Braided Crossed Modules and Loday-Pirashvili category

TESE DE DOUTORAMENTO
B aided C ossed Modules
and Loday-Pi ash ili ca ego y
ALEJANDRO FERNÁNDEZ-FARIÑA
ESCOLA DE DOUTORAMENTO INTERNACIONAL
PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS
SANTIAGO DE COMPOSTELA
2021
DECLARACIÓN DO AUTOR DA TESE
B aided C ossed Modules
and Loday-Pi ash ili ca ego y
D. Alejand o Fe nández Fa iña
P esen oamiña ese, seguindoop ocedemen oaxei ado aoRegulamen o, e decla o
que:
1) A ese aba ca os esul ados da elabo ación do meu aballo.
2) De selo caso, na ese aise e e encia ás colabo acións que i o es e aballo.
3) A ese é a e sión de ini i a p esen ada pa a a súa de ensa e coincide coa e sión
en iada en o ma o elec ónico.
4) Con i mo que a ese non inco e en ningún ipo de plaxio dou os au o es nin
de aballos p esen ados po min pa a a ob ención dou os í ulos.
En San iago de Compos ela, 11 de ab il de 2021
Asdo.: Alejand o Fe nández Fa iña
AUTORIZACIÓN DO DIRECTOR / TITOR DA TESE
B aided C ossed Modules
and Loday-Pi ash ili ca ego y
D. Manuel Lad a González
INFORMA:
Que a p esen e ese, co espóndese co aballo ealizado po D. Ale-
jand o Fe nández Fa iña, baixo amiñadi ección, e au o izoa súap esen ación,
conside ando que eúne os equisi os esixidos no Regulamen o de Es udos de
Dou o amen o da USC, e que como di ec o des a non inco e nas causas de
abs ención es ablecidas na Lei 40/2015.
De aco do co indicado no Regulamen o de Es udos de Dou o amen o,
decla a amén que a p esen e ese de dou o amen o é idónea pa a se de-
endida en base á modalidade de Monog á ica con ep oducción de publica-
ciones, nos que a pa icipación do/a dou o ando/a oi decisi a pa a a súa
elabo ación e as publicacións se axus an ao Plan de In es igación.
En San iago de Compos ela, 11 de ab il de 2021
Asdo.: Manuel Lad a González

B aided C ossed Modules
and Loday-Pi ash ili ca ego y
by
ALEJANDRO FERNÁNDEZ-FARIÑA
DISSERTATION
Submi ed o he deg ee o
DOCTOR EN MATEMÁTICAS
UNIVERSIDAD DE SANTIAGO DE COMPOSTELA
San iago de Compos ela, 2021
The esul s p esen ed in his hesis we e ob ained hanks o a g an om he Xun a
de Galicia (Spain), wi h e e ence ED481A-2017/064, he p ojec MTM2016-79661-
P om he Minis e io de Ciencia e Inno ación and Agencia Es a al de In es i-
gación (Eu opean FEDER suppo included, UE), and by he Conselle ía de Cul u a,
Educación e Uni e sidade da Xun a de Galicia, h ough he Compe i i e Re e ence
G oups (GRC), ED431C 2019/10 (Eu opean FEDER suppo included, UE).
MTM2016-79661-P
Unión Eu opea
Fondo Eu opeo de Desa ollo Regional
Una mane a de hace EUROPA
ED481A-2017/064 Xun a de Galicia
ED431C 2019/10
ca ego ies.
Keeping in mind wha is done o g oups, in his hesis we will gi e de ini ions o
b aidings o he a o emen ioned in e nal ca ego ies and c ossed modules. The case
o associa i e algeb as is no complex because he associa i i y allows us o wo k in
a na u al way wi h b aidings on semig oupal ca ego ies [17]. The no ion o b aiding
o Lie algeb as was al eady gi en by Ulualan [50]. On he o he hand, Ellis [20]
de ined he no ion o 2-c ossed module o Lie algeb as, also s udied by Ma ins and
Picken [46]. We will use a sligh ly di e en de ini ion o b aiding o c ossed mod-
ules o Lie algeb as han he one gi en by Ulualan [50], since we wan a pa allelism
be ween he examples o b aided c ossed modules o g oups and b aided c ossed mod-
ules o Lie algeb as, and we also equi e b aided c ossed modules o be a pa icula
case o 2-c ossed modules, as i happens in he case o g oups.
Leibniz algeb as appea in ma hema ics as a “non-an isymme ic” case o Lie al-
geb as. Bea ing his in mind, in his hesis, we will show how o ex end he idea o
b aiding o c ossed modules and in e nal ca ego ies o Lie algeb as o he Leibniz se -
ing. A e in oducing hese no ions, we will p o e he equi alence be ween b aided
c ossed modules o Leibniz algeb as and b aided ca ego ical Leibniz algeb as, and
we will show he pa allelism be ween i s examples and he ones gi en o g oups,
associa i e algeb as and Lie algeb as.
Al hough Lie algeb as a e a sub a ie y o he a ie y o Leibniz algeb as, Loday
and Pi ash ili ound in [44] ha Leibniz algeb as can be seen as a ull co e lec i e
subca ego y o a speci ic ype o Lie objec s. They in oduced a new enso p oduc
in he ca ego y o linea maps o ec o spaces and in e nalised he concep o Lie
algeb a in a (b aided) symme ic monoidal ca ego y. This ealisa ion p o ed o be
e y hand ul s udying di e en p oblems in Leibniz algeb as, as Lie heo y is much
be e de eloped, see [12,23,49] o example. We will use his ca ego y o ex end he
concep o b aiding om he Lie case o he Leibniz case.
The concep o cen al ex ension o g oups o Lie algeb as is highly ele an in
ma hema ics, and i plays a undamen al ole in se e al a eas o physics as well. This
no ion was ex ended o c ossed modules o g oups o Lie algeb as. The s udy o
cen al ex ensions in he ca ego ies o c ossed modules was ini ia ed in [48] o g oups
and in [13] o Lie algeb as, and i emains a cu en esea ch opic, as shown by he
di e en li e a u e ackling his issue.
Since c ossed modules o g oups and Lie algeb as a e a gene alisa ion o g oups
and Lie algeb as, i is essen ial o sea ch, in he ca ego y o c ossed modules o g oups
o Lie algeb as, ex ensions o classical esul s in he heo y o g oups o Lie algeb as.
In [26], Fukushi ga e a b aided e sion o he esul s on uni e sal cen al ex en-
x i

sions o c ossed modules o g oups p o ided by No ie in [48]. He ound a na u al
b aiding on he uni e sal cen al ex ension o a c ossed module o g oups which be-
ha es well wi h one b aided c ossed module. Howe e , i is no he a che ype o
uni e sal cen al ex ension in he ca ego y o b aided c ossed modules since, in his
ca ego y, i is necessa y o add addi ional es ic ions including he b aiding on he
no ions o cen e and commu a o .
In his wo k, we will de ise a b aided e sion o he esul s gi en by Casas and
Lad a in [13] o b aided c ossed modules o Lie 𝐾-algeb as; mo e p ecisely, we will
s udy uni e sal cen al ex ensions in he ca ego y o b aided Lie c ossed modules
BX(LieAlg𝐾). Fo ha pu pose, we will need he de ini ion o cen e and commu a o
gi en by Huq in [35] in he b aided con ex .
No e ha he amewo k o Chap e 3 is di e en om ha gi en in [14], since
he ca ego y X(LieAlg𝐾)is no a Bi kho subca ego y o BX(LieAlg𝐾).
The s udy o he in e nalisa ion o Lie algeb as is also a e y hand ul ool, as we
will see in Sec ion 2.4. I also allows p o ing di e en p ope ies in se e al kinds o
ca ego ies a he same ime, such as Lie supe algeb as, ℤ-g aded Lie algeb as, di -
e en ial g aded Lie algeb as o egula Hom-Lie algeb as [33]. Fo example, wo
impo an p ope ies ha cha ac e ise he a ie y o Lie algeb as amongs all he a i-
e ies o non-associa i e algeb as, he exis ence o algeb aic exponen s [29,30] o he
ep esen abili y o ac ions [28], hold also in he ca ego ies o Lie objec s o e ce ain
ypes o monoidal ca ego ies [27,34].
We wan o gene alise Loday and Pi ash ili cons uc ion ou o he linea maps
ca ego y, de ining a new enso p oduc in ce ain kinds o ca ego ies wi h ope a ions,
wi h he leas amoun o p ope ies needed o do so, o ob ain he Loday-Pi ash ili
ca ego y. Then, we will p o e ha he Leibniz objec s ( he in e nalisa ion o Leibniz
algeb as), can be seen as a pa icula case o Lie objec s in he Loday-Pi ash ili ca -
ego y. In he pa icula case o ec o spaces, his cons uc ion gene alises he one
gi en in [44].
Th oughou his ex , we will suppose ha 𝐾is a ield.
S uc u e o he hesis
This manusc ip is o ganized as ollows. In he p elimina ies (Chap e 1), we will
ecall some basic de ini ions, and we will gi e he no ion o b aiding o semig oupal
ca ego ies.
In Chap e 2, we will s udy he b aidings o c ossed modules and in e nal ca -
ego ies. We will s a showing he i s case o b aiding, he case o g oups (Sec-
x ii
ion 2.1), and we will ake i as a base o in oduce he no ions o b aided ca ego ical
associa i e algeb a and b aided c ossed module o associa i e algeb as (Sec ion 2.2).
We will show he equi alence o he associa i e case. Then, in Sec ion 2.3, we will
mo i a e he de ini ion gi en by Ulualan [50] o b aided c ossed modules o Lie al-
geb as using ou de ini ion o b aiding o c ossed modules o associa i e algeb as,
and we gi e a simple de ini ion when cha (𝐾)≠2. We will also discuss a di e en
de ini ion o b aided c ossed module o Lie algeb as showing i s ela ionship wi h he
associa i e case. F om he e, in Sec ion 2.4 we will s udy he Leibniz algeb as case.
We show he in e naliza ion o a c ossed module’s no ion wi h a le Lie ac ion o Lie
objec s in an a bi a y ca ego y. We will also de ine b aidings o c ossed modules o
Lie objec s and ca ego ical Lie objec s. Then we apply his de ini ion o he Loday-
Pi ash ili ca ego y 𝐾, and we will ob ain he concep s o b aiding o c ossed
modules o Leibniz algeb as and ca ego ical Leibniz algeb as. Wi h he new de i-
ni ion o b aiding, we will p o e he equi alence be ween b aided ca ego ies in he
Leibniz algeb as case, and inally, in Sec ion 2.5, we will see he non-abelian enso
p oduc o g oups as an example o a b aided c ossed module o g oups. Fu he mo e,
wi h ou de ini ion o b aiding o c ossed modules o Lie algeb as, we ob ain simi-
la ly an example o b aiding using he non-abelian enso p oduc o Lie algeb as. The
same is ue o ou de ini ion o b aiding o c ossed modules o Leibniz algeb as.
In Chap e 3, we will s udy wo ideas o uni e sal cen al ex ensions o b aid-
ing c ossed modules o Lie algeb as. In Sec ion 3.1, we p o ide he de ini ions o
cen al ex ensions in he ca ego y o Lie c ossed modules X(LieAlg𝐾)and B-cen al
ex ensions in BX(LieAlg𝐾), necessa y o de eloping he chap e . In Sec ion 3.2, we
cons uc he uni e sal B-cen al ex ension o a B-pe ec b aided Lie c ossed module
and p o e ha a b aided Lie c ossed module admi s a uni e sal B-cen al ex ension i
and only i i is B-pe ec . In Sec ion 3.3, we cons uc he uni e sal 𝔘-cen al ex en-
sion o b aided c ossed modules, which a e pe ec as Lie c ossed modules, whe e
𝔘∶BX(LieAlg𝐾)⟶X(LieAlg𝐾)is he o ge ul unc o . In Sec ion 3.4, we s udy
he ela ion be ween he uni e sal B-cen al ex ension and he uni e sal 𝔘-cen al
ex ension o a b aided Lie c ossed module. Finally, we p o e ha bo h uni e sal ex-
ensions exis and coincide o a B-pe ec b aided Lie c ossed module.
In Chap e 4, we will de ine he LP ca ego y o di e en enso ca ego ies, and
hen we s udy he Lie objec s in some kind o LP ca ego ies and hei ela ionship wi h
he Leibniz objec s in he base ca ego y. In Sec ion 4.1, we will s udy he di e en
enso ca ego ies: ca ego ies wi h ope a ions, (b aided) semig oupal ca ego ies and
(b aided) monoidal ca ego ies, and we will cons uc hei Loday-Pi ash ili ca ego y.
In Sec ion 4.2, we will alk abou addi i e ca ego ies and we will show ha , wi h
x iii
some assump ions, we can eco e many p ope ies in he maps be ween he enso
p oduc and he “+” ope a ion. The las sec ion (Sec ion 4.3) is de o ed o s udy
he in e naliza ion o a Leibniz objec and Lie objec in a ca ego y , showing ha
he Liesa ion unc o exis s be ween hese ca ego ies. Then we will p o ide a be e
unde s anding o Lie objec s in he Loday-Pi ash ili ca ego y o . To conclude, we
will p o e ha he ca ego y o Leibniz objec s in is a ull co e lec i e subca ego y
o he Lie objec s in he Loday-Pi ash ili ca ego y o .
xix
xx
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Objec i es and hypo heses
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This hesis has he ollowing hypo hesis and objec i es:
Hyp. 1 S ic monoidal ca ego ies can be seen as ca ego ical monoids in Se . Simila ly,
we can hink ha a s ic semig oupal ca ego y o e an in e nal ca ego y in
Vec 𝐾is eally an in e nal associa i e 𝐾-algeb a.
Obj. 1 We wan o use he idea o b aiding in a semig oupal ca ego y o make a b aid-
ing o ca ego ical associa i e 𝐾-algeb as using ha he same idea as he ac
ha he ca ego y o c ossed modules o g oups has a na u al idea o b aiding
u ilising he idea o monoidal ca ego y.
Hyp. 2 In he ca ego y o c ossed modules o g oups, we can de ine he concep o
b aiding. Wi h ha cons uc ion, we ha e en equi alence be ween b aided
c ossed modules o g oups and b aided ca ego ical g oups.
Obj. 2 We wan o cons uc a b aiding o c ossed modules o associa i e algeb as
such ha his new ca ego y is equi alen o b aiding ca ego ical associa i e 𝐾-
algeb as.
Hyp. 3 The ca ego y o associa i e 𝐾-algeb as and Lie 𝐾-algeb as a e ela ed wi h a
unc o (−)∶AssAlg𝐾←←→ LieAlg𝐾, which akes an associa i e algeb a 𝐴and
gi es back a Lie 𝐾-algeb a 𝐴. This Lie algeb a has he same 𝐾- ec o space
as unde lying s uc u e and has as mul iplica ion [𝑥, 𝑦] ∶= 𝑥𝑦 −𝑦𝑥.
Obj. 3 Use he idea o unc o (−) o make a new unc o om c ossed modules o
associa i e 𝐾-algeb as o c ossed modules o Lie 𝐾-algeb as. We will also
de ine a unc o om i s b aided e sions, showing he na u alness when one
de ines he b aiding o he Lie case. We will also show an equi alen (in he
1

2 Objec i es and hypo heses
sense o ca ego ies) de ini ion o b aiding c ossed modules, which will gi e us
a good example using he non-abelian enso p oduc .
Hyp. 4 TheLoday-Pi ash ili ca ego y p o idesuswi h a way oseeLeibniz𝐾-algeb as
as a pa icula case o Lie objec s when i is well known ha Lie 𝐾-algeb as
a e a pa icula case o Leibniz 𝐾-algeb as.
Obj. 4 Using he Loday-Pi ash ili ca ego y and in e naliza ion, we wan o use he
de ini ion o b aiding o he Lie case o de ine b aiding o c ossed modules o
Leibniz 𝐾-algeb as and ca ego ical Leibniz 𝐾-algeb as. Once done, we will
show ha hese new s uc u es ha e he Lie case as a pa icula example. Also,
we will ha e an excellen example o b aiding c ossed modules o Leibniz 𝐾-
algeb as using he non-abelian enso p oduc .
Hyp. 5 The b aiding o he Lie case gi es equi alen ca ego ies.
Obj. 5 We wan o show ha he b aidings o he Leibniz case gi e equi alen ca e-
go ies.
Hyp. 6 The b aiding c ossed modules o g oups ha e a uni e sal 𝔘-cen al ex ension.
Obj. 6 We wan o de ine he 𝔘-cen al ex ension o Lie algeb as ca ego y since many
esul s a e ue in he g oup case a e ue in he Lie case. We also desc ibe he
B-cen al ex ensions using he idea o cen e gi e by Huq [35]. In gene al, he
𝔘-cen al ex ensions and B-cen al ex ension do no coincide, bu we wan o
show he ela ionship be ween he uni e sal ones.
Hyp. 7 The cons uc ion o he LP-ca ego y gi en by Loday and Pi ash ili can be de-
ined in ca ego ies wi h a small se o p ope ies.
Obj. 7 We wan o de ine he Loday-Pi ash ili ca ego y using he leas p ope ies ha
a e possible. We will do ha o ca ego ies wi h ope a ions, (b aided) semi-
g oupal ca ego ies and (b aided) monoidal ca ego ies, showing ha he enso
p oduc in he Loday-Pi ash ili ca ego y gi es a enso ca ego y o he same
ype.
Hyp. 8 The e is a Liesa ion unc o om Leibniz 𝐾-algeb as o Lie 𝐾-algeb as ha is
a le adjoin o he o ge ul unc o .
Obj. 8 We wan o cons uc a Liesa ion unc o o Leibniz objec s o Lie objec s o
any ca ego y wi h a small se o p ope ies.
Objec i es and hypo heses 3
Obj. 9 We will p o e ha he ca ego y o Leibniz objec s in is a ull co e lec i e
subca ego y o he Lie objec s in he Loday-Pi ash ili ca ego y o .
4 Objec i es and hypo heses
CHAPTER 1
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P elimina ies
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In his chap e , we will gi e he basic concep s o del e in o he es o he chap e s.
1.1 In e nal ca ego ies
De ini ion 1.1.1. Le Cbe a ca ego y wi h pullbacks.
An in e nal ca ego y in Cconsis o wo objec s 𝐶1(mo phisms objec ) and 𝐶0
(objec s objec ) o C, oge he wi h he ou ollowing mo phisms 𝑠, 𝑡, 𝑒, 𝑘:
𝐶0𝑒//𝐶1
𝑠
oo
𝑡
oo𝐶1×𝐶0𝐶1,
𝑘
oo
whe e 𝐶1×𝐶0𝐶1is he pullback o 𝑡and 𝑠.
𝑠is called sou ce mo phism,𝑡is called a ge mo phism,𝑒is called iden i y map-
ping mo phism and 𝑘is called composi ion mo phism.
In addi ion, he mo phisms mus sa is y commu a i e diag ams ha exp ess he
usual ca ego y laws (see [6]):
(I1)
𝐶0𝐶1
𝐶0.
𝑒
Id𝐶0
𝑠(I2)
𝐶0𝐶1
𝐶0.
𝑒
Id𝐶0
𝑡
5
12 1 P elimina ies
Le us begin wi h he de ini ion o 𝔄on objec s. Le (𝑀𝜕
←←←←←←→ 𝑁, ∗) be a c ossed
module in AssAlg𝐾. We will ake he semidi ec p oduc 𝑀⋊𝑁wi h ∗. Conside
he ollowing diag am:
(𝑀⋊𝑁) ×𝑁(𝑀⋊𝑁)𝑀⋊𝑁 𝑁

𝑘

𝑡
𝑠
𝑒
wi h he ollowing maps: 𝑠((𝑚, 𝑛)) = 𝑏,
𝑡((𝑚, 𝑛)) = 𝜕(𝑚) + 𝑛,𝑒(𝑛) = (0, 𝑛)and

𝑘(((𝑚, 𝑛),(𝑚′, 𝜕(𝑚) + 𝑛))) = (𝑚+𝑚′, 𝑛) o all 𝑚, 𝑚′∈𝑀,𝑛∈𝑁. No e ha i
((𝑛, 𝑚),(𝑛′, 𝑚′)) ∈ (𝑀⋊𝑁) ×𝑁(𝑀⋊𝑁),𝑛′=𝑠((𝑚′, 𝑛′)) = 
𝑡((𝑚, 𝑛)) = 𝜕(𝑚) + 𝑛, so
he de ini ion o 
𝑘makes sense. I is necessa y o p o e ha 𝑠,
𝑡,𝑒 and 
𝑘a e mo phisms
in AssAlg𝐾, ha is, hey p ese e all he ope a ions. Since i is ob ious ha 𝑠 and 𝑒
p ese e he ope a ions, we will ocus on ske ching how o p o e ha 
𝑡and 
𝑘p ese e
he sum and he p oduc . Calcula ions a e qui e long, so we will no include hem,
al hough we will poin ou he c ucial ideas equi ed o comple e hem. Rega ding 
𝑡,
i p ese es he sum di ec ly om he ac ha 𝜕p ese e i . Fu he mo e, he ac
ha 𝜕is an 𝑁-equi a ian associa i e mo phism is he key o p o e ha 
𝑡p ese es
he p oduc .
Conce ning 
𝑘, no e ha he elemen s in (𝑀⋊𝑁) ×𝑁(𝑀⋊𝑁)a e o he o m
((𝑚, 𝑛),(𝑚′, 𝜕(𝑚)+𝑛)), wi h 𝑚, 𝑚′∈𝑀,𝑛∈𝑁. Immedia ely below we will show he
calcula ions equi ed op o e ha 
𝑘p ese es he sum. Le ((𝑚𝑖, 𝑛𝑖),(𝑚′
𝑖, 𝜕(𝑚𝑖)+𝑛𝑖)) ∈
(𝑀⋊𝑁) ×𝑁(𝑀⋊𝑁) o 𝑖= 1,2. On one hand we ha e ha

𝑘(((𝑚1, 𝑛1),(𝑚′
1, 𝜕(𝑚1) + 𝑛1)) + ((𝑚2, 𝑛2),(𝑚′
2, 𝜕(𝑚2) + 𝑛2)))
=
𝑘((𝑚1, 𝑛1)+(𝑚2, 𝑛2),(𝑚′
1, 𝜕(𝑚1) + 𝑛1)+(𝑚′
2, 𝜕(𝑚2) + 𝑛2))
= ((𝑚1+𝑚2, 𝑛1+𝑛2),(𝑚′
1+𝑚′
2, 𝜕(𝑚1) + 𝑛1+𝜕(𝑚2) + 𝑛2))
= (𝑚1+𝑚2+𝑚′
1+𝑚′
2, 𝑛1+𝑛2),
On he o he hand,

𝑘(((𝑚1, 𝑛1),(𝑚′
1, 𝜕(𝑚1) + 𝑛1))) + 
𝑘(((𝑚2, 𝑛2),(𝑚′
2, 𝜕(𝑚2) + 𝑛2)))

1.3.2 C ossed modules o associa i e algeb as 13
= (𝑚1+𝑚′
1, 𝑛1)+(𝑚2+𝑚′
2, 𝑛2) = (𝑚1+𝑚′
1+𝑚2+𝑚′
2, 𝑛1+𝑛2),
by making use o he de ini ion o 
𝑘and he addi ion in 𝑀⋊𝑁. Hence, 
𝑘p ese es
he sum. Calcula ions o he p oduc a e simila , bu in ol ing dis ibu i i y and he
Pei e iden i y.
Commu a i i y o he diag ams o he in e nal ca ego ies is easy.
De ining 𝔄on mo phisms is qui e ob ious. Gi en a mo phism o c ossed mod-
ules (𝑓1, 𝑓2)be ween (𝑀𝜕
←←←←←←→ 𝑁, ∗) and (𝑀′𝜕′
←←←←←←←←→ 𝑁′,∗′), i s co esponding in e -
nal unc o is gi en by 𝑓1×𝑓2∶𝑀⋊𝑁→𝑀′⋊𝑁′and 𝑓2∶𝑁→𝑁′, whe e
𝑓1×𝑓2((𝑎, 𝑏)) = (𝑓1(𝑎), 𝑓2(𝑏)). Commu a i i y o he diag ams o in e nal unc-
o s ollows om he de ini ions o 𝑠,
𝑠′,
𝑡,
𝑡′,𝑒,
𝑒′,
𝑘and 
𝑘′, along wi h he equali y
𝑓2◦𝜕=𝜕′◦𝑓1.
𝔄is clea ly a unc o wi h he p e ious assignmen s o objec s and mo phisms.
Now le us de ine he unc o 𝔄. Le 𝐶= (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)be an in e nal ca -
ego y in AssAlg𝐾. Conside ke (𝑠)and he mo phism 𝑡|ke (𝑠)∶ ke (𝑠)→𝐶0. We
will w i e 𝜕𝑡in o de o ease no a ion. We de ine an associa i e ac ion (𝑒∗,∗𝑒)wi h
𝑎𝑒∗𝑥=𝑒(𝑎)𝑥and 𝑥∗𝑒𝑎=𝑥𝑒(𝑎)wi h 𝑎∈𝐶0, 𝑥 ∈ ke (𝑠). I is easy ha he maps
a e well de ined.
I only emains o p o e ha (ke (𝑠)𝜕𝑡
←←←←←←←←→ 𝐶0,(𝑒∗,∗𝑒)) sa is ies is a c ossed module.
Gi en 𝑎∈𝐶0and 𝑥∈ ke (𝑠),
𝜕𝑡(𝑎𝑒∗𝑥) = 𝜕𝑡(𝑒(𝑎)𝑥) = 𝑡(𝑒(𝑎)𝑥) = 𝑡(𝑒(𝑎))𝑡(𝑥) = 𝑎𝜕𝑡(𝑥),
𝜕𝑡(𝑥∗𝑒𝑎) = 𝜕𝑡(𝑥𝑒(𝑎)) = 𝑡(𝑥𝑒(𝑎)) = 𝑡(𝑥)𝑡(𝑒(𝑎)) = 𝜕𝑡(𝑥)𝑎.
No e ha we use ha in an in e nal ca ego y 𝑡◦𝑒= Id𝐶0.
To p o e he Pei e iden i y, le 𝑥1, 𝑥2∈ ke (𝑠).
𝜕𝑡(𝑥1)𝑒∗𝑥2=𝑒(𝑡(𝑥1))𝑥2.
We need o show ha 𝑒(𝑡(𝑥1))𝑥2=𝑥1𝑥2. We will ake 𝑧= (𝑒(𝑡(𝑥1)) − 𝑥1).
𝑡(𝑧) = 𝑡(𝑒(𝑡(𝑥1))) − 𝑡(𝑥1) = 𝑡(𝑥1) − 𝑡(𝑥1)=0so we can compose wi h 𝑒(0)
since. 𝑡(𝑧) = 0 = 𝑠(0) = 𝑠(𝑒(0)). Since 𝑥2∈ ke (𝑠), we can ake 𝑘((𝑒(0), 𝑥2)),
14 1 P elimina ies
because 𝑠(𝑦) = 0 = 𝑡(0) = 𝑡(𝑒(0)). In addi ion we ha e ha in an in e nal ca ego y
𝑘((𝑧, 𝑒(0))) = 𝑧and 𝑘((𝑒(0), 𝑥2)) = 𝑥2.
0 = 𝑘((0,0)) = 𝑘((𝑧𝑒(0), 𝑒(0)𝑥2))
=𝑘((𝑧, 𝑒(0))(𝑒(0), 𝑥2)) = 𝑘((𝑧, 𝑒(0)))𝑘((𝑒(0), 𝑥2)) = 𝑧𝑥2.
Finally, we ha e:
0 = 𝑧𝑥2= (𝑒(𝑡(𝑥1)) − 𝑥1)𝑥2=𝑒(𝑡(𝑥1))𝑥2−𝑥1𝑥2,
which es ablishes one o he Pei e iden i ies. Simila a gumen s apply o he o he
Pei e iden i y.
De ining 𝔄on mo phisms is also qui e ob ious. Le 𝐶= (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)and
𝐶′= (𝐶′
1, 𝐶′
0, 𝑠′, 𝑡′, 𝑒′, 𝑘′)be wo in e nal ca ego ies in AssAlg𝐾and 𝐹∶𝐶→𝐶′
an in e nal unc o , wi h 𝐹1∶𝐶1→𝐶′
1and 𝐹0∶𝐶0→𝐶′
0. I s co esponding mo -
phism o c ossed modules is gi en by (𝐹𝑠
1, 𝐹0), wi h 𝐹𝑠
1(𝑥) = 𝐹1(𝑥) o 𝑥∈ ke (𝑠),
which ollows om he diag ams o in e nal unc o s. I is easy o check ha , wi h
he p e ious assignmen s, 𝔄is indeed a unc o .
𝔄and 𝔄es ablish an equi alence be ween he ca ego ies whe e he na u al
isomo phisms IdX(AssAlg𝐾)
𝛼𝔄
≅𝔄◦𝔄and IdICa (AssAlg𝐾)
𝛽𝔄
≅𝔄◦𝔄a e gi en by:
∙i = (𝑀𝜕
←←←←←←→ 𝑁, (∗1,∗2)) is a c ossed module o associa i e 𝐾-algeb as, hen
𝛼𝔄
= (𝛼𝔄
𝑀,Id𝑁), wi h 𝛼𝔄
𝑀∶𝑀←←→ (𝑀, 0) de ined as 𝛼𝑀(𝑚) = (𝑚, 0);
∙i = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)is a ca ego ical associa i e 𝐾-algeb a, hen 𝛽𝔄
=
(𝛽𝔄
𝑠,Id𝐶0), wi h 𝛽𝔄
𝐶1∶𝐶1←←→ ke (𝑠)⋊𝐶0is de ined as 𝛽𝔄
𝐶1(𝑥)=(𝑥−𝑒(𝑠(𝑥)), 𝑠(𝑥)).
1.3.3 C ossed modules o Lie algeb as
We ha e an analogous de ini ion o he case o Lie 𝐾-algeb as. C ossed modules
o Lie 𝐾-algeb as we e in oduced by Kassel and Loday in [40].
De ini ion 1.3.12. Le 𝑀and 𝑁 wo Lie 𝐾-algeb as. A Lie (le -)ac ion o 𝑁on 𝑀
is a 𝐾-bilinea map ⋅∶𝑁×𝑀⟶𝑀,(𝑛, 𝑚)⟼𝑛⋅𝑚, sa is ying:
[𝑛, 𝑛′]⋅𝑚=𝑛⋅(𝑛′⋅𝑚) − 𝑛′⋅(𝑛⋅𝑚),
1.3.3 C ossed modules o Lie algeb as 15
𝑛⋅[𝑚, 𝑚′] = [𝑛⋅𝑚, 𝑚′]+[𝑚, 𝑛 ⋅𝑚′], 𝑛, 𝑛′∈𝑁, 𝑚, 𝑚′∈𝑀.
I we deno e ⋅= [−,−], he wo iden i ies a e he wo possible ew i es o he
Jacobi iden i y by aking wo elemen s in 𝑁o wo in 𝑀.
In pa icula , i 𝑀is a Lie 𝐾-algeb a and 𝑥∈𝑀, we ha e ha he adjoin map
ad(𝑥)∶ 𝑀←←→ 𝑀,ad(𝑥)(𝑦) = [𝑥, 𝑦], is a Lie ac ion o 𝑀on i sel .
De ini ion 1.3.13. Ac ossed module o Lie 𝐾-algeb as is a pai (𝑀𝜕
←←←←←←→ 𝑁, ⋅)whe e
𝑀and 𝑁a e Lie 𝐾-algeb as, ⋅is a Lie ac ion o 𝑁on 𝑀, and 𝑀𝜕
←←←←←←→ 𝑁is a Lie
𝐾-homomo phism sa is ying:
-𝜕is an 𝑁-equi a ian Lie 𝐾-homomo phism (we suppose he adjoin ac ion o 𝑁
on i sel ), i.e.
𝜕(𝑛⋅𝑚) = ad(𝑛)(𝜕(𝑚)) = [𝑛, 𝜕(𝑚)], 𝑛 ∈𝑁, 𝑚 ∈𝑀,
-Pei e iden i y:
𝜕(𝑚)⋅𝑚′= ad(𝑚)(𝑚′) = [𝑚, 𝑚′], 𝑚, 𝑚′∈𝑀.
Example 1.3.14.
1. As in he p e ious cases we ha e he example o c ossed module o Lie 𝐾-
algeb as (𝑀Id𝑀
←←←←←←←←←←←←←←←→ 𝑀, [−,−]), wi h he adjoin ac ion, 𝑚⋅𝑚′= [𝑚, 𝑚′], whe e
𝑀is a Lie 𝐾-algeb a.
2. Any cen al ex ension o Lie algeb as 𝑀𝜕
←←←←←←→→ 𝑁is a c ossed module, wi h he
ac ion 𝜕(𝑚)⋅𝑚′= [𝑚, 𝑚′]. Con e sely, a simply connec ed c ossed module (i.e.
𝜕is su jec i e) is a cen al ex ension.
In pa icula , 𝑀ad
←←←←←←←←←→→ IDe (𝑀),𝑚↦ad(𝑚), wi h he ac ion, ad(𝑚)⋅𝑚′=
[𝑚, 𝑚′], is a Lie c ossed module, whe e IDe (𝑀)a e he inne de i a ions o a
Lie algeb a 𝑀.
De ini ion 1.3.15. Ahomomo phism o c ossed modules o Lie 𝐾-algeb as be ween
(𝑀𝜕
←←←←←←→ 𝑁, ⋅)and (𝑀′𝜕
←←←←←←→ 𝑁′,∗) is a pai o Lie 𝐾-homomo phisms, 𝑓1∶𝑀←←→ 𝑀′
and 𝑓2∶𝑁←←→ 𝑁′such ha :
𝑓1(𝑛⋅𝑚) = 𝑓2(𝑛) ∗ 𝑓1(𝑚),(XLieH1)
16 1 P elimina ies
𝜕′◦𝑓1=𝑓2◦𝜕, (XLieH1)
𝑚∈𝑀, 𝑛 ∈𝑁.
The e is a na u al way o co ela e he c ossed modules o associa i e 𝐾-algeb as
wi h he c ossed modules o Lie 𝐾-algeb as. The ollowing esul s ha ela e bo h
can be seen in [21].
Lemma 1.3.16. Le 𝑀and 𝑁be wo associa i e 𝐾-algeb as.
We deno e by 𝐴 he Lie 𝐾-algeb a associa ed o an associa i e 𝐾-algeb a 𝐴,
i.e. he Lie 𝐾-algeb a wi h he ope a ion [𝑎, 𝑎′] = 𝑎𝑎′−𝑎′𝑎.
(i) I ∗= (∗1,∗2)is an associa i e ac ion o 𝑁on 𝑀, hen we ha e ha he map
[−,−]∗∶𝑁×𝑀←←→ 𝑀, de ined as [𝑛, 𝑚]∗=𝑛∗1𝑚−𝑚∗2𝑛, is a Lie ac ion
o 𝑁on 𝑀.
(ii) I (𝑀𝜕
←←←←←←→ 𝑁, ∗) is a c ossed module o associa i e 𝐾-algeb as, hen we ha e
ha (𝑀𝜕
←←←←←←→ 𝑁,[−,−]∗)is a c ossed module o Lie 𝐾-algeb as.
Rema k 1.3.17.Wi h he p e ious p ope y we can see ha he examples gi en o he
associa i e algeb as case, (𝑀Id𝑀
←←←←←←←←←←←←←←←→ 𝑀, (∗,∗)), and o he Lie case o he associa e
Lie algeb a 𝑀,(𝑀Id𝑀
←←←←←←←←←←←←←←←←←←←←→, 𝑀,[−,−]∗), a e ela ed.
We deno e by X(LieAlg𝐾) he ca ego y o c ossed modules o Lie 𝐾-algeb as and
hei homomo phisms.
Rema k 1.3.18.The p e ious lemma gi es us a unc o
(−)
∶X(AssAlg𝐾)←←←←←←←←←←←←→ X(LieAlg𝐾).
We ha e he nex p oposi ion which ela es he ca ego ical associa i e 𝐾-algeb as
wi h he ca ego ical Lie 𝐾-algeb as.
P oposi ion 1.3.19. I (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)is a ca ego ical associa i e 𝐾-algeb a, hen
(𝐶
1, 𝐶
0, 𝑠, 𝑡, 𝑒, 𝑘)is a ca ego ical Lie 𝐾-algeb a.
P oo . Immedia e since (𝐶1×𝐶0𝐶1)=𝐶
1×𝐶
0𝐶
1. They a e he same unde lying
ec o space and ha e he same ope a ion.
1.3.4 C ossed modules o Leibniz algeb as 17
Rema k 1.3.20.The p e ious p oposi ion gi es us a unc o
(−)
∶ICa (AssAlg𝐾)←←←←←←←←←←←←→ ICa (LieAlg𝐾).
Rema k 1.3.21.As in he case o g oups and associa i e 𝐾-algeb as, he ca ego ies
ICa (LieAlg𝐾)and X(LieAlg𝐾)a e equi alen (see [5,22,25]).
I is easy o check ha he equi alence unc o s commu e wi h he unc o s (−)

and (−)
. We only need o show ha (−)p ese es he semidi ec p oduc .
De ini ion 1.3.22. Le 𝑀and 𝑁be wo Lie 𝐾-algeb as and ⋅a Lie ac ion o 𝑁on 𝑀.
We de ine i s semidi ec p oduc , deno ed by 𝑀⋊𝑁, as he 𝐾- ec o space 𝑀×𝑁
wi h he ollowing b acke :
[(𝑚, 𝑛),(𝑚′, 𝑛′)] = ([𝑚, 𝑚′] + 𝑛⋅𝑚′−𝑛′⋅𝑚, [𝑛, 𝑛′]).
P oposi ion 1.3.23. Le 𝑀and 𝑁be associa i e 𝐾-algeb as. I ∗is an associa i e
ac ion o 𝑁on 𝑀( hen [−,−]∗is a Lie ac ion o 𝑁in 𝑀), hen we ha e ha
(𝑀⋊𝑁)=𝑀⋊𝑁.
P oo . Since he unde lying ec o space is he same, we only need o p o e ha he
b acke is he same.
(𝑚, 𝑛)(𝑚′, 𝑛′)−(𝑚′, 𝑛′)(𝑚, 𝑛)
= (𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′, 𝑛𝑛′)−(𝑚′𝑚+𝑛′∗1𝑚+𝑚′∗2𝑛, 𝑛′𝑛)
= (𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′−𝑚′𝑚−𝑛′∗1𝑚−𝑚′∗2𝑛, 𝑛𝑛′−𝑛′𝑛)
= ([𝑚, 𝑚′]+[𝑛, 𝑚′]∗− [𝑛′, 𝑚]∗,[𝑛, 𝑛′]),
whe e (𝑚, 𝑛),(𝑚′, 𝑛′) ∈ 𝑀×𝑁.
1.3.4 C ossed modules o Leibniz algeb as
The de ini ion o c ossed modules o Leibniz 𝐾-algeb as, “non-an isymme ic”
case o Lie 𝐾-algeb as, was in oduced by Loday and Pi ash ili in [43].

18 1 P elimina ies
De ini ion 1.3.24. Le 𝑁and 𝑀be wo Leibniz 𝐾-algeb as. A Leibniz ac ion o 𝑁
on 𝑀is a pai ⋅= (⋅1,⋅2)whe e ⋅1∶𝑁×𝑀←←→ 𝑀and ⋅2∶𝑀×𝑁←←→ 𝑀a e 𝐾-bilinea
maps and he ollowing p ope ies a e sa is ied
𝑛⋅1[𝑚, 𝑚′]=[𝑛⋅1𝑚, 𝑚′]−[𝑛⋅1𝑚′, 𝑚],(ALeib1)
[𝑚, 𝑛 ⋅1𝑚′]=[𝑚⋅2𝑛, 𝑚′]−[𝑚, 𝑚′]⋅2𝑛, (ALeib2)
[𝑚, 𝑚′⋅2𝑛]=[𝑚, 𝑚′]⋅2𝑛− [𝑚⋅2𝑛, 𝑚′],(ALeib3)
𝑚⋅2[𝑛, 𝑛′] = (𝑚⋅2𝑛)⋅2𝑛′− (𝑚⋅2𝑛′)⋅2𝑛, (ALeib4)
𝑛⋅1(𝑚⋅2𝑛′) = (𝑛⋅1𝑚)⋅2𝑛′− [𝑛, 𝑛′]⋅1𝑚, (ALeib5)
𝑛⋅1(𝑛′⋅1𝑚) = [𝑛, 𝑛′]⋅1𝑚− (𝑛⋅1𝑚)⋅2𝑛′.(ALeib6)
𝑚, 𝑚′∈𝑀, 𝑛, 𝑛′∈𝑁.
Rema k 1.3.25.I we change he no a ion o ⋅1and ⋅2by [−,−] in bo h cases, he
axioms o he Leibniz ac ions a e all possible ew i ings o he Leibniz iden i y when
we choose wo elemen s in 𝑀and one in 𝑁( he i s h ee) o one in 𝑀and wo 𝑁
( he las h ee).
In pa icula , we ha e ha he pai ([−,−],[−,−]) whe e [−,−] is he Leibniz
b acke o he Leibniz 𝐾-algeb a 𝑀is a Leibniz ac ion o 𝑀on i sel .
De ini ion 1.3.26. A c ossed module o Leibniz 𝐾-algeb as is a pai (𝑀𝜕
←←←←←←→ 𝑁, ⋅)
whe e 𝑀and 𝑁a e Leibniz 𝐾-algeb as, ⋅= (⋅1,⋅2)is a Leibniz ac ion o 𝑁on 𝑀,
𝜕∶𝑀←←→ 𝑁is a Leibniz 𝐾-homomo phism, and he ollowing p ope ies a e sa is ied:
-𝜕is an 𝑁-equi a ian Leibniz 𝐾-homomo phism (we suppose ha he b acke gi es
he ac ion in 𝑁), i.e.
𝜕(𝑛⋅1𝑚) = [𝑛, 𝜕(𝑚)] and 𝜕(𝑚⋅2𝑛)=[𝜕(𝑚), 𝑛], 𝑛 ∈𝑁, 𝑚 ∈𝑀,
-Pei e iden i y:
𝜕(𝑚)⋅1𝑚′= [𝑚, 𝑚′] = 𝑚⋅2𝜕(𝑚′)𝑚, 𝑚′∈𝑀, 𝑛 ∈𝑁.
Example 1.3.27. As o he p e ious cases, we ha e ha i 𝑀is a Leibniz 𝐾-algeb a
hen (𝑀Id𝑀
←←←←←←←←←←←←←←←→ 𝑀, ([−,−],[−,−])) is a c ossed module o Leibniz 𝐾-algeb as.
1.3.4 C ossed modules o Leibniz algeb as 19
The nex immedia e p oposi ions gi e a ela ion be ween c ossed modules o Lie
and Leibniz 𝐾-algeb as.
P oposi ion 1.3.28. Le 𝑀and 𝑁be wo Lie 𝐾-algeb as. Then, ⋅is a Lie ac ion o
𝑁on 𝑀i and only i (⋅,⋅−)is a Leibniz ac ion o 𝑁on 𝑀, whe e ⋅−∶𝑀×𝑁←←→ 𝑀
is de ined by 𝑚⋅−𝑛∶= −𝑛⋅𝑚.
Tha is, he Lie ac ion is a pa icula case o a Leibniz ac ion when he ac ion is
“an icommu a i e”.
P oposi ion 1.3.29. Le 𝑀and 𝑁be Lie K-algeb as. Then, (𝑀𝜕
←←←←←←→ 𝑁, ⋅)is a c ossed
module o Lie 𝐾-algeb as i and only i (𝑀𝜕
←←←←←←→ 𝑁, (⋅,⋅−)) is a c ossed module o
Leibniz 𝐾-algeb as.
Rema k 1.3.30.Wi h he p e ious p ope y we can see ha he examples gi en o he
Lie algeb a case, (𝑀Id𝑀
←←←←←←←←←←←←←←←→ 𝑀, [−,−]), and o he Leibniz algeb a case, aking a Lie
algeb a 𝑀,(𝑀Id𝑀
←←←←←←←←←←←←←←←→ 𝑀, ([−,−],[−,−]), a e ela ed, since using an icommu a i i y
[−,−]−= [−,−].
De ini ion 1.3.31. Le (𝑀𝜕
←←←←←←→ 𝑁, ⋅)and (𝑀′𝜕′
←←←←←←←←→ 𝑁′,∗) be c ossed modules o Leibniz
𝐾-algeb as. A homomo phism is a pai o Leibniz 𝐾-homomo phisms, 𝑓1∶𝑀←←→ 𝑀′
and 𝑓2∶𝑁←←→ 𝑁′such ha
𝑓1(𝑛⋅1𝑚) = 𝑓2(𝑛) ∗1𝑓1(𝑚), 𝑓1(𝑚⋅2𝑛) = 𝑓1(𝑚) ∗2𝑓2(𝑛), 𝑛 ∈𝑁, 𝑚 ∈𝑀,
and
𝜕′◦𝑓1=𝑓2◦𝜕.
We will deno e by X(LeibAlg𝐾) he ca ego y o c ossed modules o Leibniz 𝐾-
algeb as and i s homomo phisms.
Rema k 1.3.32.As in he case o g oups and Lie 𝐾-algeb as, we ha e an equi alence
be ween he ca ego ies X(LeibAlg𝐾)and ICa (LeibAlg𝐾). A p oo o his can be
ound in [22].
X(LieAlg𝐾)can be seen as a ull subca ego y o he ca ego y X(LeibAlg𝐾)using
P oposi ion 1.3.29 (we ac ually ha e a unc o ial isomo phism be ween a ull subca -
ego y o X(LeibAlg𝐾)and X(LieAlg𝐾)).
20 1 P elimina ies
Since he pullbacks in LieAlg𝐾and LeibAlg𝐾a e he same, i is immedia e o
show ha ICa (LieAlg𝐾)is a ull subca ego y o ICa (LeibAlg𝐾). The equi alence in
he Leibniz case gene alizes he equi alence in he Lie case since he b acke gi es he
ac ion in he unc o s (which was p esen ed in [22]), and hen, i is an icommu a i e
when we ha e Lie 𝐾-algeb as. We only ha e o check ha he Leibniz semidi ec
p oduc gene alizes he Lie semidi ec p oduc , bu his is immedia e om de ini ion
(since 𝑚⋅2𝑛′= −𝑛′⋅1𝑚is he Lie case).
De ini ion 1.3.33. Le 𝑀and 𝑁be wo Leibniz 𝐾-algeb as and ⋅a Leibniz ac ion o
𝑁on 𝑀. The semidi ec p oduc , deno ed by 𝑀⋊𝑁, is he 𝐾- ec o space 𝑀×𝑁
wi h he b acke
[(𝑚, 𝑛),(𝑚′, 𝑛′)] ∶= ([𝑚, 𝑚′] + 𝑛⋅1𝑚′+𝑚⋅2𝑛′,[𝑛, 𝑛′]), 𝑚, 𝑚′∈𝑀, 𝑛, 𝑛′∈𝑁.
1.4 B aided semig oupal Ca ego y
A bi unc o is a unc o whose sou ce ca ego y is a p oduc ca ego y.
Le 𝐹∶C×D←←→ Ebe a bi unc o . Fo 𝐴∈ Ob(C)and 𝐵∈ Ob(D), we deno e by
𝐴𝐹and 𝐹𝐵 he unc o s:
𝐴𝐹∶D←←→ E,𝐴𝐹(𝐷𝑓
←←←←←←←→ 𝐷′) = 𝐹(𝐴, 𝐷)𝐹(Id𝐴,𝑓)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝐹(𝐴, 𝐷′),
𝐹𝐵∶C←←→ E, 𝐹𝐵(𝐶𝑔
←←←←←←→ 𝐶′) = 𝐹(𝐶, 𝐵)𝐹(𝑔,Id𝐵)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝐹(𝐶′, 𝐵).
De ini ion 1.4.1. Gi en he ca ego ies, 𝐂,𝐃and 𝐄, we ha e he unc o
𝐴𝐂,𝐃,𝐄∶ (𝐂×𝐃) × 𝐄←←→ 𝐂× (𝐃×𝐄)
de ined as
𝐴𝐂,𝐃,𝐄(((𝐴, 𝐵), 𝐶)((𝑓,𝑔),ℎ)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ ((𝐴′, 𝐵′), 𝐶′))=(𝐴, (𝐵, 𝐶))(𝑓,(𝑔,ℎ))
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ (𝐴′,(𝐵′, 𝐶′)),
called associa o unc o o he ca ego ical p oduc o he gi en ca ego ies. I is
always a unc o isomo phism wi h he ob ious in e se.
1.4 B aided semig oupal Ca ego y 21
C ane and Ye e de ined in [17] he no ion o semig oupal ca ego y.
De ini ion 1.4.2. Asemig oupal ca ego y is a iple = (𝐂, ⊗, 𝑎)whe e 𝐂is a
ca ego y, ⊗∶𝐂×𝐂→𝐂is a bi unc o , and 𝑎∶⊗◦(⊗×Id𝐂)←←→ ⊗◦(Id𝐂×⊗)◦𝐴𝐂,𝐂,𝐂
is a na u al isomo phism called he associa o , such ha o all 𝑋, 𝑌 , 𝑍, 𝑊 ∈ Ob(𝐂)
he ollowing associa i e cohe ence diag am (pen agon axiom) holds:
((𝑋 ⊗ 𝑌 )⊗ 𝑍)⊗ 𝑊
(𝑋 ⊗ (𝑌 ⊗ 𝑍)) ⊗ 𝑊
𝑋 ⊗ ((𝑌 ⊗ 𝑍)⊗ 𝑊 )𝑋 ⊗ (𝑌 ⊗ (𝑍 ⊗ 𝑊 ))
(𝑋 ⊗ 𝑌 )⊗(𝑍 ⊗ 𝑊 )
𝑎𝑋,𝑌 ,𝑍 ⊗Id𝑊
𝑎𝑋,𝑌 ⊗𝑍,𝑊
Id𝑋⊗𝑎𝑌 ,𝑍,𝑊
𝑎𝑋,𝑌 ,𝑍⊗𝑊
𝑎𝑋⊗𝑌 ,𝑍,𝑊
We say ha a semig oupal ca ego y is s ic i he isomo phism 𝑎is he iden i y
mo phism. In his case we ha e ha (𝑋 ⊗ 𝑌 )⊗ 𝑍 =𝑋 ⊗ (𝑌 ⊗ 𝑍).
I is known ha he cohe ence diag am implies ha any diag am made in he same
way o mo e enso p oduc s will be commu a i e.
The de ini ion o monoidal ca ego y was gi en in [7,45].
De ini ion1.4.3. Amonoidal ca ego y is a6- uple= (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟)whe e(𝐂, ⊗, 𝑎)
is a semig oupal ca ego y, 𝐼is an objec o 𝐂(called he enso uni ), and he pai
𝑙∶ (𝐼 ⊗ −) ←←→ Id𝐂,𝑟∶ (− ⊗ 𝐼)←←→ Id𝐂a e na u al isomo phisms (called he le and
igh uni o s, espec i ely), such ha o all 𝑋, 𝑌 ∈ Ob(𝐂) he uni cohe ence diag am
( iangle equa ion) holds:
(𝑋 ⊗ 𝐼)⊗ 𝑌 𝑎𝑋,𝐼,𝑌 //
𝑟𝑋⊗Id𝑌''
𝑋 ⊗ (𝐼 ⊗ 𝑌 )
Id𝑋⊗𝑙𝑌
ww
𝑋 ⊗ 𝑌
We say ha a monoidal ca ego y is s ic i he isomo phisms 𝑎,𝑙and 𝑟a e he iden i y
mo phisms. In his case (𝑋 ⊗ 𝑌 )⊗ 𝑍 =𝑋 ⊗ (𝑌 ⊗ 𝑍),𝑋 ⊗ 𝐼 =𝑋=𝐼 ⊗ 𝑋.
28 2 B aidings o c ossed modules and in e nal objec s
De ini ion 2.2.3. Ab aided in e nal unc o be ween wo b aided ca ego ical as-
socia i e 𝐾-algeb as is an in e nal unc o (𝐹1, 𝐹0)such ha 𝐹1(𝜏𝑎,𝑏) = 𝜏′
𝐹0(𝑎),𝐹0(𝑏),
whe e 𝜏and 𝜏′a e he b aidings and 𝑎, 𝑏 ∈𝐶0.
We deno e by BICa (AssAlg𝐾) he ca ego y o b aided ca ego ical associa i e
𝐾-algeb as and b aided in e nal unc o s be ween hem.
We will in oduce he no ion o b aiding o c ossed modules o associa i e alge-
b as looking o an equi alence be ween b aided c ossed modules and b aided in e nal
ca ego ies o associa i e algeb as, as i happens in he case o g oups.
De ini ion 2.2.4. Le (𝑀𝜕
←←←←←←→ 𝑁, ∗= (∗1,∗2)) be a c ossed module o associa i e 𝐾-
algeb as. A b aiding (o Pei e li ing) is a 𝐾-bilinea map {−,−}∶ 𝑁×𝑁←←→ 𝑀
sa is ying:
𝜕{𝑛, 𝑛′}=[𝑛, 𝑛′],(BXAs1)
{𝜕𝑚, 𝜕𝑚′}=[𝑚, 𝑚′],(BXAs2)
{𝜕𝑚, 𝑛} = −[𝑛, 𝑚]∗,(BXAs3)
{𝑛, 𝜕𝑚}=[𝑛, 𝑚]∗,(BXAs4)
{𝑛, 𝑛′𝑛′′} = 𝑛′∗1{𝑛, 𝑛′′}+{𝑛, 𝑛′} ∗2𝑛′′,(BXAs5)
{𝑛𝑛′, 𝑛′′} = 𝑛∗1{𝑛′, 𝑛′′}+{𝑛, 𝑛′′} ∗2𝑛′,(BXAs6)
wi h 𝑚, 𝑚′∈𝑀,𝑛, 𝑛′, 𝑛′′ ∈𝑁.
He e, [𝑛, 𝑚]∗=𝑛∗1𝑚−𝑚∗2𝑛and [𝑥, 𝑦] = 𝑥𝑦 −𝑦𝑥.
(𝑀𝜕
←←←←←←→ 𝑁, ∗,{−,−}) is a b aided c ossed module o associa i e 𝐾-algeb as.
Example 2.2.5. The commu a o map [−,−] is a b aiding on he c ossed module
(𝑀𝜕
←←←←←←→ 𝑀, (∗,∗)).
De ini ion 2.2.6. Ahomomo phism o b aided c ossed modules o associa i e 𝐾-
algeb as (𝑓1, 𝑓2)∶ (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) ←←←←←←←→ (𝑀′𝜕′
←←←←←←←←→ 𝑁′,∗,{−,−}′)is a homomo -
phism o c ossed modules o associa i e 𝐾-algeb as such ha
𝑓1({𝑛, 𝑛′}) = {𝑓2(𝑛), 𝑓2(𝑛′)}′, 𝑛, 𝑛′∈𝑁.

2.2 B aiding o he Associa i e Case 29
We deno e by BX(AssAlg𝐾) he ca ego y o b aided c ossed modules o associa-
i e 𝐾-algeb as and hei homomo phisms.
P oposi ion 2.2.7. Le = (𝑀𝜕
←←←←←←→ 𝑁, (∗1,∗2),{−,−}) be a b aided c ossed module
o associa i e 𝐾-algeb as.
Then ∶= (𝑀⋊𝑁, 𝑁, 𝑠, 
𝑡, 𝑒, 
𝑘, 𝜏)is a b aided ca ego ical associa i e 𝐾-
algeb a whe e 𝑠, 
𝑡, 𝑒, 
𝑘a e de ined in P oposi ion 1.3.11 and he b aiding is:
𝜏 ∶𝑁×𝑁←←→ 𝑀⋊𝑁, 𝜏𝑛,𝑛′= (−{𝑛, 𝑛′}, 𝑛𝑛′).
P oo . We only need o check he b aiding axioms o his in e nal ca ego y since
(𝑀⋊𝑁, 𝑁, 𝑠, 
𝑡, 𝑒, 
𝑘)is a ca ego ical associa i e 𝐾-algeb a by P oposi ion 1.3.11.
We will s a wi h AsB1. Le 𝑛, 𝑛′∈𝑁.
𝑠(𝜏𝑛, 𝑛′) = 𝑠((−{𝑛, 𝑛′}, 𝑛𝑛′)) = 𝑛𝑛′,

𝑡(𝜏𝑛,𝑛′) = 
𝑡((−{𝑛, 𝑛′}, 𝑛𝑛′)) = −𝜕{𝑛, 𝑛′} + 𝑛𝑛′= −[𝑛, 𝑛′] + 𝑛𝑛′=𝑛′𝑛,
whe e we use (BXAs1).
We will p o e now AsB2. Le 𝑥= (𝑚, 𝑛), 𝑦 = (𝑚′, 𝑛′) ∈ 𝑀⋊𝑁.
We need o show ha 𝜏𝑡(𝑥),𝑡(𝑦)◦𝑥𝑦 =𝑦𝑥◦𝜏𝑠(𝑥),𝑠(𝑦).
𝜏𝑡(𝑥),𝑡(𝑦)◦𝑥𝑦
=
𝑘(((𝑚, 𝑛)(𝑚′, 𝑛′),(−{
𝑡((𝑚, 𝑛)),
𝑡((𝑚′, 𝑛′))},
𝑡((𝑚, 𝑛))
𝑡((𝑚′, 𝑛′)))))
=
𝑘(((𝑚, 𝑛)(𝑚′, 𝑛′),(−{𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′},(𝜕𝑚 +𝑛)(𝜕𝑚′+𝑛′))))
=
𝑘(((𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′, 𝑛𝑛′),(−{𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′},
(𝜕𝑚 +𝑛)(𝜕𝑚′+𝑛′))))
= (𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′− {𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′}, 𝑛𝑛′)
= (𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′− {𝜕𝑚, 𝜕𝑚′}−{𝜕𝑚, 𝑛′}−{𝑛, 𝜕𝑚′}−{𝑛, 𝑛′}, 𝑛𝑛′)
= (𝑚𝑚′+𝑛∗1𝑚′+𝑚∗2𝑛′− [𝑚, 𝑚′]+[𝑛′, 𝑚]∗− [𝑛, 𝑚′]∗− {𝑛, 𝑛′}, 𝑛𝑛′)
= (𝑚′𝑚+𝑚′∗2𝑛+𝑛′∗1𝑚− {𝑛, 𝑛′}, 𝑛𝑛′),
30 2 B aidings o c ossed modules and in e nal objec s
whe e we use (BXAs2), (BXAs3) and (BXAs4) in he six h equali y. On he o he
hand,
𝑦𝑥◦𝜏𝑠(𝑥),𝑠(𝑦)
=
𝑘(((−{𝑠((𝑚, 𝑛)), 𝑠((𝑚, 𝑛′))}, 𝑠((𝑚, 𝑛))𝑠((𝑚′, 𝑛′))),(𝑚′, 𝑛′)(𝑚, 𝑛)))
=
𝑘(((−{𝑛, 𝑛′}, 𝑛𝑛′),(𝑚′, 𝑛′)(𝑚, 𝑛)))
=
𝑘(((−{𝑛, 𝑛′}, 𝑛𝑛′),(𝑚′𝑚+𝑛′∗1𝑚+𝑚′∗2𝑛, 𝑛′𝑛)))
= (−{𝑛, 𝑛′} + 𝑚′𝑚+𝑛′∗1𝑚+𝑚′∗2𝑛, 𝑛𝑛′).
We will e i y AsB3. I 𝑛, 𝑛′, 𝑛′′ ∈𝑁, hen
(𝜏𝑛,𝑛′′ 𝑒(𝑛′))◦(𝑒(𝑛)𝜏𝑛′,𝑛′′ ) = 
𝑘(( 𝑒(𝑛)𝜏𝑛′,𝑛′′ , 𝜏𝑛,𝑛′′ 𝑒(𝑛′)))
=
𝑘((0, 𝑛)(−{𝑛′, 𝑛′′}, 𝑛′𝑛′′),(−{𝑛, 𝑛′′}, 𝑛𝑛′′)(0, 𝑛′))
=
𝑘(−𝑛∗1{𝑛′, 𝑛′′}, 𝑛(𝑛′𝑛′′),(−{𝑛, 𝑛′′} ∗2𝑛′,(𝑛𝑛′′)𝑛′))
= (−𝑛∗1{𝑛′, 𝑛′′}−{𝑛, 𝑛′′} ∗2𝑛′, 𝑛(𝑛′𝑛′′)) = (−{𝑛𝑛′, 𝑛′′},(𝑛𝑛′)𝑛′′) = 𝜏𝑛𝑛′,𝑛′′ ,
whe e we ha e used (BXAs6) and associa i i y.
Finally, we will show ha AsB4 is sa is ied. I 𝑛, 𝑛′, 𝑛′′ ∈𝑁, hen
(𝑒(𝑛′)𝜏𝑛,𝑛′′ )◦(𝜏𝑛,𝑛′𝑒(𝑛′′)) = 
𝑘(𝜏𝑛,𝑛′𝑒(𝑛′′), 𝑒(𝑛′)𝜏𝑛,𝑛′′ )
=
𝑘((−{𝑛, 𝑛′}, 𝑛𝑛′)(0, 𝑛′′),(0, 𝑛′)(−{𝑛, 𝑛′′}, 𝑛𝑛′′))
=
𝑘((−{𝑛, 𝑛′} ∗2𝑛′′,(𝑛𝑛′)𝑛′′),(−𝑛′∗1{𝑛, 𝑛′′}, 𝑛′(𝑛𝑛′′)))
= (−𝑛′∗1{𝑛, 𝑛′′}−{𝑛, 𝑛′} ∗2𝑛′′,(𝑛𝑛′)𝑛′′) = (−{𝑛, 𝑛′𝑛′′}, 𝑛(𝑛′𝑛′′)) = 𝜏𝑛,𝑛′𝑛′′ ,
whe e we use (BXAs5) along wi h he associa i i y in he second equali y.
P oposi ion 2.2.8. We ha e a unc o 𝔄∶BX(AssAlg𝐾)←←→ BICa (AssAlg𝐾)de ined
by
𝔄((𝑓1,𝑓2)
←←←←←←←←←←←←←←←←←←←←←←←←→ ′) = 
(𝑓1×𝑓2,𝑓2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ ′
whe e is desc ibed in he p e ious p oposi ion.
2.2 B aiding o he Associa i e Case 31
P oo . We know ha he pai (𝑓1×𝑓2, 𝑓2)is an in e nal unc o be ween he espec-
i e in e nal ca ego ies since wha we a e doing is ex ending an exis ing unc o (see
P oposi ion 1.3.11) o he b aided case. In he same way, as i is an ex ension, we only
need o show ha i is well de ined, since i sa is ies he p ope ies o unc o because
he composi ion and iden i y a e he same as in he ca ego ies wi hou b aiding.
So, o conclude he p oo , i is enough o see ha (𝑓1×𝑓2, 𝑓2)is a b aided in e nal
unc o o b aided ca ego ical associa i e 𝐾-algeb as.
(𝑓1×𝑓2)( 𝜏𝑛,𝑛′) = (𝑓1×𝑓2)((−{𝑛, 𝑛′}, 𝑛𝑛′)) = (−𝑓1({𝑛, 𝑛′}), 𝑓2(𝑛𝑛′))
= (−{𝑓2(𝑛), 𝑓2(𝑛′)}′, 𝑓2(𝑛)𝑓2(𝑛′)) = 𝜏′
𝑓2(𝑛),𝑓2(𝑛′),
whe e we use ha (𝑓1, 𝑓2)is a homomo phism o b aided c ossed modules o asso-
cia i e algeb as.
P oposi ion 2.2.9. Le = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)be a b aided ca ego ical associa i e
𝐾-algeb a.
Then ∶= (ke (𝑠)𝜕𝑡
←←←←←←←←→ 𝐶0,(𝑒∗,∗𝑒),{−,−}𝜏)is a b aided c ossed module o
associa i e 𝐾-algeb as, whe e (𝑒∗,∗𝑒), 𝜕𝑡a e de ined in P oposi ion 1.3.11 and he
b aiding is:
{−,−}𝜏∶𝐶0×𝐶0←←→ ke (𝑠),{𝑎, 𝑏}𝜏∶=𝑒(𝑎𝑏) − 𝜏𝑎,𝑏.
P oo . We only need o show ha {−,−}𝜏is a b aiding on a c ossed module, since
unde he abo e assump ions, (ke (𝑠)𝜕𝑡
←←←←←←←←→ 𝐶0,(∗𝑒,𝑒∗)) is a c ossed module o asso-
cia i e 𝐾-algeb as by P oposi ion 1.3.11.
Fi s , we see ha i is well de ined, i.e. {𝑎, 𝑏}𝜏∈ ke (𝑠) o 𝑎, 𝑏 ∈𝐶0.
𝑠({𝑎, 𝑏}𝜏) = 𝑠(𝑒(𝑎𝑏) − 𝜏𝑎,𝑏) = 𝑎𝑏 −𝑎𝑏 = 0,
whe e we use AsB1.
Now, we will check (BXAs1). I 𝑎, 𝑏 ∈𝐶0, hen
𝜕𝑡{𝑎, 𝑏}𝜏=𝑡(𝑒(𝑎𝑏) − 𝜏𝑎,𝑏) = 𝑎𝑏 −𝑏𝑎 = [𝑎, 𝑏],
32 2 B aidings o c ossed modules and in e nal objec s
whe e we ha e used (AsB1).
We p oceed o check (BXAs2). I 𝑥, 𝑦 ∈ ke (𝑠), hen
{𝜕𝑡𝑥, 𝜕𝑡𝑦}𝜏=𝑒(𝜕𝑡𝑥𝜕𝑡𝑦) − 𝜏𝜕𝑡𝑥,𝜕𝑡𝑦=𝑒(𝑡(𝑥)𝑡(𝑦)) − 𝜏𝑡(𝑥),𝑡(𝑦).
We need o show ha 𝑒(𝑡(𝑥)𝑡(𝑦)) − 𝜏𝑡(𝑥),𝑡(𝑦)= [𝑥, 𝑦].
By axiom AsB2 we know he equali y
𝑘((𝑥𝑦, 𝜏𝑡(𝑥),𝑡(𝑦))) = 𝑘((𝜏𝑠(𝑥),𝑠(𝑦), 𝑦𝑥)).
As 𝑥∈ ke (𝑠), we ha e ha 𝑠(𝑥)=0(in he same way o 𝑦), and 𝜏𝑠(𝑥),𝑠(𝑦)= 0 by
𝐾-bilinea i y. We ha e hen ha
𝑘((𝜏𝑠(𝑥),𝑠(𝑦), 𝑦𝑥)) = 𝑘((0, 𝑦𝑥)),
and he e o e he equali y
𝑘((𝑥𝑦, 𝜏𝑡(𝑥),𝑡(𝑦))) = 𝑘((0, 𝑦𝑥)).
Using now he 𝐾-linea i y o 𝑘in he p e ious exp ession, we ob ain
0 = 𝑘((𝑥𝑦, 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥)).
Since 𝑡(𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥) = 𝑡(𝑦)𝑡(𝑥)−𝑡(𝑦)𝑡(𝑥) = 0 = 𝑠(𝑒(0)) we can w i e 𝑘((𝜏𝑡(𝑥),𝑡(𝑦)−
𝑦𝑥, 𝑒(0))). Fu he 𝑘((𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥, 𝑒(0))) = 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥 by he in e nal ca ego y
axioms.
Adding bo h equali ies and using he 𝐾-linea i y o 𝑘, we ge
𝑘((𝑥𝑦 +𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥, 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥)) = 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥.
The e o e, by g ouping, we ha e
𝑘(([𝑥, 𝑦] + 𝜏𝑡(𝑥),𝑡(𝑦), 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥)) = 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥.
As 𝑠(𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥) = 𝑡(𝑥)𝑡(𝑦) + 0 = 𝑡(𝑥)𝑡(𝑦)(we use ha 𝑥o 𝑦a e in ke (𝑠)) i makes
sense o alk abou he composi ion 𝑘((𝑒(𝑡(𝑥)𝑡(𝑦)), 𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥)), which is equal o
𝜏𝑡(𝑥),𝑡(𝑦)−𝑦𝑥.
2.2 B aiding o he Associa i e Case 33
Sub ac ing bo h equali ies and using he 𝐾-linea i y o 𝑘, we ob ain
𝑘(([𝑥, 𝑦] + 𝜏𝑡(𝑥),𝑡(𝑦)−𝑒(𝑡(𝑥)𝑡(𝑦)),0)) = 0.
Again, using he p ope ies o in e nal ca ego ies, we ha e
0 = 𝑘(([𝑥, 𝑦] + 𝜏𝑡(𝑥),𝑡(𝑦)−𝑒(𝑡(𝑥)𝑡(𝑦)),0))
=𝑘(([𝑥, 𝑦] + 𝜏𝑡(𝑥),𝑡(𝑦)−𝑒(𝑡(𝑥)𝑡(𝑦)), 𝑒(0)))
= [𝑥, 𝑦] + 𝜏𝑡(𝑥),𝑡(𝑦)−𝑒(𝑡(𝑥)𝑡(𝑦)),
which gi es us he equi ed equali y.
As an obse a ion o he abo e, in he pa o he p oo ha is ela ed o 𝑥, 𝑦 ∈
ke (𝑠), i is su icien ha one o he wo is in ha ke nel. By using his ac we ha e
he ollowing equali ies o 𝑥∈ ke (𝑠)and 𝑦∈𝐶1:
𝑒(𝑡(𝑥)𝑡(𝑦)) − 𝜏𝑡(𝑥),𝑡(𝑦)= [𝑥, 𝑦], 𝑒(𝑡(𝑦)𝑡(𝑥)) − 𝜏𝑡(𝑦),𝑡(𝑥)= [𝑦, 𝑥].
Now wi h hese equali ies, we will p o e (BXAs3) and (BXAs4).
Le 𝑎∈𝐶0and 𝑥∈ ke (𝑠). Then
{𝜕𝑡𝑥, 𝑎}𝜏=𝑒(𝑡(𝑥)𝑡(𝑒(𝑎))) − 𝜏𝑡(𝑥),𝑡(𝑒(𝑎)) = [𝑥, 𝑒(𝑎)] = 𝑥𝑒(𝑎) − 𝑒(𝑎)𝑥=𝑥∗𝑒𝑎−𝑎𝑒∗𝑥,
{𝑎, 𝜕𝑡𝑥}𝜏=𝑒(𝑡(𝑒(𝑎))𝑡(𝑥)) − 𝜏𝑡(𝑒(𝑎)),𝑡(𝑥)= [𝑒(𝑎), 𝑥] = 𝑒(𝑎)𝑥−𝑥𝑒(𝑎) = 𝑎𝑒∗𝑥−𝑥∗𝑒𝑎.
We will see now he las condi ions, s a ing wi h (BXAs5). Le 𝑎, 𝑏, 𝑐 ∈𝐶0.
{𝑎, 𝑏𝑐}𝜏=𝑒(𝑎(𝑏𝑐)) − 𝜏𝑎,𝑏𝑐 =𝑒(𝑎(𝑏𝑐)) − ((𝑒(𝑏)𝜏𝑎,𝑐)◦(𝜏𝑎,𝑏𝑒(𝑐)))
=𝑒(𝑎(𝑏𝑐)) − 𝑒(𝑏)𝜏𝑎,𝑐 −𝜏𝑎,𝑏𝑒(𝑐) + 𝑒(𝑡(𝜏𝑎,𝑏𝑒(𝑐)))
=𝑒((𝑎𝑏)𝑐) − 𝑒(𝑏)𝜏𝑎,𝑐 −𝜏𝑎,𝑏𝑒(𝑐) + 𝑒((𝑏𝑎)𝑐)
=𝑒(𝑏)𝑒(𝑎𝑐) − 𝑒(𝑏)𝜏𝑎,𝑐 +𝑒(𝑎𝑏)𝑒(𝑐) − 𝜏𝑎,𝑏𝑒(𝑐)
=𝑒(𝑏){𝑎, 𝑐}𝜏+ {𝑎, 𝑏}𝜏𝑒(𝑐) = 𝑏𝑒∗ {𝑎, 𝑐}𝜏+ {𝑎, 𝑏}𝜏∗𝑒𝑐,
whe e we ha e used (AsB4), Lemma 2.1.1 and he associa i i y.
To conclude we will check (BXAs6).
{𝑎𝑏, 𝑐}𝜏=𝑒((𝑎𝑏)𝑐) − 𝜏𝑎𝑏,𝑐 =𝑒((𝑎𝑏)𝑐) − ((𝜏𝑎,𝑐𝑒(𝑏))◦(𝑒(𝑎)𝜏𝑏,𝑐))

34 2 B aidings o c ossed modules and in e nal objec s
=𝑒((𝑎𝑏)𝑐) − 𝜏𝑎,𝑐𝑒(𝑏) − 𝑒(𝑎)𝜏𝑏,𝑐 +𝑒(𝑡(𝑒(𝑎)𝜏𝑏,𝑐))
=𝑒(𝑎(𝑏𝑐)) − 𝜏𝑎,𝑐𝑒(𝑏) − 𝑒(𝑎)𝜏𝑏,𝑐 +𝑒(𝑎(𝑐𝑏))
=𝑒(𝑎)𝑒(𝑏𝑐) − 𝑒(𝑎)𝜏𝑏,𝑐 +𝑒(𝑎𝑐)𝑒(𝑏) − 𝜏𝑎,𝑐𝑒(𝑏)
=𝑒(𝑎){𝑏, 𝑐}𝜏+ {𝑎, 𝑐}𝜏𝑒(𝑏) = 𝑎𝑒∗ {𝑏, 𝑐}𝜏+ {𝑎, 𝑐}𝜏∗𝑒𝑏,
whe e we ha e used (AsB3), Lemma 2.1.1 and associa i i y.
P oposi ion 2.2.10. We ha e a unc o 𝔄∶BICa (AssAlg𝐾)←←→ BX(AssAlg𝐾)de-
ined by
𝔄((𝐹1,𝐹0)
←←←←←←←←←←←←←←←←←←←←←←←←←→ ′) = 
(𝐹𝑠
1,𝐹0)
←←←←←←←←←←←←←←←←←←←←←←←←←←→ ′,
whe e is desc ibed in he p e ious p oposi ion and 𝐹𝑠
1∶ ke (𝑠)←←→ ke (𝑠′)is de e -
mined by 𝐹𝑠
1(𝑥) = 𝐹1(𝑥), wi h 𝑥∈ ke (𝑠).
P oo . We only need o show ha 𝔄can be ex ended o he b aided case since is
a unc o be ween he ca ego ies wi hou b aiding (see P oposi ion 1.3.11). Fo his,
we ha e o sa is y he axioms o he homomo phisms o b aided c ossed modules o
associa i e 𝐾-algeb as.
𝐹𝑠
1({𝑎, 𝑏}𝜏) = 𝐹1(𝑒(𝑎𝑏) − 𝜏𝑎,𝑏) = 𝐹1(𝑒(𝑎, 𝑏)) − 𝐹1(𝜏𝑎,𝑏)
=𝑒′(𝐹0(𝑎𝑏)) − 𝜏′
𝐹0(𝑎),𝐹0(𝑏)=𝑒′(𝐹0(𝑎)𝐹0(𝑏)) − 𝜏′
𝐹0(𝑎),𝐹0(𝑏)
= {𝐹0(𝑎), 𝐹0(𝑏)}𝜏′.
Rema k 2.2.11.No e ha , i (𝑀𝜕
←←←←←←→ 𝑁, (∗1,∗2),{−,−}) is a b aided c ossed module
o associa i e 𝐾-algeb as, hen ke (𝑠) = {(𝑚, 0) ∈ 𝑀⋊𝑁∣𝑚∈𝑀} =∶ (𝑀, 0),
whe e 𝑠 is de ined o he unc o 𝔄.
P oposi ion 2.2.12. The ca ego ies BX(AssAlg𝐾)and BICa (AssAlg𝐾)a e equi a-
len ca ego ies.
Fu he , he unc o s 𝔄and 𝔄a e in e se equi alences, whe e he na u al iso-
mo phisms IdBX(AssAlg𝐾)
𝛼𝔄
≅𝔄◦𝔄and IdBICa (AssAlg𝐾)
𝛽𝔄
≅𝔄◦𝔄a e gi en by:
∙i = (𝑀𝜕
←←←←←←→ 𝑁, (∗1,∗2),{−,−}) is a b aided c ossed module o associa i e 𝐾-
algeb as, hen 𝛼𝔄
= (𝛼𝔄
𝑀,Id𝑁), wi h 𝛼𝔄
𝑀∶𝑀←←→ (𝑀, 0) de ined as 𝛼𝑀(𝑚) = (𝑚, 0);
2.3 B aiding o he Lie Case 35
∙i = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)is a b aided ca ego ical associa i e 𝐾-algeb a, hen
𝛽𝔄
= (𝛽𝔄
𝑠,Id𝐶0), wi h 𝛽𝔄
𝐶1∶𝐶1←←→ ke (𝑠)⋊𝐶0de ined as 𝛽𝔄
𝐶1(𝑥)=(𝑥−𝑒(𝑠(𝑥)), 𝑠(𝑥)).
P oo . We only need o show ha 𝛼𝔄
and 𝛽𝔄
a e isomo phisms be ween b aided ob-
jec s since ha hey a e well-de ined maps, isomo phisms in he ca ego ies wi hou
b aiding, as well as a e na u al isomo phisms (see [22]). So, i is su icien o p o e
ha 𝛼𝔄
and 𝛽𝔄
sa is y he b aided axioms, since he bijec i e mo phisms a e isomo -
phisms in bo h ca ego ies.
Le = (𝑀𝜕
←←←←←←→ 𝑁, (⋅1,⋅2),{−,−}) a b aided c ossed module o associa i e 𝐾-
algeb as. We will check ha 𝛼𝔄
= (𝛼𝔄
𝑀,Id𝑁)is a homomo phism.
Id𝑁({𝑛, 𝑛′}𝜏) = {𝑛, 𝑛′}𝜏 =𝑒(𝑛𝑛′) − 𝜏𝑛,𝑛′= (0, 𝑛𝑛′) − (−{𝑛, 𝑛′}, 𝑛𝑛′)
= ({𝑛, 𝑛′},0) = 𝛼𝔄
𝑀({𝑛, 𝑛′}),whe e 𝑛, 𝑛′∈𝑁.
Le = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)be a b aided ca ego ical associa i e 𝐾-algeb a. We
will check ha 𝛽𝔄
= (𝛽𝔄
𝑠,Id𝐶0)is a mo phism.
I 𝑎, 𝑏 ∈𝐶0, we ha e
Id𝐶0(𝜏𝑎,𝑏) = 𝜏𝑎,𝑏 = (−{𝑎, 𝑏}𝜏, 𝑎𝑏)=(𝜏𝑎,𝑏 −𝑒(𝑎𝑏), 𝑎𝑏)
= (𝜏𝑎,𝑏 −𝑒(𝑠(𝜏𝑎,𝑏)), 𝑠(𝜏𝑎,𝑏)) = 𝛽𝔄
𝐶1(𝜏𝑎,𝑏).
The e o e, he equi alence o ca ego ies is ob ained since hey a e mo phisms,
and we know ha hey a e na u al isomo phisms.
2.3 B aiding o ca ego ical Lie algeb as and c ossed mod-
ules o Lie algeb as
In his sec ion, we will show ha he de ini ion gi en by Ulualan in [50] o b aided
ca ego ical Lie 𝐾-algeb as appea s na u ally om he p e ious one, using he ac ha
we can ans o m an associa i e 𝐾-algeb a 𝑀in a Lie 𝐾-algeb a 𝑀wi h b acke
[𝑥, 𝑦] = 𝑥𝑦 −𝑦𝑥.
Now, we will suppose ha 𝐾is a ield o cha (𝐾)≠2 o change a li le he
de ini ion o b aiding. Doing his we will ob ain he de ini ion gi en in [24], whe e
36 2 B aidings o c ossed modules and in e nal objec s
he equi alence is p o en wi h he ca ego y o b aided c ossed modules o Lie 𝐾-
algeb as when cha (𝐾)≠2.
The no ion o b aiding o ca ego ical Lie 𝐾-algeb as was in oduced by Ulualan
in [50].
De ini ion 2.3.1 ( [50]).Le = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)be a ca ego ical Lie 𝐾-algeb a.
Ab aiding on is a 𝐾-bilinea map 𝜏∶𝐶0×𝐶0←←→ 𝐶1,(𝑎, 𝑏)↦𝜏𝑎,𝑏, sa is ying:
𝜏𝑎,𝑏 ∶ [𝑎, 𝑏]←←→ [𝑏, 𝑎],(LieT1)
[𝑠(𝑥), 𝑠(𝑦)] [𝑡(𝑥), 𝑡(𝑦)]
[𝑠(𝑦), 𝑠(𝑥)] [𝑡(𝑦), 𝑡(𝑥)],
𝜏𝑠(𝑥),𝑠(𝑦)
[𝑥,𝑦]
𝜏𝑡(𝑥),𝑡(𝑦)
[𝑦,𝑥]
,(LieT2)
𝜏[𝑎,𝑏],𝑐 = [𝜏𝑎,𝑐, 𝑒(𝑏)] + [𝑒(𝑎), 𝜏𝑏,𝑐],(LieB3)
𝜏𝑎,[𝑏,𝑐]= [𝑒(𝑏), 𝜏𝑎,𝑐]+[𝜏𝑎,𝑏, 𝑒(𝑐)],(LieB4)
o 𝑎, 𝑏, 𝑐 ∈𝐶0,𝑥, 𝑦 ∈𝐶1.
We say ha (𝐶0, 𝐶1, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)is a b aided ca ego ical Lie 𝐾-algeb a.
Rema k 2.3.2.The lack o associa i i y o he Lie b acke mo i a es he use o he
addi ion in LieB3 and LieB4 ins ead o he composi ion. This choice makes sense
since he sou ce and he a ge a e he same using he Jacobi iden i y.
We wan o show ha he de ini ion o b aiding o associa i e 𝐾-algeb as is well
ela ed wi h he de ini ion o b aiding o Lie 𝐾-algeb as.
P oposi ion 2.3.3. Le (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)be a b aided ca ego ical associa i e 𝐾-
algeb a, hen (𝐶
1, 𝐶
0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏𝐿𝑖𝑒)is a b aided ca ego ical Lie 𝐾-algeb a, whe e
𝜏𝐿𝑖𝑒 ∶𝐶
0×𝐶
0←←→ 𝐶
1, 𝜏𝐿𝑖𝑒
𝑎,𝑏 ∶=𝜏𝑎,𝑏 −𝜏𝑏,𝑎.
P oo . I is easy o see ha AsB1 implies LieT1 and AsB2 implies LieT2.
By using Lemma 2.1.1 we ob ain LieB3 and LieB4 om AsB3 and AsB4, espec-
i ely.
2.3 B aiding o he Lie Case 37
Ano he de ini ion o b aided in e nal ca ego y o Lie 𝐾-algeb as was gi en
in [24] o make he equi alence wi h he b aided c ossed modules o Lie 𝐾-algeb as.
The equi alence was p o en o a ield wi h cha (𝐾)≠2, so we will show ha he
wo de ini ions a e equi alen .
The de ini ion gi en in [24] is he ollowing one.
De ini ion 2.3.4 ( [24]).Le = (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)be a ca ego ical Lie 𝐾-algeb a.
Ab aiding on is a 𝐾-bilinea map 𝜏∶𝐶0×𝐶0←←→ 𝐶1,(𝑎, 𝑏)↦𝜏𝑎,𝑏, sa is ying
LieT1, LieT2 and he ollowing equali ies:
𝜏[𝑎,𝑏],𝑐 =𝜏𝑎,[𝑏,𝑐]−𝜏𝑏,[𝑎,𝑐],(LieT3)
𝜏𝑎,[𝑏,𝑐]=𝜏[𝑎,𝑏],𝑐 −𝜏[𝑎,𝑐],𝑏,(LieT4)
o 𝑎, 𝑏, 𝑐 ∈𝐶0.
In he ollowing p oposi ion we show ha he wo de ini ions a e equi alen when
cha (𝐾)≠2.
P oposi ion 2.3.5. Le 𝐾be a ield o cha (𝐾)≠2and (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)a ca ego ical
Lie 𝐾-algeb a.
I 𝜏∶𝐶0×𝐶0←←→ 𝐶1is a 𝐾-bilinea map sa is ying LieT1 and LieT2, hen
𝜏𝑎,[𝑏,𝑐]= [𝑒(𝑎), 𝜏𝑏,𝑐]and 𝜏[𝑏,𝑐],𝑎 = [𝜏𝑏,𝑐, 𝑒(𝑎)].
In pa icula , by he an icommu a i i y, we ha e ha 𝜏𝑎,[𝑏,𝑐]= −𝜏[𝑏,𝑐],𝑎.
P oo . Using LieT1 and LieT2, we ha e he ollowing commu a i e diag am:
[𝑎, [𝑏, 𝑐]] [𝑎, [𝑐, 𝑏]]
[[𝑏, 𝑐], 𝑎] [[𝑐, 𝑏], 𝑎].
𝜏𝑎,[𝑏,𝑐]
[𝑒(𝑎),𝜏𝑏,𝑐]
𝜏𝑎,[𝑐,𝑏]
[𝜏𝑏,𝑐,𝑒(𝑎)]
Tha is, we ha e he equali y
𝑘(([𝑒(𝑎), 𝜏𝑏,𝑐], 𝜏𝑎,[𝑐,𝑏])) = 𝑘((𝜏𝑎,[𝑏,𝑐],[𝜏𝑏,𝑐, 𝑒(𝑎)])),
44 2 B aidings o c ossed modules and in e nal objec s
Example 2.4.6. I we ake he as he ca ego y o ec o spaces wi h he usual enso
p oduc , he p e ious de ini ion eco e s he de ini ion o Lie Algeb a i he cha ac e -
is ic o he p e ixed ield is no 2. Since he gene aliza ion is only ue o cha (𝐾)≠2,
we will assume i o he es o he sec ion.
We wan o explain wha a e an objec and a mo phism in Lie(𝐾).
De ini ion 2.4.7. Le 𝑉be a 𝐾- ec o space and 𝑀be a Lie 𝐾-algeb a.
We say ha (𝑉 , ⋅)is a igh 𝑀-module i ⋅∶𝑉×𝑀←←→ 𝑉is a 𝐾-bilinea map
(𝑣, 𝑚)↦𝑣⋅𝑚such ha :
𝑣⋅[𝑚1, 𝑚2] = (𝑣⋅𝑚1)⋅𝑚2− (𝑣⋅𝑚2)⋅𝑚1,
o 𝑣∈𝑉,𝑚1, 𝑚2∈𝑀.
We say ha (𝑉 , ⋅)is a le 𝑀-module i ⋅∶𝑀×𝑉←←→ 𝑉is a 𝐾-bilinea map
(𝑚, 𝑣)↦𝑚⋅𝑣such ha :
[𝑚1, 𝑚2]⋅𝑣=𝑚1⋅(𝑚2⋅𝑣) − 𝑚2⋅(𝑚1⋅𝑣),
o 𝑣∈𝑉,𝑚1, 𝑚2∈𝑀.
De ini ion 2.4.8. Le 𝛼∶𝑀←←→ 𝑁be a Lie 𝐾-homomo phism. Le (𝑉 , ⋅)be a igh
( esp. le ) 𝑀-module and (𝑊 , ∗) a igh ( esp. le ) 𝑁-module. A 𝐾-linea map
𝑉𝑓
←←←←←←←→ 𝑊is (𝛼∶𝑀←←→ 𝑁, ⋅,∗)-equi a ian i we ha e ha
𝑓(𝑣⋅𝑚) = 𝑓(𝑣) ∗ 𝛼(𝑚) ( esp. 𝑓(𝑚⋅𝑣) = 𝛼(𝑚) ∗ 𝑓(𝑣)), o 𝑣∈𝑉 , 𝑚 ∈𝑀.
When 𝑁=𝑀and 𝛼= Id𝑀we said ha 𝑓is (𝑀, ⋅,∗)-equi a ian .
Le (𝑉 , ⋅)be a le 𝑀-module and (𝑊 , ∗) a igh 𝑁-module. A 𝐾-linea map
𝑉𝑓
←←←←←←←→ 𝑊is (𝛼∶𝑀←←→ 𝑁, ⋅,∗)-equi a ian i we ha e ha
𝑓(𝑚⋅𝑣) = −𝑓(𝑣) ∗ 𝛼(𝑚), o 𝑣∈𝑉 , 𝑚 ∈𝑀.
When 𝑁=𝑀and 𝛼= Id𝑀we said ha 𝑓is (𝑀, ⋅,∗)-equi a ian .

2.4 B aiding o he Leibniz Case 45
Rema k 2.4.9.I is easy o check ha i 𝑀is a Lie 𝐾-algeb a, hen i is a igh and
le 𝑀-module.
Mo eo e , i ⋅is a Lie ac ion o 𝑁in 𝑀, we ha e ha (𝑀, ⋅)is a le 𝑁-module.
Using his, we can see in [44] ha a Lie objec in 𝐾is he ollowing da a:
De ini ion 2.4.10. A Lie objec in 𝐾is a iple ( 𝑀
𝑁
𝑓,∗𝑀
𝑁,[−,−]𝑁) whe e
•(𝑁, [−,−]𝑁)is a Lie 𝐾-algeb a.
•∗𝑀
𝑁∶𝑀×𝑁←←→ 𝑀is such ha (𝑀, ∗𝑀
𝑁)is an (𝑁, [−,−]𝑁)-module.
•𝑓is ((𝑁, [−,−]𝑁),∗𝑀
𝑁,[−,−]𝑁)-equi a ian .
As in he case o Lie 𝐾-algeb as, we will deno e a Lie objec in 𝐾using he
𝐾-linea map on which i is de ined when he e is no con usion.
Rema k 2.4.11.The “an icommu a i e” p ope y o Lie objec o 𝐾allows o
eco e he Lie p oduc 𝜇= (𝜇1, 𝜇2) o 𝑀
𝑁
𝑓wi h he maps 𝜇2= [−,−]𝑁and
𝜇1∶ (𝑀 ⊗𝑁)⊕(𝑁 ⊗𝑀)←←→ 𝑀, wi h 𝜇1((𝑚⊗𝑛)+(𝑛′⊗𝑚′)) = 𝑚∗𝑀
𝑁𝑛−𝑚′∗𝑀
𝑁𝑛′.
De ini ion 2.4.12. Le 𝑀
𝑁
𝑓and 𝐿
𝐻
𝑔be Lie objec s. A Lie mo phism in 𝐾be ween
hem is an 𝐾mo phism (𝛼1, 𝛼2)such ha :
•𝛼2∶𝑁←←→ 𝐻is a Lie 𝐾-homomo phism.
•𝛼1∶𝑀←←→ 𝐿is an (𝛼2∶𝑁←←→ 𝐻, ∗𝑀
𝑁,∗𝐿
𝐻)-equi a ian map.
In [44] is shown a way o see he Leibniz 𝐾-algeb as as a pa icula case o Lie
objec s in 𝐾. We show i in he nex example.
Example 2.4.13. Le 𝑀be a Leibniz 𝐾-algeb a.
We deno e o 𝐼𝑀 he ideal gene a ed by elemen s o he o m [𝑥, 𝑥]wi h 𝑥∈𝑀.
I is e iden ha he quo ien Leibniz 𝐾-algeb a is a Lie 𝐾-algeb a. We will deno e
i s Lie b acke as [−,−], and he elemen s o he quo ien as 𝑚wi h 𝑚∈𝑀.
46 2 B aidings o c ossed modules and in e nal objec s
Lie(𝑀)∶=𝑀
𝐼𝑀
is known as Liesa ion (no e ha i 𝑀is a Lie 𝐾-algeb a, hen
Lie(𝑀)is i ially na u ally isomo phic o 𝑀), and i is unc o ial.
We conside he ollowing Lie objec in 𝐾:
We ake 𝑀
Lie(𝑀)
𝜋𝑀whe e 𝜋(𝑚) = 𝑚is he na u al map. I is a Lie objec in 𝐾
wi h he ollowing da a:
•𝑚∗𝑀
Lie(𝑀)𝑚′= [𝑚, 𝑚′],
•[𝑚, 𝑚′]Lie(𝑀)= [𝑚, 𝑚′]∶= [𝑚, 𝑚′].
I is e iden ha 𝜋is (Lie(𝑀),∗𝑀
Lie(𝑀),[−,−]Lie(𝑀))-equi a ian .
So, we ha e a unc o Φ∶ LeibAlg𝐾←←→ Lie(𝐾), ha is i ially ull.
This unc o is also injec i e on objec s and mo phisms, because he e is a unc o
Ψ∶ Lie(𝐾)←←→ LeibAlg𝐾such ha Ψ◦Φ = IdLeibAlg𝐾(see [44]). The unc o Ψon
objec s is desc ibed in he ollowing p oposi ion.
P oposi ion 2.4.14 ( [44]).Le 𝑀
𝑁
𝑓be a Lie objec in 𝐾. Then (𝑀, [−,−]), whe e
[𝑚, 𝑚′]∶=𝑚∗𝑀
𝑁𝑓(𝑚′), is a Leibniz 𝐾-algeb a.
In [23], we can see ha he p e ious cons uc ion can be ex ended o c ossed
modules o Lie algeb as in 𝐾. They did a c ossed module wi h a igh ac ion. In
his pape , we will de ine which is a c ossed module wi h a le ac ion, o simply a
c ossed module o Lie objec s.
De ini ion 2.4.15. Le = (C, ⊗, 𝑎, )be a b aided semig oupal ca ego y whe e C
is an addi i e ca ego y.
I (𝐴, 𝜇𝐴)and (𝐵, 𝜇𝐵)a e Lie objec s, hen a (le ) Lie ac ion o (𝐵, 𝜇𝐵)on (𝐴, 𝜇𝐴)
is a mo phism 𝑝∶𝐵 ⊗ 𝐴 ←←→ 𝐴such ha
𝑝◦(𝜇𝐵⊗Id𝐴) = 𝑝◦(Id𝐵⊗𝑝)◦𝑎𝐵,𝐵,𝐴◦(Id(𝐵⊗𝐵)⊗𝐴 −(𝜏𝐵,𝐵 ⊗Id𝐴)),
𝑝◦(Id𝐵⊗𝜇𝐴)◦𝑎𝐵,𝐴,𝐴 =𝜇𝐴◦(𝑝 ⊗ Id𝐴)◦(Id(𝐵⊗𝐵)⊗𝐴 −(𝑎−1
𝐵,𝐴,𝐴◦(Id𝐵⊗𝜏𝐴,𝐴)◦𝑎𝐵,𝐴,𝐴)).
2.4 B aiding o he Leibniz Case 47
We said ha ((𝐴, 𝜇𝐴)𝜕
←←←←←←→ (𝐵, 𝜇𝐵), 𝑝)is a c ossed module o Lie objec s i 𝑝is a Lie
ac ion o (𝐵, 𝜇𝐵)on (𝐴, 𝜇𝐴)and 𝜕∶ (𝐴, 𝜇𝐴)←←→ (𝐵, 𝜇𝐵)is a Lie mo phism such ha
𝜕◦𝑝=𝜇𝐵◦(Id𝐵⊗𝜕),
𝜇𝐴=𝑝◦(𝜕 ⊗ Id𝐴).
A mo phism be ween wo c ossed modules o Lie objec s ((𝐴, 𝜇𝐴)𝜕
←←←←←←→ (𝐵, 𝜇𝐵), 𝑝, )
and ((𝐶, 𝜇𝐶)𝛿
←←←←←←→ (𝐷, 𝜇𝐷), 𝑞)is a pai o Lie mo phisms (𝛼, 𝛽),𝛼∶ (𝐴, 𝜇𝐴)←←→ (𝐶, 𝜇𝐶)
and 𝛽∶ (𝐵, 𝜇𝐵)←←→ (𝐷, 𝜇𝐷), which sa is ies he ollowing diag ams:
𝐵 ⊗ 𝐴 𝐴
𝐷 ⊗ 𝐶 𝐶,
𝛽⊗𝛼
𝑝
𝛼
𝑞
𝐴 𝐵
𝐶 𝐷.
𝛼
𝜕
𝛽
𝛿
We ha e he ca ego y XLie()wi h he usual composi ion in C×C o pai s o
mo phisms o Lie mo phisms.
Example 2.4.16. We ha e ha XLie(Vec 𝐾)and X(LieAlg𝐾)a e isomo phic ca e-
go ies wi h he usual enso p oduc in Vec 𝐾(we assume cha (𝐾)≠2).
Now, we desc ibe he ca ego y XLie(𝐾).
De ini ion 2.4.17. Le 𝑀
𝑁
𝑓and 𝐿
𝐻
𝑔be Lie objec s in 𝐾. A (le ) Lie ac ion o 𝐿
𝐻
𝑔
on 𝑀
𝑁
𝑓in 𝐾is a iple ⋅= (⋅1,⋅2, 𝜉⋅)whe e
•⋅1∶𝐻×𝑀←←→ 𝑀is a 𝐾-bilinea map such ha (𝑀, ⋅1)is a le 𝐻-module;
•⋅2∶𝐻×𝑁←←→ 𝑁is a Lie ac ion o 𝐻on 𝑁;
•𝜉⋅∶𝐿×𝑁←←→ 𝑀is a 𝐾-bilinea map;
such ha he ollowing p ope ies a e sa is ied:
48 2 B aidings o c ossed modules and in e nal objec s
•⋅1and ⋅2a e compa ible ac ions wi h ∗𝑀
𝑁. Tha is, o ℎ∈𝐻,𝑛∈𝑁,𝑚∈𝑀,
we ha e
ℎ⋅1(𝑚∗𝑀
𝑁𝑛)=(ℎ⋅1𝑚) ∗𝑀
𝑁𝑛+𝑚∗𝑀
𝑁(ℎ⋅2𝑛);
•𝑓is an (𝐻, ⋅1,⋅2)-equi a ian map;
•𝜉⋅sa is ies, o 𝑙∈𝐿,𝑛, 𝑛′∈𝑁,ℎ∈𝐻, he ollowing equali ies
𝑓(𝜉⋅(𝑙, 𝑛)) = 𝑔(𝑙)⋅2𝑛,
𝜉⋅(𝑙∗𝐿
𝐻ℎ, 𝑛) = 𝜉⋅(𝑙, ℎ ⋅2𝑛) − ℎ⋅1𝜉⋅(𝑙, 𝑛),
𝜉⋅(𝑙, [𝑛, 𝑛′]𝑁) = 𝜉⋅(𝑙, 𝑛) ∗𝑀
𝑁𝑛′−𝜉⋅(𝑙, 𝑛′) ∗𝑀
𝑁𝑛.
Rema k 2.4.18.An ac ion is, in ac , a pai ⋅= (⋅1, ⋅2), wi h he wo maps
⋅1∶ (𝐿 ⊗ 𝑁)⊕(𝐻 ⊗ 𝑁)←←→ 𝑀and ⋅2∶𝐻 ⊗ 𝑁 ←←→ 𝑁
sa is ying he gene al p ope ies, bu we can easily ob ain he p e ious de ini ion ak-
ing ⋅2∶=⋅2and eco e ing ⋅1((𝑙 ⊗ 𝑛)+(ℎ⊗𝑚)) =∶ 𝜉⋅(𝑙, 𝑛) + ℎ⋅1𝑚.
De ini ion 2.4.19. A c ossed module o Lie objec s in 𝐾is a pai ( 𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅)
whe e 𝑀
𝑁
𝑓and 𝐿
𝐻
𝑔a e Lie objec s in 𝐾,⋅is a Lie ac ion o 𝐿
𝐻
𝑔on 𝑀
𝑁
𝑓, and
𝜕= (𝜕1, 𝜕2)∶
𝑀
𝑁
𝑓←←→
𝐿
𝐻
𝑔is a Lie mo phism in 𝐾such ha
•(𝑁, 𝐻, ⋅2, 𝜕2)is a c ossed module o Lie 𝐾-algeb as;
•𝜕1is an (𝐻, ⋅1,∗𝐿
𝐻)-equi a ian map;
•𝜕1(𝜉⋅(𝑙, 𝑛)) = 𝑙∗𝐿
𝑁𝜕2(ℎ)and 𝜉⋅(𝜕1(𝑚), 𝑛) = 𝑚∗𝑀
𝑁𝑛= −𝜕2(𝑛)⋅1𝑚,ℎ∈𝐻,
𝑙∈𝐿,𝑚∈𝑀,𝑛∈𝑁.
2.4 B aiding o he Leibniz Case 49
De ini ion 2.4.20. Le ( 𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅) and ( 𝑋
𝑌
𝑘
𝛿
←←←←←←→
𝑉
𝑊
ℎ,
⋆) be c ossed modules o Lie ob-
jec s in 𝐾. A mo phism o c ossed modules o Lie objec s in 𝐾is a pai (𝛼, 𝛽)
o Lie mo phisms 𝛼= (𝛼1, 𝛼2)∶
𝑀
𝑁
𝑓←←→
𝑋
𝑌
𝑘and 𝛽= (𝛽1, 𝛽2)∶
𝐿
𝐻
𝑔←←→
𝑉
𝑊
ℎsuch ha
•(𝛼2, 𝛽2)∶ (𝑁, 𝐻, ⋅2, 𝜕2)←←→ (𝑌 , 𝑊 , ⋆2, 𝛿2)is an homomo phism o c ossed mod-
ules o Lie 𝐾-algeb as;
•𝛼1(𝜉⋅(𝑙, 𝑛)) = 𝜉⋆(𝛽1(𝑙), 𝛼2(𝑛)), o 𝑙∈𝐿,𝑛∈𝑁;
•𝛼1is an (𝐻𝛽2
←←←←←←←←←→ 𝑊 , ⋅1, ⋆1)-equi a ian map;
•𝛽1◦𝜕1=𝛿1◦𝛼1.
As in he case o Leibniz 𝐾-algeb as we wan o ha e a pai o unc o s be ween
he ca ego ies XLie(𝐾)and XLeibAlg𝐾. Fo his pu pose, we gi e he ollowing
p oposi ions o which we omi hei p oo s because hey a e immedia e. The i s is
symme ical o he cons uc ion we can see in [23] o c ossed modules wi h igh
ac ions.
P oposi ion2.4.21. Le (𝑀𝜕
←←←←←←→ 𝑁, (⋅1,⋅2)) be ac ossedmoduleo Leibniz 𝐾-algeb as.
Then (
𝑀
𝑀
[𝑀,𝑁]𝑥
𝜋𝑀,𝑁
Lie(𝑁)
𝜋𝑁,
⋅, 𝜕)is a c ossed module o Lie objec s in 𝐾, whe e
•𝑀
[𝑀,𝑁]𝑥
is he Lie 𝐾-algeb a quo ien o 𝑀by he ideal [𝑀, 𝑁]𝑥whose gene -
a o s a e [𝑚, 𝑚] o 𝑚∈𝑀and 𝑛⋅1𝑚+𝑚⋅2𝑛 o 𝑛∈𝑁,𝑚∈𝑀; we deno e
he na u al map by 𝜋𝑀∶𝑀←←→ 𝑀
[𝑀,𝑁]𝑥
, and he elemen s o 𝑀
[𝑀,𝑁]𝑥
by 𝑚,
•⋅1∶ Lie(𝑁) × 𝑀←←→ 𝑀,(𝑛, 𝑚)↦−𝑚⋅2𝑛,
•⋅2∶ Lie(𝑁) × 𝑀
[𝑀,𝑁]𝑥
←←→ 𝑀
[𝑀,𝑁]𝑥
,(𝑛, 𝑚)↦𝑛⋅1𝑚= −𝑚⋅2𝑛,
•𝜉⋅∶𝑁×𝑀
[𝑀,𝑁]𝑥
←←→ 𝑀,(𝑛, 𝑚)←←→ 𝑛⋅1𝑚,
•𝜕1∶𝑀←←→ 𝑁,𝑚↦𝜕(𝑚),

50 2 B aidings o c ossed modules and in e nal objec s
•𝜕2∶𝑀
[𝑀,𝑁]𝑥
←←→ Lie(𝑁),𝑚↦𝜕𝑚.
Rema k 2.4.22.We will say ha he bo om pa (𝑀
[𝑀,𝑁]𝑥
𝜕2
←←←←←←←←←→ Lie(𝑁), ⋅2)is he Lie-
sa ion o he c ossed module o Leibniz 𝐾-algeb as. In his way we ound a simila
ela ion wi h he Leibniz and Lie objec case.
This Liesa ion sa is ies again ha applied on a c ossed module o Lie 𝐾-algeb as,
hough as a c ossed module o Leibniz 𝐾-algeb as wi h he ac ion (⋅,⋅−), is na u ally
isomo phic o i sel . Tha occu s because, in he quo ien , he second gene a o s a e
null oo:
𝑛⋅1𝑚+𝑚⋅2𝑛=𝑛⋅𝑚+𝑚⋅−𝑛=𝑛⋅𝑚−𝑛⋅𝑚= 0.
P oposi ion 2.4.23. Le (𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅)be a c ossed module o Lie objec s in 𝐾,
hen (𝑀𝜕1
←←←←←←←←←→ 𝐿, (⋅1,⋅2)) is a c ossed module o Leibniz 𝐾-algeb as, whe e
•The Leibniz b acke s a e gi en by: [𝑚, 𝑚′] = 𝑚∗𝑀
𝑁𝑓(𝑚′), o 𝑚, 𝑚′∈𝑀and
[𝑙, 𝑙′] = 𝑙∗𝐿
𝐻𝑔(𝑙′), o 𝑙, 𝑙′∈𝑀;
•⋅1∶𝐿×𝑁←←→ 𝑀is de ined by 𝑙⋅1𝑚=𝜉⋅(𝑙, 𝑓(𝑚)) o 𝑙∈𝐿,𝑚∈𝑀;
•⋅2∶𝑀×𝐿←←→ 𝑀is de ined by 𝑚 ⋅2𝑙= −𝑔(𝑙)⋅1𝑚 o 𝑙∈𝐿,𝑚∈𝑀.
We ha e he unc o s X(LeibAlg𝐾)
𝑋Φ//XLie(𝐾)
𝑋Ψ
oosa is ying 𝑋Ψ◦𝑋Φ =
IdX(LeibAlg𝐾), and so, he unc o 𝑋Φis a ull inclusion unc o .
2.4.1 B aiding o c ossed modules o Lie objec s in 𝐾and c ossed
modules o Leibniz algeb as
We wan o de ine he no ion o b aiding o c ossed modules o Leibniz algeb as. We
will use he idea ha he b aiding o c ossed module o Leibniz 𝐾-algeb as mus be a
pa icula case o b aiding o Lie objec s in 𝐾, sa is ying symme ical p ope ies
o he p e ious ones.
2.4.1 B aiding o c ossed modules o Lie objec s in 𝐾51
De ini ion 2.4.24. Le = (C, ⊗, 𝑎, )be a b aided semig oupal ca ego y whe e C
is an addi i e ca ego y. Le = ((𝐴, 𝜇𝐴)𝜕
←←←←←←→ (𝐵, 𝜇𝐵), 𝑝)be a c ossed module o Lie
objec s in .
A b aiding (o Pei e li ing) on is a mo phism 𝔗∶𝐵 ⊗ 𝐵 ←←→ 𝐴sa is ying:
𝜕◦𝔗=𝜇𝐵,
𝔗◦(𝜕 ⊗ 𝜕) = 𝜇𝐴,
−𝔗◦(𝜕 ⊗ Id𝐵) = 𝑝◦𝐴,𝐵,
𝔗◦(Id𝐵⊗𝜕) = 𝑝,
𝔗◦(Id𝐵⊗𝜇𝐵)⊗ 𝑎𝐵,𝐵,𝐵 =𝔗◦(𝜇𝐵⊗Id𝐵)◦(Id(𝐵⊗𝐵)⊗𝐵 −(𝑎−1
𝐵,𝐵,𝐵◦(Id𝐵⊗𝐵,𝐵)◦𝑎𝐵,𝐵,𝐵)),
𝔗◦(𝜇𝐵⊗Id𝐵) = 𝔗◦(Id𝐵⊗𝜇𝐵)◦𝑎𝐵,𝐵,𝐵◦(Id(𝐵⊗𝐵)⊗𝐵 −(𝐵,𝐵 ⊗Id𝐵)).
((𝐴, 𝜇𝐴)𝜕
←←←←←←→ (𝐵, 𝜇𝐵), 𝑝, 𝔗)will be called a b aided c ossed module o Lie objec s in .
A mo phism (𝛼, 𝛽)∶ ((𝐴, 𝜇𝐴)𝜕
←←←←←←→ (𝐵, 𝜇𝐵), 𝑝, 𝔗)→((𝐶, 𝜇𝐶)𝛿
←←←←←←→ (𝐷, 𝜇𝐷), 𝑞, 𝔜)
o b aided c ossed modules o Lie objec s is a mo phism o c ossed modules o Lie
objec s in he ca ego y sa is ying he ollowing commu a i e diag am
𝐵 ⊗ 𝐵 𝐴
𝐷 ⊗ 𝐷 𝐵.
𝛽⊗𝛽
𝔗
𝛼
𝔜
We deno e his new ca ego y as BXLie().
Example 2.4.25. As in he p e iouscases, we ha e ha BXLie(Vec 𝐾)andBX(LieAlg𝐾)
a e isomo phic, aking in Vec 𝐾 he usual enso p oduc .
BXLie(𝐾)is desc ibed in he ollowing de ini ions.
De ini ion 2.4.26. Le =(𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅) be a c ossed module o Lie objec s in 𝐾.
A b aiding (o Pei e li ing) o is gi en by a iple o maps
𝑇{−,−} = ({−,−}𝐿𝐻 ,{−,−}𝐻𝐿,{−,−}2)
whe e
52 2 B aidings o c ossed modules and in e nal objec s
•{−,−}2∶𝐻×𝐻←←→ 𝑁is a 𝐾-bilinea map such ha (𝑁, 𝐻, ⋅2, 𝜕2,{−,−}2)is
a b aided c ossed module o Lie 𝐾-algeb as.
•{−,−}𝐿𝐻 ∶𝐿×𝐻←←→ 𝑀and {−,−}𝐻𝐿 ∶𝐻×𝐿←←→ 𝑀a e 𝐾-bilinea maps,
which wi h {−,−}2sa is y he ollowing p ope ies o 𝑙∈𝐿,ℎ, ℎ′∈𝐻,
𝑚∈𝑀,𝑛∈𝑁:
𝑓({𝑙, ℎ}𝐿𝐻 ) = {𝑔(𝑙), ℎ}2, 𝑓({ℎ, 𝑙}𝐻𝐿) = {ℎ, 𝑔(𝑙)}2,
𝜕1{𝑙, ℎ}𝐿𝐻 =𝑙∗𝐿
𝐻ℎ, 𝜕1{ℎ, 𝑙}𝐻𝐿 = −𝑙∗𝐿
𝐻ℎ,
{𝜕1(𝑚), 𝜕2(𝑛)}𝐿𝐻 =𝑚∗𝑀
𝑁𝑛, {𝜕2(𝑛), 𝜕1(𝑚)}𝐻𝐿 = −𝑚∗𝑀
𝑁𝑛,
{𝜕1(𝑚), ℎ}𝐿𝐻 = −ℎ⋅1𝑚, {𝜕2(𝑛), 𝑙}𝐻𝐿 = −𝜉⋅(𝑙, 𝑛),
{𝑙, 𝜕2(𝑛)} = 𝜉⋅(𝑙, 𝑛),{ℎ, 𝜕1(𝑚)} = ℎ⋅1𝑚,
{𝑙, [ℎ, ℎ′]𝐻}𝐿𝐻 = {𝑙∗𝐿
𝐻ℎ, ℎ′}𝐿𝐻 − {𝑙∗𝐿
𝐻ℎ′, ℎ}𝐿𝐻 ,
{[ℎ, ℎ′]𝐻, 𝑙}𝐻𝐿 = −{ℎ, 𝑙 ∗𝐿
𝐻ℎ′}𝐻𝐿 − {𝑙∗𝐿
𝐻ℎ, ℎ′}𝐿𝐻 ,
{𝑙, [ℎ, ℎ′]𝐻}𝐿𝐻 = {𝑙∗𝐿
𝐻ℎ, ℎ′}𝐿𝐻 + {ℎ, 𝑙 ∗𝐿
𝐻ℎ′}𝐻𝐿,
{[ℎ, ℎ′]𝐻, 𝑙}𝐻𝐿 = −{ℎ, 𝑙 ∗𝐿
𝐻ℎ′}𝐻𝐿 + {ℎ′, 𝑙 ∗𝐿
𝐻ℎ}𝐻𝐿.
We will say ha ( 𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅, 𝑇{−,−}) is a b aided c ossed module o Lie objec s in
𝐾.
Rema k 2.4.27.A b aiding is a pai 𝑇{−,−} = (𝑇1
{−,−}, 𝑇 2
{−,−}), bu o simplici y we
deno e 𝑇1
{−,−} ∶ (𝐿⊗𝐻)⊕(𝐻 ⊗𝐿)←←→ 𝑀wi h 𝑇1
{−,−}((𝑙⊗ℎ)+(ℎ′⊗𝑙′)) = {𝑙, ℎ}𝐿𝐻 +
{ℎ′, 𝑙′}𝐻𝐿 and 𝑇2
{−,−}(ℎ, ℎ′) = {ℎ, ℎ′}2.
De ini ion 2.4.28. Le ( 𝑀
𝑁
𝑓
𝜕
←←←←←←→
𝐿
𝐻
𝑔, ⋅, 𝑇{−,−}) and ( 𝑋
𝑌
𝑘
𝛿
←←←←←←→
𝑉
𝑊
ℎ,
⋆, 𝑇{−,−}′) be wo b aided
c ossed modules o Lie objec s in 𝐾. A mo phism o b aided c ossed modules o
Lie objec s in 𝐾is a mo phism (𝛼, 𝛽)o c ossed modules o Lie objec s in 𝐾
sa is ying:
•(𝛼2, 𝛽2)∶ (𝑁, 𝐻, ⋅2, 𝜕2,{−,−}2)←←→ (𝑌 , 𝑊 , ⋆2, 𝛿2,{−,−}′
2)is an mo phism o
b aided c ossed modules o Lie 𝐾-algeb as,
2.4.1 B aiding o c ossed modules o Lie objec s in 𝐾53
•𝛼1({𝑙, ℎ}𝐿𝐻 ) = {𝛽1(𝑙), 𝛽2(ℎ)}′
𝑉 𝑊 , o 𝑙∈𝐿,ℎ∈𝐻,
•𝛼1({ℎ, 𝑙}𝐻𝐿)={𝛽2(ℎ), 𝛽1(𝑙)}′
𝑊 𝑉 , o 𝑙∈𝐿,ℎ∈𝐻.
We wan o use he concep o b aiding on c ossed modules o Lie objec s in
𝐾 o ob ain a de ini ion o c ossed modules o Leibniz 𝐾-algeb as. Fo ha ,
we will ake a b aiding on (
𝑀
𝑀
[𝑀,𝑁]𝑥
𝜋𝑀
𝜕
←←←←←←→
𝑁
Lie(𝑁)
𝜋𝑁,
⋅). I we y o ake one 𝐾-bilinea map
{−,−} we would ind p oblems wi h he way o de ining he co esponding maps
because we ha e ha he i s p ope ies add one mo e quo ien ha we would like
o be i ial o Lie 𝐾-algeb as, o i we ake i o be i ial, he es o p ope ies
p e en i om being made o he gene al case o Leibniz 𝐾-algeb as (i we ake
{𝑛, 𝑛′}𝑁Lie(𝑁)= {𝑛, 𝑛′}={𝑛, 𝑛′}Lie(𝑁)𝑁 o example, he hi d and ou h p ope y
leads us o p o e ha 𝑀mus be Lie 𝐾-algeb a).
Fo his, as in he case o he wo ac ions, we will ake o b aiding wo 𝐾-
bilinea maps {−,−},⟨−,−⟩∶𝑁×𝑁←←→ 𝑀, and de ine {𝑛, 𝑛′}𝑁Lie(𝑁)= {𝑛, 𝑛′},
{𝑛, 𝑛′}Lie(𝑁)𝑁= −⟨𝑛′, 𝑛⟩and {𝑛, 𝑛′}2= {𝑛, 𝑛′} = −⟨𝑛′, 𝑛⟩, whe e we can see ha we
in oduce a new quo ien in 𝑀.
De ini ion 2.4.29. Le = (𝑀𝜕
←←←←←←→ 𝑁, (⋅1,⋅2)) be a c ossed module o Leibniz 𝐾-
algeb as.
A b aiding (o Pei e li ing) on is a pai ({−,−},⟨−,−⟩)o 𝐾-bilinea maps
{−,−},⟨−,−⟩∶𝑁×𝑁←←→ 𝑀,(𝑛, 𝑛′)↦{𝑛, 𝑛′}and (𝑛, 𝑛′)↦⟨𝑛, 𝑛′⟩, sa is ying:
𝜕{𝑛, 𝑛′} = [𝑛, 𝑛′] = 𝜕⟨𝑛, 𝑛′⟩,(BXLeib1)
{𝜕𝑚, 𝜕𝑚′} = [𝑚, 𝑚′] = ⟨𝜕𝑚, 𝜕𝑚′⟩,(BXLeib2)
{𝜕𝑚, 𝑛} = 𝑚⋅2𝑛=⟨𝜕𝑚, 𝑛⟩,(BXLeib3)
{𝑛, 𝜕𝑚} = 𝑛⋅1𝑚=⟨𝑛, 𝜕𝑚⟩,(BXLeib4)
{𝑛, [𝑛′, 𝑛′′]} = {[𝑛, 𝑛′], 𝑛′′}−{[𝑛, 𝑛′′], 𝑛′},(BXLeib5)
⟨𝑛, [𝑛′, 𝑛′′]⟩= {[𝑛, 𝑛′], 𝑛′′}−⟨[𝑛, 𝑛′′], 𝑛′⟩,(BXLeib6)
{𝑛, [𝑛′, 𝑛′′]} = {[𝑛, 𝑛′], 𝑛′′}−⟨[𝑛, 𝑛′′], 𝑛′⟩,(BXLeib7)
⟨𝑛, [𝑛′, 𝑛′′]⟩=⟨[𝑛, 𝑛′], 𝑛′′⟩−⟨[𝑛, 𝑛′′], 𝑛′⟩,(BXLeib8)
60 2 B aidings o c ossed modules and in e nal objec s
(𝐴×𝐶𝐵)⊗(𝐴×𝐶𝐵)𝐴 ⊗ 𝐴
𝐴×𝐶𝐵 𝐴
𝐵 ⊗ 𝐵 𝐵 𝐶.
𝜇𝐴×𝐶𝐵
𝜋𝐵⊗𝜋𝐵
𝜋𝐴⊗𝜋𝐴
𝜇𝐴
𝜋𝐴
𝜋𝐵𝑓
𝜇𝐵𝑔
I is s aigh o wa d o see ha 𝜇𝐴×𝐶𝐵is well de ined. Now, we will p o e ha
(𝐴×𝐶𝐵, 𝜇𝐴×𝐶𝐵)is a Lie objec checking he i s axiom. Fo simplici y o no a ion,
we will deno e 𝐷∶=𝐴×𝐶𝐵. Le 𝑋∈ {𝐴, 𝐵}. Using uni e sal p ope ies we ha e:
𝜋𝑋◦(−𝜇𝐷◦𝐷,𝐷) = −𝜋𝑋◦𝜇𝐷◦𝐷,𝐷 = −𝜇𝑋◦(𝜋𝑋⊗ 𝜋𝑋)◦𝐷,𝐷.
Since is a na u al isomo phism and ha (𝑋, 𝜇𝑋)is a Lie objec , we ge
𝜋𝑋◦(−𝜇𝐷◦𝐷,𝐷) = −𝜇𝑋◦𝑋,𝑋◦(𝜋𝑋⊗ 𝜋𝑋) = 𝜇𝑋◦(𝜋𝑋⊗ 𝜋𝑋).
We conclude ha 𝜇𝐷= −𝜇𝐷◦𝐷,𝐷 because 𝜇𝐷is he unique mo phism ha sa is ies
he p e ious equali y o 𝑋∈ {𝐴, 𝐵}.
Now, we will check he second axiom o Lie objec .
Fo he i s summand, we ha e:
𝜋𝑋◦𝜇𝐷◦(Id𝐷⊗𝜇𝐷)◦𝑎𝐷,𝐷,𝐷 =𝜇𝑋◦(𝜋𝑋⊗ 𝜋𝑋)◦(Id𝐷⊗𝜇𝐷)◦𝑎𝐷,𝐷,𝐷
=𝜇𝑋◦(𝜋𝑋⊗(𝜋𝑋◦𝜇𝐷))◦𝑎𝐷,𝐷,𝐷 =𝜇𝑋◦(𝜋𝑋⊗(𝜇𝑋◦(𝜋𝑋⊗ 𝜋𝑋)))◦𝑎𝐷,𝐷,𝐷
=𝜇𝑋◦(Id𝑋⊗𝜇𝑋)◦(𝜋𝑋⊗(𝜋𝑋⊗ 𝜋𝑋))◦𝑎𝐷,𝐷,𝐷.
Using ha 𝑎is a na u al isomo phism, we ha e
𝜋𝑋◦𝜇𝐷◦(Id𝐷⊗𝜇𝐷)◦𝑎𝐷,𝐷,𝐷 =𝜇𝑋◦(Id𝑋⊗𝜇𝑋)◦𝑎𝑋,𝑋,𝑋◦((𝜋𝑋⊗ 𝜋𝑋)⊗ 𝜋𝑋).
Doing he same o he second and hi d summands ( he na u alness o 𝑎gi es he
same na u alness o 𝑎−1), we ha e ha :
𝜋𝑋◦𝜇𝐷◦(𝜇𝐷⊗Id𝐷)◦𝑎−1
𝐷,𝐷,𝐷◦(Id𝐷⊗𝐷,𝐷)◦𝑎𝐷,𝐷,𝐷
=𝜇𝑋◦(𝜇𝑋⊗Id𝑋)◦𝑎−1
𝑋,𝑋,𝑋◦(Id𝑋⊗𝑋,𝑋)◦𝑎𝑋,𝑋,𝑋◦((𝜋𝑋⊗ 𝜋𝑋)⊗ 𝜋𝑋),

2.4.2 B aiding o ca ego ical Lie objec s in 𝐾61
𝜋𝑋◦(−𝜇𝐷◦(𝜇𝐷⊗Id𝐷)) = −𝜇𝑋◦(𝜇𝑋⊗Id𝑋)◦((𝜋𝑋⊗ 𝜋𝑋)⊗ 𝜋𝑋).
Adding he h ee las equali ies and using he dis ibu i i y o he composi ion, we
ha e 𝜋𝑋◦𝐷=𝑋◦((𝜋𝑋⊗ 𝜋𝑋)⊗ 𝜋𝑋), whe e deno e by 𝑌 he mo phism ha is
in he i s e m o he equali y o he second axiom o a Lie objec (𝑌 , 𝜇𝑌). Since
(𝑋, 𝜇𝑋)is a Lie objec , we ge 𝑋=(𝑋⊗𝑋)⊗𝑋0𝑋, and so 𝜋𝑋◦𝐷=(𝐷⊗𝐷)⊗𝐷0𝑋.
Now, by he uni e sal p ope y, we ha e ha 𝐷=(𝐷⊗𝐷)⊗𝐷0𝐷and he e o e
(𝐷, 𝜇𝐷)is a Lie objec .
To conclude he p oo i is enough o check ha he mo phism gi en by he pull-
back in Cis a Lie mo phism, bu his is a ou ine e i ica ion.
De ini ion 2.4.42. Le = (C, ⊗, 𝑎, )be a b aided semig oupal ca ego y whe e C
is an addi i e ca ego y wi h pullbacks.
Le ℭ= ((𝐶1, 𝜇𝐶1),(𝐶0, 𝜇𝐶0), 𝑠, 𝑡, 𝑒, 𝑘)be a ca ego ical Lie objec in Lie().
A b aiding on ℭis a mo phism 𝜏∶𝐶0⊗ 𝐶0←←→ 𝐶1sa is ying:
•𝑠◦𝜏=𝜇𝐶0and 𝑡◦𝜏=𝜇𝐶0◦𝐶0,𝐶0,
•We de ine 𝐶0⊗𝐶0
𝜇𝐶1×𝐶0(𝜏◦(𝑡⊗𝑡)),(𝜏◦(𝑠⊗𝑠))×𝐶0(𝜇𝐶1◦)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝐶1×𝐶0𝐶1as he wo unique
mo phisms which sa is y he uni e sal p ope y, espec i ely, in he ollowing
diag ams:
𝐶1⊗ 𝐶1
𝐶1×𝐶0𝐶1𝐶1
𝐶1𝐶0
𝜇𝐶1
𝜏◦(𝑡⊗𝑡)
𝜋1
𝜋2𝑡
𝑠
𝐶1⊗ 𝐶1
𝐶1×𝐶0𝐶1𝐶1
𝐶1𝐶0
𝜏◦(𝑠⊗𝑠)
𝜇𝐶1◦𝐶1,𝐶1
𝜋1
𝜋2𝑡
𝑠
,
and he equali y
𝑘◦(𝜇𝐶1×𝐶0(𝜏◦(𝑡 ⊗ 𝑡))) = 𝑘◦((𝜏◦(𝑠⊗𝑠)) ×𝐶0(𝜇𝐶1◦)).
62 2 B aidings o c ossed modules and in e nal objec s
•I mus sa is y
𝜏◦(Id𝐶0⊗𝜇𝐶0)⊗ 𝑎𝐶0=𝜏◦(𝜇𝐶0⊗Id𝐶0)◦(Id(𝐶0⊗𝐶0)⊗𝐶0−(𝑎−1
𝐶0
◦(Id𝐶0⊗𝐶0)◦𝑎𝐶0)),
𝜏◦(𝜇𝐶0⊗Id𝐶0) = 𝜏◦(Id𝐶0⊗𝜇𝐶0)◦𝑎𝐶0◦(Id(𝐶0⊗𝐶0)⊗𝐶0−(𝐶0⊗Id𝐶0)).
We deno e 𝑎𝐶0,𝐶0,𝐶0=∶ 𝑎𝐶0and 𝐶0,𝐶0=∶ 𝐶0.
We will say ha ((𝐶1, 𝜇𝐶1),(𝐶0, 𝜇𝐶0), 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)is a b aided ca ego ical Lie objec in
.
An in e nal unc o
((𝐶1, 𝜇𝐶1),(𝐶0, 𝜇𝐶0), 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)(𝐹1,𝐹0)
←←←←←←←←←←←←←←←←←←←←←←←←←→ ((𝐶′
1, 𝜇𝐶′
1),(𝐶′
0, 𝜇𝐶′
0), 𝑠′, 𝑡′, 𝑒′, 𝑘′, 𝜏′)
is said o be a b aided in e nal unc o o b aided ca ego ical Lie objec s in i i
sa is ies he ollowing diag am:
𝐶0⊗ 𝐶0𝐶1
𝐶′
0⊗ 𝐶′
0𝐶1.
𝐹0⊗𝐹0
𝜏
𝐹1
𝜏′
We deno e his new ca ego y as BICa (Lie()).
Example 2.4.43. Weha e ha heca ego iesBICa (Lie(Vec 𝐾)) and BICa (LieAlg𝐾)
a e isomo phic, aking in Vec 𝐾 he usual enso p oduc (we assume cha (𝐾)≠2).
De ini ion 2.4.44. Le =(
𝐶1
𝐷1
𝑓1,
𝐶0
𝐷0
𝑓0, 𝑠, 𝑡, 𝑒, 𝑘) be a ca ego ical Lie objec in 𝐾.
A b aiding on is a iple 𝜏= (𝜏𝐶0,𝐷0, 𝜏𝐷0,𝐶0, 𝜏2)whe e
•𝜏2∶𝐷0×𝐷0←←→ 𝐷1is a 𝐾-bilinea map such ha (𝐷1, 𝐷0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏2)is a
b aided c ossed module o Lie 𝐾-algeb as,
•𝜏𝐷0,𝐶0∶𝐷0×𝐶0←←→ 𝐶1and 𝜏𝐶0,𝐷0∶𝐶0×𝐷0←←→ 𝐶1a e 𝐾-bilinea maps which,
wi h 𝜏2, sa is y he ollowing p ope ies o 𝑐∈𝐶0,𝑑, 𝑑′∈𝐷0,𝑥∈𝐶1,
𝑦∈𝐷1:
𝑓1(𝜏𝐶0,𝐷0
𝑐,𝑑 ) = 𝜏2
𝑓0(𝑐),𝑑 and 𝑓1(𝜏𝐷0,𝐶0
𝑑,𝑐 ) = 𝜏2
𝑑,𝑓0(𝑐),
2.4.2 B aiding o ca ego ical Lie objec s in 𝐾63
𝜏𝐶0,𝐷0
𝑐,𝑑 ∶𝑐∗𝐶0
𝐷0𝑑←←→ −𝑐∗𝐶0
𝐷0𝑑and 𝜏𝐷0,𝐶0
𝑑,𝑐 ∶ − 𝑐∗𝐶0
𝐷0𝑑←←→ 𝑐∗𝐶0
𝐷0𝑑.
The ollowing diag ams a e sa is ied in he in e nal ca ego y:
𝑠1(𝑥) ∗𝐶0
𝐷0𝑠2(𝑦)𝑡1(𝑥) ∗𝐶0
𝐷0𝑡2(𝑦)
−𝑠1(𝑥) ∗𝐶0
𝐷0𝑠2(𝑦) −𝑡1(𝑥) ∗𝐶0
𝐷0𝑡2(𝑦)
𝜏𝐶0,𝐷0
𝑠1(𝑥),𝑠2(𝑦)
𝑥∗𝐶1
𝐶0𝑦
𝜏𝐶0,𝐷0
𝑡1(𝑥),𝑡2(𝑦)
−𝑥∗𝐶1
𝐷1𝑦
,
−𝑠1(𝑥) ∗𝐶0
𝐷0𝑠2(𝑦) −𝑡1(𝑥) ∗𝐶0
𝐷0𝑡2(𝑦)
𝑠1(𝑥) ∗𝐶0
𝐷0𝑠2(𝑦)𝑡1(𝑥) ∗𝐶0
𝐷0𝑡2(𝑦)
𝜏𝐷0,𝐶0
𝑠2(𝑦),𝑠1(𝑥)
−𝑥∗𝐶1
𝐶0𝑦
𝜏𝐷0,𝐶0
𝑡2(𝑦),𝑡1(𝑥)
𝑥∗𝐶1
𝐷1𝑦
.
Mo eo e , we ha e he ollowing p ope ies:
𝜏𝐶0,𝐷0
𝑐,[𝑑,𝑑′]𝐷0
=𝜏𝐶0,𝐷0
𝑐∗𝐶0
𝐷0𝑑,𝑑′−𝜏𝐶0,𝐷0
𝑐∗𝐶0
𝐷0𝑑′,𝑑,
𝜏𝐷0,𝐶0
[𝑑,𝑑′]𝐷0,𝑐 = −𝜏𝐷0,𝐶0
𝑑,𝑐∗𝐶0
𝐷0𝑑′−𝜏𝐶0,𝐷0
𝑐∗𝐶0
𝐷0𝑑,𝑑′,
𝜏𝐶0,𝐷0
𝑐,[𝑑,𝑑′]𝐷0
=𝜏𝐶0,𝐷0
𝑐∗𝐶0
𝐷0𝑑,𝑑′+𝜏𝐷0,𝐶0
𝑑,𝑐∗𝐶0
𝐷0𝑑′,
𝜏𝐷0,𝐶0
[𝑑,𝑑′]𝐷0,𝑐 = −𝜏𝐷0,𝐶0
𝑑,𝑐∗𝐶0
𝐷0𝑑′+𝜏𝐷0,𝐶0
𝑑′,𝑐∗𝐶0
𝐷0𝑑.
We will say ha (
𝐶1
𝐷1
𝑓1,
𝐶0
𝐷0
𝑓0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏) is a b aided ca ego ical Lie objec in 𝐾.
Rema k 2.4.45.A b aiding is a pai 𝜏= (𝜏1, 𝜏2)bu , o simplici y, we ake o he
de ini ion 𝜏2(𝑑, 𝑑′) = 𝜏2
𝑑,𝑑′and 𝜏1∶ (𝐶0⊗ 𝐷0)⊕(𝐷0⊗ 𝐶0)←←→ 𝐶1by he exp ession
𝜏1((𝑐 ⊗ 𝑑)+(𝑑′⊗ 𝑐′)) = 𝜏𝐶0,𝐷0
𝑐,𝑑 +𝜏𝐷0,𝐶0
𝑑′,𝑐′.
De ini ion2.4.46. Le (
𝐶1
𝐷1
𝑓1,
𝐶0
𝐷0
𝑓0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏) and (
𝐶′
1
𝐷′
1
𝑔1,
𝐶′
0
𝐷′
0
𝑔0, 𝑠′, 𝑡′, 𝑒′, 𝑘′, 𝜓) be b aided
ca ego ical Lie objec s in 𝐾. A b aided in e nal unc o be ween ca ego ical Lie
64 2 B aidings o c ossed modules and in e nal objec s
objec s in 𝐾is an in e nal unc o ((𝐹1
1, 𝐹0
1),(𝐹1
0, 𝐹0
0)) be ween he espec i e
ca ego ical Lie objec s which sa is ies:
•(𝐹0
1, 𝐹0
0)∶ (𝐷1, 𝐷0, 𝑠2, 𝑡2, 𝑒2, 𝑘2, 𝜏2)←←→ (𝐷′
1, 𝐷′
0, 𝑠′
2, 𝑡′
2, 𝑒′
2, 𝑘′
2, 𝜓2)is a b aided in-
e nal unc o be ween ca ego ical Lie 𝐾-algeb as.
•𝐹1
1(𝜏𝐶0,𝐷0
𝑐,𝑑 ) = 𝜓𝐶′
0,𝐷′
0
𝐹1
0(𝑐),𝐹0
0(𝑑) o 𝑐∈𝐶0,𝑑∈𝐷0.
•𝐹1
1(𝜏𝐷0,𝐶0
𝑑,𝑐 ) = 𝜓𝐷′
0,𝐶′
0
𝐹0
0(𝑑),𝐹1
0(𝑐) o 𝑐∈𝐶0,𝑑∈𝐷0.
To in oduce a b aiding o he ca ego ical Leibniz 𝐾-algeb as wi h he p e ious
scheme, we will use wo 𝐾-bilinea maps 𝜏, 𝜓 ∶𝐶0×𝐶0←←→ 𝐶1, as in he case o a
b aiding o c ossed modules o Leibniz 𝐾-algeb as. Conside o he inclusion Lie
objec in 𝐾 he b aiding 𝜏 de ined by 𝜏𝐶0,Lie(𝐶0)
𝑎,𝑏 =𝜏𝑎,𝑏,𝜏Lie(𝐶0),𝐶0
𝑎,𝑏 = −𝜓𝑏,𝑎 and
𝜏2
𝑎,𝑏 =𝜏𝑎,𝑏 = −𝜓𝑏,𝑎, whe e we in oduce a quo ien in 𝐶1whose elemen s we will
deno e as 𝑥.
De ini ion 2.4.47. A b aiding o he ca ego ical Leibniz 𝐾-algeb a (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)
is a pai (𝜏, 𝜓)o 𝐾-bilinea maps 𝜏, 𝜓 ∶𝐶0×𝐶0←←→ 𝐶1,(𝑎, 𝑏)↦𝜏𝑎,𝑏 and (𝑎, 𝑏)↦𝜓𝑎,𝑏,
sa is ying:
𝜏𝑎,𝑏 ∶ [𝑎, 𝑏]←←→ −[𝑎, 𝑏]and 𝜓𝑎,𝑏 ∶ [𝑎, 𝑏]←←→ −[𝑎, 𝑏],(LeibT1)
[𝑠(𝑥), 𝑠(𝑦)] [𝑡(𝑥), 𝑡(𝑦)]
−[𝑠(𝑥), 𝑠(𝑦)] −[𝑡(𝑥), 𝑡(𝑦)],
𝜏𝑠(𝑥),𝑠(𝑦)
[𝑥,𝑦]
𝜏𝑡(𝑥),𝑡(𝑦)
−[𝑥,𝑦]
[𝑠(𝑥), 𝑠(𝑦)] [𝑡(𝑥), 𝑡(𝑦)]
−[𝑠(𝑥), 𝑠(𝑦)] −[𝑡(𝑥), 𝑡(𝑦)],
𝜓𝑠(𝑥),𝑠(𝑦)
[𝑥,𝑦]
𝜓𝑡(𝑥),𝑡(𝑦)
−[𝑥,𝑦]
(LeibT2)
𝜏𝑎,[𝑏,𝑐]=𝜏[𝑎,𝑏],𝑐 −𝜏[𝑎,𝑐],𝑏,(LeibT3)
𝜓𝑎,[𝑏,𝑐]=𝜏[𝑎,𝑏],𝑐 −𝜓[𝑎,𝑐],𝑏,(LeibT4)
𝜏𝑎,[𝑏,𝑐]=𝜏[𝑎,𝑏],𝑐 −𝜓[𝑎,𝑐],𝑏,(LeibT5)
𝜓𝑎,[𝑏,𝑐]=𝜓[𝑎,𝑏],𝑐 −𝜓[𝑎,𝑐],𝑏, 𝑎, 𝑏, 𝑐 ∈𝐶0, 𝑥, 𝑦 ∈𝐶1.(LeibT6)
We will say ha (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, (𝜏, 𝜓)) is a b aided ca ego icalLeibniz𝐾-algeb a.
2.4.2 B aiding o ca ego ical Lie objec s in 𝐾65
De ini ion 2.4.48. Le (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, (𝜏, 𝜓)) and (𝐶′
1, 𝐶′
0, 𝑠′, 𝑡′, 𝑒′, 𝑘′,(𝜏′, 𝜓′)) be
wo b aided ca ego ical Leibniz 𝐾-algeb as.
An in e nal unc o (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘)(𝐹1,𝐹0)
←←←←←←←←←←←←←←←←←←←←←←←←←→ (𝐶′
1, 𝐶′
0, 𝑠′, 𝑡′, 𝑒′, 𝑘′)is said o be a
b aided in e nal unc o be ween wo b aided ca ego ical Leibniz 𝐾-algeb as i i sa -
is ies:
𝐹1(𝜏𝑎,𝑏) = 𝜏′
𝐹0(𝑎),𝐹0(𝑏),(LeibHT1)
𝐹1(𝜓𝑎,𝑏) = 𝜓′
𝐹0(𝑎),𝐹0(𝑏), 𝑎, 𝑏 ∈𝐶0.(LeibHT2)
We deno e he ca ego y o b aided ca ego ical Leibniz 𝐾-algeb as and b aided
in e nal unc o s be ween hem as BICa (LeibAlg𝐾).
We wan o see he b aided ca ego ical Lie 𝐾-algeb as as a pa icula case o
b aided ca ego ical Leibniz 𝐾-algeb as.
P oposi ion 2.4.49. Le 𝐶1and 𝐶0be Lie 𝐾-algeb as. Then, (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)is a
b aided ca ego ical Lie 𝐾-algeb a i and only i (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, (𝜏, 𝜏−)) is a b aided
ca ego ical Leibniz 𝐾-algeb a. 𝜏−∶𝐶0×𝐶0←←→ 𝐶1is de ined as 𝜏−
𝑎,𝑏 = −𝜏𝑏,𝑎.
P oo . (LeibT1) and (LeibT2) can be ew i en as (LieT1) and LieT2, espec i ely,
using he an icommu a i i y. Mo eo e , i is clea ha LeibT3 and LieT4 a e iden ical,
and ha (LeibT6) is equi alen o LieT3.
To see he las equi alences, (LeibT4) wi h (LieT3), and (LeibT5) wi h (BXLie4),
we mus p o e 𝜏[𝑎,𝑏],𝑐 = −𝜏𝑐,[𝑎,𝑏], o 𝑎, 𝑏, 𝑐 ∈𝐶0.
•In he Lie case, i is ue using P oposi ion 2.3.5.
•In he Leibniz case i is no ue in gene al, because we need 𝜏𝑎,𝑏 = −𝜓𝑏,𝑎; bu
using (LeibT4) and (LeibT5) we can obse e ha 𝜏𝑎,[𝑏,𝑐]=𝜓𝑎,[𝑏,𝑐]=𝜏−
𝑎,[𝑏,𝑐]=
−𝜏[𝑏,𝑐],𝑎.
P oposi ion 2.4.50. Le (𝐶1, 𝐶0, 𝑠, 𝑡, 𝑒, 𝑘, (𝜏, 𝜓)) be a b aided ca ego ical Leibniz 𝐾-
algeb a. Then

66 2 B aidings o c ossed modules and in e nal objec s
(
𝐶1
𝐶1
[𝜏𝐶0,𝐶0]
𝜋𝐶1,
𝐶0
Lie(𝐶0)
𝜋𝐶0,(𝑠, 𝑠),(𝑡, 𝑡),(𝑒, 𝑒),(𝑘,
𝑘), 𝜏)is a b aided ca ego ical Lie objec in
𝐾, whe e 𝐶1
[𝜏𝐶0,𝐶0]is he Lie 𝐾-algeb a which is a Leibniz quo ien o 𝐶1by he
ideal gene a ed by elemen s o he o m [𝑥, 𝑥]and 𝜏𝑎,𝑏 +𝜓𝑏,𝑎,𝑥∈𝐶1,𝑎, 𝑏 ∈𝐶0; and
he maps a e he ollowing ones:
•𝑠∶𝐶1
[𝜏𝐶0,𝐶0]←←→ Lie(𝐶0)de ined as 𝑠(𝑥) = 𝑠(𝑥) o 𝑥∈𝐶1
[𝜏𝐶0,𝐶0];
•𝑡∶𝐶1
[𝜏𝐶0,𝐶0]←←→ Lie(𝐶0)de ined as 𝑡(𝑥) = 𝑡(𝑥) o 𝑥∈𝐶1
[𝜏𝐶0,𝐶0];
•𝑒∶ Lie(𝐶0)←←→ 𝐶1
[𝜏𝐶0,𝐶0]de ined as 𝑒(𝑎) = 𝑒(𝑎) o 𝑎∈ Lie(𝐶0);
•
𝑘∶𝐶1
[𝜏𝐶0,𝐶0]×Lie(𝐶0)
𝐶1
[𝜏𝐶0,𝐶0]←←→ 𝐶1
[𝜏𝐶0,𝐶0]de ined as 𝑘((𝑥, 𝑦)) = 
𝑘(𝑥, 𝑦) o (𝑥, 𝑦) ∈
𝐶1
[𝜏𝐶0,𝐶0]×Lie(𝐶0)
𝐶1
[𝜏𝐶0,𝐶0], whe e 
𝑘is again he ex ension o he p oduc 
𝑘(𝑥, 𝑦) =
𝑥+𝑦−𝑒(𝑠(𝑦)) (we can ake 
𝑘′(𝑥, 𝑦) = 𝑥+𝑦−𝑒(𝑡(𝑥)) oo, because in he quo ien
i will no change any hing);
•𝜏𝐶0,Lie(𝐶0)∶𝐶0× Lie(𝐶0)←←→ 𝐶1de ined as 𝜏𝐶0,Lie(𝐶0)
𝑎,𝑏 =𝜏𝑎,𝑏 o 𝑎∈𝐶0,𝑏∈
Lie(𝐶0);
•𝜏Lie(𝐶0),𝐶0∶ Lie(𝐶0) × 𝐶0←←→ 𝐶1de ined as 𝜏Lie(𝐶0),𝐶0
𝑎,𝑏 = −𝜓𝑏,𝑎 o 𝑎∈ Lie(𝐶0),
𝑏∈𝐶0;
•𝜏2∶ Lie(𝐶0) × Lie(𝐶0)←←→ 𝐶1
[𝜏𝐶0,𝐶0]de ined as 𝜏2
𝑎,𝑏 =𝜏𝑎,𝑏 = −𝜓𝑏,𝑎 o 𝑎, 𝑏 ∈
Lie(𝐶0).
Rema k 2.4.51.The bo om pa (𝐶1
[𝜏𝐶0,𝐶0],Lie(𝐶0), 𝑠, 𝑡, 𝑒,
𝑘, 𝜏2)will be called Liesa-
ion, and i is again unc o ial.
I we apply his Liesa ion on a b aided ca ego ical Lie 𝐾-algeb a, hough as a
c ossed module o Leibniz 𝐾-algeb as wi h he ac ion wi h he b aiding (𝜏, 𝜏−), he
new gene a o is null
𝜏𝑎,𝑏 +𝜓𝑏,𝑎 =𝜏𝑎,𝑏 +𝜏−
𝑏,𝑎 =𝜏𝑎,𝑏 −𝜏𝑎,𝑏 = 0.
2.4.3 The equi alence be ween he ca ego ies o b aided c ossed modules and b aided in e nal ca ego ies in he case o Leibniz algeb as67
P oposi ion 2.4.52. Le (
𝐶1
𝐷1
𝑓1,
𝐶0
𝐷0
𝑓0, 𝑠, 𝑡, 𝑒, 𝑘, 𝜏)be a b aided ca ego ical Lie objec in
𝐾.
Then (𝐶1, 𝐶0, 𝑠1, 𝑡1, 𝑒1, 𝑘1,(𝜏𝜏, 𝜓𝜏)) is a b aided ca ego ical Leibniz 𝐾-algeb a,
whe e [𝑥, 𝑦]𝐶1=𝑥∗𝐶1
𝐷1𝑦and [𝑎, 𝑏]𝐶0=𝑎∗𝐶0
𝐷0𝑏 o 𝑥, 𝑦 ∈𝐶1, 𝑎, 𝑏 ∈𝐶0, and
𝜏𝜏
𝑎,𝑏 =𝜏𝐶0,𝐷0
𝑎,𝑓0(𝑏),𝜓𝜏
𝑎,𝑏 = −𝜏𝐷0,𝐶0
𝑓(𝑏),𝑎 o 𝑎, 𝑏 ∈𝐶0.
We ha e again he pai o unc o s BICa (LeibAlg𝐾)
𝐵𝐼Φ//BICa (Lie(𝐾))
𝐵𝐼Ψ
oo
sa is ying 𝐵𝐼Ψ◦𝐵𝐼Φ = IdBICa (LeibAlg𝐾), and so, he unc o 𝐵𝐼Φis a ull inclusion
unc o .
2.4.3 The equi alence be ween he ca ego ies o b aided c ossed mod-
ules and b aided in e nal ca ego ies in he case o Leibniz algeb as
Fi s , we will p o e ha BICa (LeibAlg𝐾)and BX(LeibAlg𝐾)a e equi alen , as in he
case o g oups and Lie 𝐾-algeb as. Mo eo e , he equi alence mus gene alize he Lie
𝐾-algeb as case (i.e. he b aidings o he Leibniz 𝐾-algeb as mus sa is y {𝑛, 𝑛′} =
−⟨𝑛′, 𝑛⟩and 𝜏𝑎,𝑏 = −𝜓𝑏,𝑎 and he unc o s o he Lie case would be eco e ed) and
mus be an ex ension o he one gi en o he non-b aiding case.
P oposi ion 2.4.53. Le = (𝑀𝜕
←←←←←←→ 𝑁, (⋅1,⋅2)({−,−},⟨−,−⟩)) be a b aided c ossed
module o Leibniz 𝐾-algeb as.
Then ∶= (𝑀⋊𝑁, 𝑁, 𝑠, 
𝑡, 𝑒, 
𝑘, (𝜏, 𝜓)) is a b aided ca ego ical Leibniz 𝐾-
algeb a whe e
•𝑠∶𝑀⋊𝑁←←→ 𝑁,𝑠((𝑚, 𝑛)) = 𝑛,
•
𝑡∶𝑀⋊𝑁←←→ 𝑁,
𝑡((𝑚, 𝑛)) = 𝜕𝑚 +𝑛,
•𝑒∶𝑁←←→ 𝑀⋊𝑁,𝑒(𝑛) = (0, 𝑛),
•
𝑘∶ (𝑀⋊𝑁) ×𝑁(𝑀⋊𝑁)←←→ 𝑀⋊𝑁, whe e he sou ce is he pullback o 
𝑡
wi h 𝑠, de ined as 𝑘(((𝑚, 𝑛),(𝑚′, 𝜕𝑚 +𝑛))) = (𝑚+𝑚′, 𝑛),
68 2 B aidings o c ossed modules and in e nal objec s
•𝜏 ∶𝑁×𝑁←←→ 𝑀⋊𝑁,𝜏𝑛,𝑛′= (−2{𝑛, 𝑛′},[𝑛, 𝑛′]),
•𝜓 ∶𝑁×𝑁←←→ 𝑀⋊𝑁,𝜓𝑛,𝑛′= (−2⟨𝑛, 𝑛′⟩,[𝑛, 𝑛′]).
P oo . We only need o check he b aiding axioms, since (𝑀⋊𝑁, 𝑁, 𝑠, 
𝑡, 𝑒, 
𝑘)is a
ca ego ical Leibniz 𝐾-algeb a (see [22]).
We will s a wi h (LeibT1). Le 𝑛, 𝑛′∈𝑁.
𝑠(𝜏𝑛,𝑛′) = 𝑠((−2{𝑛, 𝑛′},[𝑛, 𝑛′])) = [𝑛, 𝑛′],

𝑡(𝜏𝑛,𝑛′) = 
𝑡((−2{𝑛, 𝑛′},[𝑛, 𝑛′])) = −2𝜕{𝑛, 𝑛′}+[𝑛, 𝑛′] = −2[𝑛, 𝑛′]+[𝑛, 𝑛′] = −[𝑛, 𝑛′],
whe e we use (BXLeib1). In he same way we can p o e his p ope y o 𝜓 by he
symme y o he cons uc ion.
We will p o e now (LeibT2). Again, we will only check his o 𝜏.
Le 𝑥= (𝑚, 𝑛), 𝑦 = (𝑚′, 𝑛′) ∈ 𝑀⋊𝑁.
We need o show ha 𝜏𝑡(𝑥),𝑡(𝑦)◦[𝑥, 𝑦] = −[𝑥, 𝑦]◦𝜏𝑠(𝑥),𝑠(𝑦). Now, we will w i e he
equali ies in unc ion o he da a gi en by he b aided c ossed module.
𝜏𝑡(𝑥),𝑡(𝑦)◦[𝑥, 𝑦]
=
𝑘(([(𝑚, 𝑛),(𝑚′, 𝑛′)],(−2{
𝑡((𝑚, 𝑛)),
𝑡((𝑚′, 𝑛′))},[
𝑡((𝑚, 𝑛)),
𝑡((𝑚′, 𝑛′))])))
=
𝑘(([(𝑚, 𝑛),(𝑚′, 𝑛′)],(−2{𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′},[𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′])))
=
𝑘((([𝑚, 𝑚′] + 𝑛⋅1𝑚′+𝑚⋅2𝑛′,[𝑛, 𝑛′]),(−2{𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′},[𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′])))
= ([𝑚, 𝑚′] + 𝑛⋅1𝑚′+𝑚⋅2𝑛′− 2{𝜕𝑚 +𝑛, 𝜕𝑚′+𝑛′},[𝑛, 𝑛′])
= ([𝑚, 𝑚′] + 𝑛⋅1𝑚′+𝑚⋅2𝑛′− 2{𝜕𝑚, 𝜕𝑚′} − 2{𝜕𝑚, 𝑛′} − 2{𝑛, 𝜕𝑚′} − 2{𝑛, 𝑛′},[𝑛, 𝑛′])
= ([𝑚, 𝑚′] + 𝑛⋅1𝑚′+𝑚⋅2𝑛′− 2[𝑚, 𝑚′] − 2(𝑚⋅2𝑛′) − 2(𝑛⋅1𝑚′) − 2{𝑛, 𝑛′},[𝑛, 𝑛′])
= (−[𝑚, 𝑚′] − 𝑛⋅1𝑚′−𝑚⋅2𝑛′− 2{𝑛, 𝑛′},[𝑛, 𝑛′]),
whe e we use (BXLeib2), (BXLeib3) and (BXLeib4) in he six h equali y. In he o he
way,
− [𝑥, 𝑦]◦𝜏𝑠(𝑥),𝑠(𝑦)
=
𝑘(((−2{𝑠((𝑚, 𝑛)), 𝑠((𝑚, 𝑛′))},[𝑠((𝑚, 𝑛)), 𝑠((𝑚′, 𝑛′))]),−[(𝑚, 𝑛),(𝑚′, 𝑛′)]))
2.4.3 Equi alence be ween ca ego ies 69
=
𝑘(((−2{𝑛, 𝑛′},[𝑛, 𝑛′]),−[(𝑚, 𝑛),(𝑚′, 𝑛′)]))
=
𝑘(((−2{𝑛, 𝑛′},[𝑛, 𝑛′]),(−[𝑚, 𝑚′] − 𝑛⋅1𝑚′−𝑚⋅2𝑛′,−[𝑛, 𝑛′])))
= (−2{𝑛, 𝑛′}−[𝑚, 𝑚′] − 𝑛⋅1𝑚′−𝑚⋅2𝑛′,[𝑛, 𝑛′]).
We will e i y (LeibT3) below. Le 𝑛, 𝑛′, 𝑛′′ ∈𝑁. Then
𝜏𝑛,[𝑛′,𝑛′′]= (−2{𝑛, [𝑛′, 𝑛′′]},[𝑛, [𝑛′, 𝑛′′]])
= (−2({[𝑛, 𝑛′], 𝑛′′} − {[𝑛, 𝑛′′], 𝑛′}),[[𝑛, 𝑛′], 𝑛′′] − [[𝑛, 𝑛′′], 𝑛′])
= (−2{[𝑛, 𝑛′], 𝑛′′},[[𝑛, 𝑛′], 𝑛′′]) − (−2{[𝑛, 𝑛′′], 𝑛′},[[𝑛, 𝑛′′], 𝑛′])
=𝜏[𝑛,𝑛′],𝑛′′ −𝜏[𝑛,𝑛′′],𝑛′,
whe e we use (BXLeib5) and he Leibniz iden i y in he second equali y.
The same a gumen is alid o (LeibT6), using (BXLeib8) and by he symme y
o he p ope ies.
Finally, we will show ha (LeibT4) and (LeibT5) a e sa is ied.
𝜓𝑛,[𝑛′,𝑛′′]= (−2⟨𝑛, [𝑛′, 𝑛′′]⟩,[𝑛, [𝑛′, 𝑛′′]])
= (−2({[𝑛, 𝑛′], 𝑛′′} − ⟨[𝑛, 𝑛′′], 𝑛′⟩),[[𝑛, 𝑛′], 𝑛′′] − [[𝑛, 𝑛′′], 𝑛′])
= (−2{[𝑛, 𝑛′], 𝑛′′},[[𝑛, 𝑛′], 𝑛′′]) − (−2⟨[𝑛, 𝑛′′], 𝑛′⟩,[[𝑛, 𝑛′′], 𝑛′])
=𝜏[𝑛,𝑛′],𝑛′′ −𝜓[𝑛,𝑛′′],𝑛′
= (−2({[𝑛, 𝑛′], 𝑛′′} − ⟨[𝑛, 𝑛′′], 𝑛′⟩),[[𝑛, 𝑛′], 𝑛′′] − [[𝑛, 𝑛′′], 𝑛′])
= (−2{𝑛, [𝑛′, 𝑛′′]},[𝑛, [𝑛′, 𝑛′′]]) = 𝜏𝑛,[𝑛′,𝑛′′],
whe e we use (BXLeib6) along wi h he Leibniz iden i y in he second equali y; and
(BXLeib7) wi h he Leibniz iden i y in he penul ima e equali y.
Rema k 2.4.54.No e ha i is a b aided c ossed module o Lie 𝐾-algeb as, hen
𝜏𝑛,𝑛′= (−2{𝑛, 𝑛′},[𝑛, 𝑛′]) = −(−2⟨𝑛′, 𝑛⟩,[𝑛′, 𝑛]) = − 𝜓𝑛′,𝑛
and we eco e he cons uc ion o he Lie case (see [24]).
76 2 B aidings o c ossed modules and in e nal objec s
The ollowing p oposi ion is gi en o a gene al case in [10, 11], using ac ions
which a e denomina ed compa ible ac ions o make he enso p oduc .
P oposi ion 2.5.2 ( [10,11]).Le 𝐺be a g oup. Then (𝐺 ⊗ 𝐺 𝜕
←←←←←←→ 𝐺, ⋅)is a c ossed
module o g oups whe e 𝐺 ⊗ 𝐺 is he non-abelian enso p oduc o 𝐺wi h i sel
using he conjuga ion ac ion. The ac ion ⋅∶𝐺× (𝐺 ⊗ 𝐺)←←→ (𝐺 ⊗ 𝐺)and he map
𝜕∶𝐺 ⊗ 𝐺 ←←→ 𝐺a e de ined on gene a o s as 𝑔⋅(𝑔1⊗ 𝑔2) = 𝑔𝑔1𝑔−1 ⊗ 𝑔𝑔2𝑔−1 and
𝜕(𝑔1⊗ 𝑔2)=[𝑔1, 𝑔2].
The nex example shows ha his c ossed module can be associa ed wi h a na u al
b aiding (see [26]).
Example 2.5.3. Le 𝐺be a g oup. The map {−,−}∶ 𝐺×𝐺←←→ 𝐺 ⊗ 𝐺 de ined as
{𝑔1, 𝑔2} = 𝑔1⊗ 𝑔2is a b aiding on (𝐺 ⊗ 𝐺 𝜕
←←←←←←→ 𝐺, ⋅).
Using he p ope ies o he non-abelian enso p oduc o g oups (see [47, P opo-
si ion 1.2.3]) and he de ini ion, he esul ollows easily.
Once gi en he example in g oups, we look o i s analogue in Lie 𝐾-algeb as. Fo
his we need he concep o non-abelian enso p oduc o Lie 𝐾-algeb as, in oduced
by Ellis in [19].
De ini ion 2.5.4. Le 𝑀and 𝑁be wo Lie 𝐾-algeb as such ha 𝑀ac s in 𝑁by ⋅
and 𝑁ac s in 𝑀wi h ∗.
The non-abelian enso p oduc , deno ed by 𝑀 ⊗ 𝑁, is he Lie 𝐾-algeb a gene -
a ed by he symbols 𝑚⊗𝑛, whe e 𝑚∈𝑀,𝑛∈𝑁, wi h he ela ions
𝜆(𝑚 ⊗ 𝑛) = 𝜆𝑚 ⊗ 𝑛 =𝑚 ⊗ 𝜆𝑛, (T1)
(𝑚+𝑚′)⊗ 𝑛 =𝑚⊗𝑛+𝑚′⊗ 𝑛, (T2)
𝑚 ⊗ (𝑛+𝑛′) = 𝑚⊗𝑛+𝑚⊗𝑛′,
[𝑚, 𝑚′]⊗ 𝑛 =𝑚 ⊗ (𝑚′⋅𝑛) − 𝑚′⊗(𝑚⋅𝑛),(T3)
𝑚 ⊗ [𝑛, 𝑛′]=(𝑛′∗𝑚)⊗ 𝑛 − (𝑛∗𝑚)⊗ 𝑛′,
[(𝑚 ⊗ 𝑛),(𝑚′⊗ 𝑛′)] = −(𝑛∗𝑚)⊗(𝑚′⋅𝑛′),(T4)
whe e 𝑚, 𝑚′∈𝑀,𝑛, 𝑛′∈𝑁,𝜆∈𝐾.

2.5 The non-abelian enso p oduc as example o b aiding 77
The nex p oposi ion, ollowing he pa e n o he case o g oups, was p o ed mo e
gene ally in [19], bu we es ic ou sel es o he case ha in e es s us.
P oposi ion 2.5.5 ( [19]).Le 𝑀be a Lie 𝐾-algeb a. Then (𝑀 ⊗ 𝑀 𝜕
←←←←←←→ 𝑀, ⋅)is a
c ossed module o Lie 𝐾-algeb as, whe e 𝑀 ⊗ 𝑀 is he non-abelian enso p oduc
o 𝑀wi h i sel using he adjoin ac ion.
The ac ion ⋅∶𝑀× (𝑀 ⊗ 𝑀)←←→ (𝑀 ⊗ 𝑀)and he map 𝜕∶𝑀 ⊗ 𝑀 ←←→ 𝑀
a e de ined on gene a o s as 𝑚⋅(𝑚1⊗ 𝑚2)=[𝑚, 𝑚1]⊗ 𝑚2+𝑚1⊗[𝑚, 𝑚2]and
𝜕(𝑚1⊗ 𝑚2) = [𝑚1, 𝑚2], whe e [−,−] is he b acke o 𝑀.
Rema k 2.5.6.We will ew i e, o cla i y, he ela ions (T3) and (T4) o he case o
𝑀 ⊗ 𝑀 wi h he adjoin ac ion o 𝑀on i sel .
(T3) [𝑚1, 𝑚2]⊗ 𝑚3=𝑚1⊗[𝑚2, 𝑚3] − 𝑚2⊗[𝑚1, 𝑚3],
𝑚1⊗[𝑚2, 𝑚3] = [𝑚3, 𝑚1]⊗ 𝑚2− [𝑚2, 𝑚1]⊗ 𝑚3,
(T4) [(𝑚1⊗ 𝑚2),(𝑚3⊗ 𝑚4)] = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4],
whe e 𝑚1, 𝑚2, 𝑚3, 𝑚4∈𝑀. Fo he las ela ion we use he an icommu a i i y.
Now, we show an example o he case o Lie 𝐾-algeb as analogous o he case
o g oups.
Example 2.5.7. Le 𝑀be a Lie 𝐾-algeb a. The 𝐾-bilinea map {−,−}∶ 𝑀×𝑀←←→
𝑀 ⊗ 𝑀 de ined by {𝑚1, 𝑚2} = 𝑚1⊗ 𝑚2is a b aiding on he c ossed module o Lie
𝐾-algeb as (𝑀 ⊗ 𝑀 𝜕
←←←←←←→ 𝑀, ⋅).
We will check (BXLie1). I 𝑚, 𝑚′∈𝑀, hen
𝜕{𝑚, 𝑚′} = 𝜕(𝑚⊗𝑚′) = [𝑚, 𝑚′].
To check (BXLie2), we willwo kon gene a o s by he 𝐾-linea i y and 𝐾-bilinea i y,
since he gene al case is only a sum o hem.
I 𝑚1⊗ 𝑚2and 𝑚3⊗ 𝑚4a e gene a o s o 𝑀 ⊗ 𝑀, hen
{𝜕(𝑚1⊗ 𝑚2), 𝜕(𝑚3⊗ 𝑚4)} = {[𝑚1, 𝑚2],[𝑚3, 𝑚4]} = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4]
= [(𝑚1⊗ 𝑚2),(𝑚3⊗ 𝑚4)],
78 2 B aidings o c ossed modules and in e nal objec s
whe e he las equali y is gi en by (T4).
Fo he ollowing p ope ies we need a p e ious esul .
We will use (T3) o p o e 𝑚1⊗[𝑚2, 𝑚3] = −[𝑚2, 𝑚3]⊗ 𝑚1.
[𝑚1, 𝑚2]⊗ 𝑚3=𝑚1⊗[𝑚2, 𝑚3] − 𝑚2⊗[𝑚1, 𝑚3]
=𝑚1⊗[𝑚2, 𝑚3]−[𝑚3, 𝑚2]⊗ 𝑚1+ [𝑚1, 𝑚2]⊗ 𝑚3.
Simpli ying we ha e 0 = 𝑚1⊗[𝑚2, 𝑚3]+[𝑚2, 𝑚3]⊗ 𝑚1.
Now, we will show (BXLie3). Le 𝑚∈𝑀and 𝑚1⊗ 𝑚2∈𝑀 ⊗ 𝑀.
{𝜕(𝑚1⊗ 𝑚2), 𝑚} = {[𝑚1, 𝑚2], 𝑚} = [𝑚1, 𝑚2]⊗ 𝑚
=𝑚1⊗[𝑚2, 𝑚] − 𝑚2⊗[𝑚1, 𝑚] = −𝑚1⊗[𝑚, 𝑚2] + 𝑚2⊗[𝑚, 𝑚1]
= −𝑚1⊗[𝑚, 𝑚2]−[𝑚, 𝑚1]⊗ 𝑚2= −𝑚⋅(𝑚1⊗ 𝑚2),
whe e we use (T3) oge he wi h he p e ious esul .
Now, we will e i y (BXLie4).
{𝑚, 𝜕(𝑚1⊗ 𝑚2)} = 𝑚 ⊗ [𝑚1, 𝑚2] = −[𝑚1, 𝑚2]⊗ 𝑚 = −{𝜕(𝑚1⊗ 𝑚2), 𝑚}
= −(−𝑚⋅(𝑚1⊗ 𝑚2)) = 𝑚⋅(𝑚1⊗ 𝑚2),
whe e we use (BXLie3) and 𝑚 ⊗ [𝑚1, 𝑚2] = −[𝑚1, 𝑚2]⊗ 𝑚.
Now, we will e i y (BXLie5) and (BXLie6). Le 𝑚, 𝑚′, 𝑚′′ ∈𝑀.
{𝑚, [𝑚′, 𝑚′′]} = 𝑚 ⊗ [𝑚′, 𝑚′′] = [𝑚′′, 𝑚]⊗ 𝑚′− [𝑚′, 𝑚]⊗ 𝑚′′
= [𝑚, 𝑚′]⊗ 𝑚′′ − [𝑚, 𝑚′′]⊗ 𝑚′= {[𝑚, 𝑚′], 𝑚′′} − {[𝑚, 𝑚′′], 𝑚′},
{[𝑚, 𝑚′], 𝑚′′} = [𝑚, 𝑚′]⊗ 𝑚′′ =𝑚 ⊗ [𝑚′, 𝑚′′] − 𝑚′⊗[𝑚, 𝑚′′]
= {𝑚, [𝑚′, 𝑚′′]} − {𝑚′,[𝑚, 𝑚′′]}.
We use (T3) in he second equali y o bo h chains o equali ies.
So, we ha e shown ha {𝑚, 𝑚′} = 𝑚⊗𝑚′is a b aiding.
Rema k 2.5.8.No e ha he ac ion gi en in he p e ious example is ac ually gi en by
𝑚⋅(𝑚1⊗ 𝑚2) = 𝑚 ⊗ [𝑚1, 𝑚2].
2.5 The non-abelian enso p oduc as example o b aiding 79
The non-abelian enso p oduc o Leibniz 𝐾-algeb as was in oduced by Gned-
baye in [32], whe e he enso p oduc is deno ed as 𝑀 ⋆ 𝑁, and i s gene a o s as
𝑚∗𝑛and 𝑛∗𝑚. In he gene al case i does no gi e ise o con usion, bu in he case
𝑀=𝑁 hese gene a o s would be deno ed in he same way, gi ing ise o con usion.
To a oid his, we change he nomencla u e, meaning 𝑚∗𝑛as 𝑚 ⊗ 𝑛 and 𝑛∗𝑚as
𝑛⊛𝑚.
De ini ion 2.5.9. Le 𝑀and 𝑁 wo Leibniz 𝐾-algeb as oge he wi h wo Leibniz
ac ions ⋅= (⋅1,⋅2)o 𝑀on 𝑁and ∗= (∗1,∗2)o 𝑁on 𝑀.
The non-abelian enso p oduc o 𝑀and 𝑁, deno ed by 𝑀 ⋆ 𝑁, is he Leibniz
𝐾-algeb a gene a ed by he symbols 𝑚⊗𝑛and 𝑛⊛𝑚wi h 𝑚∈𝑀,𝑛∈𝑁, oge he
wi h he ela ions:
𝜆(𝑚 ⊗ 𝑛) = 𝜆𝑚 ⊗ 𝑛 =𝑚 ⊗ 𝜆𝑛, (RTLeib1)
𝜆(𝑛 ⊛ 𝑚) = 𝜆𝑛 ⊛ 𝑚 =𝑛 ⊛ 𝜆𝑚,
(𝑚+𝑚′)⊗ 𝑛 =𝑚⊗𝑛+𝑚′⊗ 𝑛, (RTLeib2)
𝑚 ⊗ (𝑛+𝑛′) = 𝑚⊗𝑛+𝑚⊗𝑛′,
(𝑛+𝑛′)⊛ 𝑚 =𝑛⊛𝑚+𝑛′⊛ 𝑚,
𝑛 ⊛ (𝑚+𝑚′) = 𝑛⊛𝑚+𝑛⊛𝑚′,
𝑚 ⊗ [𝑛, 𝑛′] = (𝑚∗2𝑛)⊗ 𝑛′− (𝑚∗2𝑛′)⊗ 𝑛, (RTLeib3)
𝑛 ⊛ [𝑚, 𝑚′] = (𝑛⋅2𝑚)⊛ 𝑚′− (𝑛⋅2𝑚′)⊛ 𝑚,
[𝑚, 𝑚′]⊗ 𝑛 = (𝑚⋅1𝑛)⊛ 𝑚′−𝑚 ⊗ (𝑛⋅2𝑚′),
[𝑛, 𝑛′]⊛ 𝑚 = (𝑛∗1𝑚)⊗ 𝑛′−𝑛 ⊛ (𝑚∗2𝑛′),
𝑚 ⊗ (𝑚′⋅1𝑛)=−𝑚 ⊗ (𝑛⋅2𝑚′),(RTLeib4)
𝑛 ⊛ (𝑛′∗1𝑚) = −𝑛 ⊛ (𝑚∗2𝑛′),
(𝑚∗2𝑛)⊗(𝑚′⋅1𝑛′) = [𝑚 ⊗ 𝑛, 𝑚′⊗ 𝑛′] = (𝑚⋅1𝑛)⊛(𝑚′∗2𝑛′),(RTLeib5)
80 2 B aidings o c ossed modules and in e nal objec s
(𝑚∗2𝑛)⊗(𝑛′⋅2𝑚′) = [𝑚 ⊗ 𝑛, 𝑛′⊛ 𝑚′] = (𝑚⋅1𝑛)⊛(𝑛′∗1𝑚′),
(𝑛∗1𝑚)⊗(𝑛′⋅2𝑚′) = [𝑛 ⊛ 𝑚, 𝑛′⊛ 𝑚′] = (𝑛⋅2𝑚)⊛(𝑛′∗1𝑚′),
(𝑛∗1𝑚)⊗(𝑚′⋅1𝑛′) = [𝑛 ⊛ 𝑚, 𝑚′⊗ 𝑛′] = (𝑛⋅2𝑚)⊛(𝑚′∗2𝑛′),
𝑚, 𝑚′∈𝑀, 𝑛, 𝑛′∈𝑁.
P oposi ion 2.5.10 ( [32]).Le 𝑀be a Leibniz 𝐾-algeb a.
Then (𝑀 ⋆ 𝑀 𝜕
←←←←←←→ 𝑀, (⋅1,⋅2)) is a c ossed module o Leibniz 𝐾-algeb as, whe e
𝑀 ⋆ 𝑀 is he non-abelian enso p oduc o 𝑀wi h i sel using he ac ions gi en by
he Leibniz b acke , whe e
• he le ac ion on gene a o s is gi en by 𝑚⋅1(𝑚1⊗ 𝑚2) = [𝑚, 𝑚1]⊗ 𝑚2−
[𝑚, 𝑚2]⊛ 𝑚1,𝑚⋅1(𝑚1⊛ 𝑚2) = [𝑚, 𝑚1]⊛ 𝑚2− [𝑚, 𝑚2]⊗ 𝑚1;
• he igh ac ion on gene a o s is gi en by (𝑚1⊗ 𝑚2)⋅2𝑚= [𝑚1, 𝑚]⊗ 𝑚2+
𝑚1⊗[𝑚2, 𝑚],(𝑚1⊛ 𝑚2)⋅2𝑚= [𝑚1, 𝑚]⊛ 𝑚2+𝑚1⊛[𝑚2, 𝑚];
• he map 𝜕is de ined on gene a o s as 𝜕(𝑚1⊗ 𝑚2) = [𝑚1, 𝑚2] = 𝜕(𝑚1⊛ 𝑚2).
Rema k 2.5.11.We will show how a e he ela ions (RTLeib3)–(RTLeib5) o he
non-abelian enso p oduc 𝑀 ⋆ 𝑀 wi h he ac ion ([−,−],[−,−]) on i sel :
𝑚1⊗[𝑚2, 𝑚3] = [𝑚1, 𝑚2]⊗ 𝑚3− [𝑚1, 𝑚3]⊗ 𝑚2,(RTLeib3)
𝑚1⊛[𝑚2, 𝑚3] = [𝑚1, 𝑚2]⊛ 𝑚3− [𝑚1, 𝑚3]⊛ 𝑚2,
[𝑚1, 𝑚2]⊗ 𝑚3= [𝑚1, 𝑚3]⊛ 𝑚2−𝑚1⊗[𝑚3, 𝑚2],
[𝑚1, 𝑚2]⊛ 𝑚3= [𝑚1, 𝑚3]⊗ 𝑚2−𝑚1⊛[𝑚3, 𝑚2],
𝑚1⊗[𝑚2, 𝑚3] = −𝑚1⊗[𝑚3, 𝑚2],(RTLeib4)
𝑚1⊛[𝑚2, 𝑚3] = −𝑚1⊛[𝑚3, 𝑚2],
[𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊗ 𝑚2, 𝑚3⊗ 𝑚4] = [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4],(RTLeib5)
[𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊗ 𝑚2, 𝑚3⊛ 𝑚4] = [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4],
[𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊛ 𝑚4] = [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4],
[𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊗ 𝑚4] = [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4],
𝑚1, 𝑚2, 𝑚3, 𝑚4∈𝑀.
2.5 The non-abelian enso p oduc as example o b aiding 81
The ollowing example shows he necessi y o a pai o b aidings o he Leibniz
𝐾-algeb as case since hey will be di e en .
Example 2.5.12. Le 𝑀be a Leibniz 𝐾-algeb a.
The pai o 𝐾-bilinea maps {−,−},⟨−,−⟩∶𝑀×𝑀←←→ 𝑀 ⋆ 𝑀 de ined as
{𝑚1, 𝑚2} = 𝑚1⊗ 𝑚2and ⟨𝑚1, 𝑚2⟩=𝑚1⊛ 𝑚2is a b aiding on he c ossed mod-
ule o Leibniz 𝐾-algeb as (𝑀 ⋆ 𝑀 𝜕
←←←←←←→ 𝑀, (⋅1,⋅2)).
Fi s , will check (BXLeib1).
𝜕{𝑚1, 𝑚2} = 𝜕(𝑚1⊗ 𝑚2) = [𝑚1, 𝑚2] = 𝜕(𝑚1⊛ 𝑚2) = 𝜕⟨𝑚1, 𝑚2⟩, 𝑚1, 𝑚2∈𝑀.
Now, we will p o e (BXLeib2).
{𝜕(𝑚1⊗ 𝑚2), 𝜕(𝑚3⊗ 𝑚4)} = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊗ 𝑚2, 𝑚3⊗ 𝑚4],
{𝜕(𝑚1⊗ 𝑚2), 𝜕(𝑚3⊛ 𝑚4)} = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4]=[𝑚1⊗ 𝑚2, 𝑚3⊛ 𝑚4],
{𝜕(𝑚1⊛ 𝑚2), 𝜕(𝑚3⊗ 𝑚4)} = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊗ 𝑚4],
{𝜕(𝑚1⊛ 𝑚2), 𝜕(𝑚3⊛ 𝑚4)} = [𝑚1, 𝑚2]⊗[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊛ 𝑚4],
⟨𝜕(𝑚1⊗ 𝑚2), 𝜕(𝑚3⊗ 𝑚4)⟩= [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4] = [𝑚1⊗ 𝑚2, 𝑚3⊗ 𝑚4],
⟨𝜕(𝑚1⊗ 𝑚2), 𝜕(𝑚3⊛ 𝑚4)⟩= [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4] = [𝑚1⊗ 𝑚2, 𝑚3⊛ 𝑚4],
⟨𝜕(𝑚1⊛ 𝑚2), 𝜕(𝑚3⊗ 𝑚4)⟩= [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊗ 𝑚4],
⟨𝜕(𝑚1⊛ 𝑚2), 𝜕(𝑚3⊛ 𝑚4)⟩= [𝑚1, 𝑚2]⊛[𝑚3, 𝑚4] = [𝑚1⊛ 𝑚2, 𝑚3⊛ 𝑚4],
whe e in all he cases we used (RTLeib5).
Be o e o check he ollowing axioms, we need o check a p ope y ha can be
p o en using (RTLeib3) and (RTLeib4).
Using (RTLeib4) in he las equali y o ela ion (RTLeib3) and ew i ing ha
equali y and he second one, we ge
𝑚1⊛[𝑚2, 𝑚3] = [𝑚1, 𝑚2]⊛ 𝑚3− [𝑚1, 𝑚3]⊛ 𝑚2,
𝑚1⊛[𝑚2, 𝑚3] = [𝑚1, 𝑚2]⊛ 𝑚3− [𝑚1, 𝑚3]⊗ 𝑚2.
Sub ac ing, we ob ain he equali y [𝑚1, 𝑚3]⊗ 𝑚2= [𝑚1, 𝑚3]⊛ 𝑚2. Using his las
equali y and he i s and second equali y o (RTLeib3), we ob ain
𝑚1⊗[𝑚2, 𝑚3] = [𝑚1, 𝑚2]⊗ 𝑚3− [𝑚1, 𝑚3]⊗ 𝑚2

82 2 B aidings o c ossed modules and in e nal objec s
= [𝑚1, 𝑚2]⊛ 𝑚3− [𝑚1, 𝑚3]⊛ 𝑚2=𝑚1⊛[𝑚2, 𝑚3].
Le us e i y now he i s equali y o (BXLeib3) wi h 𝑚, 𝑚1, 𝑚2∈𝑀,
{𝜕(𝑚1⊗ 𝑚2), 𝑚} = [𝑚1, 𝑚2]⊗ 𝑚 = [𝑚1, 𝑚]⊛ 𝑚2−𝑚1⊗[𝑚, 𝑚2]
= [𝑚1, 𝑚]⊛ 𝑚2+𝑚1⊗[𝑚2, 𝑚] = [𝑚1, 𝑚]⊗ 𝑚2+𝑚1⊗[𝑚2, 𝑚]
= (𝑚1⊗ 𝑚2)⋅2𝑚,
whe e we use (RTLeib3) and (RTLeib4).
The second equali y is analogous:
{𝜕(𝑚1⊛ 𝑚2), 𝑚} = [𝑚1, 𝑚2]⊗ 𝑚 = [𝑚1, 𝑚]⊛ 𝑚2+𝑚1⊗[𝑚2, 𝑚]
= [𝑚1, 𝑚]⊛ 𝑚2+𝑚1⊛[𝑚2, 𝑚]=(𝑚1⊛ 𝑚2)⋅2𝑚.
Using he exchange p ope ies be ween ⊗and ⊛again, we will see he emaining
equali ies:
⟨𝜕(𝑚1⊗ 𝑚2), 𝑚⟩= [𝑚1, 𝑚2]⊛ 𝑚 = [𝑚1, 𝑚2]⊗ 𝑚 = (𝑚1⊗ 𝑚2)⋅2𝑚,
⟨𝜕(𝑚1⊛ 𝑚2), 𝑚⟩= [𝑚1, 𝑚2]⊛ 𝑚 = [𝑚1, 𝑚2]⊗ 𝑚 = (𝑚1⊛ 𝑚2)⋅2𝑚.
Now we will check he nex axiom, (BXLeib4), whe e we will use again ha we
can exchange he symbols i in one side is he b acke . S a ing wi h he i s equali y,
we ha e
{𝑚, 𝜕(𝑚1⊗ 𝑚2)} = 𝑚 ⊗ [𝑚1, 𝑚2] = [𝑚, 𝑚1]⊗ 𝑚2− [𝑚, 𝑚2]⊗ 𝑚1
= [𝑚, 𝑚1]⊗ 𝑚2− [𝑚, 𝑚2]⊛ 𝑚1=𝑚⋅1(𝑚1⊗ 𝑚2),
whe e we use (RTLeib3). Analogously we ob ain he second equali y:
{𝑚, 𝜕(𝑚1⊛ 𝑚2)} = 𝑚 ⊗ [𝑚1, 𝑚2] = [𝑚, 𝑚1]⊗ 𝑚2− [𝑚, 𝑚2]⊗ 𝑚1
= [𝑚, 𝑚1]⊛ 𝑚2− [𝑚, 𝑚2]⊗ 𝑚1=𝑚⋅1(𝑚1⊛ 𝑚2).
So, he ollowing p ope ies a e immedia e:
⟨𝑚, 𝜕(𝑚1⊗ 𝑚2)⟩=𝑚 ⊛ [𝑚1, 𝑚2] = 𝑚 ⊗ [𝑚1, 𝑚2] = 𝑚⋅1(𝑚1⊗ 𝑚2),
2.5 The non-abelian enso p oduc as example o b aiding 83
⟨𝑚, 𝜕(𝑚1⊛ 𝑚2)⟩=𝑚 ⊛ [𝑚1, 𝑚2] = 𝑚 ⊗ [𝑚1, 𝑚2] = 𝑚⋅1(𝑚1⊛ 𝑚2).
To inalize, we willp o e(BXLeib5), because, i i is sa is ied, equali ies(BXLeib6)–
(BXLeib8) will be ul illed using he ollowing p ope ies:
{𝑚, [𝑚′, 𝑚′′]} = 𝑚 ⊗ [𝑚′, 𝑚′′] = 𝑚 ⊛ [𝑚′, 𝑚′′] = ⟨𝑚, [𝑚′, 𝑚′′]⟩,
{[𝑚, 𝑚′], 𝑚′′} = [𝑚, 𝑚′]⊗ 𝑚′′ = [𝑚, 𝑚′]⊛ 𝑚′′ =⟨[𝑚, 𝑚′], 𝑚′′⟩.
By using (RTLeib3), we ha e (BXLeib5):
{𝑚, [𝑚′, 𝑚′′]} = 𝑚 ⊗ [𝑚′, 𝑚′′] = [𝑚, 𝑚′]⊗ 𝑚′′ − [𝑚, 𝑚′′]⊗ 𝑚′
= {[𝑚, 𝑚′], 𝑚′′} − {[𝑚, 𝑚′′], 𝑚′},
Rema k 2.5.13.No e ha he ac ions can be w i en wi h a simple no a ion, gi en by
𝑚⋅1(𝑚1⊗ 𝑚2) = 𝑚⋅1(𝑚1⊛ 𝑚2) = 𝑚 ⊗ [𝑚1, 𝑚2] = 𝑚 ⊛ [𝑚1, 𝑚2],
(𝑚1⊗ 𝑚2)⋅2𝑚= (𝑚1⊛ 𝑚2)⋅2𝑚= [𝑚1, 𝑚2]⊗ 𝑚 = [𝑚1, 𝑚2]⊛ 𝑚.
Rema k 2.5.14.Example 2.5.12 gene alizes he Lie example, since i we ha e ha
𝑚1⊛𝑚2= −𝑚2⊗ 𝑚1as a new ela ion, we ob ain he Lie non-abelian enso p oduc
o 𝑀wi h i sel using he adjoin ac ion.
84 2 B aidings o c ossed modules and in e nal objec s
CHAPTER 3
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄
Uni e sal cen al ex ension o b aided
c ossed modules o Lie algeb as
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄
In his chap e , we will s udy wo no ions o cen e and commu a o o he b aided
c ossed modules and Lie algeb as. Wi h ha , we will de ine i s espec i e cen al
ex ensions, and we will show he ela ionship be ween hem.
3.1 Cen e and commu a o objec s
The ca ego y o Lie c ossed modules X(LieAlg𝐾)is a semi-abelian ca ego y in he
sense o [37].
Theno iono hecen eo anobjec wasde inedin [35], inaca ego ywi h speci ic
p ope ies. This cons uc ion only needs ha he ca ego y has ini e p oduc s and ze o
objec .
The ca ego y X(LieAlg𝐾)has cen es in he sense o Huq [35], and hey we e
cons uc ed in [13].
De ini ion 3.1.1. The cen e o a Lie c ossed module = (𝑀𝜕
←←←←←←→ 𝑁, ⋅)is he c ossed
submodule 𝑍()=(𝑀𝑁𝜕|𝑀𝑁
←←←←←←←←←←←←←←←←←←←←←→ s 𝑁(𝑀) ∩ 𝑍(𝑁),⋅𝑍), whe e:
•𝑀𝑁= {𝑚∈𝑀∣𝑛⋅𝑚= 0, 𝑛 ∈𝑁},
•𝑍(𝑁) = {𝑛∈𝑁∣ [𝑛, 𝑛′] = 0, 𝑛′∈𝑁}is he cen e o he Lie 𝐾-algeb a 𝑁,
•s 𝑁(𝑀) = {𝑛∈𝑁∣𝑛⋅𝑚= 0, 𝑚 ∈𝑀},
85
92 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as
whe e we ha e used (BXLie5).
Fo Φ2is ue using a simila a gumen oge he wi h he Jacobi iden i y in bo h
equali ies.
The p oo o (T4) o Φ2 ollows since bo h equali ies a e [[𝑛1, 𝑛2],[𝑛3, 𝑛4]] a e
applying Φ2.
Fo Φ1we ha e he ollowing equali ies:
Φ1([𝑛1⊗ 𝑛2, 𝑛3⊗ 𝑛4]) = [Φ1(𝑛1⊗ 𝑛2),Φ1(𝑛3⊗ 𝑛4)] = [{𝑛1, 𝑛2},{𝑛3, 𝑛4}]
= {𝜕{𝑛1, 𝑛2}, 𝜕{𝑛3, 𝑛4}} = {[𝑛1, 𝑛2],[𝑛3, 𝑛4]}
= Φ1([𝑛1, 𝑛2]⊗[𝑛3, 𝑛4]),
whe e we ha e used (BXLie2) and (BXLie1).
So, Φ1and Φ2a e well de ined and a e Lie 𝐾-homomo phisms.
Fo he second pa , we ha e ha Im Φ1=𝐵𝑁(𝑀)and Im Φ2= [𝑁, 𝑁]. The e-
o e, Φ1and Φ2a e simul aneously su jec i e i and only i he b aided Lie c ossed
module is B-pe ec .
Lemma 3.2.2. Le (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) be a b aided Lie c ossed module, and con-
side he b aided Lie c ossed module (𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, [−,−],[−,−]) (see
Example 2.3.8 (1)).
Then (Φ1,Φ2)∶ (𝑁 ⊗𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗𝑁, [−,−],[−,−]) ⟶(𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−})
is a mo phism in BXLie, wi h Φ1and Φ2de ined in Lemma 3.2.1,.
Besides, ke (Φ1)⊂(𝑁 ⊗ 𝑁)(𝑁⊗𝑁)and ke (Φ2)⊂Z𝐵(𝑁 ⊗ 𝑁).
P oo . Fo he p oo , we will deno e he ac ion [−,−] o 𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁 as
∗, and i s b aiding as ⟦−,−⟧.
Fi s , we will show (XLieH1). Le 𝑛⊗𝑛′, 𝑛′′ ⊗ 𝑛′′′ ∈𝑁 ⊗ 𝑁.
Φ1((𝑛 ⊗ 𝑛′) ∗ (𝑛′′ ⊗ 𝑛′′′)) = Φ1([𝑛 ⊗ 𝑛′, 𝑛′′ ⊗ 𝑛′′′]) = Φ1([𝑛, 𝑛′]⊗[𝑛′′, 𝑛′′′])
= {[𝑛, 𝑛′],[𝑛′′, 𝑛′′′]} = {[𝑛, 𝑛′], 𝜕{𝑛′′, 𝑛′′′}}
= [𝑛, 𝑛′]⋅{𝑛′′, 𝑛′′′} = Φ2(𝑛⊗𝑛′)⋅Φ1(𝑛′′ ⊗ 𝑛′′′),

3.2 The uni e sal B-cen al ex ension 93
whe e we ha e used (BXLie1) and (BXLie4).
Now, we will show (XLieH1).
𝜕◦Φ1(𝑛⊗𝑛′) = 𝜕{𝑛, 𝑛′} = [𝑛, 𝑛′] = Φ2(Id𝑁⊗𝑁 (𝑛⊗𝑛′)),
whe e we ha e used (BXLie2).
Now, we will p o e (BXLieH3).
Φ1(⟦𝑛⊗𝑛′, 𝑛′′ ⊗ 𝑛′′′⟧) = Φ1([𝑛 ⊗ 𝑛′, 𝑛′′ ⊗ 𝑛′′′]) = Φ1([𝑛, 𝑛′]⊗[𝑛′′, 𝑛′′′])
= {[𝑛, 𝑛′],[𝑛′′, 𝑛′′′]} = {Φ2(𝑛⊗𝑛′),Φ2(𝑛′′ ⊗ 𝑛′′′)}.
So, (Φ1,Φ2)is a mo phism in BXLie. We will now p o e ha he inclusions hold.
I 𝑛⊗𝑛′∈ ke (Φ1) hen {𝑛, 𝑛′} = 0. Using (BXLie1) we ha e ha 0 = 𝜕{𝑛, 𝑛′} =
[𝑛, 𝑛′].
Since (𝑁 ⊗𝑁)(𝑁⊗𝑁)= {𝑥∈𝑁 ⊗𝑁 ∣ (𝑛′′ ⊗𝑛′′′) ∗ 𝑥= 0, 𝑛′′ ⊗𝑛′′′ ∈𝑁 ⊗𝑁}
(i is enough o wo k on gene a o s), we ha e
(𝑛′′ ⊗ 𝑛′′′) ∗ (𝑛⊗𝑛′) = [𝑛′′ ⊗ 𝑛′′′, 𝑛 ⊗ 𝑛′] = [𝑛′′, 𝑛′′′]⊗[𝑛, 𝑛′] = [𝑛′′, 𝑛′′′]⊗0 = 0.
The e o e, we ha e ha 𝑛⊗𝑛′∈ (𝑁 ⊗ 𝑁)(𝑁⊗𝑁)and ke (Φ1)⊂(𝑁 ⊗ 𝑁)(𝑁⊗𝑁).
Fo he second inclusion, we ake 𝑛⊗𝑛′∈ ke (Φ2), i.e. [𝑛, 𝑛′] = 0.
Since i is enough o wo k on gene a o s, we ha e ha
Z𝐵(𝑁⊗𝑁) = {𝑥∈𝑁⊗𝑁 ∣⟦𝑥, 𝑛′′⊗𝑛′′′⟧= 0 = ⟦𝑛′′⊗𝑛′′′, 𝑥⟧, 𝑛′′⊗𝑛′′′ ∈𝑁⊗𝑁}.
Taking in o accoun ha
⟦𝑛′′ ⊗ 𝑛′′′, 𝑛 ⊗ 𝑛′⟧= [𝑛′′ ⊗ 𝑛′′′, 𝑛 ⊗ 𝑛′] = [𝑛′′, 𝑛′′′]⊗[𝑛, 𝑛′] = [𝑛′′, 𝑛′′′]⊗0 = 0,
⟦𝑛⊗𝑛′, 𝑛′′ ⊗ 𝑛′′′⟧= [𝑛 ⊗ 𝑛′, 𝑛′′ ⊗ 𝑛′′′] = [𝑛, 𝑛′]⊗[𝑛′′, 𝑛′′′] = 0 ⊗[𝑛′′, 𝑛′′′] = 0,
we deduce 𝑛⊗𝑛′∈𝑍𝐵(𝑁 ⊗ 𝑁), which p o es ha ke (Φ2)⊂ 𝑍𝐵(𝑁 ⊗ 𝑁).
Co olla y 3.2.3. The mo phism gi en in Lemma 3.2.2 is a B-cen al ex ension i and
only i (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is a B-pe ec b aided Lie c ossed module.
94 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as
P oo . I will be a B-cen al ex ension i and only i (Φ1,Φ2)is an ex ension, since
Lemma 3.2.2 es ablishes he wo inclusions and hey ha e he es ic ed ope a ions as
a b aided Lie c ossed module.
Mo eo e , (Φ1,Φ2)is an ex ension i and only i Φ1and Φ2a e simul aneously
su jec i e, and by Lemma 3.2.1 ha i happens i and only i he b aided Lie c ossed
module (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is B-pe ec .
P oposi ion 3.2.4. I (𝑋1
𝛿
←←←←←←→ 𝑋2,∗,⦅−,−⦆)𝑓=(𝑓1,𝑓2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→→ (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is a B-
cen al ex ension, hen we ha e a mo phism in BX(LieAlg𝐾),
ℎ∶ (𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, [−,−],[−,−]) ←←←←←←←→ (𝑋1
𝛿
←←←←←←→ 𝑋2,∗,⦅−,−⦆),
de ined by:
•ℎ1∶𝑁 ⊗ 𝑁 ←←→ 𝑋1,𝑛 ⊗ 𝜂 ↦⦅𝑛, 𝜂⦆, whe e 𝑛, 𝜂 ∈𝑋2a e elemen s such ha
𝑓2(𝑛) = 𝑛and 𝑓2(𝜂) = 𝜂;
•ℎ2∶𝑁 ⊗ 𝑁 ←←→ 𝑋2,𝑛 ⊗ 𝜂 ↦[𝑛, 𝜂], whe e 𝑛, 𝜂 ∈𝑋2a e elemen s such ha
𝑓2(𝑛) = 𝑛and 𝑓2(𝜂) = 𝜂.
Besides, 𝑓◦ℎ= Φ, i.e. ℎis a mo phism be ween he ex ensions.
P oo . We need o p o e ha ℎ1and ℎ2a e well de ined.
We will s a wi h ℎ1. We will ake 𝑛, 𝑛, 𝜂, 𝜂 ∈𝑋2such ha 𝑓2(𝑛) = 𝑓2(𝑛) = 𝑛
and 𝑓2(𝜂) = 𝑓2(𝜂) = 𝜂and p o e ha ⦅𝑛, 𝜂⦆=⦅𝑛, 𝜂⦆.
Since 𝑓2(𝑛) = 𝑓2(𝑛)and 𝑓= (𝑓1, 𝑓2)is a B-cen al ex ension, we ha e ha
𝑛−𝑛 ∈ ke (𝑓2)⊂Z𝐵(𝑋2). By he de ini ion o Z𝐵(𝑋2)we ge ha ⦅𝑛−𝑛, 𝜂⦆= 0
and so ⦅𝑛, 𝜂⦆=⦅𝑛, 𝜂⦆.
Using an analogue easoning, we ha e ha 𝜂−𝜂 ∈ ke (𝑓2)⊂Z𝐵(𝑋2), and,
⦅𝑛, 𝜂 −𝜂⦆= 0. So ⦅𝑛, 𝜂⦆=⦅𝑛, 𝜂⦆.
Wi h bo h equali ies, we ha e ha ⦅𝑛, 𝜂⦆=⦅𝑛, 𝜂⦆=⦅𝑛, 𝜂⦆, and ℎ1is independen
o he choice.
Since Z𝐵(𝑋2)⊂Z(𝑋2)we can change he p oo o ℎ1 aking he equali ies o
[−,−] ins ead o ⦅−,−⦆which p o es ha ℎ2is independen o he choice.
3.2 The uni e sal B-cen al ex ension 95
We can use an analogue a gumen as in Lemma 3.2.1 o p o e ha ℎ1and ℎ2a e
well de ined, i.e. hey p ese e he ela ions. So, hey a e Lie 𝐾-homomo phisms
since hey a e de e mined on gene a o s.
To p o e ha ℎ= (ℎ1, ℎ2)is a mo phism o b aided Lie c ossed modules, we also
use simila easoning as he one done in Lemma 3.2.2, since we can make he changes
in he choice inside he b aidings and b acke s.
To inish, i 𝑛⊗𝜂∈𝑁 ⊗ 𝑁, hen
𝑓1◦ℎ1(𝑛⊗𝜂) = 𝑓1(⦅𝑛, 𝜂⦆) = {𝑓2(𝑛), 𝑓2(𝜂)} = {𝑛, 𝜂}=Φ1(𝑛⊗𝜂),
𝑓2◦ℎ2(𝑛⊗𝜂) = 𝑓2([𝑛, 𝜂]) = [𝑓2(𝑛), 𝑓2(𝜂)] = [𝑛, 𝜂]=Φ2(𝑛⊗𝜂).
The e o e, 𝑓◦ℎ= Φ.
Lemma 3.2.5. I 𝑁is a pe ec Lie 𝐾-algeb a, i.e. 𝑁= [𝑁, 𝑁], hen
(𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, [−,−],[−,−])
is a B-pe ec b aided Lie c ossed module.
In pa icula , i (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is a B-pe ec b aided Lie c ossed module,
hen (𝑁⊗𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁⊗𝑁, [−,−],[−,−]) is a B-pe ec b aided Lie c ossed module.
P oo . Since he b aiding in (𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, [−,−],[−,−]) is he b acke ,
we ha e ha [𝑁 ⊗ 𝑁, 𝑁 ⊗ 𝑁] = 𝐵𝑁⊗𝑁 (𝑁 ⊗ 𝑁), and so i is enough o p o e ha
[𝑁 ⊗ 𝑁, 𝑁 ⊗ 𝑁] = 𝑁 ⊗ 𝑁.
Mo eo e , i is enough o p o e ha he gene a o s [𝑛1, 𝑛2]⊗[𝑛3, 𝑛4]a e inside
[𝑁 ⊗ 𝑁, 𝑁 ⊗ 𝑁]since 𝑁= [𝑁, 𝑁]. Using (T4) we ha e ha [𝑛1, 𝑛2]⊗[𝑛3, 𝑛4] =
[𝑛1⊗ 𝑛2, 𝑛3⊗ 𝑛4] ∈ [𝑁 ⊗ 𝑁, 𝑁 ⊗ 𝑁].
Fo he second pa , i (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is B-pe ec , hen 𝑁= [𝑁, 𝑁], and
we conclude using he i s pa .
P oposi ion 3.2.6. Le (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)Ψ
←←←←←←←←→ (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) be a mo phism
o b aided Lie c ossed modules such ha (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)is B-pe ec .
96 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as
I (𝑋1
𝜌
←←←←←←→ 𝑋2,∗,⦅−,−⦆)𝑓
←←←←←←←→→ (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is a B-cen al ex ension and
exis s (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)ℎ
←←←←←←→ (𝑋1
𝜌
←←←←←←→ 𝑋2,∗,⦅−,−⦆)such ha Ψ = 𝑓◦ℎ, hen h is
he unique ha sa is ies he equali y.
P oo . Suppose ha he e a e 𝑔, ℎ∶ (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)←←←←←←←→ (𝑋1
𝜌
←←←←←←→ 𝑋2,∗,⦅−,−⦆)
such ha Ψ = 𝑓◦ℎ=𝑓◦𝑔, i.e. Ψ1=𝑓1◦ℎ1=𝑓1◦𝑔1and Ψ2=𝑓2◦ℎ2=𝑓2◦𝑔2.
I 𝑦∈𝑌2 hen 𝑓2◦ℎ2(𝑦) = 𝑓2◦𝑔2(𝑦), i.e. ℎ2(𝑦) − 𝑔2(𝑦) ∈ ke (𝑓2). Then he e is
𝑘𝑦∈ ke (𝑓2)such ha ℎ2(𝑦) = 𝑔2(𝑦) + 𝑘𝑦. Since 𝑓is a B-cen al ex ension we ha e
ha ke (𝑓2)⊂Z𝐵(𝑋2)⊂Z(𝑋2). I we ake 𝑦, 𝑧 ∈𝑌2, and since 𝑘𝑦, 𝑘𝑧∈ Z(𝑋2), we
ha e
[𝑘𝑦, 𝑔2(𝑧)] = [𝑘𝑦, 𝑘𝑧] = [𝑔2(𝑦), 𝑘𝑧] = 0.
Using his ac , we ha e:
ℎ2([𝑦, 𝑧]) = [ℎ2(𝑦), ℎ2(𝑧)] = [𝑔2(𝑦) + 𝑘𝑦, 𝑔2(𝑧) + 𝑘𝑧]
= [𝑔2(𝑦), 𝑔2(𝑧)] + [𝑘𝑦, 𝑔2(𝑧)] + [𝑘𝑦, 𝑘𝑧]+[𝑔2(𝑦), 𝑘𝑧]
= [𝑔2(𝑦), 𝑔2(𝑧)] = 𝑔2([𝑦, 𝑧]).
So, 𝑔2=ℎ2since (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)is B-pe ec .
Besides, since ke (𝑓2)⊂Z𝐵(𝑋2), o 𝑦, 𝑧 ∈𝑌2, we ha e ha :
ℎ1(⟦𝑦, 𝑧⟧) = ⦅ℎ2(𝑦), ℎ2(𝑧)⦆=⦅𝑔2(𝑦) + 𝑘𝑦, 𝑔2(𝑧) + 𝑘𝑧⦆
=⦅𝑔2(𝑦), 𝑔2(𝑧)⦆+⦅𝑘𝑦, 𝑔2(𝑧)⦆+⦅𝑘𝑦, 𝑘𝑧⦆+⦅𝑔2(𝑦), 𝑘𝑧⦆
=⦅𝑔2(𝑦), 𝑔2(𝑧)⦆=𝑔1(⟦𝑦, 𝑧⟧),
whe e we ha e used ha 𝑘𝑦, 𝑘𝑧∈ Z𝐵(𝑋2).
The e o e, 𝑔1=ℎ1because (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)is B-pe ec , i.e. 𝑌1=𝐵𝑌2(𝑌1)
is gene a ed by he images o he b aiding.
Co olla y 3.2.7. I = (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) is a B-pe ec Lie b aided c ossed
module, hen
= (𝑁 ⊗ 𝑁 Id𝑁⊗𝑁
←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, [−,−],[−,−]) Φ=(Φ1,Φ2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→→ = (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−})
(UBCE)
3.2 The uni e sal B-cen al ex ension 97
is he uni e sal B-cen al ex ension o , whe e Φ1,Φ2we e de ined in Lemma 3.2.1.
P oo . Since is B-pe ec , Co olla y 3.2.3 s a es ha he mo phism Φ
←←←←←←←←→→ is
aB-cen al ex ension.
We need o p o e ha i is uni e sal.
I we ha e ano he B-cen al ex ension 𝑓
←←←←←←←→→  hen by P oposi ion 3.2.4 he e
is ℎsuch ha Φ = 𝑓◦ℎ.
The uniqueness o his mo phism is gi en by P oposi ion 3.2.6. We can use he
p e ious p oposi ion since is B-pe ec by Lemma 3.2.5 and he ac ha is
B-pe ec .
Le us see he con e se o Co olla y 3.2.7.
P oposi ion 3.2.8. Le (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)Ψ
←←←←←←←←←←←←←→→ (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) be an ex en-
sion in BXLie such ha (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)is B-pe ec . Then (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−})
is B-pe ec .
P oo . Ψ1and Ψ2a e su jec i e maps since Ψis an ex ension, and 𝑌1=𝐵𝑌2(𝑌1)and
𝑌2= [𝑌2, 𝑌2]because (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)is B-pe ec .
Since he elemen s ⟦𝑦, 𝑧⟧, wi h 𝑦, 𝑧 ∈𝑌2a e he gene a o s o 𝑌1, we ha e ha
Ψ1(⟦𝑦, 𝑧⟧)a e he gene a o s o Im Ψ1=𝑀. Since Ψ1(⟦𝑦, 𝑧⟧) = {Φ2(𝑦),Φ2(𝑧)}, we
ge ha he gene a o s o 𝑀a e b aided elemen s and 𝑀=𝐵𝑁(𝑀).
We know ha he elemen s [𝑦, 𝑧], wi h 𝑦, 𝑧 ∈𝑌2, a e he gene a o s o 𝑌2. The e-
o e, Φ2([𝑦, 𝑧]) = [Φ2(𝑦),Φ2(𝑧)] a e he gene a o s o Im Φ2=𝑁, and hen 𝑁=
[𝑁, 𝑁].
So, (𝑀𝜕
←←←←←←→ 𝑁⋅,{−,−}) is B-pe ec .
Lemma 3.2.9. Le = (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)Ψ
←←←←←←←←←←←←←→→ = (𝑀𝜕
←←←←←←→ 𝑁, ⋅,{−,−}) be a B-
cen al ex ension in BXLie such ha is no B-pe ec . Then exis s ano he ex ension
𝑓
←←←←←←←←←←←←→→ and wo di e en mo phisms ℎ, 𝑔 ∶←←→ such ha Ψ = 𝑓◦ℎ=𝑓◦𝑔.
P oo . Le (𝐵𝑌2(𝑌1)
𝜚|𝐵𝑌2(𝑌1)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ [𝑌2, 𝑌2], ⋆𝐶,⟦−,−⟧𝐶)𝑖=(𝑖1,𝑖2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ (𝑌1
𝜚
←←←←←←→ 𝑌2, ⋆, ⟦−,−⟧)be
he inclusion mo phism o he B-commu a o b aided c ossed submodule.

98 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as
Taking he coke nel o 𝑖we ha e he Lie c ossed module
= ( 𝑌1
𝐵𝑌2(𝑌1)
𝜚
←←←←←←→ 𝑌2
[𝑌2, 𝑌2], ⋆, ⟦−,−⟧).
We will deno e in he same way, by abuse o no a ion, he b aidings in and in i s
quo ien . We will ep esen he elemen s in 𝑌1
𝐵𝑌2(𝑌1)as 𝑥,𝑥∈𝑌1, and he ones in
𝑌2
[𝑌2,𝑌2]as 𝑦,𝑦∈𝑌2.
We ake now he p oduc in he ca ego y BX(LieAlg𝐾)and we cons uc ×.
We deno e as 𝜋1 he i s p ojec ion mo phism. Since 𝜋1
1and 𝜋1
2a e su jec i e maps,
we ha e ha ×𝜋1
←←←←←←←←←←→→ is an ex ension. We will deno e he b aiding in he
p oduc as ⦃−,−⦄.
We will p o e ha i is a B-cen al ex ension, i.e. we need o p o e he inclusions
ke (𝜋1
1)⊂(𝑀×𝑌1
𝐵𝑌2(𝑌2))(𝑁×𝑌2
[𝑌2,𝑌2])and ke (𝜋1
2)⊂Z𝐵(𝑁×𝑌2
[𝑌2,𝑌2]).
I 𝑎∈ ke (𝜋1
1) hen 𝑎= (0, 𝑥)wi h 𝑥∈𝑌1.
I we ake (𝑛, 𝑦) ∈ 𝑁×𝑌2
[𝑌2,𝑌2] hen:
(𝑛, 𝑦)(⋅×⋆)(0, 𝑥) = (𝑛⋅0, 𝑦 ⋆ 𝑥) = (0, 𝑦 ⋆ 𝑥).
Bu 𝑦⋆𝑥= 0 since 𝑦⋆𝑥∈𝐷𝑌2(𝑌1)⊂ 𝐵𝑌2(𝑌1).
So ke (𝜋1
1)⊂(𝑀×𝑌1
𝐵𝑌2(𝑌2))(𝑁×𝑌2
[𝑌2,𝑌2]).
I 𝑎∈ ke (𝜋1
2) hen 𝑎= (0, 𝑦)wi h 𝑦∈𝑌2. I we ake (𝑛, 𝑦1) ∈ 𝑁×𝑌2
[𝑌2,𝑌2] hen:
⦃(0, 𝑦),(𝑛, 𝑦1)⦄= ({0, 𝑛},⟦𝑦, 𝑦1⟧) = (0,⟦𝑦, 𝑦1⟧),
⦃(𝑛, 𝑦1),(0, 𝑦)⦄= ({𝑛, 0},⟦𝑦1, 𝑦⟧) = (0,⟦𝑦1, 𝑦⟧).
Mo eo e , ⟦𝑦, 𝑦1⟧=⟦𝑦1, 𝑦⟧= 0 since ⟦𝑦1, 𝑦⟧,⟦𝑦, 𝑦1⟧∈𝐵𝑌2(𝑌1).
The e o e ke (𝜋1
2)⊂Z𝐵(𝑁×𝑌2
[𝑌2,𝑌2]), and so 𝜋1is a B-cen al ex ension.
I 𝑖𝑐∶←←←←←←←→→ is he coke nel o 𝑖, hen we ha e wo mo phisms, induced by
he p oduc , wi h domain and ×as codomain. They a e ℎ= (Ψ,0) and
3.3 B aiding on a uni e sal ex ension o Lie c ossed modules 99
𝑔= (Ψ, 𝑖𝑐). Since hey a e induced by he uni e sal p ope y o he p oduc , we ha e
ha Ψ = 𝜋1◦ℎ=𝜋1◦𝑔.
To inish he p oo , we only mus p o e ha hey a e di e en . Since he b aided
Lie c ossed module is no B-pe ec and 𝑖𝑐
1and 𝑖𝑐
2a e su jec i e we know ha 𝑖𝑐
1≠0
o 𝑖𝑐
2≠0(i bo h we e he ze o mo phisms, hen would be B-pe ec ), and so
ℎ≠𝑔.
Co olla y 3.2.10. I is a b aided Lie c ossed module, hen i s uni e sal B-cen al
ex ension, i i exis s, is B-pe ec .
P oo . I he uni e sal ex ension is no B-pe ec , hen using Lemma 3.2.9 we ha e
ano he B-cen al ex ension ←←←←←←←→→  o which he e exis wo di e en mo phisms
om he uni e sal B-cen al ex ension o ←←←←←←←→→ , which con adic s he uni e sali y.
Theo em 3.2.11. A b aided Lie c ossed module admi s a uni e sal B-cen al ex en-
sion i and only i i is B-pe ec .
P oo . I is a consequence o Co olla y 3.2.7, Co olla y 3.2.10 and P oposi ion 3.2.8.
3.3 B aiding on a uni e sal ex ension o Lie c ossed mod-
ules
Uni e sal cen al ex ensions o b aided c ossed modules o g oups a e no s udied
in [26]. Howe e , he au ho cons uc ed a canonical b aiding on he uni e sal cen-
al ex ension o a c ossed module o g oups [48], when he gi en c ossed module is
b aided as well, and showed ha i was uni e sal in a sense ha we will explain in
his sec ion.
In his pa o he pape , we will conside b aided Lie c ossed modules ex ensions,
bu unlike he p e ious sec ion, we will cons uc a b aiding on he uni e sal cen al
ex ension o a b aided Lie c ossed module hough as Lie c ossed module and wi h he
cen e in X(LieAlg𝐾), which we ha e called 𝔘-cen al ex ension. In his sense, we
100 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as
will ob ain simila esul s gi en by Fukushi in [26] o c ossed modules o g oups in
he ca ego y BX(LieAlg𝐾).
Casas and Lad a in [13] p o ed ha he uni e sal cen al ex ension o a pe ec
Lie c ossed module (𝑀←←→ 𝑁, ⋅)in X(LieAlg𝐾)is gi en by:
(𝑁 ⊗ 𝑀 Id𝑁⊗𝜕
←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊗ 𝑁, ∗) 𝑐=(𝑐1,𝑐2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→→ (𝑀𝜕
←←←←←←→ 𝑁, ⋅),(UCE)
whe e 𝑁 ⊗𝑀 is gi en by he ac ions ⋅o 𝑁on 𝑀and 𝑚⋆𝑛 = [𝜕(𝑚), 𝑛]o 𝑀on 𝑁;
he ac ion o 𝑁 ⊗ 𝑁 on 𝑁 ⊗ 𝑀 is gi en by (𝑛 ⊗ 𝑛′) ∗ (𝑛′′ ⊗ 𝑚) = [[𝑛, 𝑛′], 𝑛′′]⊗ 𝑚 +
𝑛′′ ⊗[𝑛, 𝑛′]⋅𝑚 o 𝑛, 𝑛′, 𝑛′′ ∈𝑁, 𝑚 ∈𝑀; and he mo phisms a e 𝑐1(𝑛 ⊗ 𝑚) = 𝑛⋅𝑚
and 𝑐2(𝑛⊗𝑛′) = [𝑛, 𝑛′].
P oposi ion 3.3.1. I (𝑀𝜕
←←←←←←←←→ 𝑁, ⋅,{−,−}) is a b aided Lie c ossed module hen
⦃−,−⦄∶ (𝑁⊗𝑁)×(𝑁⊗𝑁)←←→ 𝑁⊗𝑀, de ined on gene a o s by ⦃𝑛⊗𝑛′, 𝑛′′⊗𝑛′′′⦄=
[𝑛, 𝑛′]⊗{𝑛′′, 𝑛′′′}, is a b aiding o he Lie c ossed module (𝑁⊗𝑀 Id𝑁⊗𝜕
←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁⊗𝑁, ∗).
P oo . The b aiding ⦃−,−⦄is well de ined since i p ese es he ela ions (T1) and
(T2) using he 𝐾-bilinea i y o [−,−] and {−,−}, and (T3) and (T4) a e ul illed oo
since {−,−} and [−,−] sa is y i .
I is su icien o p o e he axioms o b aidings. Le 𝑛, 𝑛, 𝑛′, 𝑛′′ ∈𝑁,𝑚, 𝑚′∈𝑀.
Then
(Id𝑁⊗𝜕)(⦃𝑛⊗𝑛′, 𝑛′′ ⊗ 𝑛′′′⦄) = (Id𝑁⊗𝜕)([𝑛, 𝑛′]⊗{𝑛′′, 𝑛′′′}) = [𝑛, 𝑛′]⊗ 𝜕{𝑛′′, 𝑛′′′}
= [𝑛, 𝑛′]⊗[𝑛′′, 𝑛′′′] = [𝑛⊗𝑛′, 𝑛′′ ⊗ 𝑛′′′](BXLie1),
⦃(Id𝑁⊗𝜕)(𝑛 ⊗ 𝑚),(Id𝑁⊗𝜕)(𝑛′⊗ 𝑚′)⦄
=⦃𝑛⊗𝜕(𝑚), 𝑛′⊗ 𝜕(𝑚′)⦄= [𝑛, 𝜕(𝑚)] ⊗{𝑛′, 𝜕(𝑚′)}
= −(𝑚⋆𝑛)⊗(𝑛′⋅𝑚′) = [𝑛 ⊗ 𝑚, 𝑛′⊗ 𝑚′](BXLie2),
⦃(Id𝑁⊗𝜕)(𝑛 ⊗ 𝑚), 𝑛′⊗ 𝑛′′⦄=⦃𝑛⊗𝜕(𝑚), 𝑛′⊗ 𝑛′′⦄= [𝑛, 𝜕(𝑚)] ⊗{𝑛′, 𝑛′′}
= −(𝑚⋆𝑛)⊗{𝑛′, 𝑛′′} = 𝑛 ⊗ [𝑚, {𝑛′, 𝑛′′}] − {𝑛′, 𝑛′′}⋆ 𝑛 ⊗ 𝑚
=𝑛 ⊗ {𝜕(𝑚), 𝜕({𝑛′, 𝑛′′})} − [𝜕({𝑛′, 𝑛′′}), 𝑛]⊗ 𝑚
3.3 B aiding on a uni e sal ex ension o Lie c ossed modules 101
= −𝑛 ⊗ [𝑛′, 𝑛′′]⋅𝑚− [[𝑛′, 𝑛′′], 𝑛]⊗ 𝑚 = −(𝑛′⊗ 𝑛′′) ∗ (𝑛⊗𝑚)(BXLie3),
whe e we ha e used he second ela ion o (T3) in he hi d equali y.
⦃𝑛′⊗ 𝑛′′,(Id𝑁⊗𝜕)(𝑛 ⊗ 𝑚)⦄=⦃𝑛′⊗ 𝑛′′, 𝑛 ⊗ 𝜕(𝑚)⦄= [𝑛′, 𝑛′′]⊗{𝑛, 𝜕(𝑚)}
= [𝑛′, 𝑛′′]⊗(𝑛⋅𝑚) = 𝑛 ⊗ [𝑛, 𝑛′]⋅𝑚+ [[𝑛′, 𝑛′′], 𝑛]⊗ 𝑚
= (𝑛′⊗ 𝑛′′) ∗ (𝑛⊗𝑚)(BXLie4),
whe e we ha e used he i s ela ion o (T3) in he hi d equali y.
⦃𝑛1⊗ 𝑛′
1,[𝑛2⊗ 𝑛′
2, 𝑛3⊗ 𝑛′
3]⦄=⦃𝑛1⊗ 𝑛′
1,[𝑛2⊗ 𝑛′
2]⊗[𝑛3⊗ 𝑛′
3]⦄
= [𝑛1, 𝑛′
1]⊗{[𝑛2, 𝑛′
2],[𝑛3, 𝑛′
3]} = [𝑛1, 𝑛′
1]⊗[{𝑛2, 𝑛′
2},{𝑛3, 𝑛′
3}]
= ({𝑛3, 𝑛′
3}⋆[𝑛1, 𝑛′
1]) ⊗{𝑛2, 𝑛′
2} − ({𝑛2, 𝑛′
2}⋆[𝑛1, 𝑛′
1]) ⊗{𝑛3, 𝑛′
3}
= [𝜕({𝑛3, 𝑛′
3}),[𝑛1, 𝑛′
1]] ⊗{𝑛2, 𝑛′
2}−[𝜕({𝑛2, 𝑛′
2}),[𝑛1, 𝑛′
1]] ⊗{𝑛3, 𝑛′
3}
= [[𝑛3, 𝑛′
3],[𝑛1, 𝑛′
1]] ⊗{𝑛2, 𝑛′
2} − [[𝑛2, 𝑛′
2],[𝑛1, 𝑛′
1]] ⊗{𝑛3, 𝑛′
3}
= −[[𝑛1, 𝑛′
1],[𝑛3, 𝑛′
3]] ⊗{𝑛2, 𝑛′
2} + [[𝑛1, 𝑛′
1],[𝑛2, 𝑛′
2]] ⊗{𝑛3, 𝑛′
3}
= −⦃[𝑛1, 𝑛′
1]⊗[𝑛3, 𝑛′
3], 𝑛2⊗ 𝑛′
2⦄+⦃[𝑛1, 𝑛′
1]⊗[𝑛2, 𝑛′
2], 𝑛3⊗ 𝑛′
3⦄
=⦃[𝑛1⊗ 𝑛′
1, 𝑛2⊗ 𝑛′
2], 𝑛3⊗ 𝑛′
3⦄−⦃[𝑛1⊗ 𝑛′
1, 𝑛3⊗ 𝑛′
3], 𝑛2⊗ 𝑛′
2⦄(BXLie5),
⦃[𝑛1⊗ 𝑛′
1, 𝑛2⊗ 𝑛′
2], 𝑛3⊗ 𝑛′
3⦄=⦃[𝑛1, 𝑛′
1]⊗[𝑛2, 𝑛′
2], 𝑛3⊗ 𝑛′
3⦄
= [[𝑛1, 𝑛′
1],[𝑛2, 𝑛′
2]] ⊗{𝑛3, 𝑛′
3}
= [𝑛1, 𝑛′
1]⊗[𝑛2, 𝑛′
2]⋅{𝑛3, 𝑛′
3}−[𝑛2, 𝑛′
2]⊗[𝑛1, 𝑛′
1]⋅{𝑛3, 𝑛′
3}
= [𝑛1, 𝑛′
1]⊗{[𝑛2, 𝑛′
2], 𝜕({𝑛3, 𝑛′
3})} − [𝑛2, 𝑛′
2]⊗{[𝑛1, 𝑛′
1], 𝜕({𝑛3, 𝑛′
3})}
= [𝑛1, 𝑛′
1]⊗{[𝑛2, 𝑛′
2],[𝑛3, 𝑛′
3]} − [𝑛2, 𝑛′
2]⊗{[𝑛1, 𝑛′
1],[𝑛3, 𝑛′
3]}
=⦃𝑛1⊗ 𝑛′
1,[𝑛2, 𝑛′
2]⊗[𝑛3, 𝑛′
3]⦄−⦃𝑛2⊗ 𝑛′
2,[𝑛1, 𝑛′
1]⊗[𝑛3, 𝑛′
3]⦄
=⦃𝑛1⊗ 𝑛′
1,[𝑛2⊗ 𝑛′
2, 𝑛3⊗ 𝑛′
3]⦄−⦃𝑛2⊗ 𝑛′
2,[𝑛1⊗ 𝑛′
1, 𝑛3⊗ 𝑛′
3]⦄(BXLie6).
In all equali ies, we ha e used he p ope ies o {−,−} and ela ions o he enso
p oduc .
108 3 Uni e sal cen al ex ension o b aided c ossed modules o Lie algeb as

CHAPTER 4
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄
On he Loday-Pi ash ili Ca ego y
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄
In his chap e , we gene alize he Loday-Pi ash ili ca ego y o di e en kinds o enso
ca ego ies. Then we use i o p o e ela ionships be ween in e nal Lie objec s and
in e nal Leibniz objec s.
4.1 Tenso Ca ego ies
In his sec ion, we will gene alize he Loday-Pi ash ili ca ego y ( [44]) o di e en
kinds o enso ca ego ies. The de ini ions o (b aided) semig oupal ca ego ies and
(b aided) monoidal ca ego ies a e al eady shown in Sec ion 1.4.
4.1.1 Ca ego ies wi h ope a ions
De ini ion 4.1.1. Le 𝐂be a ca ego y and ⊗∶×←←→ a bi unc o . We will say
ha he pai = (𝐂, ⊗)is a ca ego y wi h an ope a ion.
De ini ion 4.1.2. Le 𝐂be a ca ego y. We will deno e as Hom(𝐂) he ca ego y gi en
by:
•Ob (Hom(𝐂))= A w(𝐂)
•A w (Hom(𝐂))⊂A w(𝐂)×A w(𝐂)wi h
𝐴
𝐵
𝑓
𝛼=(𝛼1,𝛼2)
←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→
𝐶
𝐷
𝑔⇔
𝐴 𝐶
𝐵 𝐷.
𝑓
𝛼1
𝑔
𝛼2
109
110 4 On he Loday-Pi ash ili Ca ego y
•Id𝑓= (Id𝐴,Id𝐵)and 𝛽◦𝛼= (𝛽1◦𝛼1, 𝛽2◦𝛼2).
Since Hom(𝐂)is de ined by pai s o a ows is immedia e ha i 𝐂has ini e co-
p oduc s, hen
𝐴
𝐵
𝑓⊕
𝐶
𝐷
𝑔∶=
𝐴⊕𝐶
𝐵 ⊕ 𝐷,
𝑓⊕𝑔
and Hom(𝐂)has ini e cop oduc s. In he same way, i 0is he ze o objec o 𝐂, hen
Id0is he ze o objec o 𝐂(same o ini ial objec and inal objec ).
P oposi ion 4.1.3. Le = (𝐂, ⊗)be a ca ego y wi h an ope a ion such ha 𝐂has
ini e cop oduc s.
The co espondence 
⊗∶ Hom(𝐂) × Hom(𝐂)←←→ Hom(𝐂)de ined:
•in objec s as
𝐴
𝐵
𝑓
⊗
𝐶
𝐷
𝑔∶=
(𝐴⊗𝐷)⊕(𝐵 ⊗ 𝐶)
𝐵 ⊗ 𝐷
[(𝑓⊗Id𝐷),(Id𝐵⊗𝑔)]
•in a ows, i 𝑓𝛼
←←←←←←→ 𝑓′and 𝑔𝛽
←←←←←←→ 𝑔′,𝛼
⊗𝛽 =((𝛼1⊗ 𝛽2)⊕(𝛼2⊗ 𝛽1), 𝛼2⊗ 𝛽2)
is a unc o . The e o e, (Hom(𝐂),
⊗)is a ca ego y wi h an ope a ion.
P oo . Since ⊗and ⊕a e unc o s, i is jus a ma e o checking ha 𝛼
⊗𝛽 is a mo -
phism in he ca ego y. To do his, we ake 𝐴′𝑓′
←←←←←←←←←→ 𝐵′and 𝐶′𝑔′
←←←←←←←←→ 𝐷′, and we easily see
ha he ollowing diag am is commu a i e:
(𝐴⊗𝐷)⊕(𝐵 ⊗ 𝐶)(𝛼1⊗𝛽2)⊕(𝛼2⊗𝛽1)//
[(𝑓⊗Id𝐷),(Id𝐵⊗𝑔)]

(𝐴′⊗ 𝐷′)⊕(𝐵′⊗ 𝐶′)
[(𝑓′⊗Id𝐷′),(Id𝐵′⊗𝑔′)]

𝐵 ⊗ 𝐷 𝛼2⊗𝛽2
//𝐵′⊗ 𝐷′
De ini ion 4.1.4. Le = (𝐂, ⊗)be a ca ego y wi h an ope a ion such ha 𝐂has ini e
cop oduc s. The ca ego y wi h an ope a ion (Hom(𝐂),
⊗)de ined in P oposi ion 4.1.3
will be deno ed as LP()and i will be called he Loday-Pi ash ili ca ego y wi h an
ope a ion o .
4.1.2 Semig oupal ca ego ies 111
4.1.2 Semig oupal ca ego ies
De ini ion 4.1.5. Le = (𝐂, ⊗)be a ca ego y wi h an ope a ion and conside h ee
objec s 𝐴, 𝐵, 𝐶 in such ha he e exis s he cop oduc o 𝐴wi h 𝐵and 𝐴 ⊗ 𝐶 wi h
𝐵 ⊗ 𝐶. We de ine he R-⊗-dis ibu o on 𝐴, 𝐵, 𝐶 as he mo phism 𝜀𝐴,𝐵,𝐶 gi en by
𝐴⊗𝐶 𝜄1//
𝜄1⊗Id𝐶((
(𝐴⊗𝐶)⊕(𝐵 ⊗ 𝐶)
𝜀𝐴,𝐵,𝐶

(𝐵 ⊗ 𝐶)
𝜄2
oo
𝜄2⊗Id𝐶
(𝐴⊕𝐵)⊗ 𝐶
Simila ly, i he e exis s he cop oduc o 𝐴wi h 𝐵and 𝐶 ⊗ 𝐴 wi h 𝐶 ⊗ 𝐵 we
de ine he L-⊗-dis ibu o on 𝐴, 𝐵, 𝐶 as he mo phism 𝛾𝐴,𝐵,𝐶 gi en by
𝐶 ⊗ 𝐴 𝜄1//
Id𝐶⊗𝜄1((
(𝐶 ⊗ 𝐴)⊕(𝐶 ⊗ 𝐵)
𝛾𝐴,𝐵,𝐶

(𝐶 ⊗ 𝐵)
𝜄2
oo
Id𝐶⊗𝜄2
𝐶 ⊗ (𝐴⊕𝐵)
P oposi ion 4.1.6. The co espondences 𝜀and 𝛾de ined abo e a e unc o ial. Fu -
he mo e, hey a e na u al ans o ma ions.
P oo . To check ha 𝜀is a na u al ans o ma ion we need o see ha he ollowing
diag am is commu a i e,
(𝐴⊗𝐶)⊕(𝐵 ⊗ 𝐶)𝜀𝐴,𝐵,𝐶 //
(𝑓⊗ℎ)⊕(𝑔⊗ℎ)

(𝐴⊕𝐵)⊗ 𝐶
(𝑓⊕𝑔)⊗ℎ

(𝐴′⊗ 𝐶′)⊕(𝐵′⊗ 𝐶′)𝜀𝐴′,𝐵′,𝐶′//(𝐴′⊕ 𝐵′)⊗ 𝐶′
whe e 𝐴𝑓
←←←←←←←→ 𝐴′, 𝐵 𝑔
←←←←←←→ 𝐵′, 𝐶 ℎ
←←←←←←→ 𝐶′∈ A w(𝐂). Since he domain is a cop oduc , i eas-
ily ollows by checking ha hey coincide when composed wi h he na u al injec ions.
A simila a gumen wo ks o 𝛾.
De ini ion 4.1.7. Le = (𝐂, ⊗)be a ca ego y wi h an ope a ion whe e 𝐂has ini e
cop oduc s. Take 𝜀and 𝛾 om De ini ion 4.1.5.
112 4 On he Loday-Pi ash ili Ca ego y
•We say ha 𝐂is Righ Mul iplica ion dis ibu i e wi h ⊗o 𝑅-⊗-dis ibu i e
i 𝜀is a na u al isomo phism. In his case, we will say ha is 𝑅-dis ibu i e.
•We say ha 𝐂is Le Mul iplica ion dis ibu i e wi h ⊗o 𝐿-⊗-dis ibu i e i
𝛾is a na u al isomo phism. In his case, we will say ha is 𝐿-dis ibu i e.
•We say ha 𝐂is dis ibu i e wi h ⊗o ⊗-dis ibu i e i 𝜀and 𝛾a e na u al
isomo phisms. In his case, we will say ha is dis ibu i e.
Theo em 4.1.8. Le (𝐂, ⊗, 𝑎)be a semig oupal ca ego y such ha = (𝐂, ⊗)is ⊗-
dis ibu i e. Fo
𝐴
𝐵
𝑓,
𝐶
𝐷
𝑔,
𝐸
𝐹
ℎ∈ Ob(Hom(𝐂)) we de ine using he uni e sal p op-
e y o cop oduc s he mo phism
((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 (((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) (𝐵 ⊗ 𝐷)⊗ 𝐸
((𝐴⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹)𝐵 ⊗ (𝐷 ⊗ 𝐸)
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) 𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸))
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸)))
𝜄1
𝜀−1
𝛼𝑓,𝑔,ℎ
𝜄2
𝑎
𝑎⊕𝑎 Id ⊗𝜄2
Id ⊕(Id ⊗𝜄1)𝜄2
Then 𝑎𝑓,𝑔,ℎ = (𝛼𝑓,𝑔,ℎ, 𝑎𝐵,𝐷,𝐹 )gi es an associa o o (Hom(𝐂),
⊗). The in e se
o his na u al isomo phism is gi en by (𝜔𝑓,𝑔,ℎ, 𝑎−1
𝐵,𝐷,𝐹 )whe e 𝜔𝑓,𝑔,ℎ is de ined by he
diag am
𝐴⊗(𝐷 ⊗ 𝐹 ) (𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸))) 𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸))
(𝐴⊗ 𝐷)⊗ 𝐹 (𝐵 ⊗ (𝐶 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐷 ⊗ 𝐸))
((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐵 ⊗ 𝐶)⊗ 𝐹)⊕((𝐵 ⊗ 𝐷)⊗ 𝐸)
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸)
𝑎−1
𝜄1
𝜔𝑓,𝑔,ℎ
𝜄2
𝛾−1
𝜄1⊗Id𝐹𝑎−1⊕𝑎−1
𝜄1(𝜄2⊗Id)⊕Id
4.1.2 Semig oupal ca ego ies 113
P oo . Fi s , we will p o e ha 𝑎𝑓,𝑔,ℎ = (𝛼𝑓,𝑔,ℎ, 𝑎𝐵,𝐷,𝐹 )∶ (𝑓
⊗𝑔)
⊗ℎ ←←→ 𝑓
⊗(𝑔
⊗ℎ),
ha means, we ha e he diag am
(((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) (𝐴 ⊗ (𝐷 ⊗ 𝐹)) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹)⊕(𝐷 ⊗ 𝐸)))
(𝐵⊗ 𝐷)⊗ 𝐹 𝐵 ⊗ (𝐷 ⊗ 𝐹 ).
𝛼𝑓,𝑔,ℎ
[([(𝑓⊗Id),(Id ⊗𝑔)]⊗Id),(Id ⊗ℎ)] [(𝑓⊗Id),(Id ⊗[(𝑔⊗Id),(Id ⊗ℎ)])]
𝑎
No e ha we use he no a ion [−,−] o he na u al mo phism o he cop oduc .
The domain is a cop oduc so ha we can s udy each pa sepa a ely. The i s
one is ((𝐴 ⊗ 𝐶)⊕(𝐵 ⊗ 𝐷)) ⊗ 𝐹 and he i s mo phism going down is 𝜀−1, so we
will p o e ha he diag am is commu a i e when p ecomposed wi h 𝜀, since i is an
isomo phism. Then he domain is again a cop oduc so ha we can analyse bo h pa s
sepa a ely again. To see he diag am’s commu a i i y is jus a ma e o esol ing he
cop oduc injec ions, as we can see in he ollowing diag ams. The second pa and
he o he ones can be checked simila ly. The ou e diag am is commu a i e since he
inne diag ams a e easily commu a i e.
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
(𝐴⊗ 𝐷)⊗ 𝐹
(𝐵⊗ 𝐷)⊗ 𝐹
𝐵⊗(𝐷 ⊗ 𝐹 )𝐴 ⊗ (𝐷 ⊗ 𝐹)
[([(𝑓⊗Id) ,(Id ⊗𝑔) ] ⊗Id) ,(Id ⊗ℎ) ]
𝜄1
[(𝑓⊗Id) ,(Id ⊗𝑔) ] ⊗Id
𝜀
𝜄1⊗Id
𝜄1
𝑎
(𝑓⊗Id)⊗Id
𝑎
𝑓⊗Id

114 4 On he Loday-Pi ash ili Ca ego y
(((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 ) (𝐴 ⊗ 𝐷)⊗ 𝐹
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) 𝐴 ⊗ (𝐷 ⊗ 𝐹 )
(𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸)))
𝐵 ⊗ (𝐷 ⊗ 𝐹 )
𝛼
𝜄1
𝜀−1
𝜀
𝑎⊕𝑎
𝜄1
𝑎
Id ⊕(Id ⊗𝜄1)
𝜄1
𝜄1
𝑓⊗Id
[(𝑓⊗Id) ,(Id ⊗[(𝑔⊗Id) ,(Id ⊗ℎ) ])
No e ha he igh mos pa is he same o hese wo diag ams, so we conclude ha
he le mos pa is he same when p ecomposed wi h he injec ions and 𝜀. The e o e,
we ha e ha 𝑎𝑓,𝑔,ℎ is a mo phism.
Le us now see ha i is a na u al ans o ma ion. Le us conside he ollowing
objec s
𝐴
𝐵
𝑓
𝜆
←←←←←←←←←←←→
𝐴′
𝐵′
𝑓′,
𝐶
𝐷
𝑔
𝛽
←←←←←←←←←←←→
𝐶′
𝐷′
𝑔′,
𝐸
𝐹
ℎ
𝜎
←←←←←←←←←←←←→
𝐸′
𝐹′
ℎ′.
The lowe pa o he na u alness is sa is ied, since he e he ope a ion is he s an-
da d one. We will ocus in he uppe pa .
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹)⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) (𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸)))
(((𝐴′⊗𝐷′)⊕(𝐵′⊗ 𝐶′)) ⊗ 𝐹 ′)⊕((𝐵′⊗ 𝐷′)⊗ 𝐸′) (𝐴′⊗(𝐷′⊗ 𝐹 ′)) ⊕(𝐵′⊗((𝐶′⊗ 𝐹 ′)⊕(𝐷′⊗ 𝐸′))).
𝛼𝑓,𝑔,ℎ
(((𝜆1⊗𝛽2)⊕(𝜆2⊗𝛽1))⊗𝜎2)⊕((𝜆2⊗𝛽2)⊗𝜎1) (𝜆1⊗(𝛽2⊗𝜎2))⊕(𝜆2⊗((𝛽1⊗𝜎2)⊕(𝛽2⊗𝜎1)))
𝛼𝑓′,𝑔′,ℎ′
Again, he domain is a cop oduc so ha we can s udy each pa sepa a ely. I is
he same cop oduc as he p oo o well de ined, so we need o use he same s eps o
p o e i , using he injec ions and 𝜀, as shown in he ollowing diag ams. The o he
cases can be checked simila ly.
4.1.2 Semig oupal ca ego ies 115
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹)⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
((𝐴⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 ) (𝐴 ⊗ 𝐷)⊗ 𝐹
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) 𝐴 ⊗ (𝐷 ⊗ 𝐹 )
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸)))
(𝐴′⊗(𝐷′⊗𝐹′)) ⊕(𝐵′⊗((𝐶′⊗ 𝐹′)⊕(𝐷′⊗ 𝐸′))) 𝐴′⊗(𝐷′⊗ 𝐹′) (𝐴′⊗ 𝐷′)⊗ 𝐹 ′,
𝛼
𝜄1
𝜀−1
𝜀
𝑎⊕𝑎
𝜄1
𝑎
(𝜆1⊗𝛽2)⊗𝜎2
Id ⊕(Id ⊗𝜄1)
𝜄1
𝜄1
𝜆1⊗(𝛽2⊗𝜆2)
(𝜆1⊗(𝛽2⊗𝜎2))⊕(𝜆2⊗((𝛽1⊗𝜎2)⊕(𝛽2⊗𝜎1)))
𝜄1𝑎
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹)⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
((𝐴′⊗𝐷′)⊕(𝐵′⊗ 𝐶′)) ⊗ 𝐹 ′(𝐴 ⊗ 𝐷)⊗ 𝐹
((𝐴′⊗𝐷′)⊗ 𝐹′)⊕((𝐵′⊗ 𝐶′)⊗ 𝐹 ′) (𝐴′⊗ 𝐷′)⊗ 𝐹′
(((𝐴′⊗𝐷′)⊕(𝐵′⊗ 𝐶′)) ⊗ 𝐹 ′)⊕((𝐵′⊗ 𝐷′)⊗ 𝐸′) (𝐴′⊗(𝐷′⊗ 𝐹 ′)) ⊕(𝐵′⊗(𝐶′⊗ 𝐹 ′))
(𝐴′⊗(𝐷′⊗𝐹′)) ⊕(𝐵′⊗((𝐶′⊗ 𝐹′)⊕(𝐷′⊗ 𝐸′))) 𝐴′⊗(𝐷′⊗ 𝐹′).
(((𝜆1⊗𝛽2)⊕(𝜆2⊗𝛽1))⊗𝜎2)⊕((𝜆2⊗𝛽2)⊗𝜎1)
𝜄1
((𝜆1⊗𝛽2)⊕(𝜆2⊗𝛽2))⊗𝜎2
𝜀
𝜄1
𝜀−1 (𝜆1⊗𝛽1)⊗𝜎2
𝜄1⊗Id 𝜄1
𝑎⊕𝑎
𝜄1⊗Id
𝑎
𝜄1
𝛼Id ⊕(Id ⊗𝜄1)
𝜄1
𝜄1
The e o e, 𝑎∶
⊗◦(
⊗× IdHom(𝐂))⇒
⊗◦(IdHom(𝐂)×
⊗)◦𝐴Hom(𝐂),Hom(𝐂),Hom(𝐂)is a
na u al ans o ma ion.
We will shownow ha i isa na u al isomo phism. We need oshow ha o 𝑓, 𝑔, ℎ ∈
Hom(𝐂), he mo phism 𝑎𝑓,𝑔,ℎ is an isomo phism wi h in e se (𝜔𝑓,𝑔,ℎ, 𝑎−1
𝐵,𝐷,𝐹 ). Since
𝑎𝐵,𝐷,𝐹 is an isomo phism wi h in e se 𝑎−1
𝐵,𝐷,𝐹 we only need o p o e ha 𝛼𝑓,𝑔,ℎ is an
isomo phism wi h in e se 𝜔𝑓,𝑔,ℎ.
We will p o e 𝜔𝑓,𝑔,ℎ◦𝛼𝑓,𝑔,ℎ = Id, he o he composi ion is analogous. As in he
116 4 On he Loday-Pi ash ili Ca ego y
o he p oo s, he domain is a cop oduc . In ac , i is he same cop oduc as in he
o he p oo s, so we need o use he same s eps o p o e i , using he injec ions and 𝜀,
as can see in he ollowing diag ams. The o he cases a e checked simila ly.
(((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 ) (𝐴 ⊗ 𝐷)⊗ 𝐹
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) 𝐴 ⊗ (𝐷 ⊗ 𝐹 )
(𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸))) (𝐴 ⊗ 𝐷)⊗ 𝐹
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) (𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
𝛼
𝜄1
𝜀−1
𝜀
𝑎⊕𝑎
𝜄1
𝑎
Id ⊕(Id ⊗𝜄1)
𝜄1
𝜄1
𝑎−1
𝜔𝜄1⊗Id
𝜄1
𝜄1
𝜀
Fo comple ion we will show ano he diag am, composing i s wi h 𝜄1and hen
wi h 𝜄2, o show when he mo phism 𝛾 om he L-⊗-dis ibu i i y appea s.
(((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹)⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
((𝐴⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 ) (𝐵 ⊗ 𝐶)⊗ 𝐹
(𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) 𝐵 ⊗ (𝐶 ⊗ 𝐹)
(𝐴⊗(𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸))) 𝐵 ⊗ ((𝐶 ⊗ 𝐹)⊕(𝐷 ⊗ 𝐸))
(𝐵 ⊗ (𝐶 ⊗ 𝐹 )) ⊕(𝐵 ⊗ (𝐷 ⊗ 𝐸))
((𝐵 ⊗ 𝐶)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) (𝐵 ⊗ 𝐶)⊗ 𝐹
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹)⊕((𝐵 ⊗ 𝐶)⊗ 𝐹 )
𝛼
𝜄1
𝜀−1
𝜀
𝑎⊕𝑎
𝜄2
𝑎
Id ⊕(Id ⊗𝜄1)
𝜄2
Id ⊗𝜄1
𝑎−1
𝜄1
𝜔
𝜄2
𝛾−1
𝑎−1⊕𝑎−1
(𝜄2⊗Id)⊕Id
𝜄1
𝜄2
𝜄2⊗Id
𝜄1𝜀
4.1.2 Semig oupal ca ego ies 117
To p o e ha he na u al isomo phism 𝑎 is an associa o , we need o p o e he
associa i i y cohe ence diag am.
((𝑓
⊗𝑔)
⊗ℎ)
⊗𝑘 (𝑓
⊗𝑔)
⊗(ℎ
⊗𝑘)
(𝑓
⊗(𝑔
⊗ℎ)) 
⊗𝑘
𝑓
⊗((𝑔
⊗ℎ)
⊗𝑘)𝑓
⊗(𝑔
⊗(ℎ
⊗𝑘)).
𝑎𝑓
⊗𝑔,ℎ,𝑘
𝑎𝑓,𝑔,ℎ 
⊗Id𝑘
𝑎𝑓,𝑔,ℎ
⊗𝑘
𝑎𝑓,𝑔 
⊗ℎ,𝑘
Id𝑓
⊗𝑎𝑔,ℎ,𝑘
Tha means ha we need o ob ain wo commu a i e diag ams gi en by he domain
and codomain o he enso p oduc 
⊗. The lowe pa is immedia e since i is he
cohe ence diag am o he associa o 𝑎.
The uppe pa is he ollowing one:
𝔄 𝔈
𝔅
ℭ 𝔇,
𝛼𝑓
⊗𝑔,ℎ,𝑘
(𝛼𝑓,𝑔,ℎ⊗Id)⊕(𝑎𝐵,𝐷,𝐸 ⊗Id)
𝛼𝑓,𝑔,ℎ 
⊗𝑘
𝛼𝑓,𝑔 
⊗ℎ,𝑘
(Id ⊗𝑎𝐷,𝐹 ,𝐻 )⊕(Id ⊗𝛼𝑔,ℎ,𝑘)
Whe e, o cla i y, we will deno e:
𝔄∶= (((((𝐴⊗𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸)) ⊗ 𝐻)⊕(((𝐵 ⊗ 𝐷)⊗ 𝐹 )⊗ 𝐺),
𝔅∶= (((𝐴 ⊗ (𝐷 ⊗ 𝐹 )) ⊕(𝐵 ⊗ ((𝐶 ⊗ 𝐹)⊕(𝐷 ⊗ 𝐸)))) ⊗ 𝐻)⊕((𝐵 ⊗ (𝐷 ⊗ 𝐹)) ⊗ 𝐺),
ℭ∶= (𝐴 ⊗ ((𝐷 ⊗ 𝐹 )⊗ 𝐻)) ⊕(𝐵 ⊗ ((((𝐶 ⊗ 𝐹 )⊕(𝐷 ⊗ 𝐸)) ⊗ 𝐻)⊕((𝐷 ⊗ 𝐹)⊗ 𝐺))),
𝔇∶= (𝐴 ⊗ (𝐷 ⊗ (𝐹 ⊗ 𝐻))) ⊕(𝐵 ⊗ ((𝐶 ⊗ (𝐹 ⊗ 𝐻)) ⊕(𝐷 ⊗ ((𝐸 ⊗ 𝐻)⊕(𝐹 ⊗ 𝐺))))),
𝔈∶= (((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗(𝐹 ⊗ 𝐻)) ⊕((𝐵 ⊗ 𝐷)⊗((𝐸 ⊗ 𝐻)⊕(𝐹 ⊗ 𝐺))).
Following o he cases, his can be p o ed by esol ing injec ions o he cop oduc ,
as we can see in he ollowing diag ams aking he i s injec ion.
124 4 On he Loday-Pi ash ili Ca ego y
(((𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 )⊕((𝐵 ⊗ 𝐷)⊗ 𝐸) ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) ⊗ 𝐹 ((𝐴 ⊗ 𝐷)⊗ 𝐹 )⊕((𝐵 ⊗ 𝐶)⊗ 𝐹)
𝐹 ⊗ ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)) (𝐴 ⊗ 𝐷)⊗ 𝐹
(𝐹 ⊗ ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶))) ⊕(𝐸 ⊗ (𝐵 ⊗ 𝐷)) 𝐹 ⊗ (𝐴⊗𝐷)
(𝐸⊗(𝐵 ⊗ 𝐷)) ⊕(𝐹 ⊗ ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶))) 𝐹 ⊗ ((𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶))
𝜏⊕𝜏
𝜏1
𝑓
⊗𝑔,ℎ
𝜄1
𝜏
𝜀
𝜄1
𝜄1
𝜏
𝜄1⊗Id
𝜏⊕Id ⊗𝜄1
Id ⊗𝜄1
𝜄2
𝜄1

4.1.4 Monoidal ca ego ies 125
De ini ion 4.1.11. Le = (𝐂, ⊗, 𝑎, 𝜏)be a b aided semig oupal ca ego y such ha
𝐂is ⊗-dis ibu i e.
The b aided semig oupal ca ego y (Hom(𝐂),
⊗, 𝑎, 𝜏)de ined in Theo em 4.1.10,
will be deno ed as LP()and will be called he Loday-Pi ash ili b aided semig oupal
ca ego y o .
De ini ion 4.1.12. Le = (𝐂, ⊗, 𝑎, 𝜏)be a b aided semig oupal ca ego y. We say
ha is symme ic i o all objec s 𝐴, 𝐵 o 𝐂is sa is ied ha
𝜏−1
𝐴,𝐵 =𝜏𝐵,𝐴.
P oposi ion 4.1.13. Le = (𝐂, ⊗, 𝑎, 𝜏)be a b aided semig oupal ca ego y such
ha 𝐂is ⊗-dis ibu i e. Then LP()is symme ic i and only i is symme ic.
P oo . Le us assume i s ha LP()is symme ic. Fo any 𝐴, 𝐵 ∈ Ob(𝐂)we know
ha (𝜏Id𝐴,Id𝐵)−1 =𝜏Id𝐵,Id𝐴. Then, by aking he second componen o he mo phisms
we ha e 𝜏−1
𝐴,𝐵 =𝜏𝐵,𝐴
Assume now ha 𝐂is symme ic. Fo any
𝐴
𝐵
𝑓,
𝐶
𝐷
𝑔∈ Ob (Hom(𝐂))we ha e
o p o e ha (𝜏𝑓,𝑔)−1 =𝜏𝑔,𝑓 . Tha means ha we need o p o e ha (𝜏1
𝑓,𝑔, 𝜏𝐴,𝐵)−1 =
(𝜏1
𝑔,𝑓 , 𝜏𝐵,𝐴), i.e. (𝜏1
𝑓,𝑔)−1 =𝜏1
𝑔,𝑓 and 𝜏−1
𝐴,𝐵◦𝜏𝐵,𝐴. The second one is ue by hypo hesis.
We know ha 𝜏1
𝑓,𝑔 is an isomo phism in 𝐂, since i is he componen o an iso-
mo phism, so i is jus enough o p o e 𝜏1
𝑔,𝑓 ◦𝜏1
𝑓,𝑔 = Id(𝐴⊗𝐷)⊕(𝐵⊗𝐶).
(𝐴⊗𝐷)⊕(𝐵 ⊗ 𝐶)
𝜏1
𝑓,𝑔 //(𝐶 ⊗ 𝐵)⊕(𝐷 ⊗ 𝐴)
𝜏1
𝑔,𝑓 //(𝐴⊗𝐷)⊕(𝐵 ⊗ 𝐶)
Bu his is ob iously ue, since wha 𝜏1
𝑓,𝑔 does is inse each pa o he cop oduc
in o i s co esponden one and hen wis i . Bu hese wis s a e in e se o each o he
by assump ion.
4.1.4 Monoidal ca ego ies
De ini ion 4.1.14. Le (𝐂, ⊗)be a ca ego y wi h an ope a ion such ha 𝐂has an
ini ial objec Λ. Then:
126 4 On he Loday-Pi ash ili Ca ego y
•𝐂is said o be le -⊗-annihila ed ( igh -⊗-annihila ed) i he unique mo phism
Λ𝐴⊗Λ∶ Λ →𝐴 ⊗ Λ( espec i ely, ΛΛ⊗𝐴 ∶ Λ →Λ⊗ 𝐴) is an isomo phism.
Tha means 𝐴 ⊗ Λ(Λ⊗ 𝐴) is an ini ial objec .
•𝐂is ⊗-annihila ed i i is bo h le -⊗-annihila ed and igh -⊗-annihila ed.
Rema k 4.1.15.The de ini ion o le -⊗-annihila ed does no depend on he ini ial
objec . Assume ha Λ′is also ano he ini ial objec , we ha e ΛΛ′and Λ′
Λa e in e se
o each o he , so
Λ′
𝐴⊗Λ′= (Id𝐴⊗ΛΛ′)◦Λ𝐴⊗Λ◦Λ′
Λ
and Λ′
𝐴⊗Λ′is an isomo phism by composi ion, since ⊗p ese es isomo phisms.
Theo em 4.1.16. Le (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟)be a monoidal ca ego y such ha (𝐂, ⊗)is dis-
ibu i e and annihila ed. Le = (𝐂, ⊗, 𝑎)and conside i s Loday-Pi ash ili semi-
g oupal ca ego y LP() = (Hom(𝐂),
⊗, 𝑎).
We will deno e 
𝐼∶=
Λ
𝐼
Λ𝐼, and ake
𝐴
𝐵
𝑓∈ Ob(Hom(𝐂)).
Le 
𝑙𝑓∶= (
𝑙1
𝑓, 𝑙𝐵)and 𝑟𝑓∶= (𝑟1
𝑓, 𝑟𝐵), whe e 
𝑙1
𝑓is he ollowing composi ion:
(Λ ⊗ 𝐵)⊕(𝐼 ⊗ 𝐴)
Λ−1
Λ⊗𝐵⊕Id𝐼⊗𝐴 //Λ⊕(𝐼 ⊗ 𝐴)(𝜄2)−1
//𝐼 ⊗ 𝐴 𝑙𝐴//𝐴 ,
and 𝑟1
𝑓is he ollowing composi ion:
(𝐴⊗𝐼)⊕(𝐵 ⊗ Λ)
Id𝐴⊗𝐼 ⊕Λ−1
𝐵⊗Λ//(𝐴⊗𝐼)⊕Λ(𝜄1)−1
//𝐴⊗𝐼 𝑟𝐴//𝐴 .
Then (Hom(𝐂),
⊗, 𝑎, 
𝐼,
𝑙,𝑟)is a monoidal ca ego y.
P oo . We will i s p o e ha 𝑟𝑓is well de ined. This is he same as showing ha
he ollowing diag am is commu a i e:
(𝐴⊗𝐼)⊕(𝐵 ⊗ Λ)
[(𝑓⊗Id𝐼),(Id𝐵⊗Λ𝐼)]

𝑟1
𝑓//𝐴
𝑓

𝐵 ⊗ 𝐼 𝑟𝐵//𝐵
4.1.4 Monoidal ca ego ies 127
The composi ion wi h he second cop oduc injec ion is i ial since 𝐵⊗Λis an ini ial
objec . The i s one ollows by he commu a i e diag am:
(𝐴⊗𝐼)⊕(𝐵 ⊗ Λ)
[(𝑓⊗Id),(Id ⊗Λ)]

𝑟1
𝑓//
Id ⊗Λ−1
))
𝐴
𝑓

(𝐴⊗𝐼)⊕Λ(𝜄1)−1
//𝐴⊗𝐼
𝑟<<
𝐴⊗𝐼
Id
88
𝑓⊗Id
uu
𝜄1
cc
𝐵 ⊗ 𝐼 𝑟//𝐵
The ac ha 𝑟 is a na u al isomo phism is immedia e by being composi ion o
na u al isomo phisms. The same a gumen s wo k o 
𝑙.
So we ha e o p o e he iangle equa ion. Fo ha , we will ake
𝐶
𝐷
𝑔. The lowe
pa is immedia e since i is he iangle equa ion o he o iginal monoidal ca ego y.
We ha e o p o e he iangle equa ion o he uppe pa , i.e. we need o p o e he
ollowing diag am:
(((𝐴 ⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷)⊕((𝐵 ⊗ 𝐼)⊗ 𝐶) (𝐴 ⊗ (𝐼 ⊗ 𝐷)) ⊕(𝐵 ⊗ ((Λ ⊗ 𝐷)⊕(𝐼 ⊗ 𝐶)))
(𝐴 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)
𝛼𝑓,
𝐼,𝑔
(𝑟1
𝑓⊗Id𝐷)⊕(𝑟𝐵⊗Id𝐶)(Id𝐴⊗𝑙𝐷)⊕(Id𝐵⊗
𝑙1
𝑔)
As in he p e ious heo em, we will use ha he domain is a cop oduc o s udy
each pa sepa a ely. The i s one is ((𝐴 ⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷 and he i s mo phism
om he e is 𝜀, so we can p ecompose wi h ha isomo phism. Now we ha e again
a cop oduc ha we can s udy sepa a ely, bu he second pa is immedia e since i s
domain, (𝐵 ⊕ Λ) ⊕ 𝐷, is an ini ial objec . This is ue because, since (𝐂, ⊗)is anni-
hila ed. Le -annihila ion say ha (𝐵 ⊗ Λ) is an ini ial objec , and since annihila ion
does no depend on he ini ial objec selec ed, igh -annihila ion say ha (𝐵 ⊗ Λ) ⊗𝐶
128 4 On he Loday-Pi ash ili Ca ego y
is an ini ial objec . So, we will p o e he i s pa . The second pa o he o iginal
cop oduc is es ablished simila ly.
(((𝐴⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷)⊕((𝐵 ⊗ 𝐼)⊗ 𝐶) ((𝐴 ⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷 ((𝐴 ⊗ 𝐼)⊗ 𝐷)⊕((𝐵 ⊗ Λ) ⊗ 𝐷)
((𝐴⊗ 𝐼)⊗ 𝐷)⊕((𝐵 ⊗ Λ) ⊗ 𝐷) (𝐴 ⊗ 𝐼)⊗ 𝐷
(𝐴⊗(𝐼 ⊗ 𝐷)) ⊕(𝐵 ⊗ ((Λ ⊗ 𝐷)⊕(𝐼 ⊗ 𝐶))) (𝐴 ⊗ (𝐼 ⊗ 𝐷)) ⊕(𝐵 ⊗ (Λ ⊗ 𝐷)) 𝐴 ⊗ (𝐼 ⊗ 𝐷)
(𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)𝐴 ⊗ 𝐷
𝛼𝑓,
𝐼,𝑔
𝜄1
𝜀−1
𝜀
𝑎⊕𝑎
𝜄1
𝑟⊗Id
𝑎
(Id ⊗𝑙)⊕(Id ⊗
𝑙1
𝑔)
Id ⊕(Id ⊗𝜄1)𝜄1
𝜄1
Id ⊗𝑙
𝜄1
(((𝐴 ⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷)⊕((𝐵 ⊗ 𝐼)⊗ 𝐶) ((𝐴 ⊗ 𝐼)⊕(𝐵 ⊗ Λ)) ⊗ 𝐷 ((𝐴 ⊗ 𝐼)⊗ 𝐷)⊕((𝐵 ⊗ Λ) ⊗ 𝐷)
(𝐴⊗ 𝐷)⊕(𝐵 ⊗ 𝐶)𝐴 ⊗ 𝐷 (𝐴 ⊗ 𝐼)⊗ 𝐷
(𝑟1
𝑓⊗Id)⊕(𝑟⊗Id)
𝜄1
𝑟1
𝑓⊗Id
𝜀
𝜄1
𝜄1
𝑟⊗Id
𝜄1⊗Id
Wi h his we know ha he iangula equa ion holds and (Hom(𝐂),
⊗, 𝑎, 
𝐼,
𝑙,𝑟)is a
monoidal ca ego y.
De ini ion 4.1.17. Le = (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟)be a monoidal ca ego y such ha 𝐂is ⊗-
dis ibu i e and ⊗-annihila ed. The monoidal ca ego y (Hom(𝐂),
⊗, 𝑎, 
𝐼,
𝑙,𝑟)de ined
in Theo em 4.1.16 will be deno ed as LP()and will be called he Loday-Pi ash ili
monoidal ca ego y o .
4.1.5 B aided monoidal ca ego ies
The no ion o b aided monoidal ca ego y was in oduced by Joyal and S ee in [38].
De ini ion 4.1.18. Ab aided symme ic monoidal ca ego y is a b aided monoidal ca -
ego y = (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟, 𝜏)whe e (𝐂, ⊗, 𝑎, 𝜏)is a b aided symme ic semig oupal
ca ego y.
4.2 Addi i e Ca ego ies wi h ope a ions 129
De ini ion 4.1.19. Le = (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟, 𝜏)be a b aided (symme ic) monoidal
ca ego y such ha 𝐂is ⊗-dis ibu i e and ⊗-annihila ed. The b aided (symme -
ic) monoidal ca ego y (Hom(𝐂),
⊗, 𝑎, 
𝐼,
𝑙,𝑟, 𝜏)de ined be ween Theo em 4.1.10 and
Theo em 4.1.16, will be deno ed as LP()and will be called he Loday-Pi ash ili
b aided monoidal ca ego y o .
Lemma 4.1.20. Le = (𝐂, ⊗, 𝑎, 𝐼, 𝑙, 𝑟, 𝜏)be a b aided monoidal ca ego y such
ha 𝐂is ⊗-dis ibu i e and ⊗-annihila ed. Then LP()is symme ic i and only i 
is symme ic.
4.2 Addi i e Ca ego ies wi h ope a ions
In his sec ion, we will y o gi e p ope ies o he sum o wo mo phisms in an
addi i e ca ego y wi h an ope a ion in ha ca ego y.
Lemma 4.2.1. Le (𝐂, ⊗)be a ca ego y wi h an ope a ion, whe e 𝐂has ini e co-
p oduc s. Then he ollowing diag ams a e commu a i e:
(𝐴⊗𝐵)⊕(𝐴⊗𝐵)
𝛾𝐴,𝐵,𝐵

▽𝐴⊗𝐵
((
(𝐴⊗𝐵)⊕(𝐴⊗𝐵)
𝜀𝐴,𝐴,𝐵

▽𝐴⊗𝐵
((
𝐴 ⊗ 𝐵 𝐴 ⊗ 𝐴
𝐴 ⊗ (𝐵 ⊕ 𝐵)
Id𝐴⊗▽𝐵
66
(𝐴⊕𝐴)⊗ 𝐵
▽𝐴⊗Id𝐵
66
whe e 𝜀and 𝛾a e de ined in De ini ion 4.1.5.
P oo . Since he domain o bo h a e cop oduc s we can use he uni e sal p ope y.
Take 𝑘∈ {1,2}.
(Id𝐴⊗▽𝐵)◦𝛾𝐴,𝐵,𝐵◦𝜄𝑘= (Id𝐴⊗▽𝐵)◦(Id𝐴⊗𝜄𝑘) = Id𝐴⊗(▽𝐵◦𝜄𝑘)
= Id𝐴⊗Id𝐵= Id𝐴⊗𝐵 =▽𝐴⊗𝐵◦𝜄𝑘
The second one ollows analogously.

130 4 On he Loday-Pi ash ili Ca ego y
Theo em 4.2.2. Le (𝐂, ⊗)a ca ego y wi h an ope a ion, dis ibu i e and annihi-
la ed, whe e 𝐂has bip oduc s. Then,
(i) (𝑓+𝑔)⊗ ℎ = (𝑓 ⊗ ℎ)+(𝑔 ⊗ ℎ) o 𝐴𝑓,𝑔
←←←←←←←←←←←←←→ 𝐵,𝐶ℎ
←←←←←←→ 𝐷,
(ii) 𝑓 ⊗ (𝑔+ℎ) = (𝑓 ⊗ 𝑔)+(𝑓 ⊗ ℎ) o 𝐴𝑓
←←←←←←←→ 𝐵,𝐶𝑔,ℎ
←←←←←←←←←←←←→ 𝐷.
P oo . We will show pa (i) since he second one is comple ely analogue. We need
o p o e ha he ollowing diag am is commu a i e:
(𝐴 ⊕ 𝐴)⊗ 𝐶 (𝐵 ⊕ 𝐵)⊗ 𝐷
𝐴⊗ 𝐶 𝐵 ⊗ 𝐷
(𝐴⊗ 𝐶)⊕(𝐴⊗𝐶) (𝐵 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐷)
(𝑓⊕𝑔)⊗ℎ
▽⊗Id
△⊗Id
△
(𝑓⊗ℎ)⊕(𝑔⊗ℎ)
▽
Besides, o check ha he p e ious diag am is commu a i e is equi alen o p o e
ha he ollowing subdiag ams a e commu a i e:
(𝐴 ⊕ 𝐴)⊗ 𝐶 (𝐵 ⊕ 𝐵)⊗ 𝐷
𝐴⊗ 𝐶 𝐵 ⊗ 𝐷
(𝐴⊗ 𝐶)⊕(𝐴⊗𝐶) (𝐵 ⊗ 𝐷)⊕(𝐵 ⊗ 𝐷)
(𝑓⊕𝑔)⊗ℎ
▽⊗Id
△⊗Id𝐶
△
𝜀
(𝑓⊗ℎ)⊕(𝑔⊗ℎ)
𝜀
▽
The igh mos subdiag am has al eady appea ed in Lemma 4.2.1, whe eas he
middle subdiag am is na u alness om P oposi ion 4.1.6. Le us p o e ha he le
subdiag am is commu a i e.
Fo doing his, we will de ine 𝛿𝑋⊕𝑌
𝑖,𝑗 ∶= 𝜄𝑋⊕𝑌
𝑖◦𝜋𝑋⊕𝑌
𝑗, i.e. 𝛿𝑖,𝑗 = Id i 𝑖=𝑗and
𝛿𝑖,𝑗 = 0 i 𝑖≠𝑗.
4.2 Addi i e Ca ego ies wi h ope a ions 131
Gi en he ollowing diag am
(𝐴⊕𝐴)⊗ 𝐶
𝜀−1

𝜋1⊗Id

𝐴⊗𝐶
△⊗Id 66
△((
𝐴⊗𝐶
𝜄𝑖⊗Id
hh
𝜄𝑖
𝜄𝑖◦𝜋𝑗%%
(𝐴⊗𝐶)⊕(𝐴⊗𝐶)
𝜀
OO
𝜋𝑗//𝐴⊗𝐶
o 𝑖, 𝑗 ∈ {1,2}. No e ha 𝜄𝑖◦𝜋𝑗= Id𝑋⊗𝑌 i 𝑖=𝑗, and 𝜄𝑖◦𝜋𝑗= 0, i 𝑖≠𝑗. Then,
i is s aigh o wa d ha each subdiag am o he igh is commu a i e: i 𝑖=𝑗, i is
immedia e, and i 𝑖≠𝑗, he opmos iangle comes om le -⊗-annihila ion (0⊗
Id𝐶= 0). The e o e, using ha he objec in he bo om is a cop oduc , he ou e igh
iangle is commu a i e:
(𝐴⊕𝐴)⊗ 𝐶
𝜀−1

𝜋1⊗Id

𝐴⊗𝐶
△⊗Id 66
△((
(𝐴⊗𝐶)⊕(𝐴⊗𝐶)
𝜀
OO
𝜋𝑗//𝐴⊗𝐶
Since he objec o he bo om is also a p oduc and we ha e he ou squa e is
commu a i e, we conclude ha he le iangle is also commu a i e.
Co olla y 4.2.3. Le (𝐂, ⊗)a ca ego y wi h an ope a ion dis ibu i e and annihi-
la ed, whe e 𝐂is addi i e. Then, o 𝐴𝑓
←←←←←←←→ 𝐵,𝐶𝑔
←←←←←←→ 𝐷mo phisms and 𝐸0
←←←←←←→ 𝐹, we
ha e:
(i) 𝑓 ⊗ 0∶ 𝐴⊗𝐸→𝐵 ⊗ 𝐹 is he ze o mo phism.
(ii) 0⊗ 𝑓 ∶𝐸 ⊗ 𝐴 →𝐹 ⊗ 𝐵 is he ze o mo phism
(iii) 𝑓 ⊗ (−𝑔) = (−𝑓)⊗ 𝑔 = −(𝑓 ⊗ 𝑔)
132 4 On he Loday-Pi ash ili Ca ego y
P oo . I ollows using he p e ious heo em oge he wi h he Abelian g oup p ope -
ies o he homomo phisms.
Rema k 4.2.4.Since he de ini ion o addi i e ca ego y esides in na u al mo phisms,
i is immedia e ha i 𝐂has bip oduc s, hen he ca ego y Hom(𝐂)also has bip oduc s.
In ac , i can be easily p o ed ha he ope a ion in his ca ego y is (𝑓, 𝑔)+(ℎ, 𝑘) =
(𝑓+ℎ, 𝑔+𝑘). Thus, i 𝐂is addi i e, hen Hom(𝐂)is addi i e and −(𝑓, 𝑔) = (−𝑓, −𝑔).
4.3 Lie and Leibniz objec s in LP
4.3.1 Lie objec s and Leibniz Objec s
In his subsec ion, we will s udy he Lie objec s and he Leibniz objec s. The de ini-
ions o ha concep s can be seen in De ini ion 2.4.3 and De ini ion 2.4.5.
Lemma 4.3.1. Le = (𝐂, ⊗, 𝑎, 𝜏)be a symme ic semig oupal ca ego y whe e 𝐂is
an addi i e ca ego y. Le (𝐿, 𝜇)be a Leibniz objec . Then,
𝜇◦(Id𝐿⊗𝜇) = −𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿).
P oo . By symme y, i we compose he Leibniz iden i y wi h 𝑎−1◦(Id𝐿⊗𝜇)◦𝑎, we
ge
𝜇◦(𝜇 ⊗ Id𝐿)◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿=𝜇◦(𝜇 ⊗ Id𝐿) + 𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿.
Subs i u ing i in he Leibniz iden i y, we ob ain
𝜇◦(𝜇 ⊗ Id𝐿) = 𝜇◦(Id𝐿⊗𝜇)◦𝑎𝐿+𝜇◦(𝜇 ⊗ Id𝐿)
+𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿
Finally, sub ac ing 𝜇◦(𝜇 ⊗ Id𝐿) + 𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿in bo h sides o he
iden i y, we ge
−𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿=𝜇◦(Id𝐿⊗𝜇)◦𝑎𝐿,
which composed wi h 𝑎−1
𝐿gi es us he desi ed iden i y.
4.3.1 Lie objec s and Leibniz Objec s 133
P oposi ion 4.3.2. Le = (𝐂, ⊗, 𝑎, 𝜏)be a b aided symme ical semig oupal ca -
ego y whe e 𝐂is an addi i e ⊗-dis ibu i e ⊗-annihila ed ca ego y. The ollowing
iden i y is called he Jacobi iden i y:
𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗(𝐿⊗𝐿)+𝑎𝐿◦𝜏𝐿,𝐿⊗𝐿 +𝜏𝐿⊗𝐿,𝐿◦𝑎−1
𝐿) = 0.(Jac)
Then (𝐿, 𝜇)is a Lie objec i and only i i sa is ies (AC) and (Jac).
P oo . Le (𝐿, 𝜇)be a Lie objec and le us p o e ha (Jac) holds. We can ew i e he
Leibniz iden i y as
0 = 𝜇◦(Id𝐿⊗𝜇) + 𝜇◦(𝜇 ⊗ Id𝐿)◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿) − 𝜇◦(𝜇 ⊗ Id𝐿)◦𝑎−1
𝐿.
The i s and hi d summands a e equal o he i s and hi d summands o (Jac), e-
spec i ely, since
𝜏𝐿,𝐿◦(𝜇 ⊗ Id𝐿) = (Id𝐿⊗𝜇)◦𝜏𝐿⊗𝐿,𝐿.
To see he second one, we will use (AC), he na u alness and symme y o 𝜏, he
second hexagon equa ion and Lemma 4.3.1:
𝜇◦(𝜇 ⊗ Id𝐿)◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿) = −𝜇◦𝜏𝐿,𝐿◦(𝜇 ⊗ Id𝐿)◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿)
= −𝜇◦(Id𝐿⊗𝜇)◦𝜏𝐿⊗𝐿,𝐿◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿)
= −𝜇◦(Id𝐿⊗𝜇)◦𝜏−1
𝐿,𝐿⊗𝐿◦𝑎−1
𝐿◦(Id𝐿⊗𝜏𝐿,𝐿)
= −𝜇◦(Id𝐿⊗𝜇)◦𝑎𝐿◦(𝜏𝐿,𝐿 ⊗Id𝐿)−1◦𝑎−1
𝐿
= −𝜇◦(Id𝐿⊗𝜇)◦𝑎𝐿◦(𝜏𝐿,𝐿 ⊗Id𝐿)◦𝑎−1
𝐿
=𝜇◦(Id𝐿⊗𝜇)◦(Id𝐿⊗𝜏𝐿,𝐿)◦𝑎𝐿◦(𝜏𝐿,𝐿 ⊗Id𝐿)◦𝑎−1
𝐿
=𝜇◦(Id𝐿⊗𝜇)◦𝑎𝐿◦𝜏𝐿,𝐿⊗𝐿.
Con e sely, using Theo em 4.2.2 and Co olla y 4.2.3, he iden i y (AC) gi es us
he same equali y gi en in Lemma 4.3.1.
−𝜇◦Id𝐿⊗𝜇◦Id𝐿⊗𝜏𝐿,𝐿 =𝜇◦Id𝐿⊗(−𝜇)◦Id𝐿⊗𝜏𝐿,𝐿
=𝜇◦Id𝐿⊗(−𝜇◦𝜏𝐿,𝐿) = 𝜇◦Id𝐿⊗𝜇.
Then, he iden i y (Lb) au oma ically ollows.