# speechocean762: An Open-Source Non-native English Speech Corpus For
Pronunciation Assessment

###### Abstract

This paper introduces a new open-source speech corpus named “speechocean762”
designed for pronunciation assessment use, consisting of 5000 English
utterances from 250 non-native speakers, where half of the speakers are
children. Five experts annotated each of the utterances at sentence-level,
word-level and phoneme-level. A baseline system is released in open source to
illustrate the phoneme-level pronunciation assessment workflow on this corpus.
This corpus is allowed to be used freely for commercial and non-commercial
purposes. It is available for free download from OpenSLR, and the
corresponding baseline system is published in the Kaldi speech recognition
toolkit.

Index Terms: corpus, computer-assisted language learning (CALL), second
language (L2)

##  1 Introduction

As an indispensable part of Computer-aided language learning (CALL), computer-
aided pronunciation training (CAPT) applications with pronunciation assessment
technology are widely used in foreign language learning [1, 2] and proficiency
tests [3]. CAPT has been proved very useful to improve the pronunciation of
the foreign language learners [4]. Due to the acute shortage of qualified
teachers [5] and the increasing popularity of online learning, the research of
pronunciation assessment is being paid more attention [6].

According to the real-world CAPT applications' features, we divide the
practical pronunciation assessment tasks into three categories by the
assessment granularity: sentence-level, word-level, and phoneme-level. The
sentence-level assessment evaluates the whole sentence. Specifically, three
types of sentence-level scores frequently appear in practical CAPT systems:
accuracy, completeness, and fluency. The accuracy indicates the level of the
learner pronounce each word in the utterance correctly; the completeness
indicates the percentage of the words that are actually pronounced, and the
fluency here is in the narrow sense[7], which focuses on whether the speaker
pronounces smoothly and without unnecessary pauses. The word-level assessment
has a finer scale than the sentence-level assessment. Typical word-level
scores are accuracy and stress. Furthermore, as the finest granularity
assessment, the phoneme-level assessment evaluates each phone's pronunciation
quality in the utterance. Note that the word-level accuracy score should not
be regarded as the simple average of the phone-level accuracy scores, although
they have strong correlations. Take the word ``above'' (/bv/) as an example. A
foreign language learner may mispronounce it as /bv/ (mispronounce // to // )
or as /kv/ (mispronounce /b/ to /k/). For the two incorrect pronunciations,
the numbers of the mispronounced phones are both one, but most people may
realize that the latter mispronunciation is worse than the former.

There are some public corpora for pronunciation assessment. The ISLE Speech
Corpus [8] is an early and widely accepted [9, 10, 11] data set. It contains
mispronunciation tags at the word and phoneme level, and the speakers are all
from German and Italian. It is free for academic use, but it is charged for
commercial use. ERJ [12] is another famous non-native English corpus for
pronunciation assessment, collected from 202 Japanese students annotated with
phonemic and prosodic symbols. ATR-Gruhn [13] is a non-native English corpus
with multiple accents. The annotations of ATR-Gruhn are speaker-level
proficiency ratings. TL-school [14] is a corpus of speech utterances collected
in northern Italy schools for assessing the performance of students learning
both English and German. The data set of a spoken CALL shared task [15] is
available to download, where Swiss students answer prompts in English, and the
students' responses are manually labeled as ``accept'' or ``reject''.
L2-ARCTIC [16] is a non-native English speech corpus with manual annotations,
which has been used in some recent studies [17, 18], and it uses substitution,
deletion, and insertion to annotate for the phoneme-level scoring. Sell-corpus
[19] is another multiple accented Chinese-English speech corpus with phoneme
substitution annotations. Some corpora, such as CU-CHLOE [20], Supra-CHLOE
[21] and COLSEC [22], have been used in many studies [23, 24, 25, 26] but are
not publicly available. Corpora for languages other than English also exist.
The Tokyo-Kikuko [27] is a non-native Japanese corpus with phonemic and
prosodic annotations. The iCALL corpus [28] is a Mandarin corpus spoken by
non-native speakers of European descent with annotated pronunciation errors.
The SingaKids-Mandarin [29] corpus focuses on mispronunciation patterns in
Singapore children’s Mandarin speech.

To our knowledge, none of the existing non-native English corpora for
pronunciation assessment contains all the following features:

  * •

It is available for free download for both commercial and non-commercial
purposes.

  * •

The speaker variety encompasses young children and adults.

  * •

The manual annotations are in many aspects at sentence-level, word-level and
phoneme-level.

To meet these features, we created this corpus to support researchers in their
pronunciation assessment studies. The corpus is available on the OpenSLR
111https://www.openslr.org/101 website, and the corresponding baseline system
has been a part of the Kaldi speech recognition toolkit
222https://github.com/kaldi-asr/kaldi/tree/master/egs/gop_speechocean762.

The rest of this paper is organized as follows: Section 2 describes the audio
acquisition. Section 3 details how we annotated the data for the pronunciation
assessment tasks. In Section 4, a Kaldi recipe for this corpus is introduced,
which illustrates how to do phoneme-level pronunciation assessment, and the
experiment results are provided as well.

##  2 Audio Acquisition

This corpus's text script is selected from daily life text, containing about
2,600 common English words. As shown in Figure 1, speakers were asked to hold
their mobile phones 20cm from their mouths and read the text as accurately as
possible in a quiet 3$\times$3 meters room. The mobile phones include the
popular models of Apple, Samsung, Xiaomi, and Huawei. The number of sentences
read aloud by each speaker is 20, and the total duration of the audio is about
6 hours.

The speakers are 250 English learners whose mother tongue is Mandarin. The
training set and test set are divided randomly, with 125 speakers for each.

We carefully selected the speakers considering gender, age and proficiency of
English. The experts roughly rated the speaker's English pronunciation
proficiency into three levels: good, average, and poor. Figure 2 shows the
distributions of the speaker's English pronunciation proficiency. Figure 3
shows the distributions of the speaker's age. The gender ratio is 1:1 for both
adults and children.

Figure 1: Recording setup. Speakers read the text holding their mobile phones
in a quiet room. Figure 2: Speaker's English pronunciation proficiency
distributions. Figure 3: Speaker's age distributions. Figure 4: The
``SpeechOcean uTrans'' Application. Before this dialog is displayed, the
experts have reached an agreement on the canonical phone sequences by voting.
For the phoneme-level scoring, the expert selects the phone symbol and then
makes a score of 0 or 1. If a phone symbol is not be selected, the score would
be 2 as the default.

##  3 Manual Annotation

Manual annotations are the essential part of this corpus. The annotations are
the scores that indicate the pronunciation quality. Each utterance in this
corpus is scored manually by five experts independently under the same
metrics.

###  3.1 Manual Scoring Metrics

The experts discussed and formulated the manual scoring metrics. Table 1 shows
the detailed metrics. The phoneme-level score is the pronunciation accuracy of
each phone. The word-level scores include accuracy and stress, and the
sentence-level scores include accuracy, completeness, fluency and prosody. The
sentence-level completeness score, which is not depicted in Table 1, is the
percentage of the words in the target text that are actually pronounced.

Table 1: Manual Scoring Metrics Score |  Description
---|---
|  Phoneme-level Accuracy
2 |  The phone is pronounced correctly
1 |  The phone is pronounced with a heavy accent
0 |  The pronunciation is incorrect or missed
|  Word-level Accuracy
10 |  The pronunciation of the whole word is correct
7-9 |  Most phones in the word are pronounced correctly, but the word's pronunciation has heavy accents
4-6 |  No more than 30% phones in the word are wrongly pronounced
2-3 |  More than 30% phones in the word are wrongly pronounced, or be mispronounced into some other word
0-1 |  The whole pronunciation is hard to distinguish or the word is missed
|  Word-level Stress
10 |  The stress position is correct, or the word is a mono-syllable word
5 |  The stress position is incorrect
|  Sentence-level Accuracy
9-10 |  The overall pronunciation of the sentence is excellent without obvious mispronunciation
7-8 |  The overall pronunciation of the sentence is good, with a few mispronunciations
5-6 |  The pronunciation of the sentence has many mispronunciations but it is still understandable
3-4 |  Awkward pronunciation with many serious mispronunciations
0-2 |  The pronunciation of the whole sentence is unable to understand or there is no voice
|  Sentence-level Fluency
8-10 |  Coherent speech, without noticeable pauses, repetition or stammering
6-7 |  Coherent speech in general, with a few pauses, repetition and stammering
4-5 |  The speech is incoherent, with many pauses, repetition and stammering
0-3 |  The speaker is not able to read the sentence as a whole or there is no voice
|  Sentence-level Prosodic
9-10 |  Correct intonation, stable speaking speed and rhythm
7-8 |  Nearly correct intonation at a stable speaking speed
3-6 |  Unstable speech speed, or the intonation is inappropriate
0-2 |  The reading of the sentence is too stammering to do prosodic scoring or there is no voice
Figure 5: Building LG directly for the word ``fast'' with the canonical phone
sequence voted by the experts, with skippable silence. Figure 6: The part
related of the word ``fast'' in L.

###  3.2 The Multiple Canonical Phone Sequences Problem

The phoneme-level scoring requires determining the canonical phone sequence. A
problem in practice is that the canonical phone sequence may not be unique.
Take the word ``fast'' as an example. In middle school, most Chinese students
were taught that this word should be pronounced as /f:st/, so a proper
canonical phone sequence is ``F AA S T'' with the phone set defined by the CMU
Dictionary [30]. However, some speakers may pronounce this word as /fæst/
following the American pronunciation. If that is the case, the phone ``AA'' in
the canonical phone sequence ``F AA S T'' would be misjudged as low score. The
proper canonical phone sequence, in this case, should be ``F AE S T''.

Our solution is as follows. For each word, experts will be shown several
possible canonical phone sequences before scoring. The expert must first
select the sequence that is closest to the pronunciation in her or his belief.
Since there are five experts, the sequence chosen by each expert may be
different, so the five experts vote to determine the final canonical sequence.
Then all the experts use the same canonical phone sequence to score. The
canonical phone sequences are carried as a part of the corpus's meta-
information.

###  3.3 Scoring Workflow

We developed an application named ``SpeechOcean uTrans'' for the experts to
convieniently score the audio. The interface of the application is shown in
Figure 4.

Before the scoring, the experts read the transcript and listen to the audio to
get familiar with the utterance. Then the experts are required to listen to
the audio repeatedly at least three times. As we mentioned, some words have
more than one canonical phone sequence. For those words, experts need to
choose and vote to reach an agreement on the canonical phone sequence. Then
the experts score the audio following the scoring metrics expressed in Table
1. If the scores seem unreasonable, for example, the word-level score is high
but all the phone-level scores are low, the ``SpeechOcean uTrans'' application
would raise a warning message to remind the expert to recheck the scores.

###  3.4 Score Distribution

Figure 7 shows the distribution of the sentence-level scores. The phoneme-
level and word-level score distributions are shown in the Figure 8, where the
phoneme-level scores are mapped linearly to the range 0 to 10 for comparison.
The sentence-level scores variety encompasses 3 to 10, while most of the word-
level and phoneme-level scores are from 8 to 10. This behaviour stems from the
fact that high sentence-level scores rely on a consistently ``good'' word and
phoneme pronouncation. Even a single word mispronunciation can lead to a low
overall score. Due to limited space, we suggest readers to refer to the
available online corpus to obtain the detailed statistics.

Figure 7: Sentence-level score distribution. Figure 8: Score distribution in
different levels.

##  4 The Kaldi Recipe

For demonstrating how to use this corpus to score at phoneme-level, we
uploaded a recipe named ``gop_speechocean762'' to the Kaldi toolkit.

###  4.1 Pipeline

We believe that the classical method is more suitable for building the
baseline system than the latest methods. So the pipeline is built following
the neural network (NN) based goodness of pronunciation (GOP) method, which is
widely used and detailed in [31]. Here we only represent some specifics of
implementing it on Kaldi. The GOP method requires a pre-trained acoustic model
trained by native spoken data, which is trained by the
``egs/librispeech/s5/local/nnet3/run_tdnn.sh'' script in Kaldi. The frame-
level posterior matrix is generated through forward propagation on the native
acoustic model, and the matrix is used for the forced alignment and the
computing to obtain the GOP values and the GOP-based features, whose
definitions could be found in [31] as well. Then we train a regressor for each
phone using the GOP-based features to predict the phoneme-level scores.

###  4.2 Alignment Graph Building without Lexicon

Kaldi's default alignment setup does not guarantee the alignment output to be
identical to the canonical phone sequence voted by the experts. We continue to
use the word ``fast'' as the example. The two possible phone sequences of this
word, which are ``F AA S T'' and ``F AE S T'' specifically, are both contained
in the lexicon finite state transducer (FST), shown in Figure 6. In that case,
the phone sequence produced by the alignment is uncertain. If the experts'
canonical phone sequence differs from the alignment result, the scores will
not be comparable with the manual scores.

Therefore, we build the lexicon-to-grammar (LG) FST directly using the
canonical phone sequence voted by the experts without composing the lexicon
FST and the grammar FST. The process of directly constructing LG is simple:
first, construct a linear FST structure, whose input labels are the canonical
phone sequences voted by the experts, whereas the output labels are the
corresponding words and epsilons [32]. Then, add skippable silence between the
words, and use the disambiguation symbol to construct the tail at the end of
LG, as shown in Figure 5.

###  4.3 Supervised Training and Data Balancing

With the GOP-based features and the corresponding manual scores, we train a
regressor for each mono phone. The model structure is a support vector
regressor (SVR) [33]. Besides, we train polynomial regression models with the
GOP values directly for each phone as an alternative lightweight method.

A problem is that the data's phoneme-level scores are quite unbalanced, as
discussed in Section 3.4. We use the high-score samples of other phones as the
current phone's low-score samples to supplement the training set to address
this issue. For example, a good pronunciation sample of the phone AE can be
considered as a poor pronunciation sample of the phone AA. For the model
training of a particular phone, we randomly select the samples of other phones
with high manual scores, setting their scores as zero and add them to the
training set.

###  4.4 Results

For evaluating the recipe's performance, we compare the predicted scores with
the manual scores to calculate the mean squared error (MSE) and Pearson
correlation coefficient (PCC). The result is shown in Table 2.

As a baseline system, this recipe is based on the classical NN-based GOP
method without using latest techniques. So the result is not quite strong,
which is in line with our expectations.

Table 2: Performance of the recipe | MSE | PCC
---|---|---
GOP value | 0.69 | 0.25
GOP-based feature | 0.16 | 0.45

##  5 Conclusions

We released an open-source corpus for pronunciation assessment tasks. The
corpus includes both child and adult speech and is manually annotated by five
experts. The annotations are at sentence-level, word-level and phoneme-level.
A Kaldi recipe is released to illustrate to use of the classic GOP method for
phoneme-level scoring. In the future, we will expand the recipe to word-level
and sentence-level scoring.

##  6 Acknowledgements

The authors would like to thank Jan Trmal for uploading this corpus to
OpenSLR. The authors would also like to thank Heinrich Dinkel and Qinghua Wu
for their helpful suggestions.

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L_{a_{2}^{-1}a_{3}^{-1}}R_{a_{2}}\widetilde{v}_{3}+L_{a_{2}^{-1}}\widetilde{v}_{2},R_{h_{2}^{-1}}w_{2}+L_{h_{2}}R_{h_{1}^{-1}h_{2}^{-1}}w_{1}\rangle-(v\leftrightarrow
w).\end{split}$

Therefore, by comparing them we see
$\bar{\omega}^{\mathfrak{h}}-\Phi_{2}^{*}\Omega=\delta\beta$ as desired. ∎

######  Remark 4.9.

It was proved in [88] that for any Manin triple
$(\mathfrak{g},\mathfrak{h}_{+}.\mathfrak{h}_{-})$ there is a $2$-shifted
Lagrangian correspondence

---
$\textstyle{pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}H_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}H_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}G}$

This is closely related to our result in this section.

##  5\. Double Lie group models of $\mathcal{B}G$

The simplicial picture introduced in the previous sections is not the only
available approach to describe $\mathcal{B}G$. In this section we will use the
language of double Lie groups to define other models for $\mathcal{B}G$ and
its symplectic structure.

###  5.1. Strict Lie 2-groups

A strict Lie $2$-group [13] is a group object in the category of Lie groupoids
and (strict) Lie groupoid morphisms, that is, it is a Lie groupoid
$G_{1}\Rightarrow G_{0}$ which equipped with a multiplication functor, an
inverse functor, and an identity functor, satisfying (strictly) associativity,
and other expected axioms for groups. It is well known (and not hard to see)
that a strict Lie $2$-group $G_{1}\Rightarrow G_{0}$ gives rise to a double
Lie group (defined in (4.1))

$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt.}$

Such a strict Lie 2-group is also a special case of a Lie 2-group which is
defined previously in Definition 2.1 using simplicial manifolds. A strict Lie
2-group $G_{1}\Rightarrow G_{0}$ gives rise to a Lie 2-group

$\cdots G_{0}\times
G_{0}\times_{m_{0},G_{0},{\mathsf{t}}}G_{1}\begin{subarray}{c}\longrightarrow\\\\[-8.50006pt]
\longrightarrow\\\\[-8.50006pt]
\longrightarrow\end{subarray}G_{0}\rightrightarrows pt,$

where $m_{0}:G_{0}\times G_{0}\to G_{0}$ is the 0-th level of the
multiplication functor. We refer to [104] for more details.

###  5.2. The de Rham triple complex of a double Lie group

Recall from Section 4.1 that a double Lie group has an associated bisimplicial
manifold $\mathcal{G}_{\bullet,\bullet}$. Therefore differential forms on
$\mathcal{G}_{\bullet,\bullet}$ live in the de Rham triple complex
$(\Omega^{\bullet}(\mathcal{G}_{\bullet,\bullet}),d,\delta^{v},\delta^{h})$
where $d:\Omega^{k}(\mathcal{G}_{j,i})\to\Omega^{k+1}(\mathcal{G}_{j,i})$ is
the usual de Rham differential and
$\delta^{v}:\Omega^{k}(\mathcal{G}_{j,i})\to\Omega^{k}(\mathcal{G}_{j+1,i}),\
\delta^{h}:\Omega^{k}(\mathcal{G}_{j,i})\to\Omega^{k}(\mathcal{G}_{j,i+1})$
are the simplicial differentials

$\delta^{h}=\sum_{l=0}^{i+1}d^{h*}_{l},\quad\delta^{v}=\sum_{l=0}^{j+1}d^{v*}_{l},\quad\text{
and }\quad\widetilde{D}=\delta^{h}+(-1)^{i}\delta^{v}+(-1)^{i+j}d$

is the differential on the total complex.

######  Definition 5.1.

A $(q,p)$-shifted $k$-form on a double Lie group
$\mathcal{G}_{\bullet,\bullet}$ is

$\alpha_{\bullet,\bullet}=\sum_{j=0}^{q}\sum_{i=0}^{p}\alpha_{j,i}\quad\text{with}\quad\alpha_{j,i}\in\Omega^{k+p+q-i-j}(\mathcal{G}_{j,i}).$

We say that $\alpha_{\bullet,\bullet}$ is closed if
$\tilde{D}\alpha_{\bullet,\bullet}=0$.

######  Remark 5.2.

Following [65] (see also [67]), we call the pair
$(\mathcal{G}_{\bullet,\bullet},\omega_{\bullet,\bullet})$ a symplectic double
Lie group if $\omega_{\bullet,\bullet}$ is a $(1,1)$-shifted $2$-form
satisfying

$\omega_{\bullet,\bullet}=\omega_{1,1}\in\Omega^{2}(\mathcal{G}_{1,1}),\quad\widetilde{D}\omega_{\bullet,\bullet}=0\quad\text{and}\quad\omega^{\sharp}_{1,1}:T^{*}\mathcal{G}_{1,1}\xrightarrow{\sim}T\mathcal{G}_{1,1}.$

As we will see, the models presented in this section are not symplectic double
Lie groups. Therefore a more general definition (that we will not introduce)
seems to be missing.

###  5.3. The models

If $G$ is a Lie group then the unit groupoid of $G$ as a strict Lie 2-group
gives rise to a double Lie group as described in Section 5.1. More concretely,
we have that

(5.1)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt}$

is a double Lie group, which we denote by $G_{\bullet,\bullet}$. The vertical
structure is given by the unit groupoid of $G$ and horizontal multiplication
is given by the multiplication on the Lie group $G$.

If $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ is a quadratic Lie algebra,
Theorem 3.1 states that $(NG_{\bullet},\Omega_{\bullet})$ is a $2$-shifted
symplectic Lie $1$-group. Clearly we can define a $(0,2)$-shifted $2$-form
$\Omega_{\bullet,\bullet}$ on $G_{\bullet,\bullet}$ by

$\Omega_{\bullet,\bullet}=\begin{pmatrix}0&0&0\\\
\Omega&-\Theta&0\end{pmatrix},\quad\text{with}\quad\Omega_{0,2}=\Omega\in\Omega^{2}(G^{\times
2})\text{ and }\Omega_{0,1}=-\Theta\in\Omega^{3}(G).$

######  Proposition 5.3.

The $(0,2)$-shifted $2$-form $\Omega_{\bullet,\bullet}$ is closed.

###### Proof.

This follows directly form the fact that $\Omega_{\bullet}$ is closed in
$NG_{\bullet}$, $\delta^{h}|_{j=0}$ is the simplicial differential of
$NG_{\bullet}$, and $\delta^{v}|_{i=0}=0$ since
$s^{v}=t^{v}=\operatorname{id}$. Hence

$\widetilde{D}(\Omega_{\bullet,\bullet})=(-1)^{i}\delta^{v}(\Omega_{0,\bullet})+D(\Omega_{\bullet})=0.$

∎

For a given Lie group $G$ we define the Lie groupoid $\Omega
G\rightrightarrows P_{e}G$ with structure maps

(5.2) $\displaystyle s(\tau)(t)=\tau(\frac{t}{2}),\quad
t(\tau)(t)=\tau(1-\frac{t}{2}),\quad i(\tau)(t)=\tau(1-t),$ (5.7)
$\displaystyle
m(\tau_{1},\tau_{2})(t)=\left\\{\begin{array}[]{ll}\tau_{2}(t)&t\in[0,\frac{1}{2}],\\\
\tau_{1}(t)&t\in[\frac{1}{2},1],\end{array}\right.\quad\text{and}\quad
u(\gamma)(t)=\left\\{\begin{array}[]{ll}\gamma(2t)&t\in[0,\frac{1}{2}],\\\
\gamma(1-2t)&t\in[\frac{1}{2},1].\end{array}\right.$

As in the previous section $\Omega G$ and $P_{e}G$ are completed in an
appropriate Sobolev norm so that the structure maps are smooth. Moreover, it
is not hard to verify that $\Omega G$ and $P_{e}G$ are infinite dimensional
Lie groups under the point-wise multiplication, and this makes $\Omega
G\rightrightarrows P_{e}G$ into a strict Lie 2-group.

######  Proposition 5.4.

For a Lie group $G$ we have the double Lie group
${\mathbb{G}}_{\bullet,\bullet}$ given by the square

(5.8) $\textstyle{\Omega
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{e}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{pt}$

with vertical structure maps defined by (5.2) and (5.7) and horizontal
multiplication given by

$m^{h}(\tau_{1},\tau_{2})(t)=\tau_{1}(t)\tau_{2}(t),\quad\forall\tau_{1},\tau_{2}\in\Omega
G\;\text{or}\;P_{e}G.$

###### Proof.

The fact that ${\mathbb{G}}_{\bullet,\bullet}$ is a double Lie group, i.e.
that the vertical source and target are group morphisms and that the
multiplications commute, follows by inspection. ∎

######  Remark 5.5.

When $G$ is connected and simply connected, the quotient stack $[P_{e}G/\Omega
G]\cong G$ is representable. Thus, the strict Lie 2-group $\Omega
G\rightrightarrows P_{e}G$ is Morita equivalent to the Lie 1-group
$NG_{\bullet}$. We may hence view both (5.1) and (5.8) as double Lie group
models for $\mathcal{B}G$. More explicitly, the Morita equivalence is given
through

(5.9) $\begin{array}[]{ccc}ev_{1,1}:\Omega G\to G,&&ev_{0,1}:P_{e}G\to G,\\\
ev_{1,1}(\tau)=\tau(\frac{1}{2}),&&ev_{0,1}(\gamma)=\gamma(1),\end{array}$

which may be further extended to a double Lie group morphism
$ev_{\bullet,\bullet}:{\mathbb{G}}_{\bullet,\bullet}\to G_{\bullet,\bullet}$.

When $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ is a quadratic Lie algebra,
the double Lie group ${\mathbb{G}}_{\bullet,\bullet}$ is endowed with the
Segal’s $2$-form $\omega\in\Omega^{2}(\Omega
G)=\Omega^{2}({\mathbb{G}}_{1,1})$ defined in (3.9). But $\omega$ can not be
multiplicative with respect to the group structure (otherwise it will be an
example of a symplectic group). Therefore, in order to obtain a closed form on
${\mathbb{G}}_{\bullet,\bullet}$ we need to introduce a new term. Define the
$1$-form

(5.10) $\eta_{(\tau_{1},\tau_{2})}((a_{1},a_{2}))=\int_{0}^{1}\langle
R_{\tau_{2}(t)^{-1}}\tau_{2}^{\prime}(t),\widehat{a}_{1}(t)\rangle dt\
\in\Omega^{1}(\Omega G^{\times 2})=\Omega^{1}({\mathbb{G}}_{1,2}),$

where $\tau_{i}\in\Omega G,\ a_{i}\in T_{\tau_{i}}\Omega G$, and
$\widehat{a}_{1}(t)=L_{\tau_{1}(t)^{-1}}a_{1}(t).$

######  Proposition 5.6.

The double Lie group ${\mathbb{G}}_{\bullet,\bullet}$ has a closed
$(1,2)$-shifted $1$-form $\omega_{\bullet,\bullet}$ defined by

$\omega_{\bullet,\bullet}=\begin{pmatrix}-\eta&\omega&0\\\
0&0&0\end{pmatrix}\quad\text{with}\quad\omega_{1,2}=-\eta\in\Omega^{1}({\mathbb{G}}_{1,2})\
\text{ and }\ \omega_{1,1}=\omega\in\Omega^{2}({\mathbb{G}}_{1,1}).$

###### Proof.

The result will follow from Theorem 5.8 which states that

$\omega_{\bullet,\bullet}=-\frac{1}{2}\big{(}\widetilde{D}(\alpha_{\bullet,\bullet})+ev^{*}_{\bullet,\bullet}\Omega_{\bullet,\bullet}\big{)},$

and the fact that $\Omega_{\bullet,\bullet}$ is closed by Proposition 5.3. ∎

It is quite surprising that in the double picture, we obtain a $1$-form (5.10)
instead of a $2$-form as in Theorem 3.15 of the simplicial picture, to bridge
the finite and infinite models. We believe that the term $\eta$ should be
related to the descent equations computed in [5].

###  5.4. The equivalence

Given a Lie group with a quadratic Lie algebra we create two different double
Lie groups endowed with differential forms
$(G_{\bullet,\bullet},\Omega_{\bullet,\bullet})$ and
$({\mathbb{G}}_{\bullet,\bullet},\omega_{\bullet,\bullet})$. Here we will show
that they are equivalent.

As stated in Remark 5.5, there is a double Lie group morphism
$ev_{\bullet,\bullet}:{\mathbb{G}}_{\bullet,\bullet}\to G_{\bullet,\bullet}$
given by (5.9). As in the simplicial case, the forms
$\omega_{\bullet,\bullet}$ and
$ev^{*}_{\bullet,\bullet}\Omega_{\bullet,\bullet}$ do not agree. So we need to
introduce another form on ${\mathbb{G}}_{\bullet,\bullet}$, which we denote
$\alpha_{\bullet,\bullet}$. The $(1,2)$-shifted $0$-form
$\alpha_{\bullet,\bullet}$ is defined by

$\alpha_{\bullet,\bullet}=\begin{pmatrix}0&-\alpha&0\\\
-\operatorname{\mathbb{T}}(\Omega)&-\operatorname{\mathbb{T}}(\Theta)&0\end{pmatrix}\quad\text{where}\quad\alpha=\int_{0}^{1}\langle
L_{\tau(t)^{-1}}\tau^{\prime}(t),L_{\tau(t)^{-1}}v(t)\rangle\in\Omega^{1}(\Omega
G)$

and $\operatorname{\mathbb{T}}(\Omega)\in\Omega^{1}(P_{e}G^{\times
2})=\Omega^{1}((P_{e}G)^{\times 2})=\Omega^{1}({\mathbb{G}}_{0,2})$, and
$\operatorname{\mathbb{T}}(\Theta)\in\Omega^{2}(P_{e}G)=\Omega^{2}({\mathbb{G}}_{0,1})$
are the transgressions of the forms on $NG_{\bullet}$ introduced in (3.1).

######  Remark 5.7.

The $1$-form $\alpha\in\Omega^{1}(\Omega G)$ also has a nice interpretation in
terms of the $S^{1}$-action on $\Omega G$. The based loop group $\Omega G$
carries an $S^{1}$-action given by rotation of loops and we denote its
infinitesimal generator by $X_{S^{1}}(\tau)=\tau^{\prime}.$ Then the fixed
points of the action correspond to the critical points of the function
$\alpha(X_{S^{1}})$.

######  Theorem 5.8.

The evaluation map $ev_{\bullet,\bullet}:{\mathbb{G}}_{\bullet,\bullet}\to
G_{\bullet,\bullet}$ satisfy

$-\omega_{\bullet,\bullet}-\frac{1}{2}ev^{*}_{\bullet,\bullet}\Omega_{\bullet,\bullet}=\widetilde{D}(\frac{1}{2}\alpha_{\bullet,\bullet}).$

###### Proof.

In order to prove this result, we need to show the following equality between
matrices,

$\begin{pmatrix}0&0&0&0\\\ 0&\eta&-\omega&0\\\
0&-\frac{1}{2}ev_{0,2}^{*}\Omega&\frac{1}{2}ev^{*}_{0,1}\Theta&0\end{pmatrix}=\frac{1}{2}\begin{pmatrix}0&0&\delta^{v}\alpha&0\\\
0&-\delta^{h}\alpha-\delta^{v}\operatorname{\mathbb{T}}(\Omega)&\delta^{v}\operatorname{\mathbb{T}}(\Theta)-d\alpha&0\\\
-\delta^{h}\operatorname{\mathbb{T}}(\Omega)&-\delta^{h}\operatorname{\mathbb{T}}(\Theta)-d\operatorname{\mathbb{T}}(\Omega)&d\operatorname{\mathbb{T}}(\Theta)&0\end{pmatrix}.$

For the equality in the first row we need to compute
$\delta^{v}\alpha\in\Omega^{1}({\mathbb{G}}_{2,1})$ and show that it is zero.
Recall that ${\mathbb{G}}_{2,1}=\Omega G\times_{P_{e}G}\Omega G$, hence we
pick $\tau_{1},\tau_{2}\in\Omega G$ with
$\tau_{1}(\frac{t}{2})=\tau_{2}(1-\frac{t}{2})$ and $a_{i}\in
T_{\tau_{i}}\Omega G$ also with $a_{1}(\frac{t}{2})=a_{2}(1-\frac{t}{2})$.
Then

$\begin{split}&(\delta^{v}\alpha)_{(\tau_{1},\tau_{2})}(a_{1},a_{2})=\alpha_{\tau_{2}}(a_{2})-\alpha_{m^{v}(\tau_{1},\tau_{2})}(Tm^{v}(a_{1},a_{2}))+\alpha_{\tau_{1}}(a_{1})\\\
=&\alpha_{\tau_{2}}(a_{2})+\alpha_{\tau_{1}}(a_{1})-\int_{0}^{\frac{1}{2}}\langle
L_{\tau_{2}(t)^{-1}}\tau_{2}^{\prime}(t),L_{\tau_{2}(t)^{-1}}a_{2}(t)\rangle-\int_{\frac{1}{2}}^{1}\langle
L_{\tau_{1}(t)^{-1}}\tau_{1}^{\prime}(t),L_{\tau_{1}(t)^{-1}}a_{1}(t)\rangle\\\
=&\int_{\frac{1}{2}}^{1}\langle
L_{\tau_{2}(t)^{-1}}\tau_{2}^{\prime}(t),L_{\tau_{2}(t)^{-1}}a_{2}(t)\rangle+\int_{0}^{\frac{1}{2}}\langle
L_{\tau_{1}(t)^{-1}}\tau_{1}^{\prime}(t),L_{\tau_{1}(t)^{-1}}a_{1}(t)\rangle=0.\end{split}$

In order to verify the first equality in the middle row, we need to make
explicit computations of the differentials. We start by computing
$\delta^{h}\alpha\in\Omega^{1}({\mathbb{G}}_{1,2})$. Since
${\mathbb{G}}_{1,2}=\Omega G^{\times 2}$, we pick $\tau_{1},\tau_{2}\in\Omega
G$ and $a_{i}\in T_{\tau_{i}}\Omega G$. Then

(5.11)
$\begin{split}(\delta^{h}\alpha)_{(\tau_{1},\tau_{2})}((a_{1},a_{2}))=&\
\alpha_{\tau_{2}}(a_{2})-\alpha_{\tau_{1}\tau_{2}}(R_{\tau_{2}}a_{1}+L_{\tau_{1}}a_{2})+\alpha_{\tau_{1}}(a_{1})\\\
=&-\int_{0}^{1}\langle
L_{\tau_{1}^{-1}(t)}\tau_{1}^{\prime}(t),R_{\tau_{2}^{-1}(t)}a_{2}(t)\rangle-\eta_{(\tau_{1},\tau_{2})}((a_{1},a_{2})).\end{split}$

An easy computation shows that
$\delta^{v}\operatorname{\mathbb{T}}(\Omega)=\operatorname{\mathbb{T}}(\Omega)_{|\Omega
G}$ and by Proposition D.2 and the formula (3.2) we get that

(5.12)
$\begin{split}\delta^{v}\operatorname{\mathbb{T}}(\Omega)_{(\tau_{1},\tau_{2})}((a_{1},a_{2}))=&\operatorname{\mathbb{T}}(\Omega)_{(\tau_{1},\tau_{2})}((a_{1},a_{2}))=\int_{0}^{1}\Omega_{(\tau_{1},\tau_{2})}((\tau_{1}^{\prime},\tau_{2}^{\prime}),(a_{1},a_{2}))\\\
=&\int_{0}^{1}\langle
L_{\tau_{1}^{-1}(t)}\tau_{1}^{\prime}(t),R_{\tau_{2}^{-1}(t)}a_{2}(t)\rangle-\eta_{(\tau_{1},\tau_{2})}((a_{1},a_{2})).\end{split}$

Hence combining (5.11) and (5.12), we obtain

$\frac{1}{2}(-\delta^{v}\alpha-\delta^{h}\operatorname{\mathbb{T}}(\Omega))=\eta.$

For the second term in the middle row, we use again the fact that
$\delta^{v}\operatorname{\mathbb{T}}(\nu)=\operatorname{\mathbb{T}}(\nu)_{|\Omega
G}$, and by Lemma 3.14 we obtain that

$-\omega=-\omega^{P}|_{\Omega
G}=\frac{1}{2}(\operatorname{\mathbb{T}}(\Theta)-d\alpha^{P})|_{\Omega
G}=\frac{1}{2}(\delta^{v}\operatorname{\mathbb{T}}(\Theta)-d\alpha).$

Finally, the equality on the last row follows directly from the properties of
the transgression given in Propositions D.1 and D.3, the fact that
$ev_{0,1}(\gamma)=\gamma(1)$ and
$ev_{0,2}(\gamma_{1},\gamma_{2})=(\gamma_{1}(1),\gamma_{2}(1))$ and that
$D\Omega_{\bullet}=0$. Explicitly

$\displaystyle\delta^{h}\operatorname{\mathbb{T}}(\Omega)$ $\displaystyle=$
$\displaystyle-\operatorname{\mathbb{T}}(\delta\Omega)=0,$
$\displaystyle-\frac{1}{2}\delta^{h}\operatorname{\mathbb{T}}(\Theta)-\frac{1}{2}d\operatorname{\mathbb{T}}(\Omega)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\operatorname{\mathbb{T}}(d\Omega)-\frac{1}{2}ev_{1}^{*}\Omega+\frac{1}{2}\operatorname{\mathbb{T}}(d\Omega)=-\frac{1}{2}ev_{0,2}^{*}\Omega,$
$\displaystyle\frac{1}{2}d\operatorname{\mathbb{T}}(\Theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}ev^{*}_{1}\Theta-\frac{1}{2}\operatorname{\mathbb{T}}(d\Theta)=\frac{1}{2}ev^{*}_{0,1}\Theta.$

Therefore the two matrices coincides in all the entries, and we have proved
the statement. ∎

##  Appendix A Sobolev spaces

Here we recall some analytic facts about Sobolev spaces used in this article
(see also [29, Sect.4], [11, Sect.14]). For a finite-dimensional compact
manifold $M$, possibly with boundary and corners, let $H_{r}(M)$ denote the
order $r$ Sobolev space of functions. These functions and their weak
derivatives are $L^{2}$-functions. The $C^{\infty}$ functions are dense in
$H_{r}(M)$. The point-wise multiplication makes $H_{r}(M)$ a Banach algebra
when $r-\frac{1}{2}\dim M>0$ [2, Theorem 4.39]. If $Z\subset M$ is a
submanifold, and $r-\frac{1}{2}\operatorname{codim}(Z)>0$, then the
restriction of continuous functions from $M$ to $Z$ extends to a continuous
linear map $H_{r}(M)\to H_{r-\frac{1}{2}\operatorname{codim}(Z)}(Z)$, with a
continuous right inverse.

If $N$ is another finite-dimensional manifold and $r-\frac{1}{2}\dim(M)>0$,
one defines spaces $\operatorname{Hom}_{r}(M,N)$ of maps from $M$ to $N$ of
Sobolev class $r$ by choosing local charts for $N$. In particular, if $G$ is a
finite-dimensional Lie group and $r-\frac{1}{2}\dim M>0$, then
$\operatorname{Hom}_{r}(M,G)$ is a Banach (even Hilbert) Lie group under
point-wise multiplication [29, Sect.4].

We are particularly interested in the loop group
$LG:=\operatorname{Hom}_{r}(S^{1},G)$ and the based loop group $\Omega
G:=\\{\tau\in LG|\tau(0)=e\\}$ for a fixed $r\in\mathbb{Z}^{\geq 1}$. In
addition to the advantage of having a Banach manifold, using the version of
based loops with Sobolev completion is also helpful as sometimes our
construction of face or degeneracy maps via concatenation does not result in a
smooth map, but instead a map in the Sobolev completion.

Given a domain $U$ in $\mathbb{R}^{n}$, the map sending $f\in C^{\infty}(U)$
to the evaluation of its $m$-th derivative $f^{(m)}(x)$ at point $x\in U$ can
be extended to a bounded linear (thus smooth) map on $H_{r}(U)$ if $r>m$.
Therefore, the de Rham differentiation operator is bounded linear (thus
smooth) $H_{r+1}(U)\to H_{r}(U)$.

##  Appendix B Universal integration $\int\mathfrak{g}_{\bullet}$ and
truncations

Using the idea of Sullivan’s space realisation and a suitable truncation,
Henriques shows in [54] a procedure to integrate a Lie $n$-algebra, i.e. an
$n$-term $L_{\infty}$-algebra, to a Lie $n$-group171717Notice that the
truncation procedure creates a possible obstruction to this integration
procedure. That is, although a finite dimensional Lie algebra can be always
integrated to a Lie group, not every Lie $n$-algebra can be integrated to a
Lie $n$-group.. In general, the Lie $n$-groups obtained by this method are
infinite dimensional. Here we recall his construction in the special case of a
Lie algebra.

Let $\mathfrak{g}=(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra and
$\operatorname{CE}(\mathfrak{g})$ its Chavelley-Eilenberg differential
complex, that is
$\operatorname{CE}(\mathfrak{g})=\wedge^{\bullet}\mathfrak{g}^{*}$ with
differential

$d_{CE}\xi(x_{1},\dots,x_{k})=\sum_{i<j}(-1)^{i+j-1}\xi([x_{i},x_{j}],x_{1},\dots,\hat{x}_{i},\dots,\hat{x}_{j},\dots,x_{k}).$

The universal object integrating it, $\int\mathfrak{g}_{\bullet}$, was
constructed in [52, 54] in the following way (we also follow the treatment in
[54, 29] of the differential structure, which is also similar to that in
[19]). For any $k\in\mathbb{Z}^{\geq 0}$ consider the standard $k$-simplex

$\Delta^{k}=\\{(t_{0},\dots,t_{k})\in\mathbb{R}^{k+1}|\sum_{i=0}^{k}t_{i}=1\\}.$

Denote by $\Omega^{\bullet}(\Delta^{k})$ subspaces of de Rham forms on
$\Delta^{k}$ with Sobolev class $r$ (to be a Banach algebra we need
$r>\frac{1}{2}k$ as in Appendix A) defined by

$\Omega^{\bullet}(\Delta^{k}):=\\{\alpha|\
\alpha=\sum_{I=i_{1}<\dots<i_{k}}\alpha^{I}dt_{I}\text{ where
}\alpha^{I}\;\text{is of Sobolev class}\;r,\text{ and }d\alpha\;\text{is also
in this form}\\}.$

This makes $(\Omega^{\bullet}(\Delta^{k}),d)$ into a differential graded
algebra, which we abbreviate to d.g.a.. Denote by
$\operatorname{Hom}_{d.g.a.}$ the set of morphisms between d.g.a.s. Then
$\operatorname{Hom}_{d.g.a.}(\operatorname{CE}(\mathfrak{g}),\Omega^{\bullet}(\Delta^{k}))$
carries a natural Banach manifold structure [54, Theorem 5.10]. Notice that
$C^{\infty}$ functions do not form a Banach space, but its completion under a
Sobolev norm to a Sobolev space is a Banach space.

It can be helpful to view elements in
$\operatorname{Hom}_{d.g.a.}\big{(}\operatorname{CE}(\mathfrak{g}),\Omega^{\bullet}(\Delta^{k})\big{)}$
as Lie algebroid morphisms from $T\Delta^{k}$ to $\mathfrak{g}$, and we define

$\operatorname{Hom}_{\operatorname{algd}}(T\Delta^{k},\mathfrak{g}):=\operatorname{Hom}_{d.g.a.}\big{(}\operatorname{CE}(\mathfrak{g}),\Omega^{\bullet}(\Delta^{k})\big{)}.$

More precisely, a vector bundle morphism $\psi:T\Delta^{k}\to\mathfrak{g}$ can
be written explicitly as $\psi=\sum_{i=0}^{k}\psi_{i}dt_{i}$ with $\psi_{i}\in
C^{r+1}(\Delta^{k},\mathfrak{g})$. Moreover, $\psi$ defines an element in
$\operatorname{Hom}_{\operatorname{algd}}(T\Delta^{k},\mathfrak{g})$ if it is
further a Lie algebroid morphism, that is it satisfies the Maurer-Cartan
equation

$\frac{d\psi_{i}}{dt_{j}}-\frac{d\psi_{j}}{dt_{i}}=[\psi_{i},\psi_{j}],\quad\forall
i,j\in\\{0,\dots,k\\}.$

The Banach manifolds
$\operatorname{Hom}_{\operatorname{algd}}(T\Delta^{\bullet},\mathfrak{g})$
form a simplicial manifold, denoted by $\int\mathfrak{g}_{\bullet}$ in [54],
with face and degeneracy maps induced by the natural ones between $\Delta^{k}$
and $\Delta^{k-1}$. The simplicial manifold $\int\mathfrak{g}_{\bullet}$ is
conjectured [53, Section 7] to be a universal integration object of
$\mathfrak{g}$, i.e. the $L_{\infty}$-group integrating $\mathfrak{g}$,
partially because its 1-truncation is the universal191919which is only
universal among Lie (1-)groups integrating $\mathfrak{g}$ (connected and
simply connected) Lie group $G$ integrating $\mathfrak{g}$. Moreover, $\int$
is an exact functor with respect to a class of distinguished
fibrations—“quasi-split fibrations” [84, Section 9]. Such fibrations include
acyclic fibrations as well as fibrations that arise in string-like extensions.
In particular, $\int$ sends $L_{\infty}$ quasi-isomorphisms to weak
equivalences, quasi-split fibrations to Kan fibrations, and preserves acyclic
fibrations as well as pullbacks of acyclic/quasi-split fibrations.

Now let us give some details on truncations.

######  Definition B.1.

[See e.g. [69] for the case of simplicial sets] Given a simplicial manifold
$X_{\bullet}$ the $n$-truncation $\tau_{n}(X)_{\bullet}$ is the simplicial set
defined as

$\tau_{n}(X)_{k}=S_{k},\quad\text{if }\;k\leq
n-1,\quad\tau_{n}(X)_{k}=X_{k}/\sim^{n}_{k+1}\text{ if }\;k\geq n,$

where $x_{k}\sim^{n}_{k+1}x^{\prime}_{k}$ if and only if they are simplicial
homotopic relative to $(n-1)$-skeleton. In other words,
$sk_{n-1}(x_{k})=sk_{n-1}(x^{\prime}_{k})$ and $\exists\
x_{k+1}\in\hom(\Delta[k]\times\Delta[1],X_{\bullet})$ such that

$x_{k+1}|_{sk_{n-1}(\Delta[k])}=sk_{n-1}(x_{k})=sk_{n-1}(x^{\prime}_{k}),\quad
x_{k+1}|_{\Delta[k]\times 0}=x_{k}\quad\text{and}\quad
x_{k+1}|_{\Delta[k]\times 1}=x^{\prime}_{k}.$

We pay special attention to the $n$-truncations for the simplicial manifold
$\int\mathfrak{g}_{\bullet}$ when $n=1,2$. In this case, the homotopies that
we mod out can be understood in the following way:
$\psi^{i}\in\int\mathfrak{g}_{k}$ with
$\psi^{0}|_{T\partial\Delta^{k}}=\psi^{1}|_{T\partial\Delta^{k}}$ for $i=0,1$
are homotopic if there exists
$\Psi\in\operatorname{Hom}_{\operatorname{algd}}(T(\Delta^{k}\times
I),\mathfrak{g})$ such that

$\Psi(x,0)=\psi^{0},\quad\Psi(x,1)=\psi^{1},\quad\Psi(y,t)=\psi^{0}(y)=\psi^{1}(y)\text{
for }y\in sk_{n-1}\Delta^{k},k\geq n.$

Notice in particular that when $k=n$, we have
$sk_{n-1}\Delta^{k}=\partial\Delta^{k}$.

The $n$-truncation of an arbitrary simplicial manifold needs not be a
simplicial manifold as there are quotients involved. In our concrete case,
[54, Theorem 7.5] implies that $\tau_{1}(\int\mathfrak{g})_{\bullet}$ and
$\tau_{2}(\int\mathfrak{g})_{\bullet}$ are simplicial manifolds because
$\pi_{2}(G)=0$ for a finite dimensional Lie group. More explicitly we have the
following.

######  Proposition B.2.

([54, Example 7.2]) Let $G$ be the connected and simply connected Lie group
integrating $\mathfrak{g}$. Then
$\tau_{1}(\int\mathfrak{g})_{\bullet}=NG_{\bullet}$.

######  Proposition B.3 ([54]).

Let $G$ be the connected and simply connected Lie group integrating
$\mathfrak{g}$. Then the Lie 2-group $\tau_{2}(\int\mathfrak{g})_{\bullet}$ is
equal to ${\mathbb{G}}_{\bullet}$ where
${\mathbb{G}}_{k}=\operatorname{Hom}_{e}(sk_{1}\Delta^{k},G)$, which is the
space of maps of Sobolev class $r+1$ sending $(0,\dots,0,1)\in
sk_{1}\Delta^{k}$ to $e$. The face and degeneracy maps are those induced from
the simplices.

###### Proof.

The calculation is more or less done in [54, Sect.8]. We summarise here: let
us denote by $\operatorname{Hom}_{e}(\Delta^{k},G)$ the space of maps of
Sobolev class ${r+1}$ which send $(0,\dots,0,1)\in\Delta^{k}$ to $e\in G$.
When $k=1$, we also write $P_{e}G:=\operatorname{Hom}_{e}(\Delta^{1},G)$. Then
according to [54, Example 5.5] or [19, Remark 3.8],
$\operatorname{Hom}_{\operatorname{algd}}(T\Delta^{k},\mathfrak{g})=\operatorname{Hom}_{e}(\Delta^{k},G)$.
Thus $\int\mathfrak{g}_{1}=P_{e}G$.

The equivalence relation $\sim_{k}$ we take in truncation is exactly the
homotopy relative to the boundary in $G$. Thus
${\mathbb{G}}_{k}=\operatorname{Hom}_{e}(sk_{1}\Delta^{k},G)$. ∎

##  Appendix C Explicit formulas for the Lie $2$-group
${\mathbb{G}}_{\bullet}$

The tangent of the Lie $2$-group ${\mathbb{G}}_{\bullet}$ is also a Lie
$2$-group202020In fact the group structure is compatible with the vector
bundle structure and hence it is more than just a Lie $2$-group. and is given
by

$T{\mathbb{G}}_{\bullet}=\cdots\Omega TG\
\begin{subarray}{c}\longrightarrow\\\\[-8.50006pt]
\longrightarrow\\\\[-8.50006pt]
\longrightarrow\end{subarray}P_{(e,0_{e})}TG\rightrightarrows pt.$

The 3rd level is also important and is

(C.1) $\begin{split}T{\mathbb{G}}_{3}=\\{(a_{0},a_{1},a_{2})\in(\Omega
TG)^{\times 3}\ |\ \text{ for }t\in[0,\frac{1}{3}],\ a_{0}(t)=a_{1}(t),\\\
a_{0}(t+\frac{2}{3})=a_{2}(\frac{1}{3}-t),\
a_{1}(t+\frac{2}{3})=a_{2}(t+\frac{2}{3})\\}.\end{split}$

The face and degeneracies are obtained by taking variations of equations
(3.4), (3.5), and (3.6). More explicitly, the faces $Td_{i}:\Omega TG\to
P_{(e,0_{e})}TG$ are given by

(C.2) $\begin{split}&Td_{0}(a)=a(\frac{t}{3}),\quad
Td_{1}(a)=a(1-\frac{t}{3}),\\\
&Td_{2}(a)=R_{\tau(\frac{1}{3})^{-1}}a(\frac{1+t}{3})-L_{\tau(\frac{t+1}{3})\tau(\frac{1}{3})^{-1}}R_{\tau(\frac{1}{3})^{-1}}a(\frac{1}{3}).\end{split}$

The faces $Td_{i}:T{\mathbb{G}}_{3}\to\Omega TG$ are given by

(C.3) $Td_{i}(a_{0},a_{1},a_{2})=a_{i}\quad\text{ for }0\leq i\leq 3,$

where

(C.4)
$a_{3}(t)=\left\\{\begin{array}[]{ll}R_{\tau_{0}(\frac{1}{3})^{-1}}a_{0}(t+\frac{1}{3})-L_{\tau_{3}(t)}R_{\tau_{0}(\frac{1}{3})^{-1}}a_{0}(\frac{1}{3})&t\in[0,\frac{1}{3}],\\\
R_{\tau_{0}(\frac{1}{3})^{-1}}a_{2}(t)-L_{\tau_{3}(t)}R_{\tau_{0}(\frac{1}{3})^{-1}}a_{0}(\frac{1}{3})&t\in[\frac{1}{3},\frac{2}{3}],\\\
R_{\tau_{0}(\frac{1}{3})^{-1}}a_{1}(\frac{4}{3}-t)-L_{\tau_{3}(t)}R_{\tau_{0}(\frac{1}{3})^{-1}}a_{0}(\frac{1}{3})&t\in[\frac{2}{3},1].\end{array}\right.$

The degeneracies $Ts_{i}:P_{(e,0_{e})}TG\to\Omega TG$ are given by

(C.5) $Ts_{0}(u)(t)=\left\\{\begin{array}[]{ll}u(3t)&t\in[0,\frac{1}{3}],\\\
u(1)&t\in[\frac{1}{3},\frac{2}{3}],\\\
u(3-3t)&t\in[\frac{2}{3},1],\end{array}\right.\quad
Ts_{1}(u)(t)=\left\\{\begin{array}[]{ll}u(0)&t\in[0,\frac{1}{3}],\\\
u(3t-1)&t\in[\frac{1}{3},\frac{2}{3}],\\\
u(3-3t)&t\in[\frac{2}{3},1].\end{array}\right.$

Here we also include a direct proof that Segal’s 2-form is multiplicative.

######  Proposition C.1.

The $2$-shifted $2$-form $\omega_{\bullet}$ on ${\mathbb{G}}_{\bullet}$ is
closed, that is,

$d\omega=0\qquad\text{and}\qquad\delta\omega=0.$

###### Proof.

Since $\omega$ is a symplectic form on $\Omega G$ we know that $d\omega=0$, so
it remains to see that $\delta\omega=0$. We need to compute the pullback of
$\omega$ by the face maps (3.5). Pick
$\tau\equiv(\tau_{0},\tau_{1},\tau_{2})\in{\mathbb{G}}_{3}$ and
$a\equiv(a_{0},a_{1},a_{2}),b\equiv(b_{0},b_{1},b_{2})\in
T_{\tau}{\mathbb{G}}_{3}$ write
$\widehat{a}_{i}(t)=L_{\tau_{i}(t)^{-1}}a_{i}(t)$ and similarly for $b$. Then
using (3.5) and (C.3) we obtain

$(d_{i}^{*}\omega)_{\tau}(a,b)=\omega_{\tau_{i}}(a_{i},b_{i})=A_{i}+B_{i}+C_{i}\quad\text{for
}i=0,\cdots,3,$

where $A_{i}$ correspond to the first third of the loop, $B_{i}$ to the second
third and $C_{i}$ to the last third. Since $C^{\infty}$ maps are dense in
Sobolev space, we only need to verify our result at smooth loops. In order to
compare the terms $A_{i},B_{i}$, and $C_{i}$ for different values of $i$, we
use (C.1) and the face maps given by (C.3). Moreover, equation (C.4) implies

(C.6)
$\left\\{\begin{array}[]{ll}a_{0}(t)=R_{\tau_{0}(\frac{1}{3})}a_{3}(t-\frac{1}{3})+L_{\tau_{3}(t-\frac{1}{3})}a_{0}(\frac{1}{3})&t\in[\frac{1}{3},\frac{2}{3}],\\\
a_{2}(t)=R_{\tau_{0}(\frac{1}{3})}a_{3}(t)+L_{\tau_{3}(t)}a_{0}(\frac{1}{3})&t\in[\frac{1}{3},\frac{2}{3}],\\\
a_{1}(t)=R_{\tau_{0}(\frac{1}{3})}a_{3}(\frac{4}{3}-t)+L_{\tau_{3}(\frac{4}{3}-t)}a_{0}(\frac{1}{3})&t\in[\frac{1}{3},\frac{2}{3}],\\\
\end{array}\right.$

and the same identities holds for $b$.

For $i=0$ we see that the first term is given by

$A_{0}=\int_{0}^{\frac{1}{3}}\langle\widehat{a}^{\prime}_{0}(t),\widehat{b}_{0}(t)\rangle
dt=\int_{0}^{\frac{1}{3}}\langle\widehat{a}^{\prime}_{1}(t),\widehat{b}_{1}(t)\rangle
dt=A_{1},$

where the middle equality follows from the definition of $T{\mathbb{G}}_{3}$.
The second term can be rewritten using (C.6) as

$B_{0}=\int_{\frac{1}{3}}^{\frac{2}{3}}\langle\widehat{a}^{\prime}_{0}(t),\widehat{b}_{0}(t)\rangle
dt=\int_{\frac{1}{3}}^{\frac{2}{3}}\langle\alpha,\beta\rangle$

with

$\alpha=\Big{(}L_{\tau_{0}(\frac{1}{3})^{-1}\tau_{3}(t-\frac{1}{3})^{-1}}\big{(}R_{\tau_{0}(\frac{1}{3})}a_{3}(t-\frac{1}{3})+L_{\tau_{3}(t-\frac{1}{3})}a_{0}(\frac{1}{3})\big{)}\Big{)}^{\prime}=Ad_{\tau_{0}(\frac{1}{3})^{-1}}\widehat{a}^{\prime}_{3}(t-\frac{1}{3})$

and

$\beta=L_{\tau_{0}(\frac{1}{3})^{-1}\tau_{3}(t-\frac{1}{3})^{-1}}\Big{(}R_{\tau_{0}(\frac{1}{3})}b_{3}(t-\frac{1}{3})+L_{\tau_{3}(t-\frac{1}{3})}b_{0}(\frac{1}{3})\Big{)}=Ad_{\tau_{0}(\frac{1}{3})^{-1}}\widehat{b}_{3}(t-\frac{1}{3})+\widehat{b}_{0}(\frac{1}{3}).$

Setting $s=t-\frac{1}{3}$, and noticing that $\langle-,-\rangle$ is adjoint
invariant, we conclude that

$B_{0}=\int_{0}^{\frac{1}{3}}\langle\widehat{a}^{\prime}_{3}(s),\widehat{b}_{3}(s)\rangle+\langle
Ad_{\tau_{0}(\frac{1}{3})^{-1}}\widehat{a}^{\prime}_{3}(s),\widehat{b}_{0}(\frac{1}{3})\rangle
ds=A_{3}+\int_{0}^{\frac{1}{3}}\langle\widehat{a}^{\prime}_{3}(s),R_{\tau_{0}(\frac{1}{3})^{-1}}b_{0}(\frac{1}{3})\rangle
ds.$

Finally the third term

$C_{0}=\int_{\frac{2}{3}}^{1}\langle\widehat{a}^{\prime}_{0}(t),\widehat{b}_{0}(t)\rangle
dt=\int_{\frac{2}{3}}^{1}\langle\widehat{a}^{\prime}_{2}(1-t),\widehat{b}_{2}(1-t)\rangle
dt=-\int_{0}^{\frac{1}{3}}\langle\widehat{a}^{\prime}_{2}(s),\widehat{b}_{2}(s)\rangle
ds=-A_{2},$

where in the middle equation we use the definition of $T{\mathbb{G}}_{3}$ and
in the last we make the change of variable $s=1-t$. By similar computations
one shows that

$\begin{array}[]{rl}B_{1}=&-C_{3}-\int_{\frac{2}{3}}^{1}\langle\widehat{a}^{\prime}_{3}(s),R_{\tau_{0}(\frac{1}{3})^{-1}}b_{0}(\frac{1}{3})\rangle
ds,\\\ C_{1}=&C_{2},\\\
B_{2}=&B_{3}+\int_{\frac{1}{3}}^{\frac{2}{3}}\langle\widehat{a}^{\prime}_{3}(s),R_{\tau_{0}(\frac{1}{3})^{-1}}b_{0}(\frac{1}{3})\rangle
ds.\end{array}$

Thus we may conclude that

$\delta\omega_{\tau}(a,b)=\sum_{i=0}^{3}(-1)^{i}(d_{i}^{*}\omega)_{\tau}(a,b)=\sum_{i=0}^{3}(-1)^{i}(A_{i}+B_{i}+C_{i})=\int_{0}^{1}\langle\widehat{a}^{\prime}_{3}(s),R_{\tau_{0}(\frac{1}{3})^{-1}}w_{0}(\frac{1}{3})\rangle
ds=0,$

where in the last step we are using Stokes’ Theorem and the fact that
$\widehat{a}_{3}(0)=\widehat{a}_{3}(1)=0_{e}.$ ∎

##  Appendix D Transgression map

In this appendix we recollect some useful formulas for the transgression map
going from $k$-forms on a manifold to $(k-1)$-forms on its path space, which
should be well known to experts.

Let $M$ be a manifold and denote by $PM$ its path space, i.e.
$PM=\\{\gamma:[0,1]\to M\ |\ \gamma\text{ of class }H_{r}\\}$, and consider
$PM$ as a Banach manifold. The evaluation map $ev:[0,1]\times PM\to M,\
ev(t,\gamma)=\gamma(t)$ at the level of forms produces the map

$ev^{*}:\Omega^{k}(M)\to\Omega^{k}([0,1]\times PM),$

and since the interval has dimension $1$ we can give the more explicit
description

(D.1) $\Omega^{k}([0,1]\times
PM)=\Omega^{0}([0,1])\otimes\Omega^{k}(PM)\oplus\Omega^{1}([0,1])\otimes\Omega^{k-1}(PM).$

The transgression map is defined as the composition of evaluation with
integration:

(D.2)
$\begin{array}[]{cccl}\operatorname{\mathbb{T}}:&\Omega^{k}(M)&\to&\Omega^{k-1}(PM)\\\
&\omega&\mapsto&\operatorname{\mathbb{T}}(\omega)=\int_{0}^{1}ev^{*}\omega.\end{array}$

Now we give short proofs for the properties of the transgression map that we
use in the article.

######  Proposition D.1.

The transgression map and de Rham differential satisfy the following relation:

$\operatorname{\mathbb{T}}(d\omega)=ev_{1}^{*}\omega-ev_{0}^{*}\omega-
d_{PM}\operatorname{\mathbb{T}}(\omega),$

where $ev_{t}(\gamma)=\gamma(t)$ for $t\in[0,1].$

###### Proof.

$\begin{split}\operatorname{\mathbb{T}}(d\omega)=&\int_{0}^{1}ev^{*}d\omega=\int_{0}^{1}d_{[0,1]\times
PM}(ev^{*}\omega)=\int_{0}^{1}d_{[0,1]}ev^{*}\omega-\int_{0}^{1}d_{PM}(ev^{*}\omega)\\\
=&ev^{*}_{1}\omega-ev^{*}_{0}\omega-
d_{PM}\operatorname{\mathbb{T}}(\omega),\end{split}$

where we use the decomposition (D.1) and Stokes’ theorem. ∎

######  Proposition D.2.

In terms of vectors the transgression map has the following explicit formula:

$\operatorname{\mathbb{T}}(\omega)_{\gamma}(v_{1},\cdots,v_{k-1})=\int_{0}^{1}\omega_{\gamma(t)}\big{(}\gamma^{\prime}(t),v_{1}(t),\cdots,v_{k-1}(t)\big{)}dt,$

where $\omega\in\Omega^{k}(M)$ and $v_{i}\in T_{\gamma}PM.$

###### Proof.

With decomposition (D.1), we write

(D.3)
$ev^{*}\omega=\omega_{0}\otimes\omega_{k}+\omega_{1}\otimes\omega_{k-1},$

where $\omega_{k}\in\Omega^{k}(PM)$, $\omega_{k-1}\in\Omega^{k-1}(PM)$,
$\omega_{1}\in\Omega^{1}([0,1])$ and $\omega_{0}\in\Omega^{0}([0,1])$. Then
contracting with $[0,1]$ via $\int_{0}^{1}$, only the second term survives.
Thus for tangent vectors $x_{i}\in T_{\gamma}PM$, $i=1,\dots,k-1$, at
$\gamma\in PM$, we have

(D.4)
$\int_{0}^{1}ev^{*}\omega_{\gamma}(v_{1},\dots,v_{k-1})=\int_{0}^{1}\omega_{1}\otimes\omega_{k-1}(v_{1},\dots,v_{k-1}).$

At the same time, for a tangent vector $(w,v)\in T_{t}[0,1]\times
T_{\gamma}PM$, we have $T_{\gamma}ev(w,v)=w\gamma^{\prime}+v$. This can be
seen by taking a variation (a small path) $(t+w\epsilon,\gamma^{\epsilon})$
representing $(w,v)$ (thus $\gamma^{0}=\gamma$). Then

$T_{\gamma}ev(w,v)=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}(\gamma^{\epsilon}(t+w\epsilon))=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\gamma^{0}(t+w\epsilon)+\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\gamma^{\epsilon}(t)=w\gamma^{\prime}(t)+v(t).$

Thus

$\begin{split}&ev^{*}\omega|_{(t,\gamma)}((w_{1},v_{1}),…,(w_{k},v_{k}))=\omega_{\gamma(t)}(T_{\gamma}ev(w_{1},v_{1}),…,T_{\gamma}ev(w_{k},v_{k}))\\\
=&\omega_{\gamma(t)}(v_{1}(t),…,v_{k}(t))+\omega_{\gamma(t)}(w_{1}\gamma^{\prime}(t),v_{2}(t),…,v_{k}(t))+c.p..\end{split}$

Thus comparing with (D.3), we have $\omega_{k}=\omega$, $\omega_{0}=1$,
$\omega_{k-1}=\iota(\gamma^{\prime})\omega$, and $\omega_{1}=dt$. Combining
with (D.4) obtains the desired formula. ∎

Suppose that we have a simplicial manifold $X_{\bullet}$. Then the path space
$PX_{\bullet}$ is again a simplicial manifold given by

$(PX)_{n}=P(X_{n}),\quad(Pd)_{i}(\gamma)(t)=(d_{i}\circ\gamma)(t),\quad\text{and}\quad(Ps)_{i}(\gamma)(t)=(s_{i}\circ\gamma)(t).$

######  Proposition D.3.

Let $X_{\bullet}$ be a simplicial manifold. Then the transgression commutes
with the simplicial differentials, i.e.

$\operatorname{\mathbb{T}}(\delta^{X}\omega)=\delta^{PX}\operatorname{\mathbb{T}}(\omega).$

###### Proof.

In order to prove this, we first observe that if $X_{\bullet}$ is a simplicial
manifold then $TX_{\bullet}$ is also a simplicial manifold with faces and
degeneracies given by the corresponding tangent maps. It then follows that
there is a canonical identification between $P(TX)_{\bullet}$ and
$T(PX)_{\bullet}$ as simplicial manifolds. Using this canonical identification
and the explicit formula of the transgression given by Proposition D.2, for
$\omega\in\Omega^{k}(X_{n-1}),$ $\gamma\in PX_{n},$ and $v_{j}\in
T_{\gamma}PX_{n}$ we have

$\begin{split}&\big{(}(Pd_{i})^{*}\operatorname{\mathbb{T}}(\omega)\big{)}_{\gamma}(v_{1},\cdots,v_{k-1})=\operatorname{\mathbb{T}}(\omega)_{Pd_{i}(\gamma)}\Big{(}TPd_{i}(v_{1}),\cdots,TPd_{i}(v_{k-1})\Big{)}dt\\\
&=\int_{0}^{1}\omega_{Pd_{i}(\gamma)(t)}\Big{(}Pd_{i}(\gamma)^{\prime}(t),TPd_{i}(v_{1})(t),\cdots,TPd_{i}(v_{k-1})(t)\Big{)}dt\\\
&=\int_{0}^{1}\omega_{d_{i}(\gamma(t))}\Big{(}Td_{i}(\gamma(t))^{\prime},PTd_{i}(v_{1})(t),\cdots,PTd_{i}(v_{k-1})(t)\Big{)}dt\\\
&=\int_{0}^{1}\omega_{d_{i}(\gamma(t))}\Big{(}Td_{i}(\gamma^{\prime}(t)),Td_{i}(v_{1}(t)),\cdots,Td_{i}(v_{k-1}(t))\Big{)}\\\
&=\int_{0}^{1}(d^{*}_{i}\omega)_{\gamma(t)}\Big{(}\gamma^{\prime}(t),v_{1}(t),\cdots,v_{k-1}(t)\Big{)}dt=\operatorname{\mathbb{T}}(d^{*}_{i}\omega)_{\gamma}(v_{1},\cdots,v_{k-1}).\end{split}$

Since the simplicial differentials are alternating sums of face maps, this
prove the statement. ∎

##  Appendix E IM-form (by Florian Dorsch)

For completeness of this article, we now give proofs for some very useful
statements in the unpublished lecture note [51], which are stated on page
82-83 without proof.

######  Lemma E.1.

Let $\alpha_{\bullet}$ be a $m$-shifted 2-form on a Lie $n$-groupoid
$X_{\bullet}$. Then

  1. (1)

the IM-form $\lambda^{\alpha_{\bullet}}$ vanishes on degenerate vectors, that
is, for $m\geq 1$, for any point $x\in X_{0}$ we have

$\lambda^{\alpha_{\bullet}}_{x}(Ts_{i}u,w)=0,\quad\forall u\in T_{x}X_{p-1},\
w\in T_{x}X_{q},\quad 0\leq i\leq p-1.$

For $m=0$, this is an empty condition.

  2. (2)

if $\alpha_{m}$ is multiplicative, that is $\delta\alpha_{m}=0$, the IM-form
$\lambda^{\alpha_{\bullet}}$ is infinitesimally multiplicative. That is, when
$m\geq 1$, for $v\in\mathcal{T}_{p+1}(X_{\bullet})$,
$w\in\mathcal{T}_{q}(X_{\bullet})$ with $0\leq p\leq m-1$ and $p+q=m$, we have

(E.1) $\lambda^{\alpha_{\bullet}}(\partial
v,w)+(-1)^{p+1}\lambda^{\alpha_{\bullet}}(v,\partial w)=0;$

and when $m=0$, for $v\in\mathcal{T}_{1}(X_{\bullet})$,
$w\in\mathcal{T}_{0}(X_{\bullet})=T_{0}X_{0}$ we have

(E.2) $\lambda^{\alpha_{\bullet}}(\partial v,w)=0.$

These are equivalent to (2.6) with the correct interpretation in extreme cases
of indices explained in Remark 2.15;

  3. (3)

IM-forms are invariant under gauge transformation. That is, when $m\geq 1$,

$\lambda^{\alpha_{\bullet}+D\phi_{\bullet}}=\lambda^{\alpha_{\bullet}},\quad\text{for
any $(m-1)$-shifted 2-form $\phi_{\bullet}$}.$

###### Proof.

(1) To show that $\lambda^{\alpha_{\bullet}}$ vanishes on degeneracies, it is
enough to verify that

(E.3) $\alpha_{m}(Ts_{\pi(p+q-1)}\dots Ts_{\pi(p)}Ts_{i}u,Ts_{\pi(p-1)}\dots
Ts_{\pi(0)}w)=0$

for all $u\in T_{x}X_{p-1}$, $w\in T_{x}X_{q}$ and $s_{i}:X_{p-1}\rightarrow
X_{p},\,\,0\leq i\leq p-1$. We begin by making the following observation. Let

$\displaystyle i_{p-1}>\dots>i_{j}=\lambda>\dots>i_{0}\>\>\>\text{and}$
$\displaystyle i_{p+q-1}>\dots>i_{p+l}>\lambda>i_{p+l-1}>\dots>i_{p}$

be indices such that

$\displaystyle Ts_{i_{p-1}}\dots Ts_{\lambda}\dots
Ts_{i_{0}}w\>\>\>\text{and}\>\>\>Ts_{i_{p+q-1}}\dots
Ts_{i_{p+l}}Ts_{\lambda}Ts_{i_{p+l-1}}\dots Ts_{i_{p}}u$

are well-defined tangent vectors in $T_{x}X_{m}$. Then by simplicial
identities we have

(E.4) $\begin{split}&\alpha_{m}(Ts_{i_{p+q-1}}\dots
Ts_{i_{p+m}}Ts_{\lambda}Ts_{i_{p+m-1}}\dots Ts_{i_{p}}u,Ts_{i_{p-1}}\dots
Ts_{\lambda}\dots Ts_{i_{0}}w)\\\
=&\alpha_{m}(Ts_{\lambda}Ts_{i_{p+q-1}-1}\dots
Ts_{i_{p+m}-1}Ts_{i_{p+m-1}}\dots Ts_{i_{p}}u,Ts_{\lambda}Ts_{i_{p-1}-1}\dots
Ts_{i_{j+1}-1}\dots Ts_{i_{0}}w)\\\
=&s_{\lambda}^{\ast}\alpha_{m}(Ts_{i_{p+q-1}-1}\dots
Ts_{i_{p+m}-1}Ts_{i_{p+m-1}}\dots Ts_{i_{p}}u,Ts_{i_{p-1}-1}\dots
Ts_{i_{j+1}-1}\dots Ts_{i_{0}}w)\\\ =&0\end{split}$

as $\alpha_{m}$ is normalized. The rest of argument is essentially put down to
(E.4).
First suppose that $i=\pi(p+j)\in\\{\pi(p+q-1),\dots,\pi(p)\\}$ for $0\leq
j\leq q-1$. Since $i=\pi(p+j)>\dots>\pi(p)$, it follows from the simplicial
identities that

$\displaystyle Ts_{\pi(p+q-1)}\dots Ts_{\pi(p)}Ts_{i}u=Ts_{\pi(p+q-1)}\dots
Ts_{\pi(p+j+1)}Ts_{i+j+1}Ts_{\pi(p+j)}\dots Ts_{\pi(p)}u.$

If $\pi(p+j+1)>i+j+1$, the index $i+j+1$ is not contained in
$\\{\pi(p+q-1),\dots,\pi(p)\\}$, so $i+j+1\in\\{\pi(p-1),\dots,\pi(0)\\}$ and
(E.3) follows from (E.4).
Otherwise, we distinguish between the following two cases:

  1. a)

There exists a minimal $l\in\\{j+2,\dots,q-1\\}$ such that $\pi(p+l)>i+l$.
Then $i+l\in$ $\\{\pi(p-1),\dots,\pi(0)\\}$, so (E.4) applies.

  2. b)

For all $l\in\\{q-1,\dots,j+1\\}:\pi(p+l)=i+l$. Then $p+q-1\geq
i+q>\pi(p+q-1)$ and $i+q\in\\{\pi(p-1),\dots,\pi(0)\\}$, so (E.4) applies.

Thus $\alpha_{m}(Ts_{\pi(p+q-1)}\dots Ts_{\pi(p)}Ts_{i}u,Ts_{\pi(p-1)}\dots
Ts_{\pi(0)}w)$ vanishes for $i\in\\{\pi(p+q-1),\dots,\pi(p)\\}$. The case when
$i\in\\{\pi(p-1),\dots,\pi(0)\\}$ works similarly.

(2) We first look at the case when $m=0$, and we want to prove Eq. (E.2).
Since $\alpha_{\bullet}$ is multiplicative, we have

$0=\delta\alpha_{0}(v,Ts_{0}w)=\alpha_{0}(Td_{0}v,Td_{0}Ts_{0}w)-\alpha_{0}(Td_{1}v,Td_{1}Ts_{0}w).$

Since $Td_{0}v=0$ and $d_{1}s_{0}=\operatorname{id}$, the desired equation
(E.2) is proven.

When $m\geq 1$, we consider the sum

$\begin{split}&\sum_{\pi\in\mathsf{Shuff}(p+1,q)}\text{sgn}(\pi)\,\delta\alpha_{m}(Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Ts_{\pi(p)}\dots Ts_{\pi(0)}w)\\\
=&\sum_{i=0}^{m+1}(-1)^{i}\sum_{\pi\in\mathsf{Shuff}(p+1,q)}\text{sgn}(\pi)\,\alpha_{m}(Td_{i}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Td_{i}Ts_{\pi(p)}\dots Ts_{\pi(0)}w),\end{split}$

which vanishes because $\alpha_{m}$ is multiplicative. We begin by showing
that even

$\displaystyle\sum_{\pi\in\mathsf{Shuff}(p+1,q)}\text{sgn}(\pi)\,\alpha_{m}(Td_{i}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Td_{i}Ts_{\pi(p)}\dots Ts_{\pi(0)}w)=0$

for $i\in\\{0,\dots,m\\}$. For a fixed $i\in\\{0,\dots,m\\}$ and a shuffle
$\pi\in\mathsf{Shuff}(p+1,q)$, the index $i$ either lies in
$\\{\pi(p+q),\dots,\pi(p+1)\\}$ or in $\\{\pi(p),\dots,\pi(0)\\}$. First,
consider the case $i=\pi(p+1+j)\in\\{\pi(p+q),\dots,\pi(p+1)\\}$. From the
simplicial identities, it follows that

$\displaystyle Td_{i}Ts_{\pi(p+q)}\dots Ts_{\pi(p+1)}v=Ts_{\pi(p+q)-1}\dots
Ts_{\pi(p+1+j+1)-1}Ts_{\pi(p+j)}\dots Ts_{\pi(p+1)}v.$

Similarly, we have

(E.5) $\displaystyle\underbrace{Td_{i}Ts_{\pi(p)}\dots
Ts_{\pi(0)}w}_{(\ast\ast)}=\begin{cases}Ts_{\pi(p)}\dots
Ts_{\pi(0)}Td_{i-(p+1)}w,\>\>\>\>\>\>\text{if}\>\>i>1+\pi(p),\\\ \\\
Ts_{\pi(p)-1}\dots Ts_{\pi(0)-1}Td_{i}w,\>\>\>\>\>\>\text{if}\>\>i<\pi(0),\\\
\\\ Ts_{\pi(p)-1}\dots Ts_{\pi(l+1)-1}Ts_{\pi(l)}\dots
Ts_{\pi(0)}Td_{i-(l+1)}w,\\\
\text{if}\>\>\pi(l+1)>i>\pi(l)+1\>\>\text{for}\>\>l\in\\{0,\dots,p-1\\},\\\
\\\ Ts_{\pi(p)-1}\dots Ts_{\pi(l+1)-1}Ts_{\pi(l-1)}\dots Ts_{\pi(0)}w,\\\
\text{if}\>\>\>i=1+\pi(l)\>\>\text{for}\>\>l\in\\{0,\dots,p\\}.\end{cases}$

We consider in each different situation:

  1. a)

if $i>1+\pi(p)$ then $i-(p+1)\leq p+q-(p+1)=q-1$, so $Td_{i-(p+1)}w=0$ and
$(\ast\ast)$ vanishes,

  2. b)

if $i<\pi(0)$ then $i\leq p+q-(p+1)=q-1$, so $Td_{i}w=0$ and $(\ast\ast)$
vanishes,

  3. c)

if $\pi(l+1)>i>\pi(l)+1$ for $l\in\\{0,\dots,p-1\\}$, then $i-(l+1)\leq
q+l-(l+1)=q-1$, so $Td_{i-(l+1)}w=0$ and $(\ast\ast)$ vanishes.

Thus we are left with the last situation in (E.5). In this case,
$i=1+\pi(l),\>l\in\\{0,\dots,p\\}$, thus we have $\pi(l)=\pi(p+1+j)-1$. We
define a new $(p+1,q)$-shuffle $\tilde{\pi}$ by

$\displaystyle\tilde{\pi}(k)=\begin{cases}i-1=\pi(l),\>\>\>\>\>\>\text{if}\>\>k=p+1+j,\\\
i=\pi(p+1+j),\>\>\>\>\>\>\text{if}\>\>k=l,\\\
\pi(k),\>\>\>\>\>\>\text{otherwise,}\end{cases}$

which can be illustrated as

$\scriptsize{\begin{pmatrix}0&\dots&\mathbf{l}&\dots&p,&p+1&\dots&\mathbf{p+1+j}&\dots&p+q\\\
\pi(0)<&\dots&<\mathbf{\pi(p+1+j)-1}<&\dots&<\pi(p),&\pi(p+1)<&\dots&<\mathbf{\pi(l)}<&\dots&<\pi(p+q)\end{pmatrix}}.$

Then clearly $\text{sgn}(\pi)=-\text{sgn}(\tilde{\pi})$ and

$\displaystyle Td_{i}Ts_{\pi(p)}\dots
Ts_{\pi(0)}w=Td_{i}Ts_{\tilde{\pi}(p)}\dots Ts_{\tilde{\pi}(0)}w,$
$\displaystyle Td_{i}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v=Td_{i}Ts_{\tilde{\pi}(p+q)}\dots Ts_{\tilde{\pi}(p+1)}v.$

Notice that terms

$\displaystyle\text{sgn}(\pi)\,\alpha_{m}(Td_{i}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Td_{i}Ts_{\pi(p)}\dots Ts_{\pi(0)}w),$
$\displaystyle\text{sgn}(\tilde{\pi})\,\alpha_{m}(Td_{i}Ts_{\tilde{\pi}(p+q)}\dots
Ts_{\tilde{\pi}(p+1)}v,Td_{i}Ts_{\tilde{\pi}(p)}\dots Ts_{\tilde{\pi}(0)}w)$

cancel with each other. The case $i\in\\{\pi(p),\dots,\pi(0)\\}$ can be
treated similarly. Thus

$\displaystyle\sum_{\pi\in\mathsf{Shuff}(p+1,q)}\text{sgn}(\pi)\,\alpha_{m}(Td_{n+1}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Td_{n+1}Ts_{\pi(p)}\dots Ts_{\pi(0)}w)=0.$

We now distinguish between two types of $(p+1,q)$-shuffles: shuffles
satisfying $\pi(p+q)=m$ and shuffles satisfying $\pi(p)=m$. There exists a 1-1
correspondence between $(p+1,q)$-shuffles $\pi$ with $\pi(p+q)=m$ and
$(p+1,q-1)$-shuffles $\tau$ via $\tau(k)=\pi(k)$ for
$k\in\\{0,\dots,p+q-1\\}$. Likewise there exists a 1-1 correspondence between
$(p+1,q)$-shuffles $\pi$ with $\pi(p)=m$ and $(p,q)$-shuffles $\chi$ via

$\displaystyle\chi(k)=\begin{cases}\pi(k)\>\>\>\>\>\text{if}\>\>k\in\\{0,\dots,p-1\\},\\\
\pi(k+1)\>\>\>\>\>\text{if}\>\>k\in\\{p,\dots,p+q-1\\}.\end{cases}$

With these correspondences it follows that

(E.6)
$\begin{split}0=&\sum_{\pi\in\mathsf{Shuff}(p+1,q)}\text{sgn}(\pi)\,\alpha_{m}(Td_{m+1}Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}v,Td_{m+1}Ts_{\pi(p)}\dots Ts_{\pi(0)}w)\\\
=&\sum_{\pi\in\mathsf{Shuff}(p+1,q),\pi(p+q)=m}\text{sgn}(\pi)\,\alpha_{m}(Ts_{\pi(p+q-1)}\dots
Ts_{\pi(p+1)}v,Ts_{\pi(p)}\dots Ts_{\pi(0)}Td_{q}w)\\\
+&\sum_{\pi\in\mathsf{Shuff}(p+1,q),\pi(p)=m}\text{sgn}(\pi)\,\alpha_{m}(Ts_{\pi(p+q)}\dots
Ts_{\pi(p+1)}Td_{p+1}v,Ts_{\pi(p-1)}\dots Ts_{\pi(0)}w)\\\
=&(-1)^{q}\sum_{\tau\in\mathsf{Shuff}(p+1,q-1)}\text{sgn}(\tau)\,\alpha_{m}(Ts_{\tau(p+q-1)}\dots
Ts_{\tau(p+1)}v,Ts_{\tau(p)}\dots Ts_{\tau(0)}\partial w)\\\
+&(-1)^{p+1}\sum_{\chi\in\mathsf{Shuff}(p,q)}\text{sgn}(\chi)(-1)^{q}\,\alpha_{m}(Ts_{\chi(p+q-1)}\dots
Ts_{\chi(p)}\partial v,Ts_{\chi(p-1)}\dots Ts_{\chi(0)}w)\\\
=&(-1)^{q}\lambda^{\alpha_{\bullet}}(v,\partial
w)+(-1)^{q}(-1)^{p+1}\lambda^{\alpha_{\bullet}}(\partial v,w).\end{split}$

Thus (E.1) is proven.

(3) It is enough to show that $\lambda^{D\phi_{\bullet}}$ vanishes for any
$(m-1)$-shifted form $\phi_{\bullet}$. From (E.6) and the fact that
$(D\phi)_{m}=\delta\phi_{m-1}$, it follows that

$\displaystyle\lambda^{D\phi_{\bullet}}(v,w)=\lambda^{\phi_{\bullet}}(\partial
v,w)+(-1)^{p}\lambda^{\phi_{\bullet}}(v,\partial w),$

for tangent vectors $v\in T_{x_{0}}X_{p},\>w\in
T_{x_{0}}X_{q},\>p+q=m,\>\,p,q\geq 1$. The two summands on the right hand side
turn out to be equal to zero: we note that

$\displaystyle\underbrace{\lambda^{D\phi_{\bullet}}\big{(}(-1)^{p}Ts_{p-1}\partial
v,w\big{)}}_{=0}=\lambda^{\phi_{\bullet}}(\partial\,(-1)^{p}Ts_{p-1}\partial
v,w)+(-1)^{p}\underbrace{\lambda^{\phi_{\bullet}}((-1)^{p}Ts_{p-1}\partial
v,\partial w)}_{=0}$
$\displaystyle=\lambda^{\phi_{\bullet}}\big{(}(-1)^{p}Td_{p}(-1)^{p}Ts_{p-1}\partial
v,w\big{)}=\lambda^{\phi_{\bullet}}(\partial v,w).$

The terms $\lambda^{D\phi_{\bullet}}\big{(}(-1)^{p}Ts_{p-1}\partial
v,w\big{)}$ and $\lambda^{\phi_{\bullet}}\big{(}(-1)^{p}Ts_{p-1}\partial
v,\partial w\big{)}$ are zero thanks to item (1). Thus
$\lambda^{\phi_{\bullet}}(\partial v,w)=0$. Analogously
$\lambda^{\phi_{\bullet}}(v,\partial w)=0$.

It remains to be shown that $\lambda^{D\phi_{\bullet}}(v,w)=0$ if
$v\in\mathcal{T}_{p}(X_{\bullet}),\>w\in\mathcal{T}_{q}(X_{\bullet})$,
$p,q\in\\{0,m\\}$. We first consider the case $p=0,\>q=m$. Then

(E.7)
$\begin{split}\lambda^{D\phi_{\bullet}}(v,w)=&\delta\phi_{m-1}(Ts_{m-1}\dots
Ts_{0}v,w)\\\ =&\sum_{i=0}^{m}(-1)^{i}\phi(Td_{i}Ts_{m-1}\dots
Ts_{0}v,Td_{i}w)\\\ =&(-1)^{m}\phi_{m-1}(Td_{m}Ts_{n-1}\dots
Ts_{0}v,Td_{m}w)\\\ =&\phi_{m-1}(Ts_{m-2}\dots
Ts_{0}v,(-1)^{m}Td_{m}w)=\lambda^{\phi_{\bullet}}(v,\partial w),\end{split}$

which equals zero since

$\begin{split}0=\lambda^{D\phi_{\bullet}}\big{(}v,(-1)^{m}Ts_{m-1}\partial
w\big{)}=&\lambda^{\phi_{\bullet}}(v,\partial\,(-1)^{m}Ts_{m-1}\partial w)\\\
=&\lambda^{\phi_{\bullet}}(v,(-1)^{m}Td_{m}(-1)^{m}Ts_{m-1}\partial
w)=\lambda^{\phi_{\bullet}}(v,\partial w),\end{split}$

where we have used (E.7) for the first equal sign and the fact that
$\lambda^{D\phi_{\bullet}}$ vanishes on degeneracies. Analogously it can be
shown that $\lambda^{D\phi_{\bullet}}=0$ if $p=m$ and $q=0$. ∎

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