$. In other words, $(R\to A,C\to M)$ is a VB-algebroid.

###### Proof of Proposition 4.6.

The action Lie algebroid defined by the action (22) of $TC\to TM$ on $A\to TM$
lives on the pullback vector bundle of $TC\to TM$ across $\rho_{A}$ [19, Prop.
4.1.2]. As a manifold this is $R$, and the bundle projection is the projection
onto the second factor, $(V,a)\mapsto a$. As with any pullback vector bundle,
the sections of $R\to A$ are finite sums $\sum_{i}F_{i}\,\chi_{i}^{!}$ where
$F_{i}\in C^{\infty}(A)$ and $\chi_{i}\in\Gamma_{TM}(TC)$. Here, for
$\chi\in\Gamma_{TM}(TC)$, the section $\chi^{!}\in\Gamma_{A}(R)$ is given by
$\chi^{!}(a)=(\chi(\rho_{A}(a)),a)$ for all $a\in A$.

The anchor $\Theta_{A}\colon\thinspace R\to TA$ is then defined by
$\Theta_{A}(\sum_{i}F_{i}\,\chi_{i})=\sum_{i}F_{i}\,\Phi(\chi_{i})$. The Lie
bracket on $\Gamma_{A}(R)$ is defined by

$\left[\chi_{1}^{!},\chi_{2}^{!}\right]=[\chi_{1},\chi_{2}]^{!}$ (36)

for all $\chi_{1},\chi_{2}\in\Gamma_{TM}(TC)$.

By Definition, the section $\gamma^{\ddagger}\in\Gamma^{c}_{A}(R)$ defined by
$\gamma\in\Gamma(C)$ equals $(\gamma^{\dagger})^{!}\in\Gamma_{A}(R)$. Consider
$c\in\Gamma(C)$, and set

$\Gamma(\operatorname{Hom}(A,C))\ni\nabla_{\cdot}c=\sum_{j=1}^{n}\alpha_{j}\otimes
c_{j}$

with $\alpha_{1},\ldots,\alpha_{n}\in\Gamma(A^{*})$ and
$c_{1},\ldots,c_{n}\in\Gamma(C)$. It is then easy to check that

$\Gamma^{l}_{A}(R)\ni\widecheck{c}=(Tc)^{!}-\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot(c_{j}^{\dagger})^{!}.$

Simple computations yield
$\Theta_{A}(\gamma^{\ddagger})=\Phi(\gamma^{\dagger})=(\partial\gamma)^{\uparrow}$
and

$\Theta_{A}(\widecheck{c})=\Phi(Tc)-\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot\Phi(c_{j}^{\dagger})=\widehat{[\partial
c,\cdot]}-\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot(\partial
c_{j})^{\uparrow}=\widehat{\nabla^{\partial}_{c}}$

for $\gamma,c\in\Gamma(C)$. The Lie bracket on $\Gamma_{A}(R)$ is then given
by

$\left[\gamma_{1}^{\ddagger},\gamma_{2}^{\ddagger}\right]=\left[\gamma_{1}^{\dagger},\gamma_{2}^{\dagger}\right]^{!}=0,$
$\begin{split}\left[\widecheck{c},\gamma^{\ddagger}\right]&=\left[(Tc)^{!}-\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot(c_{j}^{\dagger})^{!},(\gamma^{\dagger})^{!}\right]=[c,\gamma]^{\ddagger}+\sum_{j=1}^{n}{{\pounds}}_{\Theta_{A}((\gamma^{\dagger})^{!})}(\ell_{\alpha_{j}})\cdot
c_{j}^{\ddagger}\\\
&=[c,\gamma]^{\ddagger}+\sum_{j=1}^{n}{{\pounds}}_{(\partial\gamma)^{\uparrow}}(\ell_{\alpha_{j}})\cdot
c_{j}^{\ddagger}=[c,\gamma]^{\ddagger}+\sum_{j=1}^{n}q_{A}^{*}\langle\partial\gamma,\alpha_{j}\rangle\cdot
c_{j}^{\ddagger}=([c,\gamma]+\nabla_{\partial\gamma}c)^{\ddagger}=(\nabla^{\partial}_{c}\gamma)^{\ddagger},\end{split}$

and for
$\widecheck{c}=(Tc)^{!}-\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot(c_{j}^{\dagger})^{!}$
and
$\widecheck{c^{\prime}}=(Tc)^{!}-\sum_{k=1}^{n}\ell_{\alpha_{k}^{\prime}}\cdot(c_{k}^{\dagger})^{!}$,

$\begin{split}\left[\widecheck{c},\widecheck{c^{\prime}}\right]&=\left[(Tc)^{!},(Tc^{\prime})^{!}\right]-\left[\widecheck{c},\sum_{k=1}^{n}\ell_{\alpha_{k}^{\prime}}\cdot
c_{k}^{\ddagger}\right]-\left[\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot
c_{j}^{\ddagger},\widecheck{c^{\prime}}\right]-\left[\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot
c_{j}^{\ddagger},\sum_{k=1}^{n}\ell_{\alpha_{k}^{\prime}}\cdot
c_{k}^{\ddagger}\right]\\\
&=(T[c,c^{\prime}])^{!}-\sum_{k=1}^{n}\ell_{(\nabla^{\partial})^{*}_{c}\alpha_{k}^{\prime}}\cdot
c_{k}^{\ddagger}-\sum_{k=1}^{n}\ell_{\alpha_{k}^{\prime}}\cdot(\nabla^{\partial}_{c}c_{k})^{\ddagger}+\sum_{j=1}^{n}\ell_{(\nabla^{\partial})^{*}_{c^{\prime}}\alpha_{j}}\cdot
c_{j}^{\ddagger}+\sum_{j=1}^{n}\ell_{\alpha_{j}}\cdot(\nabla^{\partial}_{c^{\prime}}c_{j})^{\ddagger}\\\
&\qquad\qquad-\sum_{j,k=1}^{n}\ell_{\alpha_{j}}\cdot q_{A}^{*}\langle\partial
c_{j},\alpha_{k}^{\prime}\rangle\cdot
c_{k}^{\ddagger}+\sum_{j,k=1}^{n}\ell_{\alpha_{k}^{\prime}}\cdot
q_{A}^{*}\langle\partial c_{k},\alpha_{j}\rangle\cdot
c_{j}^{\ddagger}.\end{split}$

A straightforward computation yields that this is
$\left[\widecheck{c},\widecheck{c^{\prime}}\right]=\widecheck{[c,c^{\prime}]}-\widetilde{R_{\nabla}^{\partial}(c,c^{\prime})}$.
∎

Let $A$ and $C$ be two Lie algebroids over $M$ and $\partial\colon C\to A$ a
Lie algebroid morphism over $M$. Choose an $A$-connection on $C$:

$\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C).$

(55) and (51) give the representations up to homotopy describing the VB-
algebroid structures $R\to C$ and respectively $R\to A$ in the splitting
defined by $\nabla$ as in Proposition 4.2. Via the splitting given by
$\nabla$, the Lie algebroid $R\to C$ is described by

$\text{id}\colon C\to
C,\quad\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C),\quad
R_{\nabla}\in\Omega^{2}(A,\operatorname{Hom}(C,C)).$ (37)

The Lie algebroid $R\to A$ is described by

$\partial\colon C\to
A,\quad\nabla^{\partial}\colon\Gamma(C)\times\Gamma(C)\to\Gamma(C),\quad\nabla^{\partial}\colon\Gamma(C)\times\Gamma(A)\to\Gamma(A),$
(38)

and

$R^{\partial}_{\nabla}\in\Omega^{2}(C,\operatorname{Hom}(A,C)).$ (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
$R=TC\oplus_{TM}A$, 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 $\partial\colon C\to A$ be
a morphism of Lie algebroids. Then the double vector bundle $(R;A,C;M)$
constructed in Section 4.2 inherits a double Lie algebroid structure with the
following two Lie algebroid structures on $R\to A$ and $R\to C$:

  1. 1.

As a Lie algebroid over $C$, $R$ is the pullback Lie algebroid structure of
$A$ over $q_{C}\colon\thinspace C\to M$.

  2. 2.

As a Lie algebroid over $A$, $R$ is the action Lie algebroid structure for the
action $\Phi$ of $TC\to TM$ on $\rho_{A}$ as defined in (22) and (23).

The core of $(R;A,C;M)$ is $C$ and its core diagram is as in Figure 6. As a
consequence, $R$ is transitive if and only if $\partial$ is surjective.

$\textstyle{\operatorname{kern}{\partial}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\scriptstyle{\partial}$$\textstyle{C}$$\textstyle{A}$
Figure 6: The core diagram of R.

###  4.4 Morphisms of comma-double Lie algebroids

Consider in this section a square of Lie algebroid morphisms over $M$ as in
Figure 7.

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{A}}$$\scriptstyle{\varphi_{C}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{A^{\prime}}}$$\textstyle{A^{\prime}}$
Figure 7: Square of Lie algebroid morphisms.

This section proves that such a square induces a morphism of the corresponding
comma double Lie algebroids $TC\oplus_{TM}A\to
TC^{\prime}\oplus_{TM^{\prime}}A^{\prime}$, with side morphisms $\varphi_{A}$
and $\varphi_{C}$ and with core morphism $\varphi_{C}$.

######  Proposition 4.9.

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

$\Phi\colon TC\oplus_{TM}A\to
TC^{\prime}\oplus_{TM^{\prime}}A^{\prime},\qquad(V,a)\mapsto(T\varphi_{C}V,\varphi_{A}(a))$

defines a morphism of double vector bundles with side morphisms
$\varphi_{A}\colon A\to A^{\prime}$ and $\varphi_{C}\colon C\to C^{\prime}$
and with core morphism $\varphi_{C}\colon C\to C^{\prime}$.

###### Proof.

First of all, $\Phi$ is well-defined since for $(V,a)\in TC\oplus_{TM}A$, the
image $\Phi(V,a)=(T\varphi_{C}V,\varphi_{A}(a))$ satisfies

$Tq_{C^{\prime}}T\varphi_{C}V=Tq_{C}V=\rho_{A}(a)=\rho_{A^{\prime}}\varphi_{A}(a)$

and is hence an element of $TC^{\prime}\oplus_{TM}C^{\prime}$. Choose an
$A$-connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ on $C$, and
an $A^{\prime}$-connection
$\nabla^{\prime}\colon\Gamma(A^{\prime})\times\Gamma(C^{\prime})\to\Gamma(C^{\prime})$
on $C^{\prime}$. Consider the induced linear splittings as in Proposition 4.2

$I_{\nabla}\colon A\times_{M}C\times_{M}C\to
TC\oplus_{TM}A,\qquad(a,c,\gamma)\mapsto\left(\widehat{\nabla_{a}}(c)+_{C}\gamma^{\uparrow}(c),a\right)$

and

$I_{\nabla^{\prime}}\colon
A^{\prime}\times_{M}C^{\prime}\times_{M}C^{\prime}\to
TC^{\prime}\oplus_{TM}A^{\prime},\qquad(a^{\prime},c^{\prime},\gamma^{\prime})\mapsto\left(\widehat{\nabla^{\prime}_{a^{\prime}}}(c^{\prime})+_{C^{\prime}}{\gamma^{\prime}}^{\uparrow}(c^{\prime}),a^{\prime}\right).$

Define $\omega_{\nabla,\nabla^{\prime}}\in\Gamma(A^{*}\otimes C^{*}\otimes
C^{\prime})$ by

$\omega_{\nabla,\nabla^{\prime}}(a,c)=\nabla^{\prime}_{\varphi_{A}(a)}(\varphi_{C}(c))-\varphi_{C}(\nabla_{a}c)$
(40)

for $a\in\Gamma(A)$ and $c\in\Gamma(C)$. Then an easy computation proves that

$T_{c}\varphi_{c}\widehat{\nabla_{a}}(c)-\widehat{\nabla^{\prime}_{\varphi(a)}}(\varphi_{C}(c))=\omega_{\nabla,\nabla^{\prime}}(a,c)^{\uparrow}(\varphi_{C}(c))$

for all $c\in C$ and $a\in A$. Therefore,

$\begin{split}(I_{\nabla^{\prime}}^{-1}\circ\Phi\circ
I_{\nabla})(a,c,\gamma)&=\left(I_{\nabla^{\prime}}^{-1}\circ\Phi)(\widehat{\nabla_{a}}(c)+_{C}\gamma^{\uparrow}(c),a\right)=I_{\nabla^{\prime}}^{-1}\left(T\varphi_{C}(\widehat{\nabla_{a}}(c))+_{C}T\varphi_{C}(\gamma^{\uparrow}(c)),\varphi_{A}(a)\right)\\\
&=I_{\nabla^{\prime}}^{-1}\left(\widehat{\nabla^{\prime}_{\varphi_{A}(a)}}(\varphi(c))+_{C}(\varphi_{C}\gamma+\omega_{\nabla,\nabla^{\prime}}(a,c))^{\uparrow}(\varphi_{C}(c)),\varphi_{A}(a)\right)\\\
&=(\varphi_{A}(a),\varphi_{C}(c),\varphi_{C}(\gamma)+\omega_{\nabla,\nabla^{\prime}}(a,c)).\end{split}$
(41)

This shows that $I_{\nabla^{\prime}}^{-1}\circ\Phi\circ I_{\nabla}\colon
A\times_{M}C\times_{M}C\to A^{\prime}\times_{M}C^{\prime}\times_{M}C^{\prime}$
is a double vector bundle morphism, and consequently that $\Phi$ 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

$\Phi\colon TC\oplus_{TM}A\to
TC^{\prime}\oplus_{TM^{\prime}}A^{\prime},\qquad(V,a)\mapsto(T\varphi_{C}V,\varphi_{A}(a))$

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).

$\textstyle{C^{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{B}}$$\scriptstyle{\partial=\partial_{A}}$$\textstyle{B}$$\textstyle{A}$
Figure 8: Setting of Section 5, the transitive core diagram $\mathcal{C}$.

Recall the representation $\nabla^{A}\colon\thinspace
A\to\operatorname{Der}(C^{A})$ defined in (19). For $a\in\Gamma(A)$, consider
the vector field

$\widehat{\nabla^{A}_{a}}\in\mathfrak{X}^{l}(C^{A})$

corresponding to the derivation $\nabla^{A}_{a}$ of $C^{A}$ over
$\rho_{A}(a)$.

######  Lemma 5.1.

In the situation above, choose $c\in\Gamma(C)$ and set $a=\partial_{A}c$. Then
the vector field $\widehat{[c,\cdot]}\in\mathfrak{X}^{l}(C)$ restricts to
$\widehat{\nabla^{A}_{a}}$ on points of $C^{A}$. In other words, if
$\iota\colon C^{A}\hookrightarrow C$ is the inclusion, then
$\widehat{\nabla^{A}_{a}}\sim_{\iota}\widehat{[c,\cdot]}$.

###### Proof.

Check first that $\widehat{[c,\cdot]}(\gamma_{m})\in T_{\gamma_{m}}C^{A}$ for
all $\gamma_{m}\in C^{A}$. In order to do this, it is sufficient to show
$\widehat{[c,\cdot]}(F)=0$ for all $F\in C^{\infty}(C)$ such that
$F\arrowvert_{C^{A}}$ is constant. Since $C^{A}\subseteq C$ is a subbundle, it
suffices to check this on the linear functions $\ell_{\mu}\in C^{\infty}(C)$
for $\mu\in\Gamma\left((C^{A})^{\circ}\right)\subseteq\Gamma(C^{*})$. But this
is immediate since

$\widehat{[c,\cdot]}(\ell_{\mu})(\gamma)=\rho_{C}(c)\langle\mu,\gamma\rangle-\langle\mu,[c,\gamma]\rangle=0$

for all $\gamma\in\Gamma(C^{A})$, since $[c,\gamma]\in\Gamma(C^{A})$.

Then $\iota^{*}q_{C}^{*}f=q_{C^{A}}^{*}f$ for all $f\in C^{\infty}(M)$ and
$\iota^{*}\ell_{\mu}=\ell_{\iota^{t}\mu}$ for all $\mu\in\Gamma(C^{*})$. The
definitions of $\widehat{[c,\cdot]}$ and $\widehat{\nabla^{A}_{a}}$ yield
easily $(\nabla_{a}^{A})^{*}(\iota^{t}\mu)=\iota^{t}{{\pounds}}_{c}\mu$ and so

${{\pounds}}_{\widehat{\nabla^{A}_{a}}}(\iota^{*}\ell_{\mu})={{\pounds}}_{\widehat{\nabla^{A}_{a}}}\ell_{\iota^{t}\mu}=\ell_{(\nabla^{A}_{a})^{*}\iota^{t}\mu}=\iota^{*}\ell_{{{\pounds}}_{c}\mu}=\iota^{*}(\widehat{[c,\cdot]}(\ell_{\mu}))$

for $\mu\in\Gamma(C^{*})$. Similarly, together with
$\rho_{C}(c)=\rho_{A}(\partial c)=\rho_{A}(a)$, the definitions of
$\widehat{[c,\cdot]}$ and $\widehat{\nabla^{A}_{a}}$ yield

${{\pounds}}_{\widehat{\nabla^{A}_{a}}}(\iota^{*}q_{C}^{*}f)={{\pounds}}_{\widehat{\nabla^{A}_{a}}}(q_{C^{A}}^{*}f)=q_{C^{A}}^{*}({{\pounds}}_{\rho_{A}(a)}(f))=\iota^{*}q_{C}^{*}({{\pounds}}_{\rho_{C}(c)}(f))=\iota^{*}({{\pounds}}_{\widehat{[c,\cdot]}}(q_{C}^{*}f))$

for $f\in C^{\infty}(M)$. ∎

Define the subset $\Pi\subseteq R$ by

$\Pi:=\left\\{\left.\left(\widehat{\nabla^{A}_{a}}(\gamma_{m}),a(m)\right)\in
TC^{A}\oplus_{TM}A\,\right|a\in\Gamma(A),\ \gamma_{m}\in C^{A}\right\\}.$ (42)

By Lemma 5.1, it equals

$\left\\{\left.\left(\widehat{[c,\cdot]}(\gamma_{m}),\partial c(m)\right)\in
TC^{A}\oplus_{TM}A\,\right|c\in\Gamma(C),\ \gamma_{m}\in C^{A}\right\\}.$

######  Proposition 5.2.

Choose any connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ such
that $\nabla_{a}\gamma=\nabla^{A}_{a}\gamma$ for all $a\in\Gamma(A)$ and
$\gamma\in\Gamma(C^{A})$. Then the isomorphism $I_{\nabla}\colon
A\times_{M}C\times_{M}C\to R$ of double vector bundles in Proposition 4.2 and
Corollary 4.3 does

$I_{\nabla}(A\times_{M}C^{A}\times_{M}0^{C})=\Pi.$

As a consequence, $\Pi$ is a sub-double vector bundle of $R$ with sides $A$
and $C^{A}$ and with trivial core.

###### Proof.

As in the proof of Lemma 5.1, prove that
$\mathfrak{X}^{l}(C^{A})\ni\widehat{\nabla^{A}_{a}}\sim_{\iota}\widehat{\nabla_{a}}\in\mathfrak{X}^{l}(C)$.
Then given $(a_{m},\gamma_{m},0^{C}_{m})\in A\times_{M}C^{A}\times_{M}0^{C}$
and any $a\in\Gamma(A)$ such that $a(m)=a_{m}$:

$I_{\nabla}(a_{m},\gamma_{m},0^{C}_{m})=\left(\widehat{\nabla_{a}}(\gamma_{m}),a_{m}\right)=\left(\widehat{\nabla^{A}_{a}}(\gamma_{m}),a_{m}\right)\in\Pi\subseteq
TC^{A}\oplus_{TM}A.\qed$

The following corollary is immediate, using the notation (30).

######  Corollary 5.3.

Choose any connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ such
that $\nabla_{a}\gamma=\nabla^{A}_{a}\gamma$ for all $a\in\Gamma(A)$ and
$\gamma\in\Gamma(C^{A})$. Then the subbundle $\Pi\to A$ of $R\to A$ is spanned
by the sections

$\sigma_{\nabla}^{C}(\gamma)=\widecheck{\gamma}\in\Gamma^{l}_{A}(R)$

for all $\gamma\in\Gamma(C^{A})$.

###  5.1 The quotient of $R$ by $\Pi$.

This section defines the quotient $R/\Pi$ and shows that it inherits a double
vector bundle structure. In the situation above define $\sim$ on $R$ by

$(V,a)\sim(V^{\prime},a^{\prime}):\Leftrightarrow a=a^{\prime}\text{ and
}\,\,\exists\,W\in TC^{A}:(W,a)\in\Pi\text{ and }V+_{TM}W=V^{\prime}.$

Then it is easy to check that $\sim$ defines an equivalence relation on $R$.
Two pairs $(V,a)$ and $(V^{\prime},a)\in R$ are equivalent if and only if
there exists $(W,a)\in\Pi$ such that $(V,a)+_{A}(W,a)=(V^{\prime},a)$. Hence,
it is easy to see that $R/_{\sim}=:R/\Pi$ has a vector bundle structure over
$A$: the quotient of the vector bundle $R\to A$ by its subbundle $\Pi\to A$.
The reader should think about $R/\Pi$ as this quotient. The following theorem
shows that the vector bundle structure on $R\to C$ induces a vector bundle
structure on $R/\Pi\to C/C^{A}$, such that $(R/\Pi,A,C/C^{A},M)$ becomes a
double vector bundle.

######  Theorem 5.4.

  1. 1.

The space $R/\Pi$ inherits from $R$ a double vector bundle structure with core
$C$ and with side $C/C^{A}\simeq B$.

  2. 2.

Choose any connection $\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ such
that $\nabla_{a}\gamma=\nabla^{A}_{a}\gamma$ for all $a\in\Gamma(A)$ and
$\gamma\in\Gamma(C^{A})$. Then the induced decomposition $I_{\nabla}\colon
A\times_{M}C\times_{M}C\to R$ induces a decomposition

$\overline{I_{\nabla}}\colon A\times_{M}(C/C^{A})\times_{M}C\to
R/\Pi,\qquad\overline{I_{\nabla}}(a,\overline{c},\gamma):=I_{\nabla}(a,c,\gamma)+_{A}\Pi.$

The remainder of this section proves this theorem. Define $F_{C}\subseteq TC$
as the subbundle over $C$ spanned by the vertical vector fields corresponding
to sections of $C^{A}$, i.e. for all $c_{m}\in C$:

$F_{C}(c_{m})=\left\\{\gamma^{\uparrow}(c_{m})\mid\gamma\in\Gamma(C^{A})\right\\}$

and so

$\Gamma(F_{C})=\operatorname{span}_{C^{\infty}(C)}\left\\{\gamma^{\uparrow}\mid\gamma\in\Gamma(C^{A})\right\\}.$

The subbundle $F_{C}\subseteq TC$ is involutive and simple since its leaf
space is $C/C^{A}\simeq B$.

Choose a linear $A$-connection
$\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ that restricts to
$\nabla^{A}$ on $\Gamma(C^{A})$, and consider the induced sections
$\widehat{a}\in\Gamma_{C}^{l}(R)$ for all $a\in\Gamma(A)$, as in (29). Recall
also from (31) that each section $\gamma\in\Gamma(C)$ defines a core section
$\gamma^{\times}\in\Gamma^{c}_{C}(R)$.

######  Proposition 5.5.

In the situation above, there is a unique flat connection

$\nabla^{i}\colon\Gamma_{C}(F_{C})\times\Gamma_{C}(R)\to\Gamma_{C}(R),$

such that

$\nabla^{i}\widehat{a}=0=\nabla^{i}\gamma^{\times}$

for all $a\in\Gamma(A)$ and all $\gamma\in\Gamma(C)$. The connection
$\nabla^{i}$ does not depend on the choice of $\nabla$ extending $\nabla^{A}$.

###### Proof.

The space of sections of the vector bundle $R\to C$ is spanned as a
$C^{\infty}(C)$-module by sections of the form $\widehat{a}$ and
$\gamma^{\times}$ for $a\in\Gamma(A)$ and $\gamma\in\Gamma(C)$. Define

$\nabla^{i}\colon\Gamma_{C}(F_{C})\times\Gamma_{C}(R)\to\Gamma_{C}(R)$

by

$\nabla^{i}_{\chi}\left(\sum_{j=1}^{s}F_{j}\widehat{a_{j}}+\sum_{l=1}^{t}G_{l}\gamma_{l}^{\times}\right)=\sum_{j=1}^{s}{{\pounds}}_{\chi}(F_{j})\cdot\widehat{a_{j}}+\sum_{l=1}^{t}{{\pounds}}_{\chi}(G_{l})\cdot\gamma_{l}^{\times}$

for $F_{1},\ldots,F_{s},G_{1},\ldots,G_{t}\in C^{\infty}(C)$,
$a_{1},\ldots,a_{s}\in\Gamma(A)$ and
$\gamma_{1},\ldots,\gamma_{t}\in\Gamma(C)$.

For $f\in C^{\infty}(M)$, $a\in\Gamma(A)$ and $\gamma\in\Gamma(C)$

$\widehat{fa}=q_{C}^{*}f\cdot\widehat{a}\qquad\text{ and
}\qquad(f\gamma)^{\times}=q_{C}^{*}f\cdot\gamma^{\times}.$

Hence $\nabla^{i}$ is well-defined if
$\nabla^{i}_{\cdot}\widehat{fa}=\nabla^{i}_{\cdot}\left(q_{C}^{*}f\cdot\widehat{a}\right)$
and
$\nabla^{i}_{\cdot}(f\gamma)^{\times}=\nabla^{i}_{\cdot}\left(q_{C}^{*}f\cdot\gamma^{\times}\right)$
for all $f\in C^{\infty}(M)$, $a\in\Gamma(A)$ and $\gamma\in\Gamma(C)$. On the
one hand,
$\nabla^{i}_{\chi}\widehat{fa}=0=\nabla^{i}_{\chi}(f\gamma)^{\times}$ and on
the other hand

$\nabla^{i}_{\chi}\left(q_{C}^{*}f\cdot\widehat{a}\right)={{\pounds}}_{\chi}(q_{C}^{*}f)\cdot\widehat{a}=0$

and

$\nabla^{i}_{\chi}\left(q_{C}^{*}f\cdot\gamma^{\times}\right)={{\pounds}}_{\chi}(q_{C}^{*}f)\cdot\gamma^{\times}=0$

since $\chi\in\Gamma(F_{C})$ and $F_{C}\subseteq T^{q_{C}}C$.

The connection $\nabla^{i}$ is obviously flat with the required flat sections.
Let $\nabla^{\prime}\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ be a second
extension of $\nabla^{A}\colon\Gamma(A)\times\Gamma(C^{A})\to\Gamma(C^{A})$.
Then $\varphi:=\nabla-\nabla^{\prime}\in\Omega^{1}(A,(C^{A})^{\circ}\otimes
C)$ does
$\sigma_{A}^{\nabla^{\prime}}(a)=\sigma_{A}^{\nabla}(a)-\widetilde{\varphi(a)}=\widehat{a}-\widetilde{\varphi(a)}$
for all $a\in\Gamma(A)$. Writing $\varphi(a)=\sum_{j=1}^{k}\nu_{j}\otimes
c_{j}$ with $\nu_{1},\ldots,\nu_{k}\in\Gamma((C^{A})^{\circ})$ and
$c_{1},\ldots,c_{k}\in\Gamma(C)$ yields

$\nabla^{i}_{\chi}\left(\sigma_{A}^{\nabla^{\prime}}(a)\right)=\nabla^{i}_{\chi}\left(\widehat{a}-\widetilde{\varphi(a)}\right)=\nabla^{i}_{\chi}\left(\sum_{j=1}^{k}\ell_{\nu_{j}}\cdot
c_{j}^{\times}\right)=\sum_{j=1}^{k}{{\pounds}}_{\chi}(\ell_{\nu_{j}})\cdot
c_{j}^{\times}=0$

since $\chi\in\Gamma(F_{C})$ and $\ell_{\nu_{j}}\in C^{\infty}(C)^{F_{C}}$. ∎

The last proposition is reformulated as follows.

######  Corollary 5.6.

The triple $(F_{C},0,\nabla^{i})$ is an infinitesimal ideal system in the
vector bundle $R\to C$ considered as a trivial Lie algebroid.

The quotient $R/\nabla^{i}\to C/F_{C}$ is construted as follows. Two vectors
$(V,a)$ and $(V^{\prime},a^{\prime})\in R$ are equivalent if $c_{m}=p_{C}(V)$
and $c_{m}^{\prime}=p_{C}(V^{\prime})$ lie in the same leaf of $F_{C}$, i.e.
they can be joined by an integral path of $F_{C}$ and there exists a parallel
section along this path taking values $(V,a)$ at the point $c_{m}$ and
$(V^{\prime},a^{\prime})$ at $c_{m}^{\prime}$. Note that here $c_{m}=p_{C}(V)$
and $c_{m}^{\prime}=p_{C}(V^{\prime})$ lie in the same leaf of $F_{C}$ if an
only if $c_{m}-c_{m}^{\prime}\in C^{A}$. Recall as well that $(V,a)$ and
$(V^{\prime},a^{\prime})\in R$ project to the same element in $R/\Pi$ if

$a=a^{\prime},\qquad p_{C}(V)-p_{C}(V^{\prime})\in C^{A}\quad\text{ and }\quad
V-_{TM}V^{\prime}=\widehat{\nabla_{a}^{A}}(p_{C}(V)-p_{C}(V^{\prime})).$

######  Proposition 5.7.

In the situation above, the two equivalence relations $\sim$ and
$\sim_{\nabla^{i}}$ coincide.

###### Proof.

Assume first that $(V,a)$ and $(V^{\prime},a^{\prime})$ are equivalent via
$\sim_{\nabla^{i}}$. Let $c_{m}=p_{C}(V)$ and
$d_{m^{\prime}}=p_{C}(V^{\prime})$. Since $c_{m}$ and $d_{m^{\prime}}$ are in
the same leaf of $F_{C}$, they must be in the same fiber of $q_{C}\colon C\to
M$ and hence $m=m^{\prime}$. Furthermore $c_{m}-d_{m}\in C^{A}(m)$ and there
exists without loss of generality one $\nabla^{i}$-parallel section $\chi$ of
$R\to C$ such that $\chi(c_{m})=(V,a)$ and
$\chi(d_{m})=(V^{\prime},a^{\prime})$. Write

$\chi=\sum_{i=1}^{k}F_{i}\widehat{a_{i}}+\sum_{j=1}^{l}G_{j}c_{j}^{\times}$

with $a_{1},\ldots,a_{k}\in\Gamma(A)$, $c_{1},\ldots,c_{l}\in\Gamma(C)$ and
$F_{1},\ldots,F_{k},G_{1},\ldots,G_{l}\in C^{\infty}(C)^{F_{C}}$. Since the
functions $F_{i},G_{j}$ are all $F_{C}$-invariant,
$F_{i}(c_{m})=F_{i}(d_{m})=:\alpha_{i}$ and
$G_{j}(c_{m})=G_{j}(d_{m})=\beta_{j}$ for $i=1,\ldots,k$ and $j=1,\ldots,l$.

Thus

$\displaystyle(V,a)$
$\displaystyle=\sum_{i=1}^{k}\alpha_{i}\widehat{a_{i}}(c_{m})+\sum_{j=1}^{l}\beta_{j}c_{j}^{\times}(c_{m})=\left(\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla_{a_{i}}}(c_{m})+\sum_{j=1}^{l}\beta_{j}c_{j}^{\uparrow}(c_{m}),\sum_{i=1}^{k}\alpha_{i}a_{i}(m)\right)$

and

$\displaystyle(V^{\prime},a^{\prime})$
$\displaystyle=\sum_{i=1}^{k}\alpha_{i}\widehat{a_{i}}(d_{m})+\sum_{j=1}^{l}\beta_{j}c_{j}^{\times}(d_{m})=\left(\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla_{a_{i}}}(d_{m})+\sum_{j=1}^{l}\beta_{j}c_{j}^{\uparrow}(d_{m}),\sum_{i=1}^{k}\alpha_{i}a_{i}(m)\right)$

This implies $a=\sum_{i=1}^{k}\alpha_{i}a_{i}(m)=a^{\prime}$ and the
difference $V-_{TM}V^{\prime}$ in the fiber of $TC\to TM$ over $\rho_{A}(a)$
equals by the interchange law

$\displaystyle V-_{TM}V^{\prime}$
$\displaystyle=\left(\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla_{a_{i}}}(c_{m})-_{TM}\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla_{a_{i}}}(d_{m})\right)+_{C}\left(\sum_{j=1}^{l}\beta_{j}c_{j}^{\uparrow}(c_{m})-_{TM}\sum_{j=1}^{l}\beta_{j}c_{j}^{\uparrow}(d_{m})\right)$
$\displaystyle=\left(\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla_{a_{i}}}(c_{m}-d_{m})\right)+_{C}0^{TC}_{c_{m}-d_{m}}=\left(\sum_{i=1}^{k}\alpha_{i}\widehat{\nabla^{A}_{a_{i}}}(c_{m}-d_{m})\right)=\widehat{\nabla_{a}^{A}}(c_{m}-d_{m}).$

This shows that $(V,a)$ and $(V^{\prime},a^{\prime})$ are equivalent in
$R/\Pi$.

Conversely, consider $(V,a_{m})$ and $(V^{\prime},a_{m})\in R$, set
$c_{m}=p_{C}(V)$ and $d_{m}=p_{C}(V^{\prime})$ and choose a section
$a\in\Gamma(A)$ with $a(m)=a_{m}$. If $(V,a_{m})$ and $(V^{\prime},a_{m})$ are
equivalent in $R/\Pi$, i.e. if $c_{m}-d_{m}\in C^{A}(m)$ and
$V-V^{\prime}=\widehat{\nabla^{A}_{a}}(c_{m}-d_{m})$, then, since
$T{q_{C}}V=\rho_{A}(a_{m})=T{q_{C}}\widehat{\nabla_{a}}(c_{m})$, there exists
$\gamma\in\Gamma(C)$ such that
$V=\widehat{\nabla_{a}}(c_{m})+\gamma^{\uparrow}(c_{m})$. In the same manner,
there is a section $\gamma^{\prime}\in\Gamma(C)$ such that
$V^{\prime}=\widehat{\nabla_{a}}(d_{m})+{\gamma^{\prime}}^{\uparrow}(d_{m})$.
Furthermore, since
$V-V^{\prime}=\widehat{\nabla_{a}^{A}}(c_{m}-d_{m})=\widehat{\nabla_{a}}(c_{m}-d_{m})$
by the choice of $\nabla$,
$V-V^{\prime}=\widehat{\nabla_{a}}(c_{m})-\widehat{\nabla_{a}}(d_{m})$ and
hence $\gamma(m)=\gamma^{\prime}(m)$. Set $\chi=\widehat{a}+\gamma^{\times}$.
Then $\chi$ is a $\nabla^{i}$-parallel section of $R\to C$ and

$(V,a_{m})=\chi(c_{m})\qquad\text{ and
}\qquad(V^{\prime},a_{m})=\chi(d_{m}).\qed$

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

$\pi_{B}\colon R/\Pi\to B,\qquad(V,a)+\Pi\mapsto\partial_{B}(p_{C}(V))$

and a well-defined addition

$+_{B}\colon(R/\Pi)\times_{B}(R/\Pi)\to R/\Pi$

in the fibers of $\pi_{B}$ which is defined as follows. If $(V,a_{m})$ and
$(V^{\prime},a^{\prime}_{m})\in R$ are such that
$\pi_{B}((V,a_{m})+\Pi)=\partial_{B}(p_{C}(V))=\partial_{B}(p_{C}(V^{\prime}))=\pi_{B}((V^{\prime},a^{\prime}_{m})+\Pi)$,
then $p_{C}(V)-p_{C}(V^{\prime})\in C^{A}_{m}$. Set $c_{m}=p_{C}(V)$ and
$c^{\prime}_{m}=p_{C}(V^{\prime})$. Choose a section $a^{\prime}\in\Gamma(A)$
such that $a^{\prime}(m)=a^{\prime}_{m}$, and $\gamma\in\Gamma(C)$ such that
$(V^{\prime},a^{\prime}_{m})=\widehat{a^{\prime}}(c_{m}^{\prime})+_{C}\gamma^{\times}(c_{m}^{\prime})$.
Since $\widehat{a^{\prime}}$ and $\gamma^{\times}$ are $\nabla^{i}$-flat and
$c_{m}-c_{m}^{\prime}\in C^{A}$, the vectors $(V^{\prime},a^{\prime})$ and
$(\widehat{a^{\prime}}(c_{m})+\gamma^{\times}(c_{m})$ are
$\nabla^{i}$-equivalent and so

$(V^{\prime},a^{\prime})\sim\left(\widehat{a^{\prime}}(c_{m})+\gamma^{\times}(c_{m}),a^{\prime}\right)$

by Proposition 5.7. Then set

$((V,a)+_{A}\Pi)+_{B}((V^{\prime},a^{\prime})+_{A}\Pi):=\Bigl{(}(V,a)+_{C}\left((\widehat{a^{\prime}}(c_{m})+_{C}\gamma^{\times}(c_{m})\right)\Bigr{)}+_{A}\Pi.$

###### Proof of Theorem 5.4.

Choose as before a connection
$\nabla\colon\Gamma(A)\times\Gamma(C)\to\Gamma(C)$ such that
$\nabla_{a}\gamma=\nabla^{A}_{a}\gamma$ for all $a\in\Gamma(A)$ and
$\gamma\in\Gamma(C^{A})$. Consider the induced decomposition $I_{\nabla}\colon
A\times_{M}C\times_{M}C\to R$. Since $I_{\nabla}(a,c,0)\in\Pi$ for $(a,c,0)\in
A\times_{M}C^{A}\times_{M}0$, the morphism

$\textstyle{A\times_{M}C\times_{M}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad
I_{\nabla}}$$\scriptstyle{\pi_{A}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{id}_{A}}$$\textstyle{A}$

of vector bundles over $A$ factors to a morphism

$\textstyle{A\times_{M}C/C^{A}\times_{M}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\overline{I_{\nabla}}}$$\scriptstyle{\pi_{A}}$$\textstyle{R/\Pi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{id}_{A}}$$\textstyle{A}$

which is given by

$\overline{I_{\nabla}}\colon A\times_{M}(C/C^{A})\times_{M}C\to
R/\Pi,\qquad\overline{I_{\nabla}}(a,\overline{c},\gamma):=I_{\nabla}(a,c,\gamma)+_{A}\Pi.$

Since $I_{\nabla}(A\times_{M}C^{A}\times_{M}0)=\Pi$, the reduced morphism
$\overline{I_{\nabla}}$ is again an isomorphism of vector bundles over $A$.

Consider $b_{m}=\partial_{B}c_{m}=\partial_{B}c^{\prime}_{m}\in C/C^{A}$,
$\gamma,\gamma^{\prime}\in\Gamma(C)$ and $a,a^{\prime}\in\Gamma(A)$. Then

$\begin{split}&\overline{I_{\nabla}}(a(m),\overline{c_{m}},\gamma_{m})+_{B}\overline{I_{\nabla}}(a^{\prime}(m),\overline{c_{m}^{\prime}},\gamma^{\prime}_{m})\\\
=\,&\left(\left(\widehat{a}(c_{m})+_{C}\gamma^{\times}(c_{m})\right)+_{A}\Pi\right)+_{B}\left(\left(\widehat{a^{\prime}}(c^{\prime}_{m})+_{C}{\gamma^{\prime}}^{\times}(c_{m}^{\prime})\right)+_{A}\Pi\right)\\\
=\,&\left(\left(\widehat{a}(c_{m})+_{C}\gamma^{\times}(c_{m})\right)+_{A}\Pi\right)+_{B}\left(\left(\widehat{a^{\prime}}(c_{m})+_{C}{\gamma^{\prime}}^{\times}(c_{m})\right)+_{A}\Pi\right)\\\
=\,&\left(\widehat{a+a^{\prime}}(c_{m})+_{C}(\gamma+\gamma^{\prime})^{\times}(c_{m})\right)+_{A}\Pi=\overline{I_{\nabla}}(a(m)+a^{\prime}(m),\overline{c_{m}},\gamma(m)+\gamma^{\prime}(m)).\end{split}$

Hence $\overline{I_{\nabla}}$ is compatible with the addition over $B\simeq
C/C^{A}$ in $A\times_{M}C/C^{A}\times_{M}C$ and with the addition $+_{B}$ in
$R/\Pi$. Since $\overline{I_{\nabla}}$ is bijective and linear over $A$ as
well, $R/\Pi$ is a double vector bundle, of which $\overline{I_{\nabla}}$ is a
decomposition. ∎

From now on the quotient $C/C^{A}$ is identified with $B$ via the
factorisation of the morphism $\partial_{B}\colon C\to B$ of vector bundles.
The linear splitting $\overline{I_{\nabla}}\colon A\times_{M}B\times_{M}C\to
R/\Pi$ defines a linear horizontal lift
$\overline{\sigma^{\nabla}_{B}}\colon\Gamma(B)\to\Gamma^{l}_{A}(R/\Pi)$: for
all $b\in\Gamma(B)$ and $a_{m}\in A$

$\begin{split}\widecheck{b}(a_{m}):=\overline{\sigma^{\nabla}_{B}}(b)(a_{m})&=\overline{I_{\nabla}}(a_{m},b(m),0^{C}_{m})=I_{\nabla}(a_{m},c(m),0^{C}_{m})+_{A}\Pi=\left(\widehat{\nabla_{a}}(c(m)),a(m)\right)+_{A}\Pi\\\
&=\widecheck{c}(a_{m})+\Pi\end{split}$ (43)

for any $a\in\Gamma(A)$ such that $a(m)=a_{m}$ and $c\in\Gamma(C)$ such that
$\partial_{B}c=b$. The core sections of $R/\Pi\to A$ are
$\gamma^{\ddagger}\in\Gamma_{A}^{c}(R/\Pi)$ defined by sections
$\gamma\in\Gamma(C)$: for $a_{m}\in A$

$\begin{split}\gamma^{\ddagger}(a_{m})&=\overline{I_{\nabla}}(a_{m},0^{B}_{m},\gamma(m))=I_{\nabla}(a_{m},0^{C},\gamma(m))+_{A}\Pi=\left(\gamma^{\dagger}(\rho_{A}(a_{m})),a_{m}\right)+_{A}\Pi\\\