# A geometrisation of $\mathbb{N}$-manifolds

M. Heuer<EMAIL_ADDRESS>and  M. Jotz  Institut für Mathematik, Julius-
Maximilians-Universität Würzburg, Germany<EMAIL_ADDRESS>wuerzburg.de

###### Abstract.

This paper proposes a _geometrisation_ of $\mathbb{N}$-manifolds of degree $n$
as $n$-fold vector bundles equipped with a (signed) $S_{n}$-symmetry. More
precisely, it proves an equivalence between the categories of $[n]$-manifolds
and the category of symmetric $n$-fold vector bundles, by finding that
symmetric $n$-fold vector bundle cocycles and $[n]$-manifold cocycles are
identical.

This extends the already known equivalences of $[1]$-manifolds with vector
bundles, and of $[2]$-manifolds with involutive double vector bundles, where
the involution is understood as an $S_{2}$-action.

###### 2010 Mathematics Subject Classification:

Primary: 58A50, 20B30. Secondary: 53B05, 53C05, 20B05.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and convention
  2. 2 Partitions and integer partitions
    1. 2.1 Partitions and cube-categories
    2. 2.2 Partitions and signs
    3. 2.3 Integer partitions of natural numbers
  3. 3 The category of $\mathbb{N}$-manifolds
    1. 3.1 Definitions
    2. 3.2 The category of split $\mathbb{N}$-manifolds
    3. 3.3 $[n]$-manifold cocycles
  4. 4 Multiple vector bundles and charts
    1. 4.1 Cores of an $n$-fold vector bundle
    2. 4.2 Linear splittings and decompositions of an $n$-fold vector bundle
    3. 4.3 Iterated highest order cores of an $n$-fold vector bundle
    4. 4.4 Proof of Corollary 3.6 in [HL20]
    5. 4.5 Atlases of multiple vector bundles
  5. 5 Symmetric $n$-fold vector bundles
    1. 5.1 Global definition of symmetric $n$-fold vector bundles.
    2. 5.2 Example: the pullback of a symmetric $n$-fold vector bundle
    3. 5.3 Iterated highest order cores of symmetric $n$-fold vector bundles
    4. 5.4 Decomposed symmetric $n$-fold vector bundles
    5. 5.5 Morphisms of decomposed symmetric $n$-fold vector bundles
    6. 5.6 Symmetric linear splittings and decompositions
    7. 5.7 Symmetric $n$-fold vector bundle atlases
  6. 6 The equivalence of symmetric $n$-fold vector bundles with $[n]$-manifolds
  7. A From linear splittings to decompositions
  8. B On the existence of symmetric linear splittings and symmetric decompositions
    1. B.1 Proof of Theorem 5.10

##  1\. Introduction

An $\mathbb{N}$-graded manifold over a smooth manifold $M$ is a sheaf of
$\mathbb{N}$-graded, graded commutative, associative, unital
$C^{\infty}(M)$-algebras over $M$, that is locally freely generated by
finitely many elements of strictly positive degree. $\mathbb{N}$-manifolds of
degree $1$ are easily seen to be the exterior algebras of sections of smooth
vector bundles. $\mathbb{N}$-manifolds of degree $2$, called for short
$[2]$-manifolds, have recently been geometrised to double vector bundles with
a linear _indirect_ involution [JL18]. (Such double vector bundles were called
“symmetric double vector bundles with inverse symmetry” by Pradines [Pra77].)
Before that, Lie $2$-algebroids i.e. $[2]$-manifolds with a _homological
vector field_ , were linked to VB-Courant algebroids by Li-Bland in his thesis
[LB12], see also [JL19], building up on the correspondence of Courant
algebroids with symplectic Lie 2-algebroids [Roy02, Šev05].

Generally, positively graded manifolds are the geometric objects underlying
Lie $n$-algebroids, which are widely accepted to infinitesimally describe Lie
$n$-groupoids. A Lie $n$-algebroid is an $[n]$-manifold equipped with a
homological vector field, i.e. a vector field of degree $1$ that squares to
zero. Lie $n$-algebroids appeared in the early 00’s in Voronov’s study of Lie
bialgebroids [Vor02] and in Roytenberg’s supergeometric approach to Courant
algebroids [Roy02], see also [Šev05]. Precisely, Courant algebroids are
equivalent to Lie $2$-algebroids equipped with a compatible symplectic
structure of degree $2$. This is at the origin of the interest for Lie
$n$-algebroids in the Poisson community, as this fact leads to a path towards
the _integration_ of Courant algebroids – to Lie $2$-groupoids with an
additional geometric structure. The correspondence of symplectic Lie
$2$-algebroids with Courant algebroids fits in fact in the more general
equivalence of Lie $2$-algebroids with VB-Courant algebroids [LB12, JL19],
which, in turn, is based on the underlying equivalence between double vector
bundles equipped with a linear involution, and positively graded manifolds
generated in the degrees $1$ and $2$ [JL18].

Note that involutive double vector bundles dualise to metric double vector
bundles, and the two classes of objects are therefore equivalent.
Understanding the equivalence between metric or involutive double vector
bundles and $[2]$-manifolds provides a precise dictionary, as summarised
below, for linear geometric structures on involutive double vector bundles
versus compatible geometric structures on $[2]$-manifolds.

Double vector bundles | Degree $2$ graded geometry
---|---
metric double vector bundles |  $[2]$-manifolds
metric VB-algebroids | Poisson $[2]$-manifolds
VB-Courant algebroids | Lie $2$-algebroids
LA-Courant algebroids | Poisson Lie $2$-algebroids
tangent doubles of Courant algebroids | symplectic Lie $2$-algebroids

In particular, several new explicit examples of Lie $2$-algebroids, of Poisson
$[2]$-manifolds and of Poisson Lie $2$-algebroids arise as the counterparts of
known examples of VB-Courant algebroids, of metric VB-algebroids and of LA-
Courant algebroids [JL18, JL19, JL20].

The involution of an involutive double vector bundle can be understood as an
$S_{2}$-action, while a classical vector bundle, which is commonly known as
the geometrisation of a $[1]$-manifold, is trivially acted upon by the trivial
group $S_{1}$. The goal of this paper, which can be considered a sequel of
[JL18], is to establish an equivalence between the category of _$S_{n}$
-symmetric $n$-fold vector bundles_ and the category of positively graded
manifolds of degree $n$. In other words, this paper geometrises
$[n]$-manifolds via $n$-fold vector bundles with a compatible $S_{n}$-action.
This is the groundwork needed for the geometrisation of Lie $n$-algebroids, or
of Lie $n$-algebroids with additional compatible geometric structures such as
symplectic or Poisson structures.

The approach in [JL18] is a classical one; an extension of the construction of
vector bundles over a manifold $M$ from free and locally finitely generated
sheaves of $C^{\infty}(M)$-modules, using the double vector bundle charts in
[Pra77]. Here also, the key to the geometrisation of $[n]$-manifolds is a
precise understanding of the atlases of symmetric $n$-fold vector bundles,
which, in turn, only exist once it has been proved that symmetric $n$-fold
vector bundles can always be equivariantly decomposed. The authors prove in
[HL20] that $n$-fold vector bundles always admit a linear decomposition, and
this paper proves that a _symmetric_ $n$-fold vector bundle admits a
_symmetric_ linear decomposition, and so a symmetric $n$-fold vector bundle
atlas. Unlike in the case $n=2$, adding the symmetry to the picture requires a
thorough reconsideration of the proof of the decomposition in the general
case. This, together with a little gap in [HL20], is the reason why some parts
of [HL20] are considerably revisited here.

However, it turns out that the equivalence of $[n]$-manifolds with symmetric
$n$-fold vector bundles is obtained in a more straightforward manner by
studying cocycles for both types of geometric structures. This paper
concentrates hence on showing that $[n]$-manifold cocycles and symmetric
$n$-fold vector bundle cocycles are basically the same objects, in particular
with same cocycle conditions.

Along the way, an explicit formula for morphisms of split $[n]$-manifolds and
for the composition of two such morphisms is given. The latter is heavily
inferred by the correspondence of split $[n]$-manifolds with decomposed
symmetric $n$-fold vector bundles, and, as far as the authors know, the first
explicit formula avoiding Koszul signs, or more precisely computing them
explicitly.

This paper does not describe how the graded functions of an $[n]$-manifold
correspond to special functions on a symmetric $n$-fold vector bundle. This is
part of a further project in progress. Also, the general equivalence of
$\mathbb{N}$-manifolds (or arbitrary, even infinite, degree) with symmetric
multiple vector bundles is easily deduced from the main result of this paper,
but will be carried out elsewhere.

The differentiation of Lie $n$-groupoids to Lie $n$-algebroids is considered
folklore in the research community, and goes back to Severa [Sev06]. However,
a precise differentiation process carrying out all the details, in particular
the multiple Lie brackets – or in other words the homological vector field –
has not been published yet. Recently, Kadiyan and Blohmann, as well as
independently Du, Fernandes, Ryvkin, and Zhu have announced a differentiation
process, that involves simplicial structures closely related to the $n$-cube
categories used here, and to symmetric $n$-fold vector bundles. This paper
hence builds the geometric fundaments for the infinitesimal description of Lie
$n$-groupoids as Lie $n$-algebroids, via the intermediate step of symmetric
$n$-fold vector bundles with an additional geometric structure that will
correspond to the homological vector field.

### Outline, main results and applications

This paper is organised as follows.

Section 2 collects some facts about (ordered) integer partitions, ordered
partitions of finite subsets of $\mathbb{N}$ and their sign, cube categories.
Partitions and integer partitions, as well as the interplay between the two
notions, are at the core of many constructions in this paper.

Section 3 quickly recalls the definitions of $[n]$-manifolds and their
morphisms. Then it discusses in details the split case, in particular
morphisms of split $\mathbb{N}$-manifolds and their composition.
$\mathbb{N}$-manifold cocycles are then defined as structures that are
equivalent to $\mathbb{N}$-manifolds.

Section 4 gives necessary background on $n$-fold vector bundles, as well as on
their cores, their linear splittings and their linear decompositions. Then it
introduces a new indexing for the iterated highest order cores of an $n$-fold
vector bundle, and fills a gap in the proof of Corollary 3.6 in [HL20],
establishing the existence of a linear decomposition for each $n$-fold vector
bundle [HL20]. This yields that each $n$-fold vector bundle has an $n$-fold
vector bundle atlas [HL20]. Appendix A refines the correspondence between
linear splittings and decompositions of $n$-fold vector bundles by showing a
uniqueness result, which is needed in this paper.

Section 5 introduces symmetric $n$-fold vector bundles and their morphisms. It
discusses decomposed symmetric $n$-fold vector bundles, as well as symmetric
linear splittings and decompositions of symmetric $n$-fold vector bundles.
Morphisms of decomposed symmetric $n$-fold vector bundles, as well as their
composition, are discussed in detail since this is crucial for defining
symmetric $n$-fold vector bundle cocycles. The existence of a symmetric linear
decomposition of a symmetric $n$-fold vector bundle is proved in Appendix B.

Finally, Sections 6 constructs from the equality of the cocycles the functors
from symmetric $n$-fold vector bundles to $[n]$-manifolds, and from
$[n]$-manifold to symmetric $n$-fold vector bundles, that establish together
the equivalence between the two categories.

### Relation to other work

Note that the question solved in this paper is already raised in [Vis19]. More
precisely, the author of [Vis19] asks how to extend the table on Page 1 with
the geometrisation of graded manifolds of higher degree. However, $n$-fold
vector bundles are _defined_ in this reference as special
$\mathbb{Z}^{n}$-graded manifolds, see also [Vor12] and [GR09]. The idea is
that a double vector bundle $D$ with sides $A$ and $B$ defines a bigraded
algebra $D[1]_{A}[1]_{B}$, see also [GSM10], and similarly, $n$-fold vector
bundles define $n$-graded algebras. The paper [Vis19] conversely associates to
an $[n]$-manifold $\mathcal{M}$ its $n$-times iterated tangent bundle and
establishes using this idea an equivalence between graded manifolds and
multiple vector bundles.

The approach and result here are different since $[n]$-manifolds are
geometrised by objects in “classical” differential geometry – in the sense
that the symmetric $n$-fold vector bundles considered here are not considered
in the graded setting.

### Acknowledgements

The authors thank Leonid Ryvkin and Chenchang Zhu for encouraging them to
complete this long-standing project and for discussions on the differentiation
of Lie $n$-groupoids. They thank as well Christian Blohmann and Lory Kadiyan
for interesting discussions, and again Lory Kadiyan for her careful reading of
and useful comments on an early version of this paper.

###  1.1. Notation and convention

The cardinality of a finite set $S$ is written $\\#S$. As a convention, $0$ is
not considered a natural number.

Let $E\to M$ and $F\to N$ be smooth vector bundles and let $\omega\colon E\to
F$ be a morphism of vector bundles over a smooth map $\omega_{0}\colon M\to
N$. The dual of $\omega$ is in general not a morphism of vector bundles, but a
morphism $\omega^{\star}$ of modules over the unital algebra morphism
$\omega_{0}^{*}\colon C^{\infty}(N)\to C^{\infty}(M)$:

(1)
$\omega^{\star}\colon\Gamma(F^{*})\to\Gamma(E^{*}),\qquad\omega^{\star}(\epsilon)(m)=\omega_{m}^{*}(\epsilon_{\omega_{0}(m)})$

for all $\epsilon\in\Gamma(F^{*})$ and $m\in M$. The following lemma is
immediate (see e.g. the appendix of [JL18] for a proof).

######  Lemma 1.1.

The map $\cdot^{\star}$, that sends a morphism of vector bundles $\omega\colon
E\to F$ over $\omega_{0}\colon M\to N$ to the morphism
$\omega^{\star}\colon\Gamma(F^{*})\to\Gamma(E^{*})$ of modules over
$\omega_{0}^{*}\colon C^{\infty}(N)\to C^{\infty}(M)$, is a bijection.

Let $E\to M$ be a smooth vector bundle. This paper adopts the convention that
the wedge product $\omega\wedge\eta\in\Gamma(\wedge^{k+l}E^{*})$ of two forms
$\omega\in\Gamma(\wedge^{k}E^{*})$ and $\eta\in\Gamma(\wedge^{l}E^{*})$ is
given by

$\omega\wedge\eta=\frac{(k+l)!}{k!l!}\operatorname{Alt}(\omega\otimes\eta),$

i.e.

(2) $(\omega\wedge\eta)(e_{1},\ldots,e_{k+l})=\frac{1}{k!l!}\sum_{\sigma\in
S_{k+l}}(-1)^{\sigma}\cdot\omega(e_{\sigma(1)},\ldots,e_{\sigma(k)})\cdot\eta(e_{\sigma(k+1)},\ldots,e_{\sigma(k+l)})$

for $e_{1},\ldots,e_{k+l}$ in the same fiber of $E$.

##  2\. Partitions and integer partitions

Partitions and integer partitions, together with a notion of sign that
partitions define, are at the core of the equivalence between
$\mathbb{N}$-manifolds and symmetric multiple vector bundles. Partitions of
finite subsets of $\mathbb{N}$ are also important for indexing iterated higher
order cores of multiple vector bundles. This section collects all the notions,
notations and constructions with partitions that are needed in the rest of the
paper.

In this paper the following notation is used. For
${\underline{n}}:=\\{1,\ldots,n\\}$ the standard $n$-cube category
$\square^{n}$ is the category with subsets $I$ of ${\underline{n}}$ as objects
and with arrows $I\to J\,\Leftrightarrow\,J\subseteq I$. More generally, an
$n$-cube category is a category that is isomorphic to the standard $n$-cube
category $\square^{n}$.

###  2.1. Partitions and cube-categories

Choose a finite subset $I\subseteq{\underline{n}}$. A partition of $I$ is a
set $\\{I_{1},\ldots,I_{l}\\}$ of pairwise disjoint and non-empty subsets
$I_{1},\ldots,I_{l}\subseteq I$ for some $l\in\mathbb{N}$, such that
$I=I_{1}\cup\ldots\cup I_{l}$. Since the order of the subsets _should not_
have any significance, i.e. $\\{I_{1},I_{2},\ldots,I_{l}\\}$ is naturally the
same partition of $I$ as $\\{I_{2},I_{1},\ldots,I_{l}\\}$, for clarity a
partition is always assumed to be _ordered_ , and if not mentioned otherwise,
in the canonical order. That is,
$0<\\#I_{1}\leq\\#I_{2}\leq\ldots\leq\\#I_{l}$ and the subsets of same
cardinality are ordered by lexicographic order. Later on, the fact that
partitions are always ordered is also crucial for defining _signs_ of
partitions. The set of (canonically ordered) partitions of $I$ is written
$\mathcal{P}(I)$ and a partition $\\{I_{1},\ldots,I_{l}\\}$ is written as a
list $(I_{1},\ldots,I_{l})$ if it needs to be emphasized that it is ordered –
in particular if it is non-canonically ordered, in which case this is always
clearly stated. However, in some situations, it is simpler if the indices of
the elements of a partition do not take into account that the partition is
ordered (see for instance Lemma 2.2 below), and then the partition is simply
written as a set.

Each partition $\rho=(I_{1},\ldots,I_{l})$ of ${\underline{n}}$ in $l$ subsets
defines an $l$-cube category $\lozenge^{\rho}$ with objects the subsets

(3) $J\subseteq{\underline{n}}\text{ such that for each }i=1,\ldots,l,\text{
either }I_{i}\cap J=\emptyset\text{ or }I_{i}\subseteq J$

and with arrows $J\to J^{\prime}\,\,\Leftrightarrow\,\,J^{\prime}\subseteq J$.
The $\\#\rho$-cube category $\lozenge^{\rho}$ is a full subcategory of
$\square^{n}$. The inclusion functor is written

$i^{\rho}\colon\lozenge^{\rho}\to\square^{n}.$

######  Remark 2.1.

Given $\rho=(I_{1},\ldots,I_{l})$, with $l\in\\{1,\ldots,n\\}$, then the
objects of $\lozenge^{\rho}$ are the unions of elements of $\rho$. The
canonical equivalence with the standard $l$-cube category $\square^{l}$ is
given by sending $J\in\operatorname{Obj}(\square^{l})$ to $\cup_{j\in
J}I_{j}\in\operatorname{Obj}(\lozenge^{\rho})$ and vice-versa. So
$\lozenge^{\rho}$ is really just $\square^{l}$, but with the “indices”
$I_{1},\ldots,I_{l}$ replacing the natural numbers $1,\ldots,l$. _Then
$\square^{n}$ is simply understood as the $n$-cube category over the elements
$1,\ldots,n$, while $\lozenge^{\rho}$ is the $l$-cube category over the
elements $I_{1},\ldots,I_{l}$._ This point of view is useful for expressing
some of the constructions later in the paper, in particular in Section 4.3,
and also for keeping track of them.

Note that if $\rho=(I_{1},\ldots,I_{l})$ is a canonically ordered partition of
$I\subseteq{\underline{n}}$, then $(\\#I_{1},\ldots,\\#I_{l})$ is a naturally
ordered integer partition of $\\#I$. The tuple $(\\#I_{1},\ldots,\\#I_{l})$ is
called the size of the partition $(I_{1},\ldots,I_{l})$, and also denoted by
$|I_{1},\ldots,I_{l}|=|\rho|$. In general, integer partitions are assumed to
be ordered and written as tuples. If not specified otherwise, they are
naturally ordered.

For simplicity, a partition of ${\underline{n}}$ in $l$ subsets is sometimes
called an $l$-partition of ${\underline{n}}$ and $l$ is called the length of
$\rho$. Let $\rho$ be such an $l$-partition of ${\underline{n}}$. Then a
coarsement of $\rho$ is a partition $\rho^{\prime}$ of ${\underline{n}}$ that
consists of (non-empty) unions of the elements of $\rho$. The length of
$\rho^{\prime}$ is thus necessarily less than or equal to the length of
$\rho$. The set of coarsements of $\rho$ is written
$\operatorname{coars}(\rho)$. Given
$(J_{1},\ldots,J_{l^{\prime}})\in\operatorname{coars}(\rho)$ and
$i\in\\{1,\ldots,l^{\prime}\\}$, then $\rho\cap J_{i}$ is defined to be the
(canonically ordered) partition of $J_{i}$ given by the elements of $\rho$ it
is a union of.

A partition $\rho^{\prime}$ of ${\underline{n}}$ is a refinement of $\rho$ if
$\rho$ is a coarsement of $\rho^{\prime}$. For the study of cores in Section
4.3, it is important to understand the different coarsements of length $l-2$
of an $l$-partition.

######  Lemma 2.2.

Let $\rho=\\{I_{1},\ldots,I_{l}\\}$ be a partition of ${\underline{n}}$.
Consider two different _coarsements_

$\rho_{ij}=\left\\{I_{i}\cup
I_{j},I_{1},\ldots,\widehat{I_{i}},\ldots,\widehat{I_{j}},\ldots,I_{l}\right\\}\text{
and }\rho_{rs}=\left\\{I_{r}\cup
I_{s},I_{1},\ldots,\widehat{I_{r}},\ldots,\widehat{I_{s}},\ldots,I_{l}\right\\}$

of length $l-1$ of $\rho$, with $i<j$ and $r<s$ in $\\{1,\ldots,l\\}$. Then

$\operatorname{Obj}\left(\lozenge^{\rho_{ij}}\right)\cap\operatorname{Obj}\left(\lozenge^{\rho_{rs}}\right)=\operatorname{Obj}\left(\lozenge^{\rho^{\prime}}\right)$

with $\rho^{\prime}$ the coarsement of length $l-2$ of $\rho$ given by

(4) $\rho^{\prime}=\left\\{\begin{array}[]{ll}\left\\{I_{i}\cup I_{j}\cup
I_{r};I_{t}\mid t\in{\underline{l}},t\neq i,j,r\right\\}&\text{ if }i=s\text{
or }j=s,\\\ \left\\{I_{i}\cup I_{j}\cup I_{s};I_{t}\mid
t\in{\underline{l}},t\neq i,j,s\right\\}&\text{ if }i=r\text{ or }j=r,\\\
\left\\{I_{i}\cup I_{j},I_{r}\cup I_{s};I_{t}\mid t\in{\underline{l}},t\neq
i,j,r,s\right\\}&\text{ if }i\neq r,s\text{ and }j\neq r,s.\end{array}\right.$

######  Definition 2.3.

In the situation of the previous lemma, the $(l-2)$-cube category
$\lozenge^{\rho^{\prime}}$ is denoted
$\lozenge^{\rho_{ij}}\cap\lozenge^{\rho_{rs}}$, and called the intersection of
the two $(l-1)$-cube categories $\lozenge^{\rho_{ij}}$ and
$\lozenge^{\rho_{rs}}$. It is again a full subcategory of $\square^{n}$.

The partition $\rho^{\prime}$ is itself sometimes written
$\rho_{ij}\sqcap\rho_{rs}$. It is the unique common $(l-2)$-coarsement of
$\rho_{ij}$ and $\rho_{rs}$.

###### Proof of Lemma 2.2.

A subset $K\subseteq{\underline{n}}$ is an object of $\lozenge^{\rho_{ij}}$
and of $\lozenge^{\rho_{rs}}$ if and only if $K\cap I_{t}=\emptyset$ or
$I_{t}\subseteq K$ for all $t\in\\{1,\ldots,l\\}\setminus\\{i,j,r,s\\}$ and

  1. (i)

$I_{i}\cup I_{j}\subseteq K$ and $I_{r}\cup I_{s}\subseteq K$ i.e. $I_{i}\cup
I_{j}\cup I_{r}\cup I_{s}\subseteq K$, or

  2. (ii)

$(I_{i}\cup I_{j})\cap K=\emptyset$ and $(I_{r}\cup I_{s})\cap K=\emptyset$
i.e. $(I_{i}\cup I_{j}\cup I_{r}\cup I_{s})\cap K=\emptyset$, or

  3. (iii)

$I_{i}\cup I_{j}\subseteq K$ and $(I_{r}\cup I_{s})\cap K=\emptyset$, or

  4. (iv)

$I_{r}\cup I_{s}\subseteq K$ and $(I_{i}\cup I_{j})\cap K=\emptyset$.

If $\\{i,j\\}\cap\\{r,s\\}\neq\emptyset$, the cases (iii) and (iv) are not
possible. The intersection $\\{i,j\\}\cap\\{r,s\\}$ has then one element
because $\rho_{ij}\neq\rho_{rs}$ and by (i) and (ii) $K$ is an object of
$\lozenge^{\rho^{\prime}}$ with one of the first two descriptions of
$\rho^{\prime}$ in (4). If $\\{i,j\\}\cap\\{r,s\\}=\emptyset$ then $K$ is an
object of $\lozenge^{\rho^{\prime}}$ with the third description of
$\rho^{\prime}$. ∎

###  2.2. Partitions and signs

For $\sigma\in S_{n}$ and $I\subseteq{\underline{n}}$, denote by
$\epsilon(\sigma,I)$ the _order of inversions_ (or simply the _sign_) of
$\sigma\arrowvert_{I}\colon I\to\sigma(I)$, where $I$ and $\sigma(I)$ are
equipped with the natural order inherited from the natural numbers. That is,

$\epsilon(\sigma,I):=(-1)^{\\#\\{(i,j)\in I\times I\mid
i<j,\sigma(i)>\sigma(j)\\}}=\prod_{\begin{subarray}{c}i,j\in I\\\
i<j\end{subarray}}\frac{\sigma(j)-\sigma(i)}{j-i}$

For $\sigma,\nu\in S_{n}$ and $I\subseteq{\underline{n}}$, the product formula

(5) $\epsilon(\sigma\nu,I)=\epsilon(\sigma,\nu(I))\cdot\epsilon(\nu,I)$

is immediate.

An ordered partition $\rho=(I_{1},\ldots,I_{l})$ of $I$ defines as follows a
reordering of the elements of $I$. If
$I_{j}=\\{i^{j}_{1},\ldots,i^{j}_{k_{j}}\\}$ for $j=1,\ldots,l$, with
$i^{j}_{1},\ldots,i^{j}_{k_{j}}\in\mathbb{N}$ naturally ordered, then

(6)
$i^{1}_{1},\ldots,i_{k_{1}}^{1},i^{2}_{1},\ldots,i^{2}_{k_{2}},\ldots,i^{l}_{1},\ldots,i^{l}_{k_{l}}$

is a reordering of the elements of $I$. For instance, the (non-canonically)
ordered partition $(\\{2\\},\\{1,4\\},\\{3\\})$ of $\\{1,2,3,4\\}$ defines the
reordering $2,1,4,3$ of the numbers $1,2,3,4$.

Given $n\in\mathbb{N}$ and a (not necessarily naturally) ordered integer
partition $(i_{1},\ldots,i_{l})$ of $n$, set the canonical partition
$\rho^{i_{1},\ldots,i_{l}}_{\rm can}$ of $\\{1,\ldots,n\\}={\underline{n}}$ to
be the unique ordered partition $\rho^{i_{1},\ldots,i_{l}}_{\rm
can}:=(K_{1},\ldots,K_{l})$ of ${\underline{n}}$ with $\\#K_{s}=i_{s}$ for
$s=1,\ldots,l$ and such that the order on ${\underline{n}}$ induced by
$\rho^{i_{1},\ldots,i_{l}}_{\rm can}$ is the natural one. For instance, for
the integer partition $(1,2,1,3,3)$ of the number $10$, the canonical
partition $\rho^{1,2,1,3,3}_{\rm can}$ of $\underline{10}$ is the partition
$(\\{1\\},\\{2,3\\},\\{4\\},\\{5,6,7\\},\\{8,9,10\\})$.

Given $\sigma\in S_{n}$ and an ordered partition $\rho=(I_{1},\ldots,I_{l})$
of a subset $I\subseteq{\underline{n}}$, write $\sigma(\rho)$ for the ordered
partition $(\sigma(I_{1}),\ldots,\sigma(I_{l}))$.

Given a subset $I\subseteq{\underline{n}}$ and a partition
$\rho=(I_{1},\ldots,I_{l})$ of $I$, set $\\#I_{j}=:i_{j}$ and
$i=\\#I=i_{1}+\ldots+i_{l}$. Consider $\rho_{\rm
can}^{i_{1},\ldots,i_{l}}=(K_{1},\ldots,K_{l})$ and choose a permutation
$\sigma\in S_{n}$ with

$\sigma(K_{j})=I_{j}\quad\text{ for }\quad j=1,\ldots,l,$

i.e. with $\rho=\sigma\left(\rho^{i_{1},\ldots,i_{l}}_{\rm can}\right)$. Set
then

$\operatorname{sgn}(\rho):=\frac{\prod_{j=1}^{l}\epsilon(\sigma,K_{j})}{\epsilon(\sigma,\\{1,\ldots,i\\})},$

the sign of the partition $\rho$.

######  Lemma 2.4.

In the situation above, the sign $\operatorname{sgn}(\rho)$ does not depend on
the choice of $\sigma\in S_{n}$ with $\sigma(K_{j})=I_{j}$ for $j=1,\ldots,l$.
Hence $\operatorname{sgn}(\rho)$ equals

$\epsilon(\sigma,\\{1,\ldots,i\\})$

for a permutation $\sigma\in S_{n}$ with
$\rho=\sigma\left(\rho^{i_{1},\ldots,i_{l}}_{\rm can}\right)$ and preserving
in addition the order of the sets $K_{1},\ldots,K_{l}$.

As a consequence, $\operatorname{sgn}(\rho)$ is the sign of the unique
permutation $I\to I$ sending $I$ in its canonical order to $I$ in its order
induced by $\rho$ as in (6).

###### Proof.

Assume that $\nu\in S_{n}$ is a second permutation with $\nu(K_{j})=I_{j}$ for
$j=1,\ldots,l$. Then $\nu=\lambda\circ\sigma$ with $\lambda\in S_{n}$
satisfying $\lambda(I_{j})=I_{j}$ for $j=1,\ldots,l$.

First assume that $\lambda\arrowvert_{I}$ is a transposition, say
$\lambda=(s,t)$ with $s\neq t$ in $\\{1,\ldots,n\\}$. Since $\lambda$
preserves each of the sets $I_{j}$, $j=1,\ldots,l$, there exists an
$r\in\\{1,\ldots,l\\}$ such that
$\lambda\arrowvert_{I_{j}}=\operatorname{id}_{I_{j}}$ for
$j\in\\{1,\ldots,l\\}\setminus\\{r\\}$ and $\lambda\arrowvert_{I_{r}}$ is a
transposition. Then

(7)
$\prod_{j=1}^{l}\epsilon(\lambda,I_{j})=\epsilon(\lambda,I_{r})=-1=\epsilon(\lambda,I).$

The general case follows with (5): a permutation $\lambda$ preserving each of
the sets $I_{1},\ldots,I_{l}$ can be written as a product of transpositions
$\lambda_{1},\ldots,\lambda_{s}$ preserving these sets. Then

(8)
$\begin{split}\prod_{j=1}^{l}\epsilon(\lambda,I_{j})&=\prod_{j=1}^{l}\epsilon(\lambda_{s}\circ\ldots\circ\lambda_{1},I_{j})\overset{\eqref{prod1}}{=}\prod_{j=1}^{l}\prod_{k=1}^{s}\epsilon(\lambda_{k},(\lambda_{k-1}\circ\ldots\circ\lambda_{1})(I_{j}))\\\
&=\prod_{k=1}^{s}\left(\prod_{j=1}^{l}\epsilon(\lambda_{k},(\lambda_{k-1}\circ\ldots\circ\lambda_{1})(I_{j}))\right)\\\
&\overset{\eqref{signs_transposition}}{=}\prod_{k=1}^{s}\epsilon(\lambda_{k},(\lambda_{k-1}\circ\ldots\circ\lambda_{1})(I))\overset{\eqref{prod1}}{=}\epsilon(\lambda,I).\end{split}$

Now the claim follows with a similar computation:

$\frac{\prod_{j=1}^{l}\epsilon(\nu,K_{j})}{\epsilon(\nu,\\{1,\ldots,i\\})}=\frac{\prod_{j=1}^{l}\epsilon(\lambda\sigma,K_{j})}{\epsilon(\lambda\sigma,\\{1,\ldots,i\\})}\overset{\eqref{prod1}}{=}\frac{\prod_{j=1}^{l}\epsilon(\lambda,I_{j})}{\epsilon(\lambda,I)}\frac{\prod_{j=1}^{l}\epsilon(\sigma,K_{j})}{\epsilon(\sigma,\\{1,\ldots,i\\})}\overset{\eqref{signs_general_preserving}}{=}\frac{\prod_{j=1}^{l}\epsilon(\sigma,K_{j})}{\epsilon(\sigma,\\{1,\ldots,i\\})}.$

To see the last claim of the lemma, note that there is a unique bijection
$f\colon I\to\\{1,\ldots,i\\}$ preserving the natural order of both sets. The
sign of this map is hence $1$ and its composition with $\sigma$ gives then the
bijection $\sigma\circ f\colon I\to I$, which sends $I$ in its canonical order
to $I$ in the order induced by $\rho$ as in (6). ∎

Finally, note that given $i\in{\underline{n}}$ and an ordered integer
partition $(i_{1},\ldots,i_{l})$ of $i$, the canonical partition $\rho_{\rm
can}^{i_{1},\ldots,i_{l}}$ of $\underline{i}$ has sign $1$ by construction.

###  2.3. Integer partitions of natural numbers

For $n\in\mathbb{N}$ let $\mathcal{P}(n)$ be the set of naturally ordered
integer partitions of the natural numbers111The reader should be aware that
the notations $\mathcal{P}(n)$ and $\mathcal{P}({\underline{n}})$ are not
consistent: $\mathcal{P}(n)$ is the set of ordered integer partitions of the
integers between 1 and $n$, while $\mathcal{P}({\underline{n}})$ is the set of
canonically ordered partitions of the set ${\underline{n}}$. These conventions
differ for simplicity of some of the formulas. $1\leq j\leq n$:

$\mathcal{P}(n):=\left\\{(i_{1},\ldots,i_{l})\in\mathbb{N}^{l}\left|\,\,l\in\mathbb{N},1\leq
i_{1}\leq\ldots\leq i_{l}\text{ and }1\leq i_{1}+\ldots+i_{l}\leq
n\right.\right\\}.$

Given a partition $p=(i_{1},\ldots,i_{l})\in\mathcal{P}(n)$, the sum $\Sigma
p$ of $p$ is simply

$\sum p:=\sum_{j=1}^{l}i_{j}.$

Given $p\in\mathcal{P}(n)$, the number $\sharp p$ is the length of the list
$p$, i.e. the number of its elements.

As explained in the preceding section, there is a bijection between ordered
integer partitions of a number $i\in\mathbb{N}$ and ordered partitions of
$\underline{i}$ inducing the canonical order on the elements of
$\underline{i}$. A integer partition $(i_{1},\ldots,i_{l})$ of $k$ corresponds
via this bijection to the unique partition $\rho^{i_{1},\ldots,i_{l}}_{\rm
can}$.

##  3\. The category of $\mathbb{N}$-manifolds

This section begins by recalling the definitions of $\mathbb{N}$-manifolds and
their morphisms, and by discussing in detail the split case. The reader is
referred to [Meh06, BP13] for more details.

###  3.1. Definitions

An $\mathbb{N}$-manifold $\mathcal{M}$ of degree $n\in\mathbb{N}$ and
dimension $(m;r_{1},\ldots,r_{n})$ is a smooth manifold $M$ of dimension $m$
together with a sheaf $C^{\infty}(\mathcal{M})$ of $\mathbb{N}$-graded, graded
commutative, associative, unital $C^{\infty}(M)$-algebras, that is locally
freely generated by $r_{1}+\ldots+r_{n}$ elements
$\xi_{1}^{1},\ldots,\xi_{1}^{r_{1}}$,
$\xi_{2}^{1},\ldots,\xi_{2}^{r_{2}},\ldots$,
$\xi_{n}^{1},\ldots,\xi_{n}^{r_{n}}$ with $\xi_{i}^{j}$ of degree $i$ for
$i\in\\{1,\ldots,n\\}$ and $j\in\\{1,\ldots,r_{i}\\}$.

A morphism of $\mathbb{N}$-manifolds $\mu\colon\mathcal{M}\to\mathcal{N}$ over
a smooth map $\mu_{0}\colon M\to N$ of the underlying smooth manifolds is a
morphism $\mu^{\star}\colon C^{\infty}(\mathcal{N})\to
C^{\infty}(\mathcal{M})$ of sheaves of graded algebras over $\mu_{0}^{*}\colon
C^{\infty}(N)\to C^{\infty}(M)$.

Note that the degree $0$ elements of $C^{\infty}(\mathcal{M})$ are precisely
the smooth functions on $M$. In the following an $\mathbb{N}$-manifold of
degree $n\in\mathbb{N}$ is called an $[n]$-manifold. $|\xi|$ is the degree of
a homogeneous element $\xi\in C^{\infty}(\mathcal{M})$, i.e. an element which
can be written as a sum of functions of the same degree, and
$C^{\infty}(\mathcal{M})^{i}$ is the space of elements of degree $i$ in
$C^{\infty}(\mathcal{M})$. Note that a $[1]$-manifold over a smooth manifold
$M$ is equivalent to a locally free and finitely generated sheaf of
$C^{\infty}(M)$-modules, hence to a vector bundle over $M$. The category of
$[n]$-manifolds is written $\mathsf{[n]Man}$, and the category of
$\mathbb{N}$-manifolds is written $\mathbb{N}\mathsf{Man}$.

Given an $[n]$-manifold $\mathcal{M}$ over a smooth manifold $M$, and an open
subset $U\subseteq M$, then $\mathcal{M}\arrowvert_{U}$ is the $n$-manifold
over $U$ defined by
$C^{\infty}(\mathcal{M}\arrowvert_{U}):=C^{\infty}(\mathcal{M})\arrowvert_{U}$.
For simplicity, it is called the restriction of $\mathcal{M}$ to $U$.

###  3.2. The category of split $\mathbb{N}$-manifolds

Let $E$ be a smooth vector bundle of rank $r$ over a smooth manifold $M$ of
dimension $m$. Assign the degree $n$ to the fiber coordinates of $E$. This
defines $E[-n]$, an $[n]$-manifold of dimension
$(m;r_{1}=0,\ldots,r_{n-1}=0,r_{n}=r)$ with generators
$C^{\infty}(E[-n])^{n}=\Gamma(E^{*})$ and
$C^{\infty}(E[-n])^{0}=C^{\infty}(M)$.

Now let $E_{1},E_{2},\ldots,E_{n}$ be smooth vector bundles of finite ranks
$r_{1},\ldots,r_{n}$ over $M$ and assign the degree $-i$ to the fiber
coordinates of $E_{i}$, for each $i=1,\ldots,n$. The direct sum
$E=E_{1}\oplus\ldots\oplus E_{n}$ is a graded vector bundle with grading
concentrated in degrees $-1,\ldots,-n$. The $[n]$-manifold
$E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]$ has local basis sections of
${E_{i}^{*}}$ as local generators of degree $i$, for $i=1,\ldots,n$, and so
dimension $(d;r_{1},\ldots,r_{n})$. The $[n]$-manifold
$E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]$ is called a split $[n]$-manifold.

For instance, in the case $n=3$, choose three vector bundles $E_{1}$, $E_{2}$
and $E_{3}$ of ranks $r_{1}$, $r_{2}$ and $r_{3}$ over a smooth manifold $M$.
Then the $[3]$-manifold $\mathcal{M}=E_{1}[-1]\oplus E_{2}[-2]\oplus
E_{3}[-3]$ is defined by $C^{\infty}(\mathcal{M})^{0}=C^{\infty}(M)$,
$C^{\infty}(\mathcal{M})^{1}=\Gamma(E_{1}^{*})$,
$C^{\infty}(\mathcal{M})^{2}=\Gamma(E_{2}^{*}\oplus\wedge^{2}E_{1}^{*})$,
$C^{\infty}(\mathcal{M})^{3}=\Gamma(E_{3}^{*}\oplus E_{1}^{*}\otimes
E_{2}^{*}\oplus\wedge^{3}E_{1}^{*})$, and

$C^{\infty}(\mathcal{M})^{4}=\Gamma(E_{4}^{*}\oplus E_{1}^{*}\otimes
E_{3}^{*}\oplus S^{2}E_{2}^{*}\oplus\wedge^{2}E_{1}^{*}\otimes
E_{2}^{*}\oplus\wedge^{4}E_{1}^{*}),$

etc. Set $E_{j}^{*}\odot E_{k}^{*}$ to mean

  * •

$\wedge$ if $j=k$ is an odd number,

  * •

the symmetric product $\cdot$ if $j=k$ is an even number,

  * •

$\otimes$ if $j\neq k$.

Then given an $[n]$-manifold $\mathcal{M}=E_{1}[-1]\oplus\ldots\oplus
E_{n}[-n]$, its functions of degree $k$ are elements of

$C^{\infty}(\mathcal{M})^{k}=\bigoplus_{\begin{subarray}{c}1\leq
i_{1}\leq\ldots\leq i_{l}\leq n,\\\
i_{1}+\ldots+i_{l}=k\end{subarray}}\Gamma\left(E^{*}_{i_{1}}\odot\ldots\odot
E^{*}_{i_{l}}\right)$

for all $k\in\mathbb{N}$.

The graded product $\odot$ on

$C^{\infty}(\mathcal{M})=\bigoplus_{k\in\mathbb{N}}\bigoplus_{\begin{subarray}{c}1\leq
i_{1}\leq\ldots\leq i_{l}\leq n,\\\
i_{1}+\ldots+i_{l}=k\end{subarray}}\Gamma\left(E^{*}_{i_{1}}\odot\ldots\odot
E^{*}_{i_{l}}\right)$

is given by the following lemma.

######  Lemma 3.1.

Let $\mathcal{M}=E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]$ be a split
$[n]$-manifold over a smooth manifold $M$ and consider
$\xi_{1},\ldots,\xi_{l}\in C^{\infty}(\mathcal{M})$ such that

$\xi_{j}\in\Gamma\left(E^{*}_{i^{j}_{1}}\odot\ldots\odot
E^{*}_{i^{j}_{l_{j}}}\right)$

with $1\leq i_{1}^{j}\leq\ldots\leq i^{j}_{l_{j}}\leq n$, for $j=1,\ldots,l$.
That is, $\xi_{j}$ has degree $d_{j}:=i^{j}_{1}+\ldots+i^{j}_{l_{j}}$. Set
$p_{j}=(i_{1}^{j},\ldots,i_{l_{j}}^{j})$. It is an ordered integer partition
of $d_{j}$ for $j=1,\ldots,l$.

The list $p$ obtained as the union of $p_{1},\ldots,p_{l}$, and canonically
reordered, is then an integer partition with $s:=l_{1}+\ldots+l_{l}$ elements
of its sum $d=d_{1}+\ldots+d_{l}$. Consider the canonical partition $\rho^{\rm
can}_{p}=(K_{1},\ldots,K_{s})$ of $\\{1,\ldots,d\\}$ associated to it. The
graded product

$\xi_{1}\odot\ldots\odot\xi_{l}$

a section of $E^{*}_{|K_{1}|}\odot\ldots\odot E^{*}_{|K_{s}|}$, is defined on
$e_{K_{1}}\in E_{|K_{1}|},\ldots,e_{K_{s}}\in E_{|K_{s}|}$ by

(9) $\begin{split}\sum_{\begin{subarray}{c}\rho^{p}_{\rm
can}=\rho_{1}\cup\ldots\cup\rho_{l}\\\ \text{as sets and}\\\
\rho_{1},\ldots,\rho_{l}\text{canonically ordered with}\\\
|\rho_{j}|=p_{j}\text{ for
}j=1,\ldots,l\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1})\cdot\ldots\cdot\xi_{l}(e_{K}\mid K\in\rho_{l}).\end{split}$

Here, $(\rho_{1},\ldots,\rho_{l})$ is the partition $\rho_{\rm can}^{p}$ of
$\underline{d}$, but reordered in the order of222For instance, if
$p_{1}=(1,3)$ and $p_{2}=(1,2)$, then $d=7$, $p=(1,1,2,3)$ and $\rho_{\rm
can}^{p}=(\\{1\\},\\{2\\},\\{3,4\\},\\{5,6,7\\})$. Then if
$\rho_{1}=(\\{2\\},\\{3,4\\})$ and $\rho_{2}=(\\{1\\},\\{5,6,7\\})$, then
$(\rho_{1},\rho_{2})=(\\{2\\},\\{3,4\\},\\{1\\},\\{5,6,7\\})$.
$\rho_{1},\ldots,\rho_{l}$.

Note that here, the notation $\xi_{j}(e_{K}\mid K\in\rho_{j})$ is short for
the following. If $\rho_{j}=(K_{i_{1}},\ldots,K_{i_{l_{j}}})$, then

$\xi_{j}(e_{K}\mid
K\in\rho_{j})=\xi_{j}(e_{K_{i_{1}}},\ldots,e_{K_{i_{l_{j}}}}).$

######  Example 3.2.

Before considering the proof, the reader can convince themself with a few
examples that (9) works as it should.

  1. (1)

Take $\xi_{1}\in\Gamma(E_{1}^{*})$,
$\eta_{1}\odot\eta_{2}\in\Gamma(E_{1}^{*}\odot E_{2}^{*})$ and
$\tau_{2}\odot\tau_{3}\in\Gamma(E_{2}^{*}\odot E_{3}^{*})$. Then $p_{1}=(1)$,
$p_{2}=(1,2)$ and $p_{3}=(2,3)$, and so $p=(1,1,2,2,3)$. Here

$\rho_{\rm can}^{p}=(\\{1\\},\\{2\\},\\{3,4\\},\\{5,6\\},\\{7,8,9\\})$

and so $(\rho_{1},\rho_{2},\rho_{3})$ range over

     * •

$((\\{1\\}),(\\{2\\},\\{3,4\\}),(\\{5,6\\},\\{7,8,9\\}))$ with sign $1$,

     * •

$((\\{2\\}),(\\{1\\},\\{3,4\\}),(\\{5,6\\},\\{7,8,9\\}))$ with sign $-1$,

     * •

$((\\{1\\}),(\\{2\\},\\{5,6\\}),(\\{3,4\\},\\{7,8,9\\}))$ with sign $1$,

     * •

$((\\{2\\}),(\\{1\\},\\{5,6\\}),(\\{3,4\\},\\{7,8,9\\}))$ with sign $-1$.

Therefore a quick computation shows that
$(\xi_{1}\odot(\eta_{1}\odot\eta_{2})\odot(\tau_{2}\odot\tau_{3}))(e_{1},e_{2},e_{34},e_{56},e_{789})$
equals

$\left(\xi_{1}(e_{1})\eta_{1}(e_{2})-\xi_{1}(e_{2})\eta_{1}(e_{1})\right)\cdot\left(\eta_{2}(e_{34})\tau_{2}(e_{56})+\eta_{2}(e_{56})\tau_{2}(e_{34})\right)\cdot\tau_{3}(e_{789}).$

  2. (2)

Take $\xi\in\Gamma(E_{3}^{*})$, $\eta\in\Gamma(E_{1})$ and
$\tau\in\Gamma(E_{2}^{*})$. Then $p_{1}=(3)$, $p_{2}=(1)$ and $p_{3}=(2)$ and
so $p=(1,2,3)$ and $\rho_{\rm can}^{p}=(\\{1\\},\\{2,3\\},\\{4,5,6\\})$. The
only possible list $(\rho_{1},\rho_{2},\rho_{3})$ is here

$(\\{4,5,6\\},\\{1\\},\\{2,3\\}),$

which has sign $-1$. Hence

$(\xi\odot\eta\odot\tau)(e_{1},e_{23},e_{456})=-\xi(e_{456})\cdot\eta(e_{1})\cdot\tau(e_{23}),$

which shows

$\xi\odot\eta\odot\tau=-\eta\otimes\tau\otimes\xi.$

  3. (3)

The reader is invited to check that $\xi\odot\eta$ for
$\xi\in\Gamma(\wedge^{k}E_{1}^{*})$ and $\eta\in\Gamma(\wedge^{l}E_{1}^{*})$
equals

$\xi\odot\eta=\frac{(k+l)!}{k!l!}\operatorname{Alt}(\xi\otimes\eta)=\xi\wedge\eta.$

This example illustrates nicely the compatibility of the graded tensor in
Lemma 3.1 with the usual wedge product of sections of
$\Gamma(\wedge^{\bullet}E_{1}^{*})$.

###### Proof of Lemma 3.1.

First, the graded symmetry of this ‘product’ in the entries
$\xi_{1},\ldots,\xi_{l}$ is checked as follows: It suffices to show that for
$i=1,\ldots,l-1$, exchanging $\xi_{i}$ with $\xi_{i+1}$ in (9) is the same as
a multiplication by the factor $(-1)^{d_{i}d_{i+1}}$. This exchange amounts to
exchanging $p_{i}$ with $p_{i+1}$ in the list of partitions, and so exchanging
the roles of $\rho_{i}$ and $\rho_{i+1}$ in each summand of (9), which only
amounts to replacing in each summand the factor
$\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})$ by
$\operatorname{sgn}(\rho_{1},\ldots,\rho_{i+1},\rho_{i},\ldots,\rho_{l})$.

But given a permutation $\sigma\in S_{d}$ such that
$\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})=(-1)^{\sigma}$, the composition
of $\sigma$ with $d_{i}\cdot d_{i+1}$ transpositions on both sides333Note that
the canonical partitions $\rho_{\rm can}^{(p_{1},\ldots,p_{l})}$ and
$\rho_{\rm can}^{(p_{1},\ldots,p_{i+1},p_{i},\ldots,p_{l})}$ are not equal in
general. gives a permutation $\sigma^{\prime}\in S_{d}$ such that
$\operatorname{sgn}(\rho_{1},\ldots,\rho_{i+1},\rho_{i},\ldots,\rho_{l})=(-1)^{\sigma^{\prime}}$.
Hence

$\operatorname{sgn}(\rho_{1},\ldots,\rho_{i+1},\rho_{i},\ldots,\rho_{l})=(-1)^{d_{i}d_{i+1}}\operatorname{sgn}(\rho_{1},\ldots,\rho_{l}).$

Then check that (9) is indeed an element of
$\Gamma\left(E^{*}_{|K_{1}|}\odot\ldots\odot E^{*}_{|K_{s}|}\right)$. The
following shows that if $|K_{i}|=|K_{j}|=a$ for some $i<j\in\\{1,\ldots,s\\}$,
then exchanging $e_{K_{i}}$ with $e_{K_{j}}$ in (9), amounts to multiplying it
by $(-1)^{a}=(-1)^{a\cdot a}$.

Consider a list of canonically ordered partitions $\rho_{1},\dots,\rho_{l}$
the union of which is $\rho_{\rm can}^{p}$ and such that $\rho_{j}$ has size
$p_{j}$ for $j=1,\ldots,l$. Write
$(\rho_{1},\ldots,\rho_{l})=(I_{1},\ldots,I_{s})$. Exchanging $K_{i}$ and
$K_{j}$ in this list defines a new list
$(\rho_{1},\ldots,\rho_{l})^{ij}=(\rho_{1}^{*},\ldots,\rho_{l}^{*})=(I_{1}^{*},\ldots,I_{d}^{*})$
with size $p$. First assume that the partitions
$\rho_{1}^{*},\ldots,\rho_{l}^{*}$ are all still (canonically) ordered. By the
last claim of Lemma 2.4,
$\operatorname{sgn}(\rho_{1},\ldots,\rho_{d})^{ij}=(-1)^{a}\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})$,
since the set $\underline{d}$ in its order induced by
$(\rho_{1},\ldots,\rho_{l})$ is sent to $\underline{d}$ in its order induced
by $(\rho_{1}^{*},\ldots,\rho_{l}^{*})$ by $a$ permutations exchanging the
elements of $K_{i}$ with the elements of $K_{j}$. As a consequence, exchanging
$e_{K_{i}}$ with $e_{K_{j}}$ in the summand of (9) defined by
$(\rho_{1},\ldots,\rho_{l})$ yields

$\begin{split}&\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1}^{*})\cdot\ldots\cdot\xi_{l}(e_{K}\mid K\in\rho_{l}^{*})\\\
&=(-1)^{a}\operatorname{sgn}(\rho_{1}^{*},\ldots,\rho_{l}^{*})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1}^{*})\cdot\ldots\cdot\xi_{l}(e_{K}\mid
K\in\rho_{l}^{*}),\end{split}$

which is $(-1)^{a}$ times the term of (9) defined by
$(\rho_{1}^{*},\ldots,\rho_{l}^{*})$.

Assume now that $K_{i}\in\rho_{r}$ and $K_{j}\in\rho_{t}$ for some
$r,t\in\\{1,\ldots,l\\}$, and that exchanging $K_{i}$ with $K_{j}$ leads to
one or both of $\rho_{r}^{*}$ and $\rho_{t}^{*}$ not being canonically ordered
anymore. If $r=t$, then exchanging $e_{K_{i}}$ and $e_{K_{j}}$ in the summand
of (9) defined by $(\rho_{1},\ldots,\rho_{l})$ multiplies the same term by
$(-1)^{a}$ since $\xi_{r}$ is graded-symmetric.

Assume next that $r\neq t$ and replacing $K_{i}$ by $K_{j}$ in $\rho_{r}$
leads to a partition which is not ordered anymore. Without loss of generality,
$K_{i}$ needs to be exchanged with one element $K_{h}$ in this partition for
it becoming canonically ordered again. The obtained ordered partition is then
denoted by $\rho_{r}^{*}$ as above. Assume without loss of generality that
after this step, $\rho_{1}^{*},\ldots,\rho_{l}^{*}$ are all ordered. By Lemma
2.4,
$\operatorname{sgn}(\rho_{1}^{*},\ldots,\rho_{d}^{*})=(-1)^{2a}\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})=\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})$.
Then exchanging $e_{K_{i}}$ with $e_{K_{j}}$ in the summand of (9) defined by
$(\rho_{1},\ldots,\rho_{l})$ yields

$\begin{split}&\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1}^{*})\cdot\ldots\cdot(-1)^{a}\cdot\xi_{r}(e_{K}\mid
K\in\rho_{r}^{*})\cdot\ldots\cdot\xi_{l}(e_{K}\mid K\in\rho_{l}^{*})\\\
&=(-1)^{a}\cdot\operatorname{sgn}(\rho_{1}^{*},\ldots,\rho_{l}^{*})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1}^{*})\cdot\ldots\cdot\xi_{r}(e_{K}\mid
K\in\rho_{r}^{*})\cdot\ldots\cdot\xi_{l}(e_{K}\mid
K\in\rho_{l}^{*}).\end{split}$

The remaining cases are treated similarly.

The above shows that exchanging $e_{K_{i}}$ with $e_{K_{j}}$ in (9) multiplies
all summands of (9) by $(-1)^{a}=(-1)^{a\cdot a}$. As a consequence

$\begin{split}&(\xi_{1}\odot\ldots\odot\xi_{l})(e_{K_{1}},\ldots,e_{K_{j}},\ldots,e_{K_{i}},\ldots,e_{K_{d}})=(-1)^{a}\cdot(\xi_{1}\odot\ldots\odot\xi_{l})(e_{K_{1}},\ldots,e_{K_{d}}).\end{split}$

That is, $\xi_{1}\odot\ldots\odot\xi_{l}$ is graded symmetric.

Then show that the graded symmetric product $\odot$ coincides with the graded-
symmetric product on generators. Take $\xi_{j}\in\Gamma(E^{*}_{i_{j}})$ for
$j=1,\ldots,l$. Set for $i=1,\ldots,n$

$M_{i}=\\{i_{j}\mid j\in\\{1,\ldots,l\\}\text{ and }i_{j}=i\\}.$

Set $\mathcal{S}_{i}$ to be the set of permutations of $M_{i}$ for
$i=1,\ldots,n$.

Assume without loss of generality (see the first step of this proof) that
$1\leq i_{1}\leq\ldots\leq i_{l}\leq n$. Then $p=(i_{1},\ldots,i_{l})$ and (9)
reads as follows on $e_{K_{1}},\ldots,e_{K_{l}}$, with $\rho_{\rm
can}^{p}=(K_{1},\ldots,K_{l})$.

$\begin{split}(\xi_{1}\odot\ldots\odot\xi_{l})(e_{K_{1}},\ldots,e_{K_{l}})&=\sum_{\begin{subarray}{c}\rho_{\rm
can}^{(i_{1},\ldots,i_{l})}=(I_{1},\ldots,I_{l})\text{ as sets}\\\
\\#I_{j}=i_{j}\text{ for
}j=1,\ldots,l\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}(I_{1},\ldots,I_{l})\cdot\xi_{1}(e_{I_{1}})\cdot\ldots\cdot\xi_{l}(e_{I_{l}}).\end{split}$

A choice of a (non-canonically) ordered partition $(I_{1},\ldots,I_{l})$ of
$\underline{d}$ such that $\rho_{\rm
can}^{(i_{1},\ldots,i_{l})}=(I_{1},\ldots,I_{l})$ as sets and $\\#I_{j}=i_{j}$
for $j=1,\ldots,l$ amounts to a choice of order of each of the tuples

$\left(K_{j}\,\mid\,j\in M_{i}\right),$

hence of a permutation $\sigma_{i}\in\mathcal{S}_{i}$ for $i=1,\ldots,n$. It
is then easy to see that

$\begin{split}&\sum_{\begin{subarray}{c}\rho_{\rm
can}^{(i_{1},\ldots,i_{l})}=(I_{1},\ldots,I_{l})\text{ as sets}\\\
|I_{j}|=i_{j}\text{ for
}j=1,\ldots,l\end{subarray}}\operatorname{sgn}(I_{1},\ldots,I_{l})\cdot\xi_{1}(e_{I_{1}})\cdot\ldots\cdot\xi_{l}(e_{I_{l}})\\\
&=\prod_{i=1}^{n}\left(\sum_{\sigma\in\mathcal{S}_{i}}((-1)^{\sigma})^{i}\cdot\prod_{j=1+\sum_{r=1}^{i-1}N_{r}}^{\sum_{r=1}^{i}N_{r}}\xi_{j}(e_{K_{\sigma(j)}})\right).\end{split}$

That is, $\xi_{1}\odot\ldots\odot\xi_{l}$ is the block-wise graded skew-
symmetrisation of

$\xi_{1}\otimes\ldots\otimes\xi_{l}=\bigotimes_{i=1}^{n}\underset{\in\Gamma\left((E_{i}^{*})^{\otimes
N_{i}}\right)}{\underbrace{\left(\bigotimes_{j=1+\sum_{r=1}^{i-1}N_{r}}^{\sum_{r=1}^{i}N_{r}}\xi_{j}\right)}}.$

Finally the associativity (9) is proved. Consider without loss of generality
the case $l=3$, and show as follows that

$(\xi_{1}\odot\xi_{2})\odot\xi_{3}=\xi_{1}\odot\xi_{2}\odot\xi_{3}=\xi_{1}\odot(\xi_{2}\odot\xi_{3}).$

By the graded-symmetry of $\odot$, it is enough to prove the first equality on
the left-hand side. Let $q$ be the partition obtained as the reordered union
of $p_{1}$ and $p_{2}$. Then $p$ is the reordered union of $q$ and $p_{3}$.
With the same notation as above, the form $(\xi_{1}\odot\xi_{2})\odot\xi_{3}$
applied to $e_{K_{1}},\ldots,e_{K_{s}}$ reads

$\begin{split}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\rho^{p}_{\rm
can}=\rho\cup\rho_{3}\\\ \text{as sets and}\\\ \rho,\rho_{3}\text{canonically
ordered with}\\\ |\rho|=q\text{ and
}|\rho_{3}|=p_{3}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}(\rho,\rho_{3})\cdot(\xi_{1}\odot\xi_{2})(e_{K}\mid
K\in\rho)\cdot\xi_{3}(e_{K}\mid K\in\rho_{3})\\\
=&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\rho^{p}_{\rm
can}=\rho\cup\rho_{3}\\\ \text{as sets and}\\\ \rho,\rho_{3}\text{canonically
ordered with}\\\ |\rho|=q\text{ and
}|\rho_{3}|=p_{3}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}(\rho,\rho_{3})\cdot\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\rho=\rho_{1}\cup\rho_{2}\\\
\text{as sets and}\\\ \rho_{1},\rho_{2}\text{canonically ordered with}\\\
|\rho_{1}|=p_{1}\text{ and
}|\rho_{2}|=p_{2}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}^{\rho}(\rho_{1},\rho_{2})\cdot\xi_{1}(e_{K}\mid
K\in\rho_{1})\cdot\xi_{2}(e_{K}\mid K\in\rho_{2})\cdot\xi_{3}(e_{K}\mid
K\in\rho_{3}),\end{split}$

where $\operatorname{sgn}^{\rho}(\rho_{1},\rho_{2})$ is the sign of the unique
permutation of $I:=\cup\rho$ sending $I$ in its order induced by $\rho$ to $I$
in its order induced by $(\rho_{1},\rho_{2})$, see Lemma 2.4. This is
$(\xi_{1}\odot\xi_{2}\odot\xi_{3})(e_{K_{1}},\ldots,e_{K_{s}})$ if

(10)
$\operatorname{sgn}(\rho,\rho_{3})\cdot\operatorname{sgn}^{\rho}(\rho_{1},\rho_{2})=\operatorname{sgn}(\rho_{1},\rho_{2},\rho_{3})$

for ordered partitions $\rho,\rho_{1},\rho_{2},\rho_{3}$ such that
$\rho^{p}_{\rm can}=\rho\cup\rho_{3}$ and $\rho=\rho_{1}\cup\rho_{2}$ as sets,
and $|\rho|=q$, $|\rho_{3}|=p_{3}$, $|\rho_{1}|=p_{1}$, $|\rho_{2}|=p_{2}$.

In order to show (10), use the last claim of Lemma 2.4. Let $I$ be the union
of the elements of $\rho$, i.e. let $\rho$ be an ordered partition of
$I\subseteq\underline{d}$, and write the elements of $I$ as
$i_{1},\ldots,i_{d_{1}+d_{2}}$, in the order defined by $\rho$. The elements
of $I_{j}:=\cup\rho_{j}$ are $i^{j}_{1},\ldots,i^{j}_{d_{j}}$ in the order
defined by $\rho_{j}$, for $j=1,2,3$. Note that $I=I_{1}\cup I_{2}$. Then
$\operatorname{sgn}(\rho,\rho_{3})$ is the sign of the unique permutation
$\sigma$ of $\underline{d}$ sending $\underline{d}$ in its canonical order to
$\underline{d}$ in the order

$i_{1},\ldots,i_{d_{1}+d_{2}},i^{3}_{1},\ldots,i^{3}_{d_{3}}.$

The number $\operatorname{sgn}^{\rho}(\rho_{1},\rho_{2})$ is the sign of the
unique permutation $\tau$ of $I_{1}\cup I_{2}=I$ sending $I$ in the order
$i_{1},\ldots,i_{d_{1}+d_{2}}$ to $I$ in the order

$i_{1}^{1},\ldots,i_{d_{1}}^{1},i^{2}_{1},\ldots,i^{2}_{d_{2}},$

and the sign of $(\rho_{1},\rho_{2},\rho_{3})$ is the sign of the unique
permutation $\eta$ of $\underline{d}$ sending $\underline{d}$ in its canonical
order to $\underline{d}$ in the order

$i_{1}^{1},\ldots,i^{1}_{d_{1}},i^{2}_{1},\ldots,i^{2}_{d_{2}},i^{3}_{1},\ldots,i^{3}_{d_{3}}.$

Extend $\tau$ to a permutation of $\underline{d}$ by setting
$\tau\arrowvert_{I_{3}}=\operatorname{id}_{I_{3}}$, which does not change its
sign. Then $\eta$ equals $\tau\circ\sigma$, and so
$\operatorname{sgn}(\rho_{1},\rho_{2},\rho_{3})=(-1)^{\eta}=(-1)^{\tau}(-1)^{\sigma}=\operatorname{sgn}^{\rho}(\rho_{1},\rho_{2})\cdot\operatorname{sgn}(\rho,\rho_{3})$.
∎

In this paper the category of split $[n]$-manifolds is written
$\mathsf{s[n]Man}$. A morphism

$\mu\colon E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]\to
F_{1}[-1]\oplus\ldots\oplus F_{m}[-m]$

of split $\mathbb{N}$-manifolds over the bases $M$ and $N$, respectively,
consists of a smooth map $\mu_{0}\colon M\to N$, and for each
$p=(i_{1},\ldots,i_{l})\in\mathcal{P}(m)$ a morphism

$\mu_{p}\colon E_{i_{1}}\odot E_{i_{2}}\odot\ldots\odot E_{i_{l}}\to
F_{i_{1}+\ldots+i_{l}}$

of vector bundles over $\mu_{0}$. The map $\mu^{\star}$ sends a degree $k$
generator $\xi\in\Gamma(F_{k}^{*})$ to

$\sum_{\begin{subarray}{c}p=(i_{1},\ldots,i_{l})\in\mathcal{P}(m)\\\ \sum
p=k\end{subarray}}\quad\underset{\in\Gamma(E_{i_{1}}^{*}\odot
E_{i_{2}}^{*}\odot\ldots\odot
E^{*}_{i_{l}})}{\underbrace{{\mu_{(i_{1},\ldots,i_{l})}}^{\star}(\xi)}}\,\in
C^{\infty}(E_{1}[-1]\oplus\ldots\oplus E_{n}[-n])^{k}.$

The morphism $\mu$ is therefore written here

$\mu=\left(\mu_{p}\right)_{p\in\mathcal{P}(m)},$

the smooth map $\mu_{0}$ being here implicit as the common base map of all the
vector bundle morphisms $\mu_{p}$, $p\in\mathcal{P}(m)$.

In the following, given an ordered integer partition $(i_{1},\ldots,i_{l})$ of
a natural number and a list of vectors $e_{1}\in E_{i_{1}},\ldots,e_{l}\in
E_{i_{l}}$, it is useful as before to index this list by

$K_{1},\ldots,K_{l}$

where $(K_{1},\ldots,K_{l})=\rho_{\rm can}^{(i_{1},\ldots,i_{l})}$. Hence

$e_{K_{1}}\in E_{\\#K_{1}},\ldots,e_{K_{l}}\in E_{\\#K_{l}}.$

######  Lemma 3.3.

Let

$\mu\colon E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]\to
F_{1}[-1]\oplus\ldots\oplus F_{m}[-m]$

be a morphism of split $\mathbb{N}$-manifolds over base manifolds $M$ and $N$.
Let $\xi_{1}\in\Gamma(F^{*}_{d_{1}}),\ldots,\xi_{l}\in\Gamma(F^{*}_{d_{l}})$,
with $d_{1}\leq\ldots\leq d_{l}$.

Then for each list of ordered integer partitions $p_{1},\ldots,p_{l}$ such
that $\Sigma p_{j}=d_{j}$, $j=1,\ldots,l$ the image
$\mu^{\star}(\xi_{1}\odot\ldots\odot\xi_{l})$ has a term in

$\Gamma\left(E_{i_{1}}^{*}\odot\ldots\odot E_{i_{s}}^{*}\right),$

if $p=(i_{1},\ldots,i_{s})$ is the canonically reordered integer partition
$p_{1}\cup\ldots\cup p_{l}$.

More precisely, set $\rho^{p}_{\rm can}=(K_{1},\ldots,K_{s})$. Then for all
$e_{K_{1}}\in\Gamma(E^{*}_{\\#K_{1}}),\ldots,e_{K_{s}}\in\Gamma(E^{*}_{\\#K_{s}})$,
$\mu^{\star}(\xi_{1}\odot\ldots\odot\xi_{l})(e_{K_{1}},\ldots,e_{K_{s}})$
equals

(11)
$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\rho_{1},\ldots,\rho_{l}\text{
ordered partitions}\\\ \rho_{\rm can}^{p}=\rho_{1}\cup\ldots\cup\rho_{l}\text{
as sets}\\\ \cup\rho_{1}<\ldots<\cup\rho_{l}\\\
\\#\cup\rho_{1}=d_{1},\ldots,\\#\cup\rho_{l}=d_{l}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})\cdot(\xi_{1}\odot\ldots\odot\xi_{l})\left(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1}),\ldots,\mu_{|\rho_{l}|}(e_{K}\mid K\in\rho_{l})\right).$

Here, for each $j$ the section $\mu_{|\rho_{j}|}$ is fed the list of vectors
$(e_{K}\mid K\in\rho_{j})$. The latter is a notation for the _ordered_ tuple
with order given by the (canonical) order of the partition $\rho_{j}$.

###### Proof.

Since $\mu^{\star}$ is a morphism of graded algebras, the image of
$\xi_{1}\odot\ldots\odot\xi_{l}$ under $\mu^{\star}$ is given by

(12)
$\begin{split}\mu^{\star}(\xi_{1}\odot\ldots\odot\xi_{l})&=\sum_{\begin{subarray}{c}p_{1},\ldots,p_{l}\in\mathcal{P}(n)\\\
\sum p_{j}=d_{j}\text{ for }j=1,\ldots,l\\\
\end{subarray}}\mu^{\star}_{p_{1}}(\xi_{1})\odot\ldots\odot\mu^{\star}_{p_{l}}(\xi_{l}).\end{split}$

Assume for simplicity that $l=2$ and take first $d_{1}<d_{2}$. Choose as in
the claim two ordered integer partitions $p_{1},p_{2}$ such that $\sum
p_{1}=d_{1}$ and $\sum p_{2}=d_{2}$. Then

$\begin{split}\mu^{\star}(\xi_{1}\odot\xi_{2})(e_{K_{1}},\ldots,e_{K_{s}})&=\sum_{\begin{subarray}{c}q_{1},q_{2}\in\mathcal{P}(n)\\\
\sum p_{j}=d_{j}\text{ for }j=1,2\\\ q_{1}\cup q_{2}=p\text{ as
sets}\end{subarray}}\left(\mu^{\star}_{q_{1}}(\xi_{1})\odot\mu^{\star}_{q_{2}}(\xi_{2})\right)(e_{K_{1}},\ldots,e_{K_{s}}).\end{split}$

By Lemma 3.1 this equals

(13) $\begin{split}&\sum_{\begin{subarray}{c}q_{1},q_{2}\in\mathcal{P}(n)\\\
\sum p_{j}=d_{j}\text{ for }j=1,2\\\ q_{1}\cup q_{2}=p\text{ as
sets}\end{subarray}}\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered
}\\\ \rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=q_{1},|\rho_{2}|=q_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{q_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{q_{2}}(e_{K}\mid
K\in\rho_{2})\right)\\\ &=\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{
ordered }\\\ \rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
\\#\cup\rho_{1}=d_{1},\\#\cup\rho_{2}=d_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{|\rho_{2}|}(e_{K}\mid
K\in\rho_{2})\right).\end{split}$

But since $\\#\cup\rho_{1}=d_{1}<d_{2}=\\#\cup\rho_{2}$ for each pair of
ordered partitions $\rho_{1},\rho_{2}$ indexing this sum, the sum equals

$\begin{split}\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered }\\\
\rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
\\#\cup\rho_{1}=d_{1},\\#\cup\rho_{2}=d_{2}\\\
\cup\rho_{1}<\cup\rho_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{|\rho_{2}|}(e_{K}\mid
K\in\rho_{2})\right).\end{split}$

Next take $l=2$ but $d_{1}=d_{2}=:a$. Choose as above two different ordered
integer partitions $p_{1},p_{2}$ such that $\sum p_{1}=\sum p_{2}=a$ and
assume for simplicity444In general, $\mu^{\star}(\xi_{1}\odot\xi_{2})$ has
more terms, that need to be considered alone or pairwise, as is done here.
that under all candidates for these two partitions, only $p_{1}$ and $p_{2}$
give non-vanishing vector bundle morphisms $\mu_{p_{1}}$ and $\mu_{p_{2}}$.
Then $\mu^{\star}(\xi_{1}\odot\xi_{2})$ is given by

$\begin{split}\mu^{\star}(\xi_{1}\odot\xi_{2})=\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})+\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2})+\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})+\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2}).\end{split}$

Since $p_{1}\neq p_{2}$, the reordered unions $p_{1}\cup p_{1}$, $p_{2}\cup
p_{2}$ and $p_{1}\cup p_{2}$ are all different. Consider first the term
$\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})$. (The term
$\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2})$ is treated in
the same manner.) Let again $p$ be the reordered union $p_{1}\cup p_{1}$ and
let $\rho^{p}_{\rm can}=(K_{1},\ldots,K_{s})$. In this case, by Lemma 3.1 the
value of
$\left(\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})\right)(e_{K_{1}},\ldots,e_{K_{s}})$
reads

$\begin{split}&\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered }\\\
\rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=|\rho_{2}|=p_{1}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{2})\right)\\\ &=\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{
ordered }\\\ \rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=p_{1}=|\rho_{2}|\\\
\cup\rho_{1}<\cup\rho_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{2})\right)\\\ &\hskip
113.81102pt+\operatorname{sgn}(\rho_{2},\rho_{1})\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{2})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\\\ &=\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{
ordered }\\\ \rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=p_{1}=|\rho_{2}|\\\
\cup\rho_{1}<\cup\rho_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\Bigl{(}\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{2})\right)\\\ &\hskip
142.26378pt\qquad+(-1)^{a}\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{2})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\Bigr{)}\\\
&=\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered }\\\ \rho^{p}_{\rm
can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\ |\rho_{1}|=p_{1}=|\rho_{2}|\\\
\cup\rho_{1}<\cup\rho_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot(\xi_{1}\odot\xi_{2})\left(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{2}),\mu_{|\rho_{2}|}(e_{K}\mid K\in\rho_{1})\right).\end{split}$

Consider the terms
$\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2})+\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})$.
Let $p$ be the reordered union of $p_{1}$ and $p_{2}$ and set $\rho_{\rm
can}^{p}=(K_{1},\ldots,K_{s})$. Here

$\left(\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2})+\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})\right)(e_{K_{1}},\ldots,e_{K_{s}})$

reads

$\begin{split}&\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered }\\\
\rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=p_{1},|\rho_{2}|=p_{2}\\\
\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{p_{2}}(e_{K}\mid
K\in\rho_{2})\right)\\\ &+\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{
ordered }\\\ \rho^{p}_{\rm can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
|\rho_{1}|=p_{1},|\rho_{2}|=p_{2}\\\
\end{subarray}}\operatorname{sgn}(\rho_{2},\rho_{1})\cdot\xi_{1}\left(\mu_{p_{2}}(e_{K}\mid
K\in\rho_{2})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right).\end{split}$

For a choice of two partitions $\rho_{1}$ and $\rho_{2}$ indexing these sums,
either555In fact, for some choices of $p_{1}$ and $p_{2}$,
$\cup\rho_{1}<\cup\rho_{2}$ is automatically true. For instance if
$p_{1}=(1,3)$ and $p_{2}=(2,2)$ the partition $p$ is $p=(1,2,2,3)$ and the
partitions must be $\rho_{1}=(\\{1\\},\\{6,7,8\\})$ and
$\rho_{2}=(\\{2,3\\},\\{4,5\\})$. But for some choices of $p_{1}$ and $p_{2}$
the order of $\cup\rho_{1}$ and $\cup\rho_{2}$ cannot be predicted. For
instance if $p_{1}=(1,3)$ and $p_{2}=(1,1,2)$ since then $p=(1,1,1,2,3)$ and
$\rho_{1}=(\\{1\\},\\{6,7,8\\})$ and $\rho_{2}=(\\{2\\},\\{3\\},\\{4,5\\})$
satisfy $\cup\rho_{1}<\cup\rho_{2}$ while $\rho_{1}=(\\{2\\},\\{6,7,8\\})$ and
$\rho_{2}=(\\{1\\},\\{3\\},\\{4,5\\})$ satisfy $\cup\rho_{1}>\cup\rho_{2}$.
$\cup\rho_{1}<\cup\rho_{2}$ or $\cup\rho_{1}>\cup\rho_{2}$. In both cases the
corresponding terms in the sum give together

$\begin{split}&\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\xi_{1}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right)\cdot\xi_{2}\left(\mu_{p_{2}}(e_{K}\mid
K\in\rho_{2})\right)\\\
&+\operatorname{sgn}(\rho_{2},\rho_{1})\cdot\xi_{1}\left(\mu_{p_{2}}(e_{K}\mid
K\in\rho_{2})\right)\cdot\xi_{2}\left(\mu_{p_{1}}(e_{K}\mid
K\in\rho_{1})\right),\end{split}$

which is written

$\operatorname{sgn}(\rho_{1},\rho_{2})(\xi_{1}\odot\xi_{2})(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1}),\mu_{|\rho_{2}|}(e_{K}\mid K\in\rho_{2}))$

if $\cup\rho_{1}<\cup\rho_{2}$ and

$\operatorname{sgn}(\rho_{2},\rho_{1})(\xi_{1}\odot\xi_{2})(\mu_{|\rho_{2}|}(e_{K}\mid
K\in\rho_{2}),\mu_{|\rho_{1}|}(e_{K}\mid K\in\rho_{1}))$

if $\cup\rho_{2}<\cup\rho_{1}$.

As a summary

$\left(\mu_{p_{1}}^{\star}(\xi_{1})\odot\mu_{p_{2}}^{\star}(\xi_{2})+\mu_{p_{2}}^{\star}(\xi_{1})\odot\mu_{p_{1}}^{\star}(\xi_{2})\right)(e_{K_{1}},\ldots,e_{K_{s}})$

equals

$\sum_{\begin{subarray}{c}\rho_{1},\rho_{2}\text{ ordered }\\\ \rho^{p}_{\rm
can}=\rho_{1}\cup\rho_{2}\text{ as sets }\\\
\\#\cup\rho_{1}=a=\\#\cup\rho_{2}\\\
\cup\rho_{1}<\cup\rho_{2}\end{subarray}}\operatorname{sgn}(\rho_{1},\rho_{2})\cdot\operatorname{sgn}(\rho_{1},\rho_{2})(\xi_{1}\odot\xi_{2})(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1}),\mu_{|\rho_{2}|}(e_{K}\mid K\in\rho_{2})).$

The above proves the claim for $l=2$. The general case for $l\geq 3$ works in
the same manner, but with more different cases to consider. ∎

The following result is then an immediate corollary of the preceding lemma.
The proof is left to the reader.

######  Corollary 3.4.

Given two morphisms

$\mu\colon E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]\to
F_{1}[-1]\oplus\ldots\oplus F_{m}[-m]$

and

$\nu\colon F_{1}[-1]\oplus\ldots\oplus F_{m}[-m]\to
G_{1}[-1]\oplus\ldots\oplus G_{q}[-q]$

of split $[n]$-manifolds over smooth maps $\mu_{0}\colon M\to N$ and
$\nu_{0}\colon N\to Q$, respectively, the composition

$\nu\circ\mu=\left((\nu\circ\mu)_{p}\right)_{p\in\mathcal{P}(q)}$

is defined by

(14)
$(\nu\circ\mu)_{p}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\rho_{1},\ldots,\rho_{l}\text{
ordered}\\\ \text{such that }\rho_{1}\cup\ldots\cup\rho_{l}=\rho^{p}_{\rm
can}\\\ \text{ as sets and}\\\
\cup\rho_{1}<\ldots<\cup\rho_{l}\end{subarray}}\operatorname{sgn}(\rho_{1},\ldots,\rho_{l})\cdot\nu_{\left(\Sigma|\rho_{1}|,\ldots,\Sigma|\rho_{l}|\right)}\circ\left(\mu_{|\rho_{1}|},\ldots,\mu_{|\rho_{l}|}\right)$

for all $p\in\mathcal{P}(q)$.

Here, $(\nu\circ\mu)_{p}$ is fed a list of vectors

$(e_{K_{1}},\ldots,e_{K_{s}})$

with $\rho^{p}_{\rm can}=(K_{1},\ldots,K_{s})$. The term
$\nu_{\left(\Sigma|\rho_{1}|,\ldots,\Sigma|\rho_{l}|\right)}\circ\left(\mu_{|\rho_{1}|},\ldots,\mu_{|\rho_{l}|}\right)$
of (14) indexed by $(\rho_{1},\ldots,\rho_{l})$ applied to this list is then
precisely

$\nu_{\left(\Sigma|\rho_{1}|,\ldots,\Sigma|\rho_{l}|\right)}\left(\mu_{|\rho_{1}|}(e_{K}\mid
K\in\rho_{1}),\ldots,\mu_{|\rho_{l}|}(e_{K}\mid K\in\rho_{l})\right),$

where, as before, $(e_{K}\mid K\in\rho_{1})$ is a notation for the _ordered_
tuple with order given by the partition $\rho_{1}$, etc.

A morphism $\mu\colon E_{1}[-1]\oplus\ldots\oplus E_{n}[-n]\to
F_{1}[-1]\oplus\ldots\oplus F_{n}[-n]$ of split $[n]$-manifolds is an
isomorphism of split $[n]$-manifolds if it has an inverse. More precisely, if
$\mu_{0}$ is a diffeomorphism and there exists a morphism $\nu\colon
F_{1}[-1]\oplus\ldots\oplus F_{n}[-n]\to E_{1}[-1]\oplus\ldots\oplus
E_{n}[-n]$ of split $[n]$-manifolds such that $\nu_{0}$ is the smooth inverse
of $\mu_{0}$ and

$(\nu\circ\mu)_{p}=0\text{ and }(\mu\circ\nu)_{p}=0$

for $p\in\mathcal{P}(n)$ with $\sharp p\geq 2$, and

$(\nu\circ\mu)_{p}=\operatorname{id}_{E_{\Sigma p}}$

for $\sharp p=1$. (This implies as always
$(\mu\circ\nu)_{p}=\operatorname{id}_{F_{\Sigma p}}$ for $\sharp p=1$.) For
instance, a collection of $n$-isomorphisms $\mu_{(1)}\colon E_{1}\to F_{1}$,
…, $\mu_{(n)}\colon E_{n}\to F_{n}$ over a smooth diffeomorphism
$\mu_{0}\colon M\to N$ gives an isomorphism of $E_{1}[-1]\oplus\ldots\oplus
E_{n}[-n]$ with $F_{1}[-1]\oplus\ldots\oplus F_{n}[-n]$ by setting $\mu_{p}=0$
for $\sharp p\geq 2$.

Any $\mathbb{N}$-manifold is non-canonically isomorphic to a split
$\mathbb{N}$-manifold of the same degree. More precisely, the embedding of the
category of split $[n]$-manifolds in the one of $[n]$-manifolds is fully
faithful and essentially surjective.

This is true locally, per definition, and proved globally for instance in
[BP13], following the proof of the $\mathbb{Z}/2\mathbb{Z}$-graded version of
this theorem, which is called there _Batchelor’s theorem_ [Bat79], see also
[Ber87] and [Vor02].

######  Proposition 3.5.

Any $[n]$-manifold is non-canonically isomorphic to a split $[n]$-manifold.

Note that $[1]$-manifolds are automatically split since they are just vector
bundles with a degree shifting in the fibers, i.e. a $[1]$-manifold over $M$
is $E[-1]$ for some vector bundle $E\to M$ and
$C^{\infty}(E[-1])=\Gamma(\wedge^{\bullet}E^{*})$, the exterior algebra of
$E$.

###  3.3. $[n]$-manifold cocycles

Let $\mathcal{M}$ be an $[n]$-manifold over a smooth manifold $M$ and choose
an open cover $(U_{\alpha})_{\alpha\in\Lambda}$ of $M$ by open sets
trivialising $C^{\infty}(\mathcal{M})$. That is, for each $\alpha\in\Lambda$,
the $\mathbb{N}$-graded, graded commutative, associative, unital
$C^{\infty}(M)$-algebra $C^{\infty}_{U_{\alpha}}(\mathcal{M})$ is freely
generated by its elements

$\xi_{\alpha,1}^{1},\ldots,\xi_{\alpha,1}^{r_{1}},\xi_{\alpha,2}^{1},\ldots,\xi_{\alpha,2}^{r_{2}},\ldots,\xi_{\alpha,n}^{1},\ldots,\xi_{\alpha,n}^{r_{n}}$

with $\xi_{\alpha,i}^{j}$ of degree $i$ for $i\in\\{1,\ldots,n\\}$ and
$j\in\\{1,\ldots,r_{i}\\}$. In other words, for each $\alpha\in\Lambda$ the
restriction $\mathcal{M}\arrowvert_{U_{\alpha}}$ is isomorphic via a morphism
$\phi_{\alpha}$ over the identity on $U_{\alpha}$ to the split $[n]$-manifold

$(U_{\alpha}\times\mathbb{R}^{r_{1}})[-1]\oplus\ldots\oplus(U_{\alpha}\times\mathbb{R}^{r_{n}})[-n],$

where the spaces $U_{\alpha}\times\mathbb{R}^{r_{j}}$ carry the canonical
trivial vector bundle structures over $U_{\alpha}$, for $j=1,\ldots,n$.

Take $\alpha,\beta\in\Lambda$ and set $U_{\alpha\beta}:=U_{\alpha}\cap
U_{\beta}$. Then the following diagram of isomorphisms of $[n]$-manifold
commutes

${\mathcal{M}\arrowvert_{U_{\alpha\beta}}}$${(U_{\alpha\beta}\times\mathbb{R}^{r_{1}})[-1]\oplus\ldots\oplus(U_{\alpha\beta}\times\mathbb{R}^{r_{n}})[-n]}$${(U_{\alpha\beta}\times\mathbb{R}^{r_{1}})[-1]\oplus\ldots\oplus(U_{\alpha\beta}\times\mathbb{R}^{r_{n}})[-n]}$$\scriptstyle{\phi_{\alpha}\arrowvert_{\mathcal{M}\arrowvert_{U_{\alpha\beta}}}}$$\scriptstyle{\phi_{\beta}\arrowvert_{\mathcal{M}\arrowvert_{U_{\alpha\beta}}}}$$\scriptstyle{\phi_{\alpha}\circ\phi_{\beta}^{-1}}$

and $\phi^{\alpha\beta}:=\phi_{\alpha}\circ\phi_{\beta}^{-1}$ is an
isomorphism of split $[n]$-manifolds. By construction,

$\phi^{\alpha\gamma}=\phi^{\alpha\beta}\circ\phi^{\beta\gamma}$

over $U_{\alpha\beta\gamma}:=U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$, i.e.

$\phi^{\alpha\gamma}_{p}=(\phi^{\alpha\beta}\circ\phi^{\beta\gamma})_{p}$

for all $p\in\mathcal{P}(n)$. The open cover
$\\{U_{\alpha}\\}_{\alpha\in\Lambda}$ of $M$ together with the collection of
isomorphisms

$(\phi^{\alpha\beta}\mid\alpha,\beta\in\Lambda)$

satisfying

  1. (1)

$\phi^{\alpha\gamma}=\phi^{\alpha\beta}\circ\phi^{\beta\gamma}$ over
$U_{\alpha\beta\gamma}$ for all $\alpha,\beta,\gamma\in\Lambda$ and

  2. (2)

$\phi^{\alpha\alpha}=\operatorname{id}_{(U_{\alpha}\times\mathbb{R}^{r_{1}})[-1]\oplus\ldots\oplus(U_{\alpha}\times\mathbb{R}^{r_{n}})[-n]}$
for all $\alpha\in\Lambda$

is an $[n]$-manifold cocycle on $M$. By their very definition, $[n]$-manifold
cocycles on $M$ are equivalent to $[n]$-manifolds over $M$.

##  4\. Multiple vector bundles and charts

This section recalls the definitions, notation and results of [HL20] which are
needed in the rest of the paper, and studies in more detail the _iterated
higher order cores_ of an $n$-fold vector bundle.

In [HL20] the authors define as follows an $n$-fold vector bundle. This is
just a different, in their opinion more convenient, formulation for the
$n$-fold vector bundles defined by Mackenzie and Gracia-Saz in [GSM09].

######  Definition 4.1.

An $n$-fold vector bundle, is a covariant functor
$\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ – to the category of
smooth manifolds, such that, writing $p^{I}_{J}:=\mathbb{E}(I\to J)$ for
$J\subseteq I\subseteq{\underline{n}}$,

  1. (a)

for all $I\subseteq{\underline{n}}$ and all $i\in I$,
$p^{I}_{I\setminus\\{i\\}}\colon\mathbb{E}(I)\to\mathbb{E}(I\setminus\\{i\\})$
has a smooth vector bundle structure, and

  2. (b)

for all $I\subseteq{\underline{n}}$ and $i\neq j\in I$,

${\mathbb{E}(I)}$${\mathbb{E}(I\setminus\\{i\\})}$${\mathbb{E}(I\setminus\\{j\\})}$${\mathbb{E}(I\setminus\\{i,j\\})}$$\scriptstyle{p^{I}_{I\setminus\\{i\\}}}$$\scriptstyle{p^{I}_{I\setminus\\{j\\}}}$$\scriptstyle{p^{I\setminus\\{i\\}}_{I\setminus\\{i,j\\}}}$$\scriptstyle{p^{I\setminus\\{j\\}}_{I\setminus\\{i,j\\}}}$

is a double vector bundle.

The smooth manifold $\mathbb{E}(\emptyset)$ is denoted $M$ if not mentioned
otherwise.

Given two $n$-fold vector bundles
$\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ and
$\mathbb{F}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$, a morphism of
$n$-fold vector bundles from $\mathbb{E}$ to $\mathbb{F}$ is a natural
transformation $\Phi\colon\mathbb{E}\to\mathbb{F}$ such that the commutative
diagrams

${\mathbb{E}(I)}$${\mathbb{F}(I)}$${\mathbb{E}(I\setminus\\{i\\})}$${\mathbb{F}(I\setminus\\{i\\})}$$\scriptstyle{\Phi(I)}$$\scriptstyle{p^{I}_{I\setminus\\{i\\}}}$$\scriptstyle{p^{I}_{I\setminus\\{i\\}}}$$\scriptstyle{\Phi(I\setminus\\{i\\})}$

are vector bundle homomorphisms for all $I\subseteq{\underline{n}}$ and $i\in
I$. The morphism $\tau$ is surjective (respectively injective) if each of its
components $\tau(I)$, $I\subseteq{\underline{n}}$ is fibrewise surjective
(respectively fibrewise injective). It is then called an epimorphism
(respectively a monomorphism) of $n$-fold vector bundles.

Note that such a morphism $\Phi\colon\mathbb{E}\to\mathbb{F}$ is completely
determined by its top map
$\Phi({\underline{n}})\colon\mathbb{E}({\underline{n}})\to\mathbb{F}({\underline{n}})$.
However, the definition above is convenient because it can be extended to
morphisms of $\infty$-fold vector bundles [HL20].

A subset $S\subseteq{\underline{n}}$ defines a full subcategory $\square^{S}$
of the $n$-cube category $\square^{n}$, with objects the subsets of $S$. Given
an $n$-fold vector bundle
$\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$, the $S$-side of
$\mathbb{E}$ is then the restriction $\mathbb{E}_{S}$ of $\mathbb{E}$ to the
subcategory $\square^{S}$. The functors obtained in this manner are the sides
of $\mathbb{E}$. Given two $n$-fold vector bundles $\mathbb{E}$ and
$\mathbb{F}$ and a morphism $\Phi\colon\mathbb{E}\to\mathbb{F}$ of $n$-fold
vector bundles, the restriction of $\Phi$ to the sides $\mathbb{E}_{S}$ and
$\mathbb{F}_{S}$ is written
$\Phi\arrowvert_{S}\colon\mathbb{E}_{S}\to\mathbb{F}_{S}$. It is defined by
$\Phi\arrowvert_{S}(J)=\Phi(J)\colon\mathbb{E}(J)\to\mathbb{F}(J)$ for all
$J\subseteq S$.

###  4.1. Cores of an $n$-fold vector bundle

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle, and choose two subsets $J\subseteq S\subseteq{\underline{n}}$.
Then by Proposition 2.18 in [HL20], the space

$\mathbb{E}^{S}_{J}:=\bigcap_{j\in
J}(p^{S}_{S\setminus\\{j\\}})^{-1}\left(\mathbf{0}^{S\setminus\\{j\\}}_{S\setminus
J}\right)=\left\\{e\in\mathbb{E}(S)\,\left|\,p^{S}_{S\setminus\\{j\\}}(e)=\mathbf{0}^{S\setminus\\{j\\}}_{p^{S}_{S\setminus
J}(e)}\text{ for all }j\in J\right.\right\\}$

with $\mathbf{0}^{S\setminus\\{j\\}}_{S\setminus J}\colon\mathbb{E}(S\setminus
J)\to\mathbb{E}(S\setminus\\{j\\})$ the composition of zero sections, is an
embedded submanifold of $\mathbb{E}(S)$. (For $J=\emptyset$ the empty
intersection is $\mathbb{E}(S)$ by convention, and for $\\#J=1$ the “zero
section” $\mathbf{0}^{S\setminus\\{j\\}}_{S\setminus J}=\mathbf{0}^{S\setminus
J}_{S\setminus J}$ is the identity on $\mathbb{E}(S\setminus J)$, so
$\mathbb{E}^{S}_{J}=\mathbb{E}(S)$.) It has further a vector bundle structure
over $\mathbb{E}(S\setminus J)$ with projection

$p^{S}_{S\setminus J}\arrowvert_{\mathbb{E}^{S}_{J}}=\mathbb{E}(S\rightarrow
S\setminus
J)\arrowvert_{\mathbb{E}^{S}_{J}}\colon\mathbb{E}^{S}_{J}\to\mathbb{E}(S\setminus
J)$

and with addition defined by

$e_{1}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle S\setminus
J}}]{{\scriptscriptstyle}}{+}e_{2}:=e_{1}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
S\setminus\\{j\\}}}]{{\scriptscriptstyle}}{+}e_{2}$

for any $j\in J$. $\mathbb{E}^{S}_{J}$ is the total space of an
$(\\#S-\\#J+1)$-fold vector bundle $\mathbb{E}^{(S,J)}$, called here the
$(S,J)$-core of $\mathbb{E}$ and defined as follows. Consider the
$(\\#S-\\#J+1)$-cube subcategory $\lozenge^{S}_{J}$ of $\square^{S}$ with
objects the subsets $I\subseteq S$ with

$I\cap J=\emptyset\text{ or }J\subseteq I$

and with arrows

$I\rightarrow I^{\prime}\,\,:\Leftrightarrow\,\,I^{\prime}\subseteq I.$

That is, $\lozenge^{S}_{J}$ is a full subcategory of $\square^{S}$. Note that
if $S={\underline{n}}$ the $(n-\\#J+1)$-cube category
$\lozenge^{\underline{n}}_{J}$ equals the $(n-\\#J+1)$-cube category
$\lozenge^{\rho_{J}}$ with $\rho_{J}$ the partition of ${\underline{n}}$ in
the subset $J$ and ${\underline{n}}-\\#J$ subsets with one element each.

The functor
$\mathbb{E}^{(S,J)}\colon\lozenge^{S}_{J}\to{\mathbf{Man}^{\infty}}$ sends an
object $I$ as above

(15) $\text{ to }\,\,\mathbb{E}(I)=\mathbb{E}_{S}(I)\,\,\text{ if }\,\,I\cap
J=\emptyset\,\,\text{ and to }\,\,\mathbb{E}^{I}_{J}\,\,\text{ if }J\subseteq
I,$

and an arrow $I\rightarrow I^{\prime}$ to

(16) $\left\\{\begin{array}[]{lc}\mathbb{E}(I\rightarrow
I^{\prime})\arrowvert_{\mathbb{E}^{I}_{J}}\colon\mathbb{E}^{I}_{J}\to\mathbb{E}^{I^{\prime}}_{J}&\text{
if }J\subseteq I^{\prime}\subseteq I,\\\ \mathbb{E}(I\rightarrow
I^{\prime})\colon\mathbb{E}(I)\to\mathbb{E}(I^{\prime})&\text{ if }I\cap
J=\emptyset\text{ and }\\\ \mathbb{E}(I\to
I^{\prime})\arrowvert_{\mathbb{E}^{I}_{J}}\colon\mathbb{E}^{I}_{J}\to\mathbb{E}(I^{\prime})&\text{
if }I^{\prime}\cap J=\emptyset\text{ but }J\subseteq I.\end{array}\right.$

Write $i^{S}_{J}\colon\lozenge^{S}_{J}\to\square^{S}$ for the inclusion
functor. Then the assignment

$\tau\colon\operatorname{Obj}\left(\lozenge^{S}_{J}\right)\to\operatorname{Mor}({\mathbf{Man}^{\infty}})$

sending $I\subseteq S$ with $I\cap J=\emptyset$ or $J\subseteq I$ to the
smooth embedding

$\tau(I):=\iota_{\mathbb{E}^{(S,J)}(I)}\colon\mathbb{E}^{(S,J)}(I)\hookrightarrow\mathbb{E}_{S}(I)=\mathbb{E}(I)$

defines a natural transformation

$\mathbb{E}^{(S,J)}\longrightarrow\mathbb{E}_{S}\circ i^{S}_{J}.$

In the following a little abuse of notation is made, and such a natural
transformation by embeddings between functors from a subcategory of
$\square^{n}$ is called a _natural transformation by embeddings in
$\mathbb{E}$._

Consider two $n$-fold vector bundles $\mathbb{E}$ and $\mathbb{F}$. Choose
again $J\subseteq S\subseteq{\underline{n}}$ and build the $(S,J)$-cores
$\mathbb{E}^{(S,J)}$ and $\mathbb{F}^{(S,J)}$. Then a morphism
$\Phi\colon\mathbb{E}\to\mathbb{F}$ of $n$-fold vector bundles induces as
follows a core morphism

(17) $\Phi^{(S,J)}\colon\mathbb{E}^{(S,J)}\to\mathbb{F}^{(S,J)}.$

For all $I\in\operatorname{Obj}(\lozenge^{S}_{J})$ the map
$\Phi^{(S,J)}(I)\colon\mathbb{E}^{(S,J)}(I)\to\mathbb{F}^{(S,J)}(I)$ is simply
the (necessarily smooth) restriction of $\Phi(I)$ to the embedded domain and
codomain $\mathbb{E}^{(S,J)}(I)\subseteq\mathbb{E}(I)$ and
$\mathbb{F}^{(S,J)}(I)\subseteq\mathbb{F}(I)$, which is well-defined because
$\Phi$ preserves the $n$-fold vector bundle structure and so in particular the
zeros.

Finally note that by definition, the face $\mathbb{E}_{S\setminus J}$ of
$\mathbb{E}$ is also a face of the core $\mathbb{E}^{(S,J)}$ since

$\operatorname{Obj}(\square^{S\setminus
J})\subseteq\operatorname{Obj}(\lozenge^{S}_{J})$

($\square^{S\setminus J}$ is a full subcategory of $\lozenge^{S}_{J}$) and

$\mathbb{E}^{(S,J)}(I)=\mathbb{E}(I)=\mathbb{E}_{S\setminus J}(I)$

for all $I\subseteq S\setminus J$.

###  4.2. Linear splittings and decompositions of an $n$-fold vector bundle

Consider a collection $\mathcal{A}=\\{A_{I}\mid I\subseteq{\underline{n}}\\}$
of vector bundles $A_{I}$ over a smooth manifold $M$, where as a convention,
$A_{\emptyset}$ is taken to be the trivial vector bundle $M\to M$. Set

$\mathbb{E}^{\mathcal{A}}\colon\square^{n}\to{\mathbf{Man}^{\infty}},$
$\mathbb{E}^{\mathcal{A}}\colon\square^{n}\to{\mathbf{Man}^{\infty}},\qquad\mathbb{E}^{\mathcal{A}}(J):=\prod^{M}_{I\subseteq
J}A_{I}$

for $J\subseteq{\underline{n}}$, where $\Pi^{M}$ are the fibered products over
$M$. Set $\mathbb{E}^{\mathcal{A}}(J\rightarrow J^{\prime})$ to be the
canonical projection $\prod^{M}_{I\subseteq J}A_{I}\to\prod^{M}_{I\subseteq
J^{\prime}}A_{I}$. Then $\mathbb{E}^{\mathcal{A}}$ with the obvious vector
bundle structures is an $n$-fold vector bundle, the decomposed $n$-fold vector
bundle defined by $\mathcal{A}$ [HL20]. The reader is invited to check that

$(\mathbb{E}^{\mathcal{A}})^{(S,J)}(I)=\prod^{M}_{K\in\operatorname{Obj}(\lozenge^{I}_{J})}A_{K}$

for all $J\subseteq I\subseteq S\subseteq{\underline{n}}$. Here, the right-
hand face is an embedded submanifold of $\mathbb{E}^{\mathcal{A}}(I)$ by
taking its fibered products with the zero sections of the “missing” bundles.

The vacant decomposed $n$-fold vector bundle defined by $\mathcal{A}$ is

$\overline{\mathbb{E}^{\mathcal{A}}}\colon\square^{n}\to{\mathbf{Man}^{\infty}},\quad\overline{\mathbb{E}^{\mathcal{A}}}(J)=\prod^{M}_{i\in
J}A_{\\{i\\}}.$

(Note that it uses only the vector bundles in $\mathcal{A}$ indexed by one-
element sets. Hence it can also be defined by a family of $n$ vector bundles
over $M$ indexed by the number $1$ to $n$.) For each
$J\subseteq{\underline{n}}$ the manifold
$\overline{\mathbb{E}^{\mathcal{A}}}(J)$ is clearly also embedded in
$\mathbb{E}^{\mathcal{A}}(J)$. Denote the embedding by
$\iota(J)\colon\overline{\mathbb{E}^{\mathcal{A}}}(J)\rightarrow\mathbb{E}^{\mathcal{A}}(J)$.
The collection of these embeddings defines a monomorphism

(18)
$\iota\colon\overline{\mathbb{E}^{\mathcal{A}}}\to\mathbb{E}^{\mathcal{A}}$

of $n$-fold vector bundles.

Write $\overline{\mathbb{E}^{\mathcal{A}}}=\overline{\mathbb{E}}$ for
simplicity. For $\\#J\geq 2$, $J\subseteq S\subseteq{\underline{n}}$ and all
$I\in\operatorname{Obj}(\lozenge^{S}_{J})$ the space
$\overline{\mathbb{E}}^{(S,J)}(I)$ is here further

$\overline{\mathbb{E}}^{(S,J)}(I)=\prod^{M}_{i\in I\setminus J}A_{\\{i\\}}$

since

$\overline{\mathbb{E}}^{(S,J)}(I)=\left\\{\begin{array}[]{ll}\overline{\mathbb{E}}^{I}_{J}&\text{
if }J\subseteq I\\\ \overline{\mathbb{E}}(I)&\text{ if }I\subseteq S\setminus
J,\end{array}\right.$

with

$\overline{\mathbb{E}}^{I}_{J}=\left\\{e\in\prod^{M}_{i\in
I}A_{\\{i\\}}\,\left|\
p^{I}_{I\setminus\\{j\\}}(e)=\mathbf{0}^{I\setminus\\{j\\}}_{p^{I}_{I\setminus
J}(e)}\text{ for all }j\in
J\right.\right\\}=\prod^{M}_{\\{i\\}\in\operatorname{Obj}(\lozenge^{I}_{J})}A_{\\{i\\}}=\prod^{M}_{i\in
I\setminus J}A_{\\{i\\}},$

The name _vacant_ [Mac92] comes from the fact that the $(I,I)$-cores of
$\overline{\mathbb{E}}$ are consequently all trivial, for
$I\subseteq{\underline{n}}$ with $\\#I\geq 2$.

The induced monomorphism
$\tau^{(S,J)}\colon\overline{\mathbb{E}}^{(S,J)}\rightarrow(\mathbb{E}^{\mathcal{A}})^{(S,J)}$
is clearly the one defined by the natural embeddings

$\overline{\mathbb{E}}^{(S,J)}(I)=\prod^{M}_{i\in I\setminus
J}A_{\\{i\\}}\hookrightarrow\prod^{M}_{K\in\operatorname{Obj}(\lozenge^{I}_{J})}A_{K}=(\mathbb{E}^{\mathcal{A}})^{(S,J)}(I).$

Let $\mathbb{E}$ be an $n$-fold vector bundle and consider the collection of
vector bundles $\mathbb{E}^{J}_{J}\to M=\mathbb{E}(\emptyset)$ for
$J\subseteq{\underline{n}}$. In particular,
$\mathbb{E}^{\\{i\\}}_{\\{i\\}}=\mathbb{E}(\\{i\\})=:E_{i}$ are the lower
sides of $\mathbb{E}$ and
$\mathbb{E}^{\emptyset}_{\emptyset}=\mathbb{E}(\emptyset)=M$. The collection
$\mathcal{A}_{\mathbb{E}}=\left\\{\left.\mathbb{E}_{I}^{I}\right|\emptyset\neq
I\subseteq{\underline{n}}\right\\}$ is the family of _building bundles_ of
$\mathbb{E}$ in the sense that $\mathbb{E}$ is non-canonically isomorphic to
the $n$-fold vector bundle $\mathbb{E}^{\mathcal{A}_{\mathbb{E}}}$, or in
other words, $\mathbb{E}$ is _decomposed by $\mathcal{A}$_. The proof of the
existence of such an isomorphism, called a _decomposition_ , is the subject of
the following two sections.

######  Definition 4.2.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle.

  1. (1)

The decomposed $n$-fold vector bundle $\mathbb{E}^{\mathcal{A}_{\mathbb{E}}}$
defined as above by $\mathbb{E}$ is denoted by $\mathbb{E}^{\rm dec}$. The
vacant decomposed $n$-fold vector bundle
$\overline{\mathbb{E}^{\mathcal{A}_{\mathbb{E}}}}$ is written
$\overline{\mathbb{E}}$.

  2. (2)

A linear splitting of the $n$-fold vector bundle $\mathbb{E}$ is a
monomorphism $\Sigma\colon\overline{\mathbb{E}}\to\mathbb{E}$ of $n$-fold
vector bundles, such that for $i=1,\ldots,n$, $\Sigma(\\{i\\})\colon E_{i}\to
E_{i}$ is the identity.

  3. (3)

A decomposition of the $n$-fold vector bundle $\mathbb{E}$ is a natural
isomorphism $\mathcal{S}\colon\mathbb{E}^{\rm dec}\to\mathbb{E}$ of $n$-fold
vector bundles over the identity maps
$\mathcal{S}(\\{i\\})=\operatorname{id}_{E_{i}}\colon E_{i}\to E_{i}$ such
that additionally the induced core morphisms $\mathcal{S}^{(I,I)}(\\{I\\})$
are the identities $\operatorname{id}_{\mathbb{E}^{I}_{I}}$ for all
$I\subseteq{\underline{n}}$.

By Corollary 3.6 in [HL20], any $n$-fold vector bundle admits a decomposition.
Since there appears to be a little gap in the proof of this result in [HL20],
it is revisited in Section 4.4.

Let $\mathcal{A}=(A_{I}\mid I\subseteq{\underline{n}})$,
$\mathcal{B}=(B_{I}\mid I\subseteq{\underline{n}})$ and
$\mathcal{C}=(C_{I}\mid I\subseteq{\underline{n}})$ be three families of
vector bundles over smooth manifolds $M$, $N$ and $P$ respectively. A morphism
$\tau\colon\mathbb{E}^{\mathcal{A}}\to\mathbb{E}^{\mathcal{B}}$ is easily seen
to amount to a collection of vector bundle morphisms

$\tau_{\rho}\colon A_{I_{1}}\otimes\ldots\otimes A_{I_{l}}\to B_{I}$

over $\tau(\emptyset)\colon M\to N$ for all $\emptyset\neq
I\subseteq{\underline{n}}$ and all
$\rho=(I_{1},\ldots,I_{l})\in\mathcal{P}(I)$, see [HL20]. Given this
collection, the map
$\tau({\underline{n}})\colon\mathbb{E}^{\mathcal{A}}({\underline{n}})\to\mathbb{E}^{\mathcal{B}}({\underline{n}})$
is given by

$\left(a_{I}\right)_{\emptyset\neq
I\subseteq{\underline{n}}}\quad\mapsto\quad\left(\sum_{\rho=(I_{1},\ldots,I_{l})\in\mathcal{P}(I)}\tau_{\rho}(a_{I_{1}},\ldots,a_{I_{l}})\right)_{\emptyset\neq
I\subseteq{\underline{n}}}.$

Given a second morphism
$\mu\colon\mathbb{E}^{\mathcal{B}}\to\mathbb{E}^{\mathcal{C}}$ over $n$-fold
vector bundles, the composition
$\mu\circ\tau\colon\mathbb{E}^{\mathcal{A}}\to\mathbb{E}^{\mathcal{C}}$ is
given by

$(\mu\circ\tau)_{\rho}\left((a_{J})_{J\in\rho}\right)=\sum_{(J_{1},\ldots,J_{l})\in\operatorname{coars}(\rho)}\mu_{(J_{1},\ldots,J_{l})}\Bigl{(}\tau_{\rho\cap
J_{1}}\bigl{(}(a_{J})_{J\in\rho\cap J_{1}}\bigr{)},\ldots,\tau_{\rho\cap
J_{l}}\bigl{(}(a_{J})_{J\in\rho\cap J_{l}}\bigr{)}\Bigr{)}$

for all $\emptyset\neq I\subseteq{\underline{n}}$ and all
$\rho\in\mathcal{P}(I)$. Here, $\rho\cap J_{k}$ is the canonically ordered
partition $\\{I_{s}\in\rho\mid I_{s}\subseteq J_{k}\\}$ for $k=1,\ldots,l$.

###  4.3. Iterated highest order cores of an $n$-fold vector bundle

This section studies in detail the _iterated highest order cores_ of an
$n$-fold vector bundle.

######  Definition 4.3.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle. Then the highest order cores
$\mathbb{E}^{({\underline{n}},J)}$, written simply $\mathbb{E}^{J}$, for
$J\subseteq{\underline{n}}$ with $\\#J=2$ are also called $(n-1)$-cores of
$\mathbb{E}$, since they are modelled on the $(n-1)$-cube categories
$\lozenge^{\underline{n}}_{J}$. The $(n-2)$-cores $\mathbb{E}$ are then
defined to be the highest order cores of the $(n-1)$-cores of $\mathbb{E}$,
hence the $(n-2)$-cores of the $(n-1)$-cores of $\mathbb{E}$.

This construction can then be iterated: for $l\in\\{1,\ldots,n-2\\}$, the
$l$-cores of $\mathbb{E}$ are defined to be the $l$-cores of the $(l+1)$-cores
of $\mathbb{E}$. Further, the (unique) $n$-core of $\mathbb{E}$ is set by
convention to be $\mathbb{E}$ itself.

The $l$-cores for $l=1,\ldots,n-1$ are generally called the iterated highest
order cores of $\mathbb{E}$.

By construction an iterated higher order core of $\mathbb{E}$ is a functor
from a full subcategory of $\square^{n}$ to ${\mathbf{Man}^{\infty}}$. As
above, an $l$-core has a natural transformation by embeddings in the former
$(l+1)$-core in its recursive construction, and all morphisms are restrictions
of the morphisms of this $(l+1)$-core. Then the $l$-core has a natural
transformation by embeddings in $\mathbb{E}$. A priori the recursively defined
subcategory of $\square^{n}$ indexing an $l$-core is complicated to write
down. In addition, two different chains of construction of $l$-cores can lead
to the same $l$-core. The goal of the following theorem is to remedy to these
problems, by understanding that the collection of $l$-cores of $\mathbb{E}$,
for $l\in\\{1,\ldots,n\\}$, is simply parametrised by the partitions of
${\underline{n}}$ in $l$ subsets.

######  Proposition 4.4.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle and choose $l\in\\{1,\ldots,n\\}$.

  1. (1)

For each $l$-core $\mathbb{F}$ of $\mathbb{E}$ there is a partition
$\rho:=\\{I_{1},\ldots,I_{l}\\}$ of ${\underline{n}}$ in $l$ (non-empty)
subsets such that $\mathbb{F}$ is a functor from the $l$-cube subcategory
$\lozenge^{\rho}$ of $\square^{n}$ to ${\mathbf{Man}^{\infty}}$. Conversely,
for each partition $\rho$ of ${\underline{n}}$ in $l$ elements there is an
$l$-core $\mathbb{F}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}}$ of
$\mathbb{E}$. If two $l$-cores are defined on the same subcategory
$\lozenge^{\rho}$, they are equal.

  2. (2)

Choose an $l$-partition $\rho$ of ${\underline{n}}$, and consider the
corresponding $l$-core $\mathbb{E}^{\rho}$ of $\mathbb{E}$. Then the $i$-cores
of $\mathbb{E}^{\rho}$, for $i=1,\ldots,l$, are the $i$-cores of $\mathbb{E}$
indexed by coarsements of $\rho$.

  3. (3)

Assume that $\rho=\\{I_{1},\ldots,I_{l}\\}$ is a partition of
${\underline{n}}$ in $l$ subsets and that $\rho_{1}$ and $\rho_{2}$ are two
different $(l-1)$-coarsements of $\rho$ as in Lemma 2.2. Then the $(l-2)$-core
$\mathbb{E}^{\rho_{1}\sqcap\rho_{2}}$ is a common $(l-2)$-core of
$\mathbb{E}^{\rho_{1}}$ and $\mathbb{E}^{\rho_{2}}$.

######  Definition 4.5.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle and let $\rho$ be a partition of ${\underline{n}}$ in $l$
elements. The unique $l$-core of $\mathbb{E}$ corresponding to $\rho$ as in
the previous theorem is denoted by $\mathbb{E}^{\rho}$.

Note that an $n$-fold vector bundle has only one $1$-core, since there is only
one partition of $n$ with $1$ element. It is the ultracore.

Note also that the $2$-cores of $\mathbb{E}$ are indexed by pairs of nonempty
subsets $I,J\subseteq{\underline{n}}$ with $I\cap J=\emptyset$, and $I\cup
J={\underline{n}}$. The corresponding $2$-cube category $\lozenge^{\\{I,J\\}}$
has then the objects $\emptyset,I,J,I\cup J={\underline{n}}$.

The characterisation above shows as well that an $n$-fold vector bundle can be
naturally understood as its own $n$-core, since the partition of
${\underline{n}}$ in $n$ subsets gives with (3) the $n$-cube category
$\square^{n}$.

###### Proof of Proposition 4.4.

For $l=n$, the first statement is clearly true since there is only one
partition of ${\underline{n}}$ in $n$ subsets and
$\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ is its own $n$-core.

Choose a subset $J\subseteq{\underline{n}}$ of cardinality $2$. Then by
Theorem 2.20 in [HL20], the core $\mathbb{E}^{({\underline{n}},J)}$ is an
$(n-1)$-fold vector bundle

(19)
$\mathbb{E}^{J}:=\mathbb{E}^{({\underline{n}},J)}\colon\lozenge^{\rho_{J}}\to{\mathbf{Man}^{\infty}},$

where as before $\rho_{J}$ is the partition of ${\underline{n}}$ with $J$ and
one-element sets, hence a partition of ${\underline{n}}$ with $n-1$ subsets.
By definition there is a one-to-one correspondence between subsets of
${\underline{n}}$ of cardinality $2$ and $(n-1)$-cores, hence between
partitions of ${\underline{n}}$ with $n-1$ elements and $(n-1)$-cores. So (1)
is also true for $l=n-1$, while (2) is true for $l=n$ and $i=n-1$.

A partition of ${\underline{n}}$ in $n-2$ subsets has either one subset with
$3$ elements and $n-3$ subsets with one element each, or two subsets with $2$
elements each and $n-4$ subsets with one elements each. Of course, here $n\geq
3$ and if $n=3$ only the first case is possible, while for $n\geq 4$ both
cases are possible.

Consider first the first case, and assume without loss of generality that the
partition is $\rho:=\\{\\{1,2,3\\},\\{4\\},\ldots,\\{n\\}\\}$. Consider
$J_{1}:=\\{1,2\\}$ and $J_{2}:=\\{1,3\\}$ and the two $(n-1)$-cores
$\mathbb{E}^{J_{1}}$ and $\mathbb{E}^{J_{2}}$ as in (19). Then
$\lozenge^{\rho_{J_{1}}}$ is the cube category over the elements
$J_{1},\\{3\\},\ldots,\\{n\\}$, and $\lozenge^{\rho_{J_{2}}}$ is the cube
category over the elements $J_{2},\\{2\\},\\{4\\},\\{5\\},\ldots,\\{n\\}$.

Build the $\\{J_{1},\\{3\\}\\}$-core of $\mathbb{E}^{J_{1}}$, which is an
$(n-2)$-core of $\mathbb{E}^{J_{1}}$ and of $\mathbb{E}$. By definition, it is
modeled on the full subcategory $\lozenge$ of $\lozenge^{\rho_{J_{1}}}$ with
objects $J\subseteq n$ consisting in unions of the sets
$J_{1}\cup\\{3\\},\\{4\\},\ldots,\\{n\\}$. Hence $\lozenge=\lozenge^{\rho}$.
This shows the existence of an $(n-2)$-core modeled on $\rho$.

The $\\{J_{2},\\{2\\}\\}$-core of $\mathbb{E}^{J_{2}}$ is then similarly
modeled on the same category $\lozenge^{\rho}$. These two $(n-2)$-cores are
hence modeled on the same partition of ${\underline{n}}$. Choose
$I\in\operatorname{Obj}(\lozenge^{\rho})$. If $I\cap\\{1,2,3\\}=\emptyset$,
then also $I\cap J_{1}=I\cap J_{2}=\emptyset$ and

$(\mathbb{E}^{J_{1}})^{\\{J_{1},\\{3\\}\\}}(I)=\mathbb{E}^{J_{1}}(I)=\mathbb{E}(I)=\mathbb{E}^{J_{2}}(I)=(\mathbb{E}^{J_{2}})^{\\{J_{2},\\{2\\}\\}}(I).$

If $\\{1,2,3\\}\subseteq I$ then also $J_{1},J_{2}\subseteq I$ and so666There
is a little abuse of notation here; in the formula below $I$ should formally
be replaced by $\\{J_{1};\\{t\\}\mid t\in I\setminus J_{1}\\}$ and
$\\{J_{2};\\{t\\}\mid t\in I\setminus J_{2}\\}$, depending on which chain of
cores is considered. For the convenience of the reader, it is just written $I$
and understood from the context.

$\begin{split}(\mathbb{E}^{J_{1}})^{\\{J_{1},\\{3\\}\\}}(I)&=(\mathbb{E}^{J_{1}})^{I}_{\\{J_{1},\\{3\\}\\}}=(c^{I}_{J_{1}})^{-1}\left(\mathbf{0}^{I\setminus
J_{1}}_{I\setminus\\{1,2,3\\}}\right)\cap(p^{I}_{3})^{-1}\left(\mathbf{0}^{I\setminus\\{3\\}}_{I\setminus\\{1,2,3\\}}\right)\\\
&=\mathbb{E}^{I}_{\\{1,2,3\\}}\\\
&=(c^{I}_{J_{2}})^{-1}\left(\mathbf{0}^{I\setminus
J_{2}}_{I\setminus\\{1,2,3\\}}\right)\cap(p^{I}_{2})^{-1}\left(\mathbf{0}^{I\setminus\\{2\\}}_{I\setminus\\{1,2,3\\}}\right)\\\
&=(\mathbb{E}^{J_{2}})^{I}_{\\{J_{2},\\{2\\}\\}}=(\mathbb{E}^{J_{2}})^{\\{J_{2},\\{2\\}\\}}(I),\end{split}$

where the third and fourth equalities are applications of [HL20, Lemma 2.19].
Since the projections in each $l$-core are restrictions of the projections of
the $n$-fold vector bundle $\mathbb{E}$, it suffices to show that the images
of the two functors $(\mathbb{E}^{J_{1}})^{\\{J_{1},\\{3\\}\\}}$ and
$(\mathbb{E}^{J_{2}})^{\\{J_{2},\\{2\\}\\}}$ on objects of $\lozenge^{\rho}$
are the same to get that the two functors are equal.

Next assume that $n\geq 4$ and consider the partition
$\rho:=\\{\\{1,2\\},\\{3,4\\},\\{5\\},\ldots,\\{n\\}\\}$ of ${\underline{n}}$
in $(n-2)$ elements. Set $J_{1}:=\\{1,2\\}$ and $J_{2}:=\\{3,4\\}$ and
consider as before the two $(n-1)$-cores $\mathbb{E}^{J_{1}}$ and
$\mathbb{E}^{J_{2}}$ of $\mathbb{E}$. Then $\lozenge^{\rho_{J_{1}}}$ is the
square category over the elements $J_{1},\\{3\\},\ldots,\\{n\\}$, and
$\lozenge^{\rho_{J_{2}}}$ is the square category over the elements
$\\{1\\},\\{2\\},J_{2},\\{5\\},\\{6\\},\ldots,\\{n\\}$.

Build the $\\{\\{3\\},\\{4\\}\\}$-core of $\mathbb{E}^{J_{1}}$, which is an
$(n-2)$-core of $\mathbb{E}^{J_{1}}$ and of $\mathbb{E}$. By definition, it is
modeled on the full subcategory $\lozenge$ of $\lozenge^{\rho_{J_{1}}}$ with
objects $I\subseteq n$ consisting in unions of the sets
$J_{1},J_{2},\\{5\\},\\{6\\},\ldots,\\{n\\}$. Hence
$\lozenge=\lozenge^{\rho}$. This shows the existence of an $(n-2)$-core
modeled on $\rho$.

The $\\{\\{1\\},\\{2\\}\\}$-core of $\mathbb{E}^{J_{2}}$ is then similarly
modeled on the same category $\lozenge^{\rho}$. These two $(n-2)$-cores are
hence modeled on the same partition of ${\underline{n}}$. In order to show
that

$(\mathbb{E}^{J_{1}})^{\\{\\{3\\},\\{4\\}\\}}=(\mathbb{E}^{J_{2}})^{\\{\\{1\\},\\{2\\}\\}}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}},$

it is again enough to show that the functors are equal on objects of
$\lozenge^{\rho}$. First take $I\in\operatorname{Obj}(\lozenge^{\rho})$ with
$I\cap J_{1}=\emptyset$ and $I\cap J_{2}=\emptyset$. Then

$(\mathbb{E}^{J_{1}})^{\\{\\{3\\},\\{4\\}\\}}(I)=\mathbb{E}^{J_{1}}(I)=\mathbb{E}(I)=\mathbb{E}^{J_{2}}(I)=(\mathbb{E}^{J_{2}})^{\\{\\{1\\},\\{2\\}\\}}(I).$

If $I\in\operatorname{Obj}(\lozenge^{\rho})$ does $J_{1}\subseteq I$ and
$I\cap J_{2}=\emptyset$, then

$(\mathbb{E}^{J_{1}})^{\\{\\{3\\},\\{4\\}\\}}(I)=\mathbb{E}^{J_{1}}(I)=E^{I}_{J_{1}}$

while

$(\mathbb{E}^{J_{2}})^{\\{\\{1\\},\\{2\\}\\}}(I)=(\mathbb{E}^{J_{2}})^{I}_{\\{\\{1\\},\\{2\\}\\}}=E^{I}_{J_{1}}$

since the $I$-face of $\mathbb{E}^{J_{2}}$ is the $I$-face of $\mathbb{E}$.
Finally if $J_{1}\cup J_{2}\subseteq I$, then

$\begin{split}(\mathbb{E}^{J_{1}})^{\\{\\{3\\},\\{4\\}\\}}(I)&=(\mathbb{E}^{J_{1}})^{I}_{\\{\\{3\\},\\{4\\}\\}}\\\
&=\left\\{e\in\mathbb{E}^{J_{1}}(I)\left|p^{I}_{I\setminus\\{3\\}}(e)=\mathbf{0}^{I\setminus\\{3\\}}_{p^{I}_{I\setminus
J_{2}}(e)}\text{ and
}p^{I}_{I\setminus\\{4\\}}(e)=\mathbf{0}^{I\setminus\\{4\\}}_{p^{I}_{I\setminus
J_{2}}(e)}\right.\right\\}\\\
&=\left\\{e\in\mathbb{E}(I)\left|\begin{array}[]{c}p^{I}_{I\setminus\\{1\\}}(e)=\mathbf{0}^{I\setminus\\{1\\}}_{p^{I}_{I\setminus
J_{1}}(e)},\quad
p^{I}_{I\setminus\\{2\\}}(e)=\mathbf{0}^{I\setminus\\{2\\}}_{p^{I}_{I\setminus
J_{1}}(e)}\\\
p^{I}_{I\setminus\\{3\\}}(e)=\mathbf{0}^{I\setminus\\{3\\}}_{p^{I}_{I\setminus
J_{2}}(e)}\text{ and
}p^{I}_{I\setminus\\{4\\}}(e)=\mathbf{0}^{I\setminus\\{4\\}}_{p^{I}_{I\setminus
J_{2}}(e)}\end{array}\right.\right\\}\\\
&=(\mathbb{E}^{J_{2}})^{\\{\\{1\\},\\{2\\}\\}}(I),\end{split}$

since the description in the third line is symmetric in $J_{1}$ and $J_{2}$.

Given an arbitrary $(n-1)$-core $\mathbb{E}^{J}$ for $J\subset{\underline{n}}$
with $\\#J=2$, its $(n-2)$-cores are either
$(\mathbb{E}^{J})^{\\{\\{i\\},\\{j\\}\\}}$ for $i,j\in{\underline{n}}\setminus
J$ or $(\mathbb{E}^{J})^{\\{J,\\{i\\}\\}}$ for $i\in{\underline{n}}\setminus
J$. As explained above, in the first case the $(n-2)$-core is indexed by the
partition $\\{J,\\{i,j\\};\\{t\\}\mid
t\in{\underline{n}}\setminus(J\cup\\{i,j\\})\\}$, and in the second case on
the partition $\\{J\cup\\{i\\};\\{t\\}\mid
t\in{\underline{n}}\setminus(J\cup\\{i\\})\\}$.

Hence (1) is proved for $l=n-2$. (2) is true for $l=n$ and $i=n-1,n-2$, and
$l=n-1$ and $i=n-2$. (3) for $l=n$ was proved above at the same time as the
equality of the two $(n-2)$-cores indexed by the same partition, since in the
first case

$\\{\\{1,2,3\\},\\{4\\},\ldots,\\{n\\}\\}=\\{J_{1},\\{3\\},\ldots,\\{n\\}\\}\sqcap\\{J_{2},\\{2\\},\\{4\\},\\{5\\},\ldots,\\{n\\}\\}.$

and in the second case

$\\{\\{1,2\\},\\{3,4\\},\\{5\\},\ldots,\\{n\\}\\}=\\{J_{1},\\{3\\},\ldots,\\{n\\}\\}\sqcap\\{\\{1\\},\\{2\\},J_{2},\\{5\\},\\{6\\},\ldots,\\{n\\}\\}.$

The proof now works recursively since an $l$-core of $\mathbb{E}$ is always an
$l$-core of an $(l+1)$-core of $\mathbb{E}$: Assume that (1) is true for some
fixed $l\in\\{2,\ldots,n\\}$. Then an $(l-1)$-core $\mathbb{F}$ of
$\mathbb{E}$ is an $(l-1)$-core of an $l$-core of $\mathbb{E}$, hence of an
$l$-core $\mathbb{E}^{\rho}$ for some partition
$\rho=\\{I_{1},\ldots,I_{l}\\}$ of ${\underline{n}}$. The $(l-1)$-core
$\mathbb{F}$ is then defined by a choice of $i<j$ in $\\{1,\ldots,l\\}$ such
that

$\mathbb{F}=(\mathbb{E}^{\rho})^{\\{I_{i},I_{j}\\}}.$

$\mathbb{F}$ is then modeled on $\rho^{\prime}=\\{I_{i}\cup
I_{j},I_{1},\ldots,\widehat{i},\ldots,\widehat{j},\ldots,I_{l}\\}$, which is
an $(l-1)$-partition of ${\underline{n}}$. Conversely, choose a partition
$\rho=\\{I_{1},\ldots,I_{l-1}\\}$ of ${\underline{n}}$ in $(l-1)$ subsets.
Then since $l-1\in\\{1,\ldots,n-2\\}$ one of the subsets
$I_{1},\ldots,I_{l-1}$ of ${\underline{n}}$ must contain more than one
element. Without generality, $I_{l-1}$ does. Write $I_{l-1}=J_{l-1}\cup J_{l}$
with $J_{l-1},J_{l}\subseteq{\underline{n}}$ disjoint and non-empty. Set
$\rho^{\prime}=\\{I_{1},\ldots,I_{l-2},J_{l-1},J_{l}\\}$. Then

$(\mathbb{E}^{\rho^{\prime}})^{\\{J_{l-1},J_{l}\\}}$

is an $(l-1)$-core of $\mathbb{E}$ indexed by $\rho$. Computations as above
show that this $(l-1)$-core of $\mathbb{E}$ does not depend on the choice of
$\rho^{\prime}$ above $\rho$. Precisely, the considerations above show that
for $I=I_{i_{1}}\cup\ldots\cup
I_{i_{k}}\in\operatorname{Obj}(\lozenge^{\rho})$,

(20)
$\begin{split}\mathbb{E}^{\rho}(I)&=\left\\{e\in\mathbb{E}(I)\left|\begin{array}[]{c}\text{
For }s=1,\ldots,k\text{ and all }j\in I_{i_{s}}:\\\
p^{I}_{I\setminus\\{j\\}}(e)=\mathbf{0}^{I\setminus\\{j\\}}_{p^{I}_{I\setminus
I_{i_{s}}}(e)}.\end{array}\right.\right\\}\\\ &=\bigcap_{s=1}^{k}\bigcap_{j\in
I_{i_{s}}}(p^{I}_{I\setminus\\{j\\}})^{-1}\left(\mathbf{0}^{I\setminus\\{j\\}}_{I\setminus
I_{i_{s}}}\right)\ \end{split}$

which does not depend on the choice of $\rho^{\prime}$. Therefore (1) holds
for all $l=1,\ldots,n$. By construction (2) holds as well for all
$l=1,\ldots,n$ and all $i=1,\ldots,l$, and (3) follows again from the one-to-
one correspondence between $l$-partitions and $l$-cores, for all
$l=1,\ldots,n$. ∎

######  Corollary 4.6.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle and let $\rho=\\{I_{1},\ldots,I_{l}\\}$ be a partition of
${\underline{n}}$ in $l$ elements. The lower sides of $\mathbb{E}^{\rho}$ are
the vector bundles $\mathbb{E}^{I_{i}}_{I_{i}}\to M$ for $i=1,\ldots,l$, and
the other building bundles of $\mathbb{E}^{\rho}$ are the bundles
$\mathbb{E}_{J}^{J}$ for $J\in\operatorname{Obj}(\lozenge^{\rho})$.

###### Proof.

The first statement is clear since $\lozenge^{\rho}$ is the $l$-cube category
over the elements $I_{1},\ldots,I_{l}$. It is also a special case of the
second statement. For the second statement, choose
$J\in\operatorname{Obj}(\lozenge^{\rho})$, without loss of generality
$J=I_{1}\cup\ldots\cup I_{k}$ for $1\leq k\leq l$ and compute with (20)

$\begin{split}(\mathbb{E}^{\rho})^{J}_{J}&=\left\\{e\in\mathbb{E}^{\rho}(J)\left|\text{
for all }s=1,\ldots,k,\,\,p^{J}_{J\setminus I_{s}}(e)=\mathbf{0}^{J\setminus
I_{s}}_{p^{J}_{\emptyset}(e)}\right.\right\\}\\\
&=\left\\{e\in\mathbb{E}(J)\left|\begin{array}[]{c}\text{ For
}s=1,\ldots,k\text{ and all }j\in I_{s}:\\\
p^{J}_{J\setminus\\{j\\}}(e)=\mathbf{0}^{J\setminus\\{j\\}}_{p^{J}_{J\setminus
J_{s}}(e)}\\\ \text{ for all }s=1,\ldots,k,\,\,p^{J}_{J\setminus
I_{s}}(e)=\mathbf{0}^{J\setminus
I_{s}}_{p^{J}_{\emptyset}(e)}\end{array}\right.\right\\}\\\
&=\left\\{e\in\mathbb{E}(J)\left|\begin{array}[]{c}\text{ For all }j\in
J:\,\,p^{J}_{J\setminus\\{j\\}}(e)=\mathbf{0}^{J\setminus\\{j\\}}_{p^{J}_{\emptyset}(e)}\\\
\end{array}\right.\right\\}=\mathbb{E}^{J}_{J}.\end{split}$

∎

######  Lemma 4.7.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle and let $\rho$ be a partition of ${\underline{n}}$ in $l$
elements.

  1. (1)

For each $J\in\operatorname{Obj}(\lozenge^{\rho})$ the manifold
$\mathbb{E}^{\rho}(J)$ is an embedded submanifold of $\mathbb{E}(J)$ and the
collection of these embeddings defines a natural transformation

$\tau^{\rho}\colon\mathbb{E}^{\rho}\longrightarrow\mathbb{E}\circ i^{\rho}$

since

  2. (2)

for each arrow $(J\rightarrow
J^{\prime})\in\operatorname{Mor}(\lozenge^{\rho})$ the arrow
$\mathbb{E}^{\rho}(J\rightarrow J^{\prime})$ is the restriction777Therefore,
for simplicity, the restriction $\mathbb{E}^{\rho}(J\to
J^{\prime})\colon\mathbb{E}^{\rho}(J)\to\mathbb{E}^{\rho}(J^{\prime})$ of
$\mathbb{E}(J\to J^{\prime})$ is written as well $p^{J}_{J^{\prime}}$ in the
following. of $\mathbb{E}(J\rightarrow
J^{\prime})=p^{J}_{J^{\prime}}\colon\mathbb{E}(J)\to\mathbb{E}(J^{\prime})$ to
the domain and codomain
$\mathbb{E}^{\rho}(J)\rightarrow\mathbb{E}^{\rho}(J^{\prime})$.

  3. (3)

Let $\mathbb{F}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be a second
$n$-fold vector bundle and $\Phi\colon\mathbb{E}\to\mathbb{F}$ a morphism of
$n$-fold vector bundles. Then $\Phi$ restricts to a morphism
$\Phi^{\rho}\colon\mathbb{E}^{\rho}\to\mathbb{F}^{\rho}$ of $l$-fold vector
bundles. Precisely, for each $J\in\operatorname{Obj}(\lozenge^{\rho})$

$\Phi^{\rho}(J)=\Phi(J)\arrowvert_{\mathbb{E}^{\rho}(J)}\colon\mathbb{E}^{\rho}(J)\to\mathbb{F}^{\rho}(J)\subseteq\mathbb{F}(J)$

is the restriction on the embedded submanifold $\mathbb{E}^{\rho}(J)$ of the
domain $\mathbb{E}(J)$ of $\Phi(J)$ and with restricted codomain the embedded
submanifold $\mathbb{F}^{\rho}(J)\subseteq\mathbb{F}(J)$.

###### Proof.

This is immediate from the definition of the highest order cores in (15) and
(16) and the recursive construction of the iterated highest order cores, as
well as the recursive restriction of morphisms as in (17). ∎

######  Example 4.8.

Consider as before an $n$-fold vector bundle $\mathbb{E}$ and the induced
collection of vector bundles

$\mathcal{A}=\left\\{\mathbb{E}_{K}^{K}\mid
K\subseteq{\underline{n}}\right\\}.$

A partition $\rho$ of ${\underline{n}}$ defines then a $\\#\rho$-fold vector
bundle

$\mathbb{E}^{\mathcal{A},\rho}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}},\qquad\mathbb{E}^{\mathcal{A},\rho}(J):=\prod^{M}_{\begin{subarray}{c}I\subseteq
J\\\ I\in\operatorname{Obj}(\lozenge^{\rho})\end{subarray}}\mathbb{E}^{I}_{I}$

for $J\in\operatorname{Obj}(\lozenge^{\rho})$, and a vacant $\\#\rho$-fold
vector bundle

$\overline{\mathbb{E}^{\rho}}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}},\qquad\overline{\mathbb{E}^{\rho}}(J):=\prod^{M}_{\begin{subarray}{c}I\subseteq
J\\\ I\in\rho\end{subarray}}\mathbb{E}^{I}_{I}$

for $J\in\operatorname{Obj}(\lozenge^{\rho})$. Here
$\overline{\mathbb{E}^{\rho}}(J)$ is clearly embedded in
$\mathbb{E}^{\mathcal{A},\rho}(J)$, which is embedded in
$\mathbb{E}^{\mathcal{A}}(J)$ for all
$J\in\operatorname{Obj}(\lozenge^{\rho})$ and the collection of these
embeddings define natural transformations

$\begin{split}\overline{\mathbb{E}^{\rho}}\overset{\overline{\tau}^{\rho}}{\longrightarrow}\mathbb{E}^{\mathcal{A},\rho}\overset{\tau^{\mathcal{A},\rho}}{\longrightarrow}\mathbb{E}^{\mathcal{A}}\circ
i^{\rho}.\end{split}$

(Recall that $i^{\rho}\colon\lozenge^{\rho}\to\square^{n}$ is the inclusion
functor.)

The $l$-cores of $\mathbb{E}^{\mathcal{A}}$ coincide with the functors
$\mathbb{E}^{\mathcal{A},\rho}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}}$
given by partitions $\rho$ of ${\underline{n}}$ with $l$ elements. A
decomposition $\mathcal{S}\colon\mathbb{E}^{\mathcal{A}}\to\mathbb{E}$ of
$\mathbb{E}$ restricts as in Lemma 4.7 to a decomposition
$\mathcal{S}^{\rho}\colon\mathbb{E}^{\mathcal{A},\rho}\to\mathbb{E}^{\rho}$ of
the $l$-core $\mathbb{E}^{\rho}$ such that the following diagram of natural
transformations commutes.

${{\mathbb{E}^{\mathcal{A}}\circ i^{\rho}}}$${{\mathbb{E}\circ
i^{\rho}}}$${{\mathbb{E}^{\mathcal{A},\rho}}}$${{\mathbb{E}^{\rho}}}$$\scriptstyle{\mathcal{S}^{\rho}}$$\scriptstyle{\mathcal{S}\arrowvert_{\lozenge^{\rho}}}$$\scriptstyle{\tau^{\mathcal{A},\rho}}$$\scriptstyle{\tau^{\rho}}$

Consider an $n$-fold vector bundle $\mathbb{E}$. Choose a partition
$\rho=\\{I_{1},\ldots,I_{l}\\}$ of ${\underline{n}}$ in $l$ subsets and
consider two distinct coarsements $\rho_{1}$ and $\rho_{2}$ of $\rho$ in
$(l-1)$ subsets. Let
$\mathcal{S}^{i}\colon\mathbb{E}^{\mathcal{A},\rho_{i}}\to\mathbb{E}^{\rho_{i}}$
be a decomposition of the $(l-1)$-core $\mathbb{E}^{\rho_{i}}$ for $i=1,2$.
The two decompositions $\mathcal{S}^{\rho_{1}}$ and $\mathcal{S}^{\rho_{2}}$
are compatible if

(21)
$(\mathcal{S}^{\rho_{1}})^{\rho_{1}\sqcap\rho_{2}}=(\mathcal{S}^{\rho_{2}})^{\rho_{1}\sqcap\rho_{2}}\colon\mathbb{E}^{\mathcal{A},\rho_{1}\sqcap\rho_{2}}\rightarrow\mathbb{E}^{\rho_{1}\sqcap\rho_{2}}.$

Take now more precisely the coarsement $\underline{\rho}:=\\{I_{1}\cup
I_{2},I_{3},\ldots,I_{4}\\}$ of $\rho$. Then
$(\mathbb{E}^{\rho})_{\\{I_{3},\ldots,I_{4}\\}}$ is a face of
$\mathbb{E}^{\rho}$ and of $\mathbb{E}^{\underline{\rho}}$. Consider a linear
splitting $\Sigma\colon\overline{\mathbb{E}^{\rho}}\to\mathbb{E}^{\rho}$ of
$\mathbb{E}^{\rho}$. Then $\Sigma$ is compatible with a decomposition
$\mathcal{S}^{\underline{\rho}}$ of $\mathbb{E}^{\underline{\rho}}$ if the
following diagram commutes

(22)
${{(\overline{\mathbb{E}^{\underline{\rho}}})_{\\{I_{3},\ldots,I_{4}\\}}}}$${{(\mathbb{E}^{\underline{\rho}})_{\\{I_{3},\ldots,I_{4}\\}}}}$${{(\overline{\mathbb{E}^{\rho}})_{\\{I_{3},\ldots,I_{4}\\}}}}$${{(\mathbb{E}^{\rho})_{\\{I_{3},\ldots,I_{4}\\}}}}$$\scriptstyle{(\mathcal{S}^{\underline{\rho}}\circ\iota)\arrowvert_{\\{I_{3},\ldots,I_{4}\\}}}$$\scriptstyle{\Sigma\arrowvert_{\\{I_{3},\ldots,I_{4}\\}}}$

where $\iota$ is the canonical natural transformation

$\iota\colon\overline{\mathbb{E}^{\underline{\rho}}}\to\mathbb{E}^{\mathcal{A},\underline{\rho}}.$

The compatibility of $\Sigma$ with a decomposition of
$\mathbb{E}^{\underline{\rho}}$ for any other coarsement $\underline{\rho}$ of
$\rho$ with $(l-1)$ elements is defined similarly.

###  4.4. Proof of Corollary 3.6 in [HL20]

The following theorem is proved in [HL20, Theorem 3.3].

######  Theorem 4.9.

  1. (a)

Let $\mathcal{S}$ be a decomposition of an $n$-fold vector bundle
$\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$. Then the composition
$\Sigma=\mathcal{S}\circ\tau\colon\overline{\mathbb{E}}\to\mathbb{E}$, with
$\tau$ defined as in (18), is a splitting of $\mathbb{E}$. Furthermore, the
core morphisms
$\mathcal{S}^{\rho_{J}}\colon\mathbb{E}^{\mathcal{A},\rho_{J}}\to\mathbb{E}^{\rho_{J}}$
are decompositions of $\mathbb{E}^{{\underline{n}}}_{J}$ for all
$J\subseteq{\underline{n}}$ with $\\#J=2$ and these decompositions and the
linear splitting are compatible.

  2. (b)

Conversely, given a linear splitting $\Sigma$ of $\mathbb{E}$ and
compatible888Pairwise compatible and all compatible with $\Sigma$.
decompositions of the $(n-1)$-cores $\mathbb{E}^{\rho_{J}}$ for
$J\subseteq{\underline{n}}$ with $\\#J=2$, there exists a unique decomposition
$\mathcal{S}$ of $\mathbb{E}$ such that $\Sigma=\mathcal{S}\circ\tau$ and such
that the core morphisms of $\mathcal{S}$ are given by
$\mathcal{S}^{\rho_{J}}=\mathcal{S}^{J}$ for all $J$.

A symmetric version of this result is proved later on (see Proposition B.1)
and the details of the proof of this theorem are discussed in Appendix A.

Theorem 3.5 in [HL20] states then that for each $n$-fold vector bundle
$\mathbb{E}$, there is a linear splitting

$\Sigma\colon\overline{\mathbb{E}}\to\mathbb{E}\,,$

that is a monomorphism of $n$-fold vector bundles from the vacant, decomposed
$n$-fold vector bundle $\overline{\mathbb{E}}$ associated to $\mathbb{E}$.
[HL20] shows this result by proving inductively (over $n$) the following two
claims:

  1. (a)

Given an $n$-fold vector bundle $\mathbb{E}$, there exist $n$ linear
splittings $\Sigma_{{\underline{n}}\setminus\\{k\\}}$ of
$\mathbb{E}_{{\underline{n}}\setminus\\{k\\}}$ for $k\in{\underline{n}}$, such
that
$\Sigma_{{\underline{n}}\setminus\\{i\\}}(I)=\Sigma_{{\underline{n}}\setminus\\{j\\}}(I)$
for any $I\subseteq{\underline{n}}\setminus\\{i,j\\}$, i.e. such that

$\Sigma_{{\underline{n}}\setminus\\{i\\}}\arrowvert_{{\underline{n}}\setminus\\{i,j\\}}=\Sigma_{{\underline{n}}\setminus\\{j\\}}\arrowvert_{{\underline{n}}\setminus\\{i,j\\}}$

for all $i,j\in{\underline{n}}$.

  2. (b)

Given a family of splittings as in (a), there exists a linear splitting
$\Sigma$ of $\mathbb{E}$ with

$\Sigma\arrowvert_{{\underline{n}}\setminus\\{k\\}}=\Sigma_{{\underline{n}}\setminus\\{k\\}}$

for each $k\in{\underline{n}}$.

The following proposition arises as a corollary of the second claim, which is
the missing piece to the proof of Corollary 3.6 in [HL20].

######  Proposition 4.10.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be an $n$-fold
vector bundle. Assume that each $(n-1)$-core
$\mathbb{E}^{\rho_{J}}\colon\lozenge^{\rho_{J}}\to{\mathbf{Man}^{\infty}}$ of
$\mathbb{E}$, for $J\subseteq{\underline{n}}$ with $\\#J=2$, has a
decomposition
$\mathcal{S}^{J}\colon\mathbb{E}^{\mathcal{A},\rho_{J}}\to\mathbb{E}^{\rho_{J}}$,
such that for all $J_{1},J_{2}\subseteq{\underline{n}}$ with
$\\#J_{1}=\\#J_{2}=2$ the decompositions $\mathcal{S}^{J_{1}}$ and
$\mathcal{S}^{J_{2}}$ are compatible as in (21). Then there exists a splitting

$\Sigma\colon\overline{\mathbb{E}}\to\mathbb{E}$

of $\mathbb{E}$ that is compatible with all decompositions $\mathcal{S}^{J}$
as in (22): for all $J\subseteq{\underline{n}}$ with $\\#J=2$,

(23) $\Sigma\arrowvert_{{\underline{n}}\setminus
J}=\mathcal{S}^{J}\arrowvert_{{\underline{n}}\setminus
J}\circ\tau\arrowvert_{{\underline{n}}\setminus
J}\colon\overline{\mathbb{E}}_{{\underline{n}}\setminus
J}\to\mathbb{E}_{{\underline{n}}\setminus J}.$

###### Proof.

For each $J\subseteq{\underline{n}}$ with $\\#J=2$, the decomposition
$\mathcal{S}^{J}$ of $\mathbb{E}^{\rho_{J}}$ defines a decomposition
$\mathcal{S}^{J}\arrowvert_{{\underline{n}}\setminus J}$ of
$\mathbb{E}_{{\underline{n}}\setminus J}$ since
$\mathbb{E}_{{\underline{n}}\setminus J}$ is a face of
$\mathbb{E}^{\rho_{J}}$. Denote by
$\Sigma^{J}\colon\overline{\mathbb{E}}_{{\underline{n}}\setminus
J}\to\mathbb{E}_{{\underline{n}}\setminus J}$ the induced linear splitting of
$\mathbb{E}_{{\underline{n}}\setminus J}$, i.e.

$\Sigma^{J}:=\mathcal{S}^{J}\arrowvert_{{\underline{n}}\setminus
J}\circ\tau\arrowvert_{{\underline{n}}\setminus
J}\colon\overline{\mathbb{E}}_{{\underline{n}}\setminus
J}\rightarrow\mathbb{E}_{{\underline{n}}\setminus J}.$

Consider $i\in{\underline{n}}$. Then the $(n-2)$-fold vector bundles
$\mathbb{E}_{{\underline{n}}\setminus J}$, for $i\in
J\subseteq{\underline{n}}$ with $\\#J=2$, are sides of the $(n-1)$-fold vector
bundle $\mathbb{E}_{{\underline{n}}\setminus\\{i\\}}$. Choose two such subsets
$J_{1}$ and $J_{2}$. Then $\rho_{J_{1}}\sqcap\rho_{J_{2}}=\\{J_{1}\cup
J_{2};\\{t\\}\mid t\in{\underline{n}}\setminus(J_{1}\cup J_{2})\\}$. Since
$\mathcal{S}^{J_{1}}$ and $\mathcal{S}^{J_{2}}$ coincide on the common core
$\mathbb{E}^{\rho_{J_{1}}\sqcap\rho_{J_{2}}}$ of $\mathbb{E}^{J_{1}}$ and
$\mathbb{E}^{J_{2}}$, they coincide in particular on its face
$\mathbb{E}_{{\underline{n}}\setminus(J_{1}\cup J_{2})}$. Hence
$\mathcal{S}^{J_{1}}\arrowvert_{{\underline{n}}\setminus(J_{1}\cup
J_{2})}=\mathcal{S}^{J_{2}}\arrowvert_{{\underline{n}}\setminus(J_{1}\cup
J_{2})}$ and the induced splittings $\Sigma^{J_{1}}$ and $\Sigma^{J_{2}}$
satisfy consequently

$\Sigma^{J_{1}}(I)=\Sigma^{J_{2}}(I)$

for all $I\subseteq({\underline{n}}\setminus
J_{1})\cap({\underline{n}}\setminus J_{2})={\underline{n}}\setminus(J_{1}\cup
J_{2})$.

That is, the splittings $\Sigma^{J}$ for $\\#J=2$ and $i\in J$ satisfy (a)
above. By (b) there is consequently a linear splitting

$\Sigma_{{\underline{n}}\setminus\\{i\\}}\colon\overline{\mathbb{E}}_{{\underline{n}}\setminus\\{i\\}}\to\mathbb{E}_{{\underline{n}}\setminus\\{i\\}}$

of $\mathbb{E}_{{\underline{n}}\setminus\\{i\\}}$, such that

$\Sigma_{{\underline{n}}\setminus\\{i\\}}\arrowvert_{{\underline{n}}\setminus
J}=\Sigma^{J}.$

for all $J\subseteq{\underline{n}}$ with $\\#J=2$ and $i\in J$. Now choose
$K\subseteq({\underline{n}}\setminus\\{i\\})\cap({\underline{n}}\setminus\\{j\\})={\underline{n}}\setminus\\{i,j\\}$
for $i\neq j\in{\underline{n}}$. Set $I:=\\{i,j\\}$. Then

$\Sigma_{{\underline{n}}\setminus\\{i\\}}(K)=\Sigma^{I}(K)=\Sigma_{{\underline{n}}\setminus\\{j\\}}(K)$

shows that (a) then still holds for the $(n-1)$-sides
$\mathbb{E}_{{\underline{n}}\setminus\\{i\\}}$ of $\mathbb{E}$, for all
$i\in{\underline{n}}$. As a consequence, there exists with (b) a linear
splitting

$\Sigma\colon\overline{\mathbb{E}}\to\mathbb{E}$

of $\mathbb{E}$ with

$\Sigma\arrowvert_{{\underline{n}}\setminus\\{i\\}}=\Sigma_{{\underline{n}}\setminus\\{i\\}}$

for $i=1,\ldots,n$. The compatibility in (23) with the decompositions of the
highest order cores is immediate by construction. ∎

Now the complete proof of [HL20, Corollary 3.6] can be given.

######  Corollary 4.11.

Every $n$-fold vector bundle $\mathbb{E}$ is non-canonically isomorphic to the
associated decomposed $n$-fold vector bundle $\mathbb{E}^{\mathcal{A}}$.

###### Proof.

First consider all $2$-cores of $\mathbb{E}$. As explained in Proposition 4.4,
these are indexed by all possible partitions of ${\underline{n}}$ in two (non-
empty) subsets. Consider such a partition $\rho=\\{I,{\underline{n}}\setminus
I\\}$ for $\emptyset\neq I\subsetneq{\underline{n}}$ and choose a linear
decomposition

${{\mathbb{E}_{I}^{I}\times_{M}\mathbb{E}^{{\underline{n}}\setminus
I}_{{\underline{n}}\setminus
I}\times_{M}\mathbb{E}_{{\underline{n}}}^{{\underline{n}}}}}$${{\mathbb{E}^{\rho}}}$${{\mathbb{E}_{{\underline{n}}\setminus
I}^{{\underline{n}}\setminus I}}}$${{\mathbb{E}_{{\underline{n}}\setminus
I}^{{\underline{n}}\setminus
I}}}$${{\mathbb{E}_{I}^{I}}}$${{\mathbb{E}_{I}^{I}}}$${M}$${M}$$\scriptstyle{\mathcal{S}^{\rho}}$$\scriptstyle{\operatorname{id}\qquad}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$

of the $2$-core $\mathbb{E}^{\rho}$, which is a double vector bundle with
sides $\mathbb{E}_{I}^{I}$ and $\mathbb{E}^{{\underline{n}}\setminus
I}_{{\underline{n}}\setminus I}$ and with core
$\mathbb{E}_{{\underline{n}}}^{{\underline{n}}}$. (Double vector bundles
always admit decompositions, see [GR09, dCM15]. See also the introduction of
[HL20] for more historical details.)

Choose now for each partition $\rho=\\{I_{1},I_{2},I_{3}\\}$ of
${\underline{n}}$ in three subsets a linear splitting
$\Sigma^{\rho}\colon\overline{\mathbb{E}^{\rho}}\to\mathbb{E}^{\rho}$ of the
3-core $\mathbb{E}^{\rho}$ of $\mathbb{E}$. Then $\Sigma^{\rho}$ is
_automatically_ compatible with the decompositions

$\mathcal{S}^{\\{I_{1}\cup I_{2},I_{3}\\}},\quad\mathcal{S}^{\\{I_{1}\cup
I_{3},I_{2}\\}},\quad\text{ and }\quad\mathcal{S}^{\\{I_{2}\cup
I_{3},I_{1}\\}}$

as in (22) since (22) reads here

$\Sigma^{\rho}(I_{i})=\mathcal{S}^{\\{I_{j}\cup
I_{k},I_{i}\\}}(I_{i})\colon\mathbb{E}_{I_{i}}^{I_{i}}\to\mathbb{E}_{I_{i}}^{I_{i}},$

for $\\{i,j,k\\}=\\{1,2,3\\}$, which is immediate since both
$\Sigma^{\rho}(I_{i})$ and $\mathcal{S}^{\\{I_{j}\cup I_{k},I_{i}\\}}(I_{i})$
are the identity on $\mathbb{E}_{I_{i}}^{I_{i}}$. Secondly, the decompositions
$\mathcal{S}^{\\{I_{1}\cup I_{2},I_{3}\\}}$, $\mathcal{S}^{\\{I_{1}\cup
I_{3},I_{2}\\}}$ and $\mathcal{S}^{\\{I_{2}\cup I_{3},I_{1}\\}}$ of the
$2$-cores of $\mathbb{E}^{\rho}$ are compatible since e.g.

$\\{I_{1}\cup I_{2},I_{3}\\}\sqcap\\{I_{1}\cup
I_{3},I_{2}\\}=\\{{\underline{n}}\\},$

so (21) reduces here to

$\left(\mathcal{S}^{\\{I_{1}\cup
I_{2},I_{3}\\}}\right)^{\\{{\underline{n}}\\}}=\left(\mathcal{S}^{\\{I_{1}\cup
I_{3},I_{2}\\}}\right)^{\\{{\underline{n}}\\}}\colon\mathbb{E}_{{\underline{n}}}^{{\underline{n}}}\to\mathbb{E}_{{\underline{n}}}^{{\underline{n}}},$

which again is immediate since both decompositions restrict to the identity on
the common core $\mathbb{E}_{{\underline{n}}}^{{\underline{n}}}$ of
$\mathbb{E}^{\\{I_{1}\cup I_{2},I_{3}\\}}$ and $\mathbb{E}^{\\{I_{1}\cup
I_{3},I_{2}\\}}$.

By Theorem 4.9, there exists a decomposition $\mathcal{S}^{\rho}$ of
$\mathbb{E}^{\rho}$ that restricts to $\Sigma^{\rho}$ and to
$\mathcal{S}^{\\{I_{1}\cup I_{2},I_{3}\\}}$, $\mathcal{S}^{\\{I_{1}\cup
I_{3},I_{2}\\}}$ and $\mathcal{S}^{\\{I_{2}\cup I_{3},I_{1}\\}}$ on the
$2$-cores of $\mathbb{E}^{\rho}$. Since $\rho$ was an arbitrary partition of
${\underline{n}}$ in three subsets, this yields decompositions of all
$3$-cores, such that the restrictions to two refinements of a same $2$-core
are automatically compatible: If $\rho_{1}$ and $\rho_{2}$ are two
3-partitions of ${\underline{n}}$ such that $\rho_{1}\sqcap\rho_{2}$ is a
2-partition of ${\underline{n}}$, then $\rho_{1}\sqcap\rho_{2}$ indexes the
common $2$-core of $\mathbb{E}^{\rho_{1}}$ and $\mathbb{E}^{\rho_{2}}$, and by
the construction above of $\mathcal{S}^{\rho_{1}}$ and
$\mathcal{S}^{\rho_{2}}$

$\left(\mathcal{S}^{\rho_{1}}\right)^{\rho_{1}\sqcap\rho_{2}}=\mathcal{S}^{\rho_{1}\sqcap\rho_{2}}=\left(\mathcal{S}^{\rho_{2}}\right)^{\rho_{1}\sqcap\rho_{2}}.$

Use Proposition 4.10 and choose linear splittings of the $4$-cores which are
compatible with all decompositions of the corresponding $3$-cores. By Theorem
4.9 there are decompositions of the $4$-cores with are compatible with the
chosen splittings and with all decompositions of the $3$-cores constructed
above. As above, the decompositions of the $4$-cores are therefore
automatically compatible, and there exist by Proposition 4.10 linear
splittings of the $5$-cores that are compatible with the decompositions of the
$4$-cores. Repeat these steps until a decomposition of the $n$-core
$\mathbb{E}$ is constructed. ∎

###  4.5. Atlases of multiple vector bundles

[HL20] shows that an $n$-fold vector bundle can be equivalently defined as a
smooth manifold endowed with an $n$-fold vector bundle atlas.

The following definition [HL20] is a straightforward generalisation of
Pradines’ definition of a double vector bundle atlas [Pra77].

######  Definition 4.12.

Let $M$ be a smooth manifold and $E$ a topological space together with a
continuous map $\pi\colon E\to M$. An n-fold vector bundle chart on $E$ is a
tuple

$c=(U,\Theta,(V_{I})_{\emptyset\neq I\subseteq{\underline{n}}}),$

where $U$ is an open set in $M$, for each $\emptyset\neq
I\subseteq{\underline{n}}$ the space $V_{I}$ is a fixed (finite dimensional)
real vector space, and $\Theta\colon\pi^{-1}(U)\to U\times\prod_{\emptyset\neq
I\subseteq\bar{n}}V_{I}$ is a homeomorphism such that
$\pi=\operatorname{pr}_{1}\circ\Theta$.

Two $n$-fold vector bundle charts $c$ and $c^{\prime}$ are smoothly compatible
if the “change of chart” $\Theta^{\prime}\circ\Theta^{-1}$ over $U\cap
U^{\prime}$ has the following form999Hence if it is a morphism of decomposed
$n$-fold vector bundles, see [HL20].:

$\bigl{(}p,(v_{I})_{\emptyset\neq
I\subseteq{\underline{n}}}\bigr{)}\mapsto\left(p,\left(\sum_{\rho=(I_{1},\ldots,I_{k})\in\mathcal{P}(I)}\omega_{\rho}(p)(v_{I_{1}},\ldots,v_{I_{k}})\right)_{\emptyset\neq
I\subseteq{\underline{n}}}\right)$

with $p\in U\cap U^{\prime}$, $v_{I}\in V_{I}$ and $\omega_{\rho}\in
C^{\infty}(U\cap U^{\prime},\operatorname{Hom}(V_{I_{1}}\otimes\ldots\otimes
V_{I_{k}},V_{I}))$ for $\rho=(I_{1},\ldots,I_{k})\in\mathcal{P}(I)$.

A smooth n-fold vector bundle atlas $\mathfrak{A}$ on $E$ is a set of n-fold
vector bundle charts of $E$ that are pairwise smoothly compatible and such
that the set of underlying open sets in $M$ is a covering of $M$. As usual,
$E$ is then a smooth manifold and two smooth $n$-fold vector bundle atlases
$\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ are equivalent if their union is a
smooth n-fold vector bundle atlas. A smooth $n$-fold vector bundle structure
on $E$ is an equivalence class of smooth n-fold vector bundle atlases on $E$.
The pair of $E$ and a smooth $n$-fold vector bundle structure on $E$ is
written $\mathbb{E}$.

Consider a smooth $n$-fold vector bundle atlas, and write
$\\{U_{\alpha}\\}_{\alpha\in\Lambda}$ for the underlying open covering of $M$.
For $\alpha,\beta,\gamma\in\Lambda$ the identity
$\Theta_{\gamma}\circ\Theta_{\alpha}^{-1}=\Theta_{\gamma}\circ\Theta_{\beta}^{-1}\circ\Theta_{\beta}\circ\Theta_{\alpha}^{-1}$
on $\pi^{-1}(U_{\alpha}\cap U_{\beta}\cap U_{\gamma})$ yields the following
cocycle conditions. For $\emptyset\neq I\subseteq{\underline{n}}$ and
$\rho=(I_{1},\ldots,I_{k})\in\mathcal{P}(I)$:

(24)
$\begin{split}\omega^{\gamma\alpha}_{\rho}&(p)(v_{I_{1}},\ldots,v_{I_{k}})=\\\
&\sum_{(J_{1},\ldots,J_{l})\in\operatorname{coars}(\rho)}\omega^{\gamma\beta}_{(J_{1},\ldots,J_{l})}(p)\Bigl{(}\omega^{\beta\alpha}_{\rho\cap
J_{1}}(p)\bigl{(}(v_{I})_{I\in\rho\cap
J_{1}}\bigr{)},\ldots,\omega^{\beta\alpha}_{\rho\cap
J_{l}}(p)\bigl{(}(v_{I})_{I\in\rho\cap J_{l}}\bigr{)}\Bigr{)}\,.\end{split}$

Since the involved constructions need to be refined, this sections recalls in
detail the proof of Corollary 3.10 in [HL20]:

######  Corollary 4.13.

Definition 4.1 of an $n$-fold vector bundle as a functor from the $n$-cube
category is equivalent to Definition 4.12 of an $n$-fold vector bundle as a
space with a maximal $n$-fold vector bundle atlas.

###### Proof.

Let $\mathbb{E}$ be an $n$-fold vector bundle. By Corollary 4.11 (see also
Theorem 3.2 and Theorem 3.3 in [HL20]), there is a decomposition
$\mathcal{S}\colon\mathbb{E}^{\rm dec}\to\mathbb{E}$ of $\mathbb{E}$, with
$\mathbb{E}^{\rm dec}$ the decomposed $n$-fold vector bundle defined by the
family $(\mathbb{E}^{I}_{I})_{I\subseteq{\underline{n}}}$ of vector bundles
over $M$. Set $E:=\mathbb{E}({\underline{n}})$, as usual
$M:=\mathbb{E}(\emptyset)$, and
$\pi=\mathbb{E}({\underline{n}}\to\emptyset)\colon E\to M$. For each
$\emptyset\neq I\subseteq{\underline{n}}$, set $V_{I}:=\mathbb{R}^{\dim
E^{I}_{I}}$, the vector space on which $E^{I}_{I}$ is modelled. Take a
covering $\\{U_{\alpha}\\}_{\alpha\in\Lambda}$ of $M$ by open sets
trivialising all the vector bundles $E^{I}_{I}$;

$\phi^{\alpha}_{I}\colon
q_{I}^{-1}(U_{\alpha})\overset{\sim}{\longrightarrow}U_{\alpha}\times V_{I}$

for all $\emptyset\neq I\subseteq{\underline{n}}$ and all $\alpha\in\Lambda$,
where $q_{I}\colon E^{I}_{I}\to M$ is the projection (the restriction of
$p^{I}_{\emptyset}$). Then define $n$-fold vector bundle charts
$\Theta_{\alpha}\colon\pi^{-1}(U_{\alpha})\to
U_{\alpha}\times\prod_{I\subseteq{\underline{n}}}V_{I}$ by

$\Theta_{\alpha}=\left(\pi\times\left(\phi^{\alpha}_{I}\right)_{I\subseteq{\underline{n}}}\right)\circ\mathcal{S}({\underline{n}})^{-1}\arrowvert_{\pi^{-1}(U_{\alpha})}\colon\pi^{-1}(U_{\alpha})\to
U_{\alpha}\times\prod_{\emptyset\neq I\subseteq{\underline{n}}}V_{I}.$

Given $\alpha,\beta\in\Lambda$ with $U_{\alpha}\cap U_{\beta}\neq\emptyset$,
the change of chart

$\Theta_{\alpha}\circ\Theta_{\beta}^{-1}\colon(U_{\alpha}\cap
U_{\beta})\times\prod_{\emptyset\neq
I\subseteq{\underline{n}}}V_{I}\to(U_{\alpha}\cap
U_{\beta})\times\prod_{\emptyset\neq I\subseteq{\underline{n}}}V_{I}$

is given by

(25) $\left(p,(v_{I})_{\emptyset\neq
I\subseteq{\underline{n}}}\right)\mapsto\left(p,(\rho^{\alpha\beta}_{I}(p)v_{I})_{\emptyset\neq
I\subseteq{\underline{n}}}\right),$

with $\rho_{I}^{\alpha\beta}\in C^{\infty}(U_{\alpha}\cap
U_{\beta},\operatorname{Gl}(V_{I}))$ the cocycle defined by
$\phi^{\alpha}_{I}\circ(\phi_{I}^{\beta})^{-1}$. The two charts are hence
smoothly compatible. Hence his defines an $n$-fold vector bundle atlas
$\mathfrak{A}=\\{(U_{\alpha},\Theta_{\alpha},(V_{I})_{I\subseteq{\underline{n}}})\mid\alpha\in\Lambda\\}$
on $E$.

Conversely, given a space $E$ with an $n$-fold vector bundle structure over a
smooth manifold $M$ as in Definition 4.12, define
$\mathbb{E}\colon\square^{\mathbb{N}}\to{\mathbf{Man}^{\infty}}$ as follows.
Take a maximal atlas
$\mathfrak{A}=\\{(U_{\alpha},\Theta_{\alpha},(V_{I})_{I\subseteq{\underline{n}}})\mid\alpha\in\Lambda\\}$
of $E$; in particular $\\{U_{\alpha}\\}_{\alpha\in\Lambda}$ is an open cover
of $M$. For $\alpha,\beta,\gamma\in\Lambda$ the identity
$\Theta_{\gamma}\circ\Theta_{\alpha}^{-1}=\Theta_{\gamma}\circ\Theta_{\beta}^{-1}\circ\Theta_{\beta}\circ\Theta_{\alpha}^{-1}$
on $\pi^{-1}(U_{\alpha}\cap U_{\beta}\cap U_{\gamma})$ yields the cocycle
conditions (24).

Set $\mathbb{E}({\underline{n}}):=E$, $\mathbb{E}(\emptyset):=M$, and more
generally for $\emptyset\neq I\subseteq{\underline{n}}$,

$\mathbb{E}(I)=\left.\left(\bigsqcup_{\alpha\in\Lambda}\left(U_{\alpha}\times\prod_{\emptyset\neq
J\subseteq I}V_{J}\right)\right)\right/\sim$

with $\sim$ the equivalence relation defined on
$\bigsqcup_{\alpha\in\Lambda}(U_{\alpha}\times\prod_{\emptyset\neq J\subseteq
I}V_{J})$ by

$U_{\alpha}\times\prod_{\emptyset\neq J\subseteq
I}V_{J}\quad\ni\quad\left(p,(v_{J})_{\emptyset\neq J\subseteq
I}\right)\quad\sim\quad\left(q,(w_{J})_{\emptyset\neq J\subseteq
I}\right)\quad\in\quad U_{\beta}\times\prod_{\emptyset\neq J\subseteq I}V_{J}$

if and only if $p=q$ and

$(v_{J})_{\emptyset\neq J\subseteq
I}=\left(\sum_{\rho=(J_{1},\ldots,J_{k})\in\mathcal{P}(J)}\omega^{\alpha\beta}_{\rho}(p)(w_{J_{1}},\ldots,w_{J_{k}})\right)_{\emptyset\neq
J\subseteq I}.$

The relations (24) show the symmetry and transitivity of this relation. As in
the construction of a vector bundle from vector bundle cocycles,
$\mathbb{E}(I)$ has a unique smooth manifold structure such that
$\pi_{I}\colon\mathbb{E}(I)\to M$, $\pi_{I}[p,(v_{I})_{\emptyset\neq
I\subseteq J}]=p$ is a surjective submersion and such that the maps

$\Theta^{I}_{\alpha}\colon\operatorname{pr}_{I}\left(U_{\alpha}\times\prod_{\emptyset\neq
J\subseteq I}V_{J}\right)\to U_{\alpha}\times\prod_{\emptyset\neq J\subseteq
I}V_{J},\qquad\left[p,(v_{I})_{\emptyset\neq I\subseteq
J}\right]\mapsto\left(p,(v_{I})_{\emptyset\neq I\subseteq J}\right)$

are diffeomorphisms, where
$\operatorname{pr}_{I}\colon\bigsqcup_{\alpha\in\Lambda}(U_{\alpha}\times\prod_{\emptyset\neq
J\subseteq I}V_{J})\to\mathbb{E}(I)$ is the projection to the equivalence
classes.

$\mathbb{E}(I)$ comes equipped with $\\#I$ surjective submersions

$p^{I}_{I\setminus\\{i\\}}\colon\mathbb{E}(I)\to\mathbb{E}(I\setminus\\{i\\})$

for $i\in I$, defined in charts by

$U_{\alpha}\times\prod_{\emptyset\neq J\subseteq
I}V_{J}\,\ni\,\left(p,(v_{J})_{\emptyset\neq J\subseteq