a Xi :1111.2723 4 [ma h.AT] 30 Ap 2014
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS
FERNANDO MURO
Abs ac . We show ha he canonical map om he associa i e ope ad o
he uni al associa i e ope ad is a homo opy epimo phism o a wide class o
symme ic monoidal model ca ego ies. As a consequence, he space o uni al
associa i e algeb a s uc u es on a gi en objec is up o homo opy a subse o
connec ed componen s o he space o non-uni al associa i e algeb a s uc u es.
1. In oduc ion
I is well known ha monoids in a monoidal ca ego y, a.k.a. algeb as, may ha e
a mos one uni . Hence, being uni al can be ega ded as a p ope y, a he han
a s uc u e. In o he wo ds, he se o uni al monoid s uc u es on a gi en objec
embeds as a subse o he se o non-uni al monoid s uc u es. This ac can be
deduced om he ollowing s onge and ancie s a emen .
P oposi ion 1.1. Gi en a closed symme ic monoidal ca ego y Vwi h an ini ial
objec , he canonical mo phism φV:AssV→uAssV om he associa i e ope ad
o he uni al associa i e ope ad is an epimo phism in he ca ego y Op(V)o non-
symme ic ope ads in V.
The canonical mo phism φVmodels he o ge ul unc o om uni al monoids
o non-uni al monoids.
I Vis also a model ca ego y, one is o en mo e in e es ed in homo opy algeb a
s uc u es a he han s ic algeb a s uc u es. This is because, gi en a monoid M
and a weak equi alence ϕ:X∼
→Min V, he e need no be a monoid s uc u e on
Xcompa ible wi h ϕ, bu he e is always a compa ible homo opy monoid s uc u e
on X, a leas i Xis ib an and co ib an .
Homo opy (uni al) associa i e algeb as a e known as (uni al) A-in ini y algeb as.
They a e o mally de ined as algeb as o e co ib an esolu ions o he ope ads
AssVand uAssV. I V= Top is he ca ego y o opological spaces, he e a e nice
esolu ions o hese ope ads gi en by associahed a [S a63] and uni al associahed a
[MT14]. The cellula homology o (uni al) associahed a yield esolu ions o V=
Ch(k) he ca ego y o chain complexes o e a commu a i e ing k.
The s onges possible homo opical gene aliza ion o P oposi ion 1.1 is he ol-
lowing esul , which is he main heo em o his pape .
Theo em 1.2. Le Vbe a simplicial o complicial closed symme ic monoidal
model ca ego y. Assume ha Vsa is ies he monoid axiom and he s ong uni
axiom. Suppose u he ha Vis co ib an ly gene a ed and has se s o gene a ing
( i ial) co ib a ions wi h p esen able sou ces. Then he mo phism φV:AssV→
uAssVis a homo opy epimo phism in Op(V).
1991 Ma hema ics Subjec Classi ica ion. 18D50, 18G55.
Key wo ds and ph ases. Ope ad, A-in ini y algeb a, uni , model ca ego y, mapping space.
1
2 FERNANDO MURO
This means ha aking de i ed mapping spaces in he model ca ego y Op(V)
[Mu 11a] ou o φV,
(φV)∗: MapOp(V)(uAssV,O)−→ MapOp(V)(AssV,O),
is essen ially an inclusion o connec ed componen s o any ope ad O, i.e. an injec-
ion on π0and an isomo phism in all homo opy g oups πn,n > 0, wi h all possible
base poin s. Pu ing O=EndV(X), he endomo phism ope ad o an objec X
in V, we deduce ha he homo opical moduli space [Rez96] o uni al A-in ini y
algeb a s uc u es on Xembeds as a subse o connec ed componen s o he homo-
opical moduli space o all A-in ini y algeb a s uc u es on X. In [Mu 11b] we go
beyond, showing ha i Xis pe ec hen φVinduces an a ine Za iski open imme -
sion o geome ic moduli spaces in many homo opical algeb aic geome y con ex s,
including de i ed, complicial, and b a e new algeb aic geome y.
A iendly cha ac e iza ion o he image o he injec i e map π0(φV)∗when
O=EndV(X) is an endomo phism ope ad is possible o many V’s hanks o
esul s o Lyubashenko–Manzyuk and Lu ie, see Rema ks 5.11 and 6.5.
Le us commen on he hypo heses o Theo em 1.2. A symme ic monoidal model
ca ego y V[SS00] is simplicial i i is equipped wi h a symme ic monoidal Quillen
adjunc ion om he ca ego y o simplicial se s,
Se ∆op F//W.
G
oo
The uppe a ow will always be he le adjoin in his kind o diag am. Simila ly,
Vis complicial i i is equipped wi h a symme ic monoidal Quillen adjunc ion
Ch(k)F//W.
G
oo
The s ong uni axiom, in oduced in [Mu 14, De ini ion A.9], says ha enso ing
wi h a co ib an eplacemen ˜
Io he enso uni Ip ese es all weak equi alences.
This ob iously holds i Iis co ib an , bu i is also ue in many o he cases o in-
e es , such as diag am spec a wi h he posi i e s able model s uc u e [MMSS01].
The es o he hypo heses a e needed o ha e a model s uc u e on Op(V) wi h
ib a ions and weak equi alences de ined as in V, see [Mu 11a, Theo em 1.1].
The pape is s uc u ed as ollows. Sec ion 2 s udies homo opy epimo phisms
in a bi a y model ca ego ies. In Sec ion 3 we ecall wha we need abou ope ads
and hei homo opy heo y. Sec ions 4 and 5 con ain he p oo o Theo em 1.2 o
wo special ca ego ies V: g oupoids and unbounded complexes o e a commu a i e
ing. In he las sec ion, Sec ion 6, we deduce he main heo em om hese wo
speci ic cases.
We assume he eade amilia i y wi h ca ego y heo y and abs ac homo opy
heo y. Some s anda d e e ences a e [Mac98, Ho 99, Hi 03]. Fo monoidal ca e-
go ies, unc o s, and adjunc ions, we e e o [AM10, Chap e 3].
Acknowledgemen s. A p e ious e sion o his pape only con ained Theo em 5.1,
wi h a subs an ially mo e complica ed p oo . Lec u ing abou his esul a he Ho-
mo opical Algeb a Summe Day in Ba celona 2012, I ealised o he possibili y o
simpli ying he p oo , as i is gi en in Sec ion 5. The simpli ica ion needs he esul s
in [Mu 14], which a e o independen in e es . I’m g a e ul o he o ganize s o ha
Summe Day, Imma G´al ez and Ja ie Gu i´e ez, o p o iding such an inspi ing
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 3
en i onmen . The esul s in [Mu 14] also alowed me o ex end Theo em 5.1 o a
wide class o model ca ego ies, see Theo em 1.2. I wished o do his since I ob ained
he i s p oo o Theo em 5.1, and I’m g a e ul o Joe Hi sh o encou aging me o
do so du ing he Summe Day.
I was pa ially suppo ed by he Andalusian Minis y o Economy, Inno a ion
and Science unde he g an FQM-5713, by he Spanish Minis y o Educa ion and
Science unde he MEC-FEDER g an MTM2010-15831, and by he Go e nmen
o Ca alonia unde he g an SGR-119-2009.
2. Homo opy epimo phisms
Recall ha a mo phism :X→Yin a ca ego y Cis an epimo phism i
C( , Z): C(Y, Z)→C(X, Z) is an injec i e map o any objec Zin C. The
ollowing cha ac e iza ion o epimo phisms is well known and easy o check.
P oposi ion 2.1. Le :X→Ybe a mo phism in a ca ego y C. Assume he
push-ou
X //

push
Y
i2

Yi1
//Y∪XY
exis s. The ollowing s a emen s a e equi alen :
(1) is an epimo phism.
(2) i1is an isomo phism.
(3) i2is an isomo phism.
(4) The codiagonal ∇= (1Y,1Y): Y∪XY→Yis an isomo phism.
I hey hold, hen i1=i2=∇−1.
The s onges homo opy in a ian p ope y which gene alizes he no ion o in-
jec i e map is he ollowing one.
De ini ion 2.2. A map g:K→Lbe ween simplicial se s is a homo opy monomo -
phism i i gi es ise o an injec ion on connec ed componen s,
π0(g): π0(K)֒→π0(L),
and isomo phisms on homo opy g oups o all possible base poin s x∈K0,
πn(g): πn(K, x)∼
=
−→ πn(L, g(x)), n ≥1.
The e a e o he ob ious cha ac e iza ions o homo opy monomo phisms o sim-
plicial se s.
Lemma 2.3. Gi en a map g:K→Lbe ween simplicial se s, he ollowing s a e-
men s a e equi alen :
(1) gis a homo opy monomo phism.
(2) gco es ic s o a weak equi alence be ween Kand a subse o connec ed
componen s o L.
(3) Fo any x∈K0, he homo opy ibe o ga g(x)is con ac ible.
(4) The homo opy ibe s o ga e emp y o con ac ible.
4 FERNANDO MURO
When we say ha a simplicial se is con ac ible we mean ha i is weakly
equi alen o a poin . The usual e minology is ‘weakly con ac ible’ bu we p e e
o sho en i .
The e a e also less ob ious cha ac e iza ions along he lines o he dual o P opo-
si ion 2.1.
P oposi ion 2.4. Le g:K։Lbe a Kan ib a ion be ween Kan complexes. Con-
side he pull-back squa e
K×LKp2////
p1

pull
K
g

Kg////L
The ollowing s a emen s a e equi alen :
(1) gis a homo opy monomo phism.
(2) p1is a weak equi alence.
(3) p2is a weak equi alence.
(4) The diagonal ∆ = 1K
1K:K→K×LKis a weak equi alence.
I hey hold, hen p1=p2= ∆−1in he homo opy ca ego y o simplicial se s.
P oo . Since pj∆ = 1K,j= 1,2, he equi alences (2) ⇔(3) ⇔(4) and he inal
s a emen a e clea .
In o de o show (1) ⇔(2), no ice ha pa allel a ows in he squa e o he
s a emen ha e essen ially he same ibe s. Mo e p ecisely, i Fxdeno es he ibe
o p1o e a base poin x∈K0, hen Fxis isomo phic o he ibe o go e g(x).
The map π0(p1) is su jec i e since p1∆ = 1K. The e o e, by he long exac sequence
in homo opy g oups, Fxis con ac ible o any x∈K0i and only i p1is a weak
equi alence. 
This p oposi ion is ac ually use ul o cha ac e ize when an a bi a y mo phism
go simplicial se s is a homo opy monomo phism, since his p ope y is homo opy
in a ian , so we can eplace gby a weakly equi alen mo phism which is a Kan
ib a ion be ween Kan complexes.
We now de ine homo opy epimo phisms in model ca ego ies ia mapping spaces
and homo opy monomo phisms o simplicial se s. This de ini ion is dual o he
no ion o homo opy monomo phism in [To¨e07].
De ini ion 2.5. A mo phism :X→Yin a model ca ego y Mis said o be a
homo opy epimo phism i o any objec Zin M, he induced mo phism on de i ed
mapping spaces,
∗= MapM( , Z): MapM(Y, Z)−→ MapM(X, Z),
is a homo opy monomo phism o simplicial se s.
Rema k 2.6.This de ini ion is compa ible wi h De ini ion 2.2, i.e. a mo phism
g:K→Lo simplicial se s is a homo opy monomo phism in he sense o De ini ion
2.2 i and only i i is a homo opy epimo phism in he opposi e o he model ca ego y
o simplicial se s in he sense o De ini ion 2.5. This ollows om P oposi ion 2.4.
The cons uc ion o de i ed mapping spaces we ha e in mind is he simplicial
se
MapM(X, Z) = M(˜
X, Z•),
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 5
whe e ˜
Xis a co ib an esolu ion o Xand Z•is a simplicial esolu ion o Z. In
pa icula , ∗= MapM( , Z) = M(˜
, Z•), whe e ˜
:˜
X→˜
Yis a li ing o o
co ib an esolu ions o Xand Y,
˜
X˜
//
∼

˜
Y
∼

X
//Y
I is usual o equi e co ib an esolu ions ˜
X∼
→X o be i ial ib a ions om a
co ib an objec . Howe e , o us i is enough o ha e a weak equi alence wi h
co ib an sou ce.
No ice ha :X→Ybeing a homo opy epimo phism only depends on he
image o in he homo opy ca ego y Ho M. Ac ually, i only depends on he
isomo phism class o in Ho M.
The ollowing esul cha ac e izes homo opy epimo phisms along he lines o
P oposi ion 2.1. The dual ca ac e iza ion o homo opy monomo phisms was no iced
in [To¨e07].
P oposi ion 2.7. Le :X֌Ybe a co ib a ion be ween co ib an objec s in a
model ca ego y M. Conside he push-ou squa e
X//
//


push
Y

i2

Y//i1
//Y∪XY
The ollowing s a emen s a e equi alen :
(1) is a homo opy epimo phism.
(2) i1is a weak equi alence.
(3) i2is a weak equi alence.
(4) The codiagonal ∇is a weak equi alence.
I hey hold, hen i1=i2=∇−1in Ho M.
P oo . Since ∇ij= 1Y,j= 1,2, he equi alences (2) ⇔(3) ⇔(4) and he inal
s a emen a e clea .
I we apply MapM(−, Z) = M(−, Z•) o he push-ou in he s a emen , we
ob ain a pull-back o simplicial se s consis ing o Kan ib a ions be ween Kan com-
plexes,
M(X, Z•)oooo
∗
OOOO
∗pull
M(Y, Z•)
OOOO
i∗
2
M(Y, Z•)ooooi∗
1
M(Y∪XY, Z•)
Hence (1) ⇔(2) ollows om P oposi ion 2.4 and he ac ha i∗
1is a weak equi a-
lence o simplicial se s o all objec s Zin Mi and only i i1is a weak equi alence
in M.

6 FERNANDO MURO
Rema k 2.8.P oposi ion 2.7 is ac ually use ul o check whe he any mo phism
in Mis a homo opy epimo phism. A mo phism :X→Yin Mis a homo opy
epimo phism i and only i a co ib an esolu ion ˜
:˜
X֌˜
Yo is. Such a co ib an
esolu ion is a co ib a ion be ween co ib an objec s i ing in o a commu a i e
diag am
˜
X//
˜
//
∼

˜
Y
∼

X
//Y
The s a emen s (2), (3) and (4) in P oposi ion 2.7 only depend on he isomo -
phism class o he commu a i e squa e
˜
X//
˜
//

˜

push
˜
Y

i2

˜
Y//i1
//˜
Y∪˜
X˜
Y
in he homo opy ca ego y Ho(M) o commu a i e squa es in M. The isomo -
phism class o his squa e only depends on he isomo phism class o :X→Y
in Ho M. Ac ually, i can be cons uc ed using he de i a o DMo M, which
consis s o all homo opy ca ego ies o diag ams in Mwi h he shape o a ini e
di ec ca ego y, such as ,• → •, o • ← • → •, see [Cis10]. A ca ego y is ini e
and di ec i i s ne e has ini ely-many non-degene a e simplices. Le Ca be he
ca ego y o ca ego ies and unc o s and Di ⊂Ca he ull subca ego y o ini e
di ec ca ego ies. The de i a o DMis he 2- unc o ,
DM: Di op −→ Ca ,
I7→ Ho(MIop ).
Appa en ly, he e is a p oblem he e wi h he size o Ca . Mo phism ‘se s’ in Ca
may be p ope classes. Ne e heless, he e is eally no ouble, since mo phism se s
in Di a e hones se s, so he de i a o DMis insensi i e o he p oblems o Ca .
Le D: Di op →Ca be an abs ac de i a o , mo e p ecisely, a igh de i a o
sa is ying [Cis10, De 5], i.e. [Mal07, De 5]. I edeno es he ca ego y wi h only
one objec and one mo phism ( he iden i y), one can gi e a de ini ion o homo opy
epimo phism in D(e) along he lines o (2), (3) and (4) abo e, ex ending he no ion
o homo opy epimo phism in DM(e) = Ho M. Homo opy epimo phisms a e p e-
se ed by cocon inuous mo phisms o igh de i a o s, in pa icula by equi alences
o de i a o s. This obse a ion yields a quick jus i ica ion o he ollowing co ol-
la y. The i s pa also ollows easily om he elemen a y p ope ies o mapping
spaces.
Co olla y 2.9. Le F:M⇄N:Gbe a Quillen adjunc ion be ween model ca -
ego ies and le LF: Ho M⇄Ho N:RGbe he de i ed adjoin pai be ween ho-
mo opy ca ego ies. The unc o LFp ese es homo opy epimo phisms. Mo eo e ,
i F⊣Gis a Quillen equi alence hen RGalso p ese es homo opy epimo phisms.
Fu he mo e, i LF e lec s isomo phisms hen i also e lec s homo opy epimo -
phisms.
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 7
Rema k 2.10.This is a con inua ion o he p e ious ema k. Le X
¯
֌˜
Y∼
→Y
be a ac o iza ion o in o a co ib a ion ollowed by a weak equi alence. I Mis
le p ope he gluing lemma holds, see [Hi 03, P oposi ion 13.5.4]. The e o e, he
p e ious push-ou squa e is isomo phic o
X//
¯
//

push
˜
Y
i2

Y//i1
//Y∪X˜
Y
in Ho(M). In pa icula , is a homo opy epimo phism i and only i his i1is a
weak equi alence.
The gluing lemma also holds in co ib a ion ca ego ies [Bau89, II.1.2 (b)]. Hence,
he same is ue i X,Yand ˜
Ybelong o a ull subca ego y o Mwhich is a
co ib a ion ca ego y wi h co ib a ions and weak equ ialences de ined as in M.
3. Ope ads
All ope ads conside ed in his pape a e non-symme ic.
De ini ion 3.1. Le Vbe a symme ic monoidal ca ego y wi h enso p oduc ⊗
and enso uni I. An ope ad Oin Vis a sequence O={O(n)}n≥0o objec s in
Vequipped wi h an iden i y,
idO:I→ O(1),
and composi ion laws, 1 ≤i≤p,q≥0,
◦i:O(p)⊗ O(q)−→ O(p+q−1),
sa is ying ce ain associa i i y and uni equa ions, see [Mu 11a, Rema k 2.6]. We
e e o O(n) as he a i y ncomponen o O.
Amo phism o ope ads :O → P is a sequence o mo phisms (n): O(n)→
P(n) in V,n≥0, compa ible wi h he iden i ies and composi ion laws in he
ob ious way. We usually d op he a i y om he no a ion (n) in o de o simpli y.
We deno e by Op(V) he ca ego y o ope ads in V.
Rema k 3.2.I V= Se is he ca ego y o se s, he iden i y is simply an elemen
idO∈ O(1) and he associa i i y and uni equa ions a e:
(1) (a◦ib)◦jc= (a◦jc)◦i+q−1bi 1 ≤j < i and c∈ O(q).
(2) (a◦ib)◦jc=a◦i(b◦j−i+1 c) i b∈ O(p) and i≤j < p +i.
(3) idO◦1a=a.
(4) a◦iidO=a.
The same happens i V= Top is he ca ego y o opological spaces o he ca ego y
Mod(k) o modules o e a commu a i e ing k.
I V= Mod(k)Zis he ca ego y o Z-g aded k-modules hen idOmus be in
deg ee 0, idO∈ O(1)0, and (1) mus be eplaced wi h
(1′) (a◦ib)◦jc= (−1)|b||c|(a◦jc)◦i+q−1bi 1 ≤j < i and c∈ O(q).
This e lec s he use o he Koszul sign ule in he de ini ion o he symme y
cons ain o he enso p oduc in Mod(k)Z.
8 FERNANDO MURO
Fu he mo e, i V= Ch(k) is he ca ego y o di e en ial g aded k-modules he
iden i y mus be a cycle, d(idO) = 0, and he di e en ial mus beha e as a de i a ion
wi h espec o all composi ion laws,
d(a◦ib) = d(a)◦ib+ (−1)|a|a◦id(b).
In his pape di e en ials ha e deg ee |d|=−1, i.e. we conside chain complexes.
The ca ego y V= G d o g oupoids wi h he ca esian symme ic monoidal
s uc u e beha es essen ially as Se . The iden i y idOis an objec o he g oupoid
O(1).
Rema k 3.3.Ope ads can be al e na i ely (and a e usually) desc ibed in e ms o
mul iplica ion mo phisms,n≥1, p1,...pn≥0,
O(n)⊗ O(p1)⊗ · · · ⊗ O(pn)−→ O(p1+···+pn),
de ined by i e a ing composi ion laws, e.g. i Vis any o he ca ego ies in he
p e ious ema k, his mo phism is gi en by
(a, b1,...,bn)7→ a(b1,...,bn)
= (···((a◦1b1)◦p1+1 b2)◦p1+p2+1 ···)◦p1+···+pn−1+1 bn.
This i e a ed composi ion can be exp essed in many di e en ways, o ins ance, i
V= Se o Mod(k),
a(b1,...,bn) = (···((a◦nbn)◦n−1bn−1)◦n−2···)◦1b1.
I V= Mod(k)Zo Ch(k) his o mula would be ue up o a sign de e mined by
he Koszul ule.
The mul iplica ion mo phisms oge he wi h he iden i y and ce ain associa i -
i y and uni equa ions yield an equi alen de ini ion o ope ad, see [Mu 11a, Rema k
2.5]. I Vis he ca ego y o se s o k-modules hese equa ions a e:
a(b1(c11,...,c1p1),......,bn(cn1,...,cnpn))
=a(b1,...,bn)(c11,...,c1p1,......,cn1,...,cnpn),
idO(a) = a,
a(idO,...,idO) = a.
I Vis he ca ego y o g aded modules we mus al e he i s equa ion wi h a
sign, acco ding o he Koszul ule. In he di e en ial g aded case, in addi ion,
he di e en ial mus beha e like a de i a ion wi h espec o he mul iplica ion
mo phisms, i.e.
d(a(b1,...,bn)) = d(a)(b1,...,bn) +
n
X
i=1
(−1)
|a|+
i−1
P
j=1
|bj|
a(b1,...,d(bi),...,bn).
Example 3.4.The uni al associa i e ope ad uAssVin V, whose algeb as a e uni al
monoids, is gi en by uAssV(n) = I o all n≥0. The iden i y o his ope ad
iduAssV:I→uAssV(1) is simply he iden i y mo phism in I, and all composi ion
laws a e gi en by he uni isomo phism I⊗I∼
=I, wich is pa o he symme ic
monoidal s uc u e o V.
Example 3.5.Suppose Vis closed and has an ini ial objec ∅. The associa i e
ope ad AssVin V, whose algeb as a e non-uni al monoids, is gi en by AssV(n) = I
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 9
o all n≥1 and AssV(0) = ∅. The ope ad s uc u e is de e mined by he ac
ha he sequence o mo phisms
φV:AssV−→ uAssV
gi en by he iden i y in Iin all posi i e a i ies, φV(n) = idI,n≥1, is a mo phism
o ope ads.
Rema k 3.6.Suppose Vis ca esian closed and has an ini ial objec ∅. I ⊗=×is
he ca esian p oduc hen Iis he inal objec in Vand uAssVis he inal ope ad,
i.e. he inal objec in Op(V). Hence, φVis he only possible map. Mo eo e , AssV
is he inal objec o he ull subca ego y o ope ads which a e ∅in a i y 0. This
happens when Vis Se , Top, o G d.
Rema k 3.7.I Vis cocomple e, hen so is Op(V). In his case, he o ge ul
unc o om Op(V) o he ca ego y VNo sequences V={V(n)}n≥0o objec s in
Vhas a le adjoin , he ee ope ad unc o ,
VNF//Op(V).
o ge
oo
We some imes w i e F=FV. This adjunc ion is monadic, i.e. Op(V) is he
ca ego y o algeb as o e he ee ope ad monad. These ac s allow he cons uc ion
o ope ads by p esen a ions. A p esen a a ion o an ope ad Oconsis s o desc ibing
Oas he coequalize o wo pa allel a ows be ween ee ope ads,
F(U)////F(V)//O.
The ee ope ad unc o Fcan be explici ly desc ibed in e ms o plan ed plana
ees wi h lea es, see [Mu 11a, §3 and §5]. Plan ing a ee consis s o choosing a
deg ee 1 e ex, called oo . The deg ee o a e ex is he numbe o adjacen edges.
The plana s uc u e is gi en by an o de in he se o e ices which indica es how
o d aw hem om le o igh . The lea es a e speci ied deg ee 1 e ices di e en
om he oo . They can be dis inguished in pic u es since we do no d aw hem.
We do no d aw he oo ei he , bu he e is no con usion since he oo is placed a
he bo om. Ve ices a e dis ubu ed in ascending laye s acco ding o he dis ance
o he oo . We call inne e ices hose which a e d awn, i.e. he e ices which
a e nei he lea es no he oo . A co k is an inne e ex o deg ee 1. An inne
edge is an edge which is no adjacen o he oo o o a lea . These no ions a e
be e illus a ed wi h a pic u e,
b
b
b
b
b
This is a plan ed plana ee wi h ou lea es and i e inne e ices, including wo
co ks. The e a e ou inne edges. Some imes, abusing language, we also call lea
o oo o he adjacen edge, which is wha we eally depic . F om now on, in
he whole pape , whene e we alk abou ees we mean plan ed plana ees wi h
lea es.
16 FERNANDO MURO
P oo . The ‘unde lying se ’ unc o V(I,−): V→Se is pa o a lax-lax symme ic
monoidal adjoin pai
Se −⊗I
//V.
V(I,−)
oo
The le adjoin sends a se S o he cop oduc o copies o he enso uni indexed
by his se S⊗I=∐s∈SI. This adjoin pai induces an adjoin pai be ween
ca ego ies o ope ads
Op(Se ) −⊗I
//Op(V).
V(I,−)
oo
No ice ha AssSe ⊗I=AssV,uAssSe ⊗I=uAssV, and φSe ⊗I=φV. Hence, his
p oposi ion ollows om he p e ious one, since le adjoin s p ese e epimo phisms.

P oposi ion 3.12 is ue e en i Vdoes no ha e cop oduc s. The p oo o
P oposi ion 3.11 can be ansla ed in o diag ams in o de o check his gene al
case, P oposi on 1.1.
I Vhas a sui able model s uc u e, compa ible wi h he monoidal s uc u e,
hen he ca ego y o ope ads Op(V) ca ies an induced model s uc u e.
Theo em 3.13 ([Mu 11a, Theo em 1.1]).Le Vbe a co ib an ly gene a ed closed
symme ic monoidal model ca ego y. Assume ha Vsa is ies he monoid axiom.
Mo eo e , suppose ha he e a e se s o gene a ing co ib a ions Iand gene a ing
i ial co ib a ions Jin Vwi h p esen able sou ces. Then he ca ego y Op(V)o
ope ads in Vis a co ib an ly gene a ed model ca ego y such ha a mo phism :O →
Pin Op(V)is a weak equi alence ( esp. ib a ion) i and only i (n): O(n)→ P(n)
is a weak equi alence ( esp. ib a ion) in V o all n≥0.
We e e he eade o [Ho 99, §4] and [SS00] o he heo y o symme ic
monoidal model ca ego ies. All ca ego ies Vin his pape will sa is y he as-
sump ions in his heo em, and his will be he only model s uc u e conside ed on
Op(V).
Rema k 3.14.Le us desc ibe se s o gene a ing ( i ial) co ib a ions in Op(V).
The model s uc u e in he p e ious heo em is ans e ed along he ee ope ad
adjunc ion in Rema k 3.7. The ca ego y o sequences VNis endowed wi h he
p oduc model s uc u e.
Recall ha gi en an objec Xin Vand n≥0, we deno e by X[n] he sequence
consis ing o Xconcen a ed in a i y nand he ini ial objec ∅elsewhe e. Gi en
a mo phism :X→Yin Vwe deno e by [n]: X[n]→Y[n] he mo phism o
sequences de ined by in a i y nand he iden i y in ∅elsewhe e. Fo any se So
mo phisms in V, we w i e
SN=[
n≥0
{ [n] ; ∈S}.
The se s INand JNa e se s o gene a ing co ib a ions and gene a ing i ial co i-
b a ions in VN, espec i ely. Hence, F(IN) and F(JN) a e se s o gene a ing co i-
b a ions and gene a ing i ial co ib a ions in Op(V).

HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 17
De ini ion 3.15. Le Vbe a symme ic monoidal model ca ego y sa is ying he
hypo heses o Theo em 3.13. An A-in ini y ope ad AV
∞in Op(V) is a co ib an
esolu ion o AssV,
AV
∞
∼
−→ AssV.
Simila ly, a uni al A-in ini y ope ad uAV
∞is a co ib an esolu ion o uAssVin
Op(V),
uAV
∞
∼
−→ uAssV.
Au-in ini y associa i e ope ad u∞AVis he middle e m o a ac o iza ion o φV
as a co ib a ion ¯
φV
∞ ollowed by a weak equi alence,
AssV¯
φV
∞
֌u∞AV∼
−→ uAssV.
Rema k 3.16.Fo each speci ic V, we may choose a co ib an esolu ion o φV,
φV
∞:AV
∞֌uAV
∞.
I Vsa is ies he s ong uni axiom [Mu 14, De ini ion A.9], e.g. i he enso uni
is co ib an , hen we can de ine a u-in ini y associa i e ope ad as he ollowing
push-ou ,
AV
∞//
φV
∞//
∼

push
uAV
∞
∼

AssV//
¯
φV
∞//u∞AV
He e, he igh e ical map is a weak equi alence by [Mu 14, Co olla y C.3 and
Theo em C.7]. Hence, he map u∞AV→uAssVinduced by he uni e sal p ope y
o he push-ou is a weak equi alence by he 2-ou -o -3 axiom.
De ini ion 3.17. Au-in ini y uni al associa i e ope ad u∞uAVis an ope ad i ing
in o a push-ou squa e as ollows,
AssV//
¯
φV
∞//
φV

push
u∞AV
ψV

uAssV//
ϕV
//u∞uAV
We will p o e Theo em 1.2 using he ollowing lemma.
Lemma 3.18. Le Vbe a symme ic monoidal model ca ego y as in Theo em
3.13. Assume u he ha Vsa is ies he s ong uni axiom. The mo phism
φV:AssV→uAssVis a homo opy epimo phism in Op(V)i and only i ϕVis
a weak equi alence.
P oo . The ope ads AssVand uAssVbelong o he ull subca ego y Oppc(V)⊂
Op(V) spanned by he ope ads Owhose componen s O(n) a e pseudo-co ib an o
all n≥0. Recall om [Mu 14, De ini ion A.1] ha an objec Xin Vis pseudo-
co ib an i he unc o X⊗ − p ese es co ib a ions. The enso uni Iand he
ini ial objec ∅a e ob iously pseudo-co ib an . The ca ego y Oppc(V) inhe i s
om Op(V) he s uc u e o a co ib a ion ca ego y, see [Mu 14, P oposi ion C.8].
The ope ads u∞AVand u∞uAVa e also in Oppc(V), see [Mu 14, Co olla y C.2].
Hence, his lemma ollows om Rema k 2.10. 
18 FERNANDO MURO
The ollowing lemma is use ul o check ha some symme ic monoidal ca ego ies
ca y a compa ible model s uc u e.
Lemma 3.19. Le F:V⇄W:Gbe a lax-lax symme ic monoidal adjunc ion
be ween symme ic monoidal ca ego ies. Suppose ha Vis a co ib an ly gene a ed
model ca ego y sa is ying he push-ou p oduc axiom in [SS00, De ini ion 3.1]. As-
sume u he ha Wpossesses a ans e ed model s uc u e along his adjunc ion,
in he sense o [Hi 03, Theo em 11.3.2]. Then Walso sa is ies he push-ou p oduc
axiom.
P oo . Le Iand Jbe se s o gene a ing ( i ial) co ib a ions o V. Then F(I) and
F(J) a e se s o gene a ing ( i ial) co ib a ions o W. Deno e by
⊙g:U⊗Y[
U⊗X
V⊗X−→ V⊗Y
he push-ou p oduc o wo mo phisms :U→Vand g:X→Y. In o de
o check he push-ou p oduc axiom o W, i is enough o p o e ha he se s
F(I)⊙F(I) and F(I)⊙F(J) consis o co ib a ions and i ial co ib a ions in W,
espec i ely, compa e [Ho 99, Co olla y 4.2.5]. The monoidal unc o Fis s ong,
see [AM10, P oposi ion 3.96]. I also p ese es push-ou s, since i is a le adjoin .
Hence, Fp ese es push-ou p oduc s. In pa icula ,
F(I)⊙F(I) = F(I⊙I), F(I)⊙F(J) = F(I⊙J).
These se s consis o co ib a ions and i ial co ib a ions, espec i ely, since V
sa is ies he push-ou p oduc axiom and Fp ese es ( i ial) co ib a ions. 
4. Main heo em o ope ads o g oupoids
Le Gpd be he model ca ego y o small g oupoids. Mo phisms a e unc o s and
weak equi alences a e equi alences o ca ego ies. A co ib a ion is a unc o which
is injec i e on objec s. Fib a ions a e unc o s sa is ying he isomo phism li ing
p ope y. Recall ha ϕ:G→Hhas he isomo phism li ing p ope y i o any
objec xin Gand any isomo phism :ϕ(x)→yin H he e exis s an isomo phism
′:x→x′in Gwi h ϕ( ′) = , in pa icula ϕ(x′) = y.
T i ial ib a ions ha e a simple cha ac e iza ion.
Lemma 4.1. A i ial ib a ion in Gpd is a ully- ai h ul unc o su jec i e on
objec s.
I is enough o no ice ha an equi alence o ca ego ies sa is ies he isomo phism
li ing p ope y i and only i i is su jec i e on objec s.
P oposi ion 4.2. The ca ego y Gpd wi h he ca esian p oduc is a combina o ial
closed symme ic monoidal model ca ego y sa is ying he monoid axiom whe e all
objec s a e co ib an .
P oo . The model s uc u e on Gpd is ans e ed along he ollowing adjoin pai
Se ∆op Π1//G d .
Ne
oo
He e, Ne is he ne e unc o and Π1is he undamen al g oupoid unc o . We
ega d Se ∆op as a symme ic monoidal model ca ego y wi h he usual model s uc-
u e and he ca esian p oduc monoidal s uc u e, see [Ho 99, P oposi ion 4.2.8].
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 19
Since Se ∆op is co ib an ly gene a ed, hen so is Gpd. Mo eo e , Gpd is locally
p esen able ( his is an elemen a y ac om ca ego y heo y). Hence, Gpd is a
combina o ial model ca ego y.
The unc o Π1is known o p ese e p oduc s. The e o e, he push-ou p oduc
axiom o G d ollows om Lemma 3.19. All objec s a e co ib an by he e y de -
ini ion o co ib a ion. Hence, he monoid axiom ollows om he push-ou p oduc
axiom, see [SS00, Rema k 3.4]. 
The main esul o his sec ion is he ollowing heo em.
Theo em 4.3. The mo phism φGpd :AssGpd →uAssGpd in Example 3.5 is a
homo opy epimo phism in Op(Gpd).
This heo em ollows om Lemma 3.18 abo e and Lemma 4.15 below.
Rema k 4.4.The se s o gene a ing ( i ial) co ib a ions o Gpd ob ained by ak-
ing undamen al g oupoids on he usual se s o gene a ing ( i ial) co ib a ions o
Se ∆op a e oo big. We can al e na i ely ake
I={∅֌e, i:{e, e′}֌E, p:Z֌e}, J ={j:e∼
֌E}.
He e eis he inal g oupoid, which consis s o only one objec and one mo phism
( he iden i y), Zis he g oupoid wi h one objec wi h au omo phism g oup Z, and
Eis he g oupoid wi h wo isomo phic objec s, e∼
=e′, wi h i ial au omo phism
g oups. The unc o iis he inclusion o he disc e e subg oupoid o med by he
wo objec s, and jis he inclusion o an objec . Recall ha a g oupoid Gis disc e e
i he only mo phisms in Ga e he inden i ies.
Indeed, a unc o sa is ying he igh li ing p ope y wi h espec o ∅֌e,i,
o p, is a unc o su jec i e on objec s, ull, o ai h ul, espec i ely.
Rema k 4.5.Limi s a e easie han colimi s in he ca ego y o g oupoids, a leas
easie han non- il e ed colimi s. Howe e , colimi s beha e well on objec s, in he
sense ha he se o objec s o he colimi o a diag am o g oupoids is he colimi
o he diag am o objec se s. This ollows om he ac ha he ‘se o objec s’
unc o om g oupoids o se s has a igh adjoin ,
G d Ob //Se .
con ac ible
oo
The igh adjoin , called ‘con ac ible g oupoid’ unc o , sends he emp y se o he
emp y g oupoid, and any non-emp y se S o he con ac ible g oupoid wi h objec
se S. Recall ha a g oupoid Gis con ac ible i i is equi alen o e, i.e. i i has
a non-emp y se o objec s and he e exis s a unique isomo phism be ween any o
objec s o G. Hence, mo phisms in o con ac ible g oupoids a e usually deno ed by
simply indica ing he sou ce, he a ge , and he map be ween objec se s.
The ‘con ac ible g oupoid’ and he ‘se o objec s’ unc o s p ese e p oduc s,
hence hey induce an adjoin pai on ope ads,
Op(G d) Ob //Op(Se ).
con ac ible
oo
In pa icula , he ‘se o objec s’ unc o also p ese es colimi s a he le el o
ope ads. This ac is used in he p oo o he ollowing lemma.
20 FERNANDO MURO
In his sec ion we conside ee ope ads o se s and ee ope ads o g oupoids.
Fo he sake o simplici y, we omi he subsc ip om he ee ope ad unc o o
se s F=FSe , bu no o g oupoids FG d, in o de o a oid con usion. These ee
ope ad unc o s and he objec se unc o commu e, Ob FG d =FOb, since he
la e p ese es ca esian p oduc s, compa e Rema k 3.7.
Lemma 4.6. A mo phism is a co ib a ion in Op(G d) i and only i i is a e ac
o a mo phism :O → P such ha Ob( ): Ob(O)→Ob(P) = Ob(O)∐ F(V)is
an inclusion o a ac o o a bina y cop oduc such ha he o he ac o is a ee
ope ad in Op(Se ).
P oo . Any ela i e FG d(IN)-cell complex is as in he s a emen . Indeed, on he
one hand, he unc o s i, p ∈Ia e he iden i y on objec s, hence a push-ou along
FG d(i[n]) o FG d(p[n]) is he iden i y on objec s. On he o he hand, a push-ou
along FG d(∅֌e[n]) adds eely a new objec in a i y n. Hence, he ‘only i ’ pa
ollows.
The con e se is also ue, i.e. any mo phism as is a ela i e FG d(IN)-cell
complex, bu his is complica ed o show di ec ly. In o de o p o e he ‘i ’ pa ,
i is easie o check ha in he s a emen sa is ies he le li ing p ope y wi h
espec o i ial ib a ions. We he e o e conside a commu a i e diag am o solid
a ows in Op(G d) as ollows,
Og//

Q
∼q

Ph
//
l
??
⑦
⑦
⑦
⑦R
He e, qis a i ial ib a ion. In o de o ob ain a li ing l, we i s conside he
diag am o objec s
Ob(O)Ob(g)//
Ob( )

Ob(Q)
Ob(q)

Ob(O)∐ F(V)Ob(h)
//
l′77
♣
♣
♣
♣
♣
♣Ob(R)
He e, Ob(q) is le elwise su jec i e by Lemma 4.1. Hence, i is easy o ob ain a li ing
l′in Op(Se ). De ine l′as Ob(g) on he i s ac o . On he second ac o , we choose
p eimages o he objec s h(V(n)) along q(n), n≥0, and ex end o a mo phism om
he ee ope ad F(V). Finally, since q(n) is ully ai h ul, he e is a unique unc o
l(n): P(n)→ Q(n) gi en by l′(n) on objec s and such ha q(n)l(n) = h(n), n≥0.
One can easily check ha he sequence o unc o s {l(n)}n≥0is an ope ad mo phism
lwhich also sa is ies l =g.
Co olla y 4.7. An ope ad Oin Op(G d) is co ib an i and only i he ope ad o
objec se s Ob(O)is a e ac o a ee ope ad in Op(Se ).
We can now easily de ine a u-in ini y associa i e ope ad o g oupoids.
De ini ion 4.8. The u-in ini y associa i e ope ad u∞AG d in Op(G d) is he le el-
wise con ac ible ope ad wi h ope ad o objec s Ob(u∞AG d) = AssSe ∐ F({u}[0]).
Lemma 4.9. The ope ad u∞AG d in he p e ious de ini ion is indeed a u-in ini y
associa i e ope ad in he sense o De ini ion 3.15. The mo phism ¯
φG d
∞:AssG d ֌
u∞AG d is gi en on objec s by he inclusion o he i s ac o o he cop oduc .
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 21
P oo . The mo phism ¯
φG d
∞is a co ib a ion by Lemma 4.6. Mo eo e , he unique
mo phism u∞AG d →uAssG d is a weak equi alence since u∞AG d is le elwise con-
ac ible. The composi ion o hese mo phisms is φG d since uAssG d is inal in
Op(G d). 
Rema k 4.10.We now desc ibe he ope ad o objec s o u∞AG d ollowing Rema ks
3.9 and 3.10. The se Ob(u∞AG d)(n), o n > 1, can be iden i ied wi h he se o
co ollas wi h nlea es, a leas wo b anches, and possibly co ks, e.g.
µ4(id, u, u, id,id) =
b
b b
∈Ob(u∞AG d(3)),
Ab anch o a ee is an edge adjacen o a lea o a co k. Fo n= 0,1, in addi ion
o he co ollas wi h nlea es, a leas wo b anches, and possibly co ks, we ha e
u=
b
∈Ob(u∞AG d(0)),id = ∈Ob(u∞AG d(1)).
By equi ing a leas wo b anches we a e explici ly excluding he ollowing wo
co ollas,
b
,
b
b
.
The composi ion laws (be ween ees di e en om id = |) a e gi en by g a ing
and hen con ac ing he newly c ea ed inne edge,
b
b
◦2
b
b b
b
b b
b b
b
b b b
,
µ2(id, u, id) ◦2µ4(id, u, u, id,id) = µ6(id, u, id, u, u, id,id),
excep when he g a ed ee is u. In ha case, he new inne edge is no con ac ed,
µ2(id, u, id) ◦2u=
b
b
◦2
b
=
b
b b
=µ2(id, u, u).
Compa e Example 3.10.
Lemma 4.11. The ope ad u∞AG d is gene a ed by he objec s µand uand by he
isomo phisms µ(u, id) ∼
=id and µ(id, u)∼
=id.
P oo . The p e ious ema k shows ha any objec in u∞AG d can be ob ained om
µand u. We mus show ha he unique exis ing isomo phism be ween any wo
objec s can be ob ained om µ(u, id) ∼
=id and µ(id, u)∼
=id. I is enough o p o e
ha we can ge all mo phisms wi h a ge µn−1, he co olla wi h nlea es and no
co ks, n≥2, all mo phisms wi h a ge id = |, and all mo phisms wi h a ge u.

22 FERNANDO MURO
S a ing wi h an objec o posi i e a i y ep esen ed by a co olla as in he p e ious
ema k, he isomo phisms in he s a emen , ha we can espec i ely deno e
b
b
λ
−→ ,
b
b
ρ
−→ ,
allow o dele e one co k a a ime, ending up wi h a co olla wi h no co ks o wi h
id = |, depending on he a i y, e.g.
b
b b
b
b
ρ
//
b
b
b
λ
//
b
.
In a i y 0, we can use he isomo phism
λ◦1u=ρ◦1u:
b
b b
−→
b
,
o educe he numbe o co ks, ending up wi h u. Hence, we a e done. 
We now de ine an ope ad ha we will la e show o be a u-in ini y uni al asso-
cia i e ope ad o g oupoids, see Lemma 4.15 below.
De ini ion 4.12. The ope ad o g oupoids Uis de ined as he le elwise con ac ible
ope ad wi h objec s
Ob(U) = uAssSe ∐ F({u′}[0]).
Rema k 4.13.Following Rema ks 3.9 and 4.10, we he e desc ibe he ope ad o se s
Ob(U). We can conside he mo phism in Op(G d)
ψ:u∞AG d −→ U
gi en on objec s by
Ob(ψ) = φSe ∐(iso. u7→ u′): AssSe ∐ F({u}[0]) −→ uAssSe ∐ F({u′}[0]).
The mo phism ψis an isomo phism in posi i e a i ies, ha we use as an iden i i-
ca ion. In a i y 0, ψin an inclusion o objec s (and hence mo phisms). We also
iden i y u∞AG d(0) wi h i s image in U(0) h ough ψ(0). The ex a objec o U(0)
is u, which is ep esen ed by a i ial co olla wi h a whi e co k, as in Example 3.10.
The e o e, black co k means u′and whi e co k means u,
µ4(id, u′, u′,id,id) =
b
b b
, u′=
b
, u =
bc
.
The composi ion laws a e de ined in e ms o ees as in Rema k 4.10 when uis no
in ol ed. I uappea s, i is almos always gi en by g a ing and hen con ac ing
he newly c ea ed inne edge,
b
b b
◦2
bc
b
b b bc
b
b b
,
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 23
µ4(id, u′, u′,id,id) ◦2u=µ4(id, u′, u′, u, id) = µ3(id, u′, u′,id).
The e a e ou excep ions, he wo excep ions in Example 3.10 and
(µ(id, u′))◦1u
b
bc b
b
=u′,(µ(u′,id))◦1u
b
bcb
b
=u′.
Lemma 4.14. The ope ad U i s in o he ollowing push-ou diag am,
FG d(e[0]) //
FG d(j[0])
∼//
ζ

push
FG d(E[0])
ζ′

uAssG d //∼
ϕ//U
whe e ϕis gi en on objec s by he inclusion o he i s ac o o he cop oduc , ζis
de ined by ζ(e) = u, and ζ′is de ined by ζ′(e) = uand ζ′(e′) = u′.
P oo . Deno e by
FG d(e[0]) //
FG d(j[0])
∼//
ζ

push
FG d(E[0])
¯
ζ

uAssG d //∼
¯//P
he push-ou in Op(G d). The squa e in he s a emen is clea ly commu a i e, so
i induces a unique compa ible mo phism χ:P → U. We a e going o show ha χ
is an isomo phism.
The squa e in he s a emen , on objec s, is a push-ou in Op(Se ),
F({e}[0]) incl. //
Ob(ζ)

push
F({e, e′}[0]) F({e}[0]) ∐ F({e′}[0])
Ob(ζ′)=Ob(ζ)∐(iso. e′7→u′)

uAssSe
incl. //uAssSe ∐ F({u′}[0])
Hence, χis bijec i e on objec s. The e o e, in o de o show ha χis an iso-
mo phism i is enough o p o e ha Pis le elwise con ac ible. This is ob ious.
Indeed FG d(j[0]) is a gene a ing i ial co ib a ion, so ¯is a i ial co ib a ion, in
pa icula a weak equi alence, i.e. ¯(n): e=uAssG d(n)→ P(n) is an equi alence
o ca ego ies o all n≥0. 
Lemma 4.15. Conside he commu a i e squa e
AssG d //
¯
φG d
∞//
φG d

u∞AG d
ψ
∼

uAssG d //∼
ϕ//U
whe e ϕand ψwe e de ined in Lemma 4.14 and Rema k 4.13, espec i ely. The
mo phisms ϕand ψa e weak equi alences since hei sou ces and hei a ge a e
le elwise con ac ible by de ini ion. We asse ha he p e ious commu a i e squa e
is a push-ou in Op(G d).
24 FERNANDO MURO
P oo . In o de o wa m up, he eade can easily check ha he squa e in he
s a emen is a push-ou on objec s. We ackle di ec ly he s a emen . We a e going
o p o e ha he squa e sa is ies he uni e sal p ope y o a push-ou . Wi h his
pu pose, we conside a commu a i e diag am o solid a ows in Op(G d),
AssG d //
¯
φG d
∞//
φG d

u∞AG d
ψ
∼
g

uAssG d //∼
ϕ//
//
U
h
##
●
●
●
●
●
O
whe e φG d =g¯
φG d
∞. We will show ha he e exis s a unique mo phism hcom-
ple ing he diag am in a commu a i e way, i.e. wi h wo new commu a i e iangles,
=hϕ and g=hψ.
The ollowing equa ion holds in U,
b
−→
bc
=
b
b
−→ !◦1
bc
.
The e o e, i hexis ed, i should sa is y
h
b
−→
bc
=g
b
b
−→ !◦1 
bc
.
By Lemma 4.14, he e exis s a unique hsa is ying his equa ion and =hϕ (no ice
ha he equa ion implies h(u) = (u)). Hence, i is only le o p o e ha g=hψ.
I is enough o show ha his equa ion holds o he gene a o s in Lemma 4.11.
This is ob ious o µ, since i comes om AssG d. Fo u∈u∞AG d(0), which is he
black co k,
h
b
=g
b
b
!◦1 
bc
= g
b
!◦2g
b
!◦1 
bc

=
b
!◦1 
bc
!◦1g
b
=   ◦1g
b

= idO◦1g
b
=g
b
.
The gene a ing isomo phisms sa is y he ollowing equa ions in U,
b
b
λ
−→ =
b
◦1
b
−→
bc
,
b
b
ρ
−→ =
b
◦2
b
−→
bc
.
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 25
Hence,
h(λ) = h
b
!◦1h
b
−→
bc

=g
b
!◦1 g
b
b
−→ !◦1 
bc
!
= g
b
!◦1g
b
b
−→ !! ◦1 
bc

=g
b
b
−→
b
!◦1 
bc

= g
b
!◦2g
b
b
−→ !! ◦1 
bc

=
b
!◦1 
bc
!◦1g
b
b
−→ !
=g
b
b
−→ !=g(λ),
h(ρ) = h
b
!◦2h
b
−→
bc

=g
b
!◦2 g
b
b
−→ !◦1 
bc
!
= g
b
!◦2g
b
b
−→ !! ◦2 
bc

=g
b
b
−→
b
!◦2 
bc

= g
b
b
−→ !◦1g
b
!! ◦2 
bc

=g
b
b
−→ !◦1
b
!◦2 
bc
!
=g
b
b
−→ !=g(ρ).
This concludes he p oo . 
5. Main heo em o DG-ope ads
Le Ch(k) be he ca ego y o DG-modules o e a g ound commu a i e ing k.
Weak equi alences in Ch(k) a e quasi-isomo phisms and ib a ions a e le elwise
su jec i e maps. In his way, Ch(k) wi h he usual enso p oduc becomes a
combina o ial closed symme ic monoidal model ca ego y wi h co ib an enso uni
32 FERNANDO MURO
i (n, m)6= (2,1),(1,1),
d(νS
n) = (−1)nµ◦1νS
n−1unless lm=n
+µ◦2νS−1
n−1unless l1= 1
+X
1≤ ≤m+1
l −1<i+ −1<l −1
(−1)i+ −1νS ∪(S′
−1)
n−1◦iµ;
and i m= 1 also
d(ν{1}
1) = 0, d(ν{1}
2) = µ◦1ν{1}
1−id, d(ν{2}
2) = µ◦2ν{1}
1−id.
No ice ha P, whose unde lying g aded ope ad is uAssCh(k)∐ F(k· {νS
n}n,S), has
a linea di e en ial ega ded as an ope ad unde uAssCh(k).
As a g aded ope ad Q=uAssCh(k)∐ F(k· {σS
n, d(σS
n)}n,S). He e σS
nis he op
gene a o o he copy o Dn+m−1indexed by nand S, so i has deg ee n+m−1
and a i y n−m. The di e en ial o Qis linea oo.
The mo phism o DG-pope ads Φ unde uAssCh(k)is de ined by
Φ(σS
n) = (−1)l1+1νS+1
n+1 ◦l1u.
No ice ha Φ is linea .
The de ini ion o Ψ is mo e complica ed. Le h:uAssCh(k)∐P1→uAssCh(k)∐Q1
be he deg ee +1 mo phism o uAssCh(k)-modules de ined by
h(νS
n) = σS
n, h(id) = 0.
Mo eo e , le p:uAssCh(k)∐P1→uAssCh(k)∐Q1be he deg ee 0 mo phism gi en by
p(id) = id,
p(νS
n) = 0 i n > 1,
p(ν{1}
1) = ui m= 1.
We de ine Ψ as a mo phism o g aded ope ads unde uAssCh(k)by
Ψ(νS
n) = dh(νS
n) + hd(νS
n) + p(νS
n) = d(σS
n) + hd(νS
n)
+ui (n, m) = (1,1).
Le us check ha Ψ is compa ible wi h di e en ials. No ice ha Ψ is linea by
de ini ion and he o mula
Ψ = dh +hd +p
holds on he uAssCh(k)-module gene a o s id and νS
no uAssCh(k)∐ P1, hence i
holds a e (co) es ic ing o linea pa s, i.e. when we e alua e each side a an
elemen in uAssCh(k)∐ P1we ob ain an equali y in uAssCh(k)∐ Q1. Then,
Ψd(νS
n) = dhd(νS
n) + hd2(νS
n) + pd(νS
n) = dhd(νS
n),
dΨ(νS
n) = d2h(νS
n) + dhd(νS
n) + dp(νS
n) = dhd(νS
n).
He e we use ha pd(νS
n) = 0 = dp(νS
n) o any νS
n. Indeed, p(νS
n) = 0 i (n, m)6=
(1,1) and dp(ν{1}
1) = d(u) = 0, so dp(νS
n) = 0 in any case. Mo eo e , as poin ed ou

HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 33
abo e, i νS
n6=ν{1}
1, ν{1}
2, ν{2}
2each summand in d(νS
n) con ains a ce ain νS′
n′6=ν{1}
1,
so p(νS′
n′) = 0 and pd(νS) = 0. Fu he mo e,
pd(ν{1}
1) = p(0) = 0,
pd(ν{1}
2) = p(µ◦1ν{1}
1−id) = µ◦1p(ν{1}
1)−id = µ◦1u−id = id −id = 0,
and simila ly pd(ν{2}
2) = 0.
We inally p o e ha ΦΨ = 1P. I is enough o check he equa ion on gene a o s
νS
nsince Φ and Ψ a e mo phisms o DG-ope ads unde uAssCh(k). We s a wi h
he gene al case (n, m)6= (1,1),(2,1):
ΦΨ(νS
n) = Φ(d(σS
n) + hd(νS
n))
=dΦ(σS
n) + Φhd(νS
n)
= (−1)l1+1d(νS+1
n+1 ◦l1u)
+
(a) unless lm=n
z}| {
(−1)n+l1+1µ◦1(νS+1
n◦l1u) +
(b) unless l1=1
z}| {
(−1)l1µ◦2(νS
n◦l1−1u)
+X
0<i<l1−1
(ci)
z}| {
(−1)i+l1(νS
n◦l1−1u)◦iµ
+X
1< ≤m+1
l −1<i+ −1<l −1
(di)
z}| {
(−1)i+ +l1(ν(S +1)∪S′
n◦l1u)◦iµ,
(−1)l1+1d(νS+1
n+1 ◦l1u) =
−(a) unless lm=n
z}| {
(−1)n+l1(µ◦1νS+1
n)◦l1u+
(b′)
z}| {
(−1)l1+1(µ◦2νS
n)◦l1u
+X
0<i<l1
−(ci)
z}| {
(−1)i+l1+1(νS
n◦iµ)◦l1u
+X
1< ≤m+1
l −1+1<i+ −1<l
(d′
i)
z}| {
(−1)i+ +l1(ν(S +1)∪S′
n◦iµ)◦l1u .
The summands (a) and −(a) ei he do no occu (i lm=n) o cancel. Mo eo e ,
(di) = −(d′
i+1), so all he (di) and he (d′
i) cancel. I l1= 1, he e a e no (ci), (b)
does no occu , and
(b′) = (µ◦2νS
n)◦1u= (µ◦1u)◦1νS
n= id ◦1νS
n=νS
n.
I l1>1 hen (b) = −(b′) and all he (ci) cancel excep o he las one:
−(cl1−1) = (νS
n◦l1−1µ)◦l1u=νS
n◦l1−1(µ◦2u) = νS
n◦l1−1id = νS
n.
The e o e ΦΨ(νS
n) = νS
n.
34 FERNANDO MURO
Le us inally check he special cases (n, m) = (1,1),(2,1):
ΦΨ(ν{1}
1) = Φ(d(σ{1}
1) + hd(ν{1}
1) + u)
=dΦ(σ{1}
1) + u
=d(ν{2}
2◦1u) + u
= (µ◦2ν{1}
1−id) ◦1u+u
= (µ◦1u)◦1ν{1}
1−u+u
= id ◦1ν{1}
1
=ν{1}
1,
ΦΨ(ν{1}
2) = Φ(d(σ{1}
2) + hd(ν{1}
2))
=dΦ(σ{1}
2) + Φhd(ν{1}
2)
=d(ν{2}
3◦1u) + Φh(µ◦1ν{1}
1−id)
= (−µ◦1ν{2}
2+µ◦2ν{1}
2)◦1u+µ◦1(ν{2}
2◦1u)
= (µ◦1u)◦1ν{1}
2
=ν{1}
2,
ΦΨ(ν{2}
2) = Φ(d(σ{2}
2) + hd(ν{2}
2))
=dΦ(σ{2}
2) + Φhd(ν{2}
2)
=−d(ν{3}
3◦2u) + Φh(µ◦2ν{1}
1−id)
=−(µ◦2ν{2}
2−ν{2}
2◦1µ)◦2u+µ◦2(ν{2}
2◦1u)
=ν{2}
2◦1(µ◦2u)
=ν{2}
2.

Co olla y 5.10. The mo phism ϕCh(k)in Rema k 5.6 is a i ial co ib a ion.
P oo . By Rema k 5.8 and Lemma 5.9, ϕCh(k)is a ans ini e (coun able) compo-
si ion o i ial co ib a ions. Hence ϕCh(k)is i sel a i ial co ib a ion. 
Rema k 5.11.The e a e iendly cha ac e iza ions o he image o he injec i e map
π0(φCh(k))∗:π0MapOp(Ch(k))(uAssCh(k),O)֒→π0MapOp(Ch(k))(AssCh(k),O)
when O=EndCh(k)(X) is he endomo phism ope ad o a co ib an complex X.
Since all complexes a e ib an , hese se s a e he se s o homo opy classes o maps
om he (uni al) A-in ini y DG-ope ad o EndCh(k)(X), i.e. homo opy classes o
(uni al) A-in ini y s uc u es on X.
Recall om Rema k 5.5 he desc ip ion o he (uni al) A- inini y DG-ope ad
(u)ACh(k)
∞and he mo phism φCh(k)
∞:ACh(k)
∞֌uACh(k)
∞. An A-in ini y s uc u e on
Xis gi en by g aded mo phisms mn:X⊗n→Xo deg ee n−2 sa is ying ce ain
equa ions. A uni al A-in ini y s uc u e is simila ly de ined by g aded mo phisms
mS
n:X⊗n−|S|→Xo deg ee n−2 + |S|. The unde lying A-in ini y s uc u e is
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 35
gi en by he mo phisms mn=m∅
n:X⊗n→X. A uni al A-in ini y s uc u e is
s ic i mS
n= 0 o S6=∅and n > 1.
Following he e minology in [Lu 12, De ini ion 5.2.3], we say ha an A-in ini y
s uc u e is quasi-uni al i he e exis s a cycle ∈X0,d( ) = 0, such ha he
chain maps m2( , −), m2(−, ): X→Xa e chain homo opic o he iden i y. The
unde lying A-in ini y s uc u e o a uni al A-in ini y s uc u e is quasi-uni al. We
can ake =m{1}
1∈X0and he chain homo opies m{1}
2, m{2}
2:X→X.
I kis a ield we can suppose ha Xhas i ial di e en ial. Then quasi-uni al
means ha m2has a uni . I is well known ha such a quasi-uni al A-in ini y
s uc u e is quasi-isomo phic, in he A-in ini y sense, o a s ic ly uni al A-in ini y
s uc u e, ia a quasi-isomo phism whose linea e m is he iden i y [LH03, §3.2.1].
One can check ha wo A-in ini y s uc u es which a e quasi-isomo phic in his way
ep esen he same elemen in π0MapOp(Ch(k))(AssCh(k),EndCh(k)(X)). The e o e,
he image o π0(φCh(k))∗is o med exac ly by he homo opy classes o quasi-uni al
A-in ini y s uc u es. A deepe esul o Lyubashenko and Manzyuk [LM08, The-
o em 3.7] shows ha his s a emen is ac ually ue o e any commu a i e ing
k.
We do no hink ha a simila esul holds o any a ge ope ad O. Ou opinion
is based in he ollowing ac s.
The gene aliza ion o a quasi-uni al A-in ini y s uc u e can be s aigh o wadly
de ined as ollows. A mo phism ξ:ACh(k)
∞→ O is quasi-uni al i he e exis s a cycle
∈ O(0)0,d( ) = 0, such ha he ollowing h ee homology classes coincide
[idO] = [ξ(µ2)◦1 ] = [ξ(µ2)◦2 ]∈H0(O(1)).
I ξex ends o uACh(k)
∞by a mo phism ¯
ξ:uACh(k)
∞→ O hen ξis quasi-uni al. We
can ake =¯
ξ(µ{1}
1) since o j= 1,2,
d(¯
ξ(µ{j}
2)) = ¯
ξ(d(µ{j}
2)) = ¯
ξ(µ∅
2◦jµ{1}
1−iduACh(k)
∞) = ξ(µ2)◦j¯
ξ(µ{1}
1)−idO.
Ac ually, i is enough ha ξex ends o he subope ad P ⊂ uACh(k)
∞spanned by µ∅
n,
n≥2, µ{1}
1,µ{1}
2, and µ{2}
2. In pa icula , he inclusion ACh(k)
∞⊂ P is quasi-uni al.
The image o he injec i e map π0(φCh(k))∗consis s o homo oy classes wi h a
quasi-uni al ep esen a i e. Suppose ha , con e sely, all homo oy classes wi h a
quasi-uni al ep esen a i e whe e in he image o any a ge ope ad O. Taking
O=Pwe would ob ain a mo phism ¯
ξ:uACh(k)
∞→ P whose es ic ion o ACh(k)
∞
would be homo opic o he inclusion. Below we show ha he composi ion o
¯
ξ ollowed by he inclusion P ⊂ uACh(k)
∞would be a homo opy au omo phism o
uACh(k)
∞, so uACh(k)
∞would be a homo opy e ac o P. We hink his is e y
unlikely o happen since Pseems oo small in e ms o bo h size and cohe ence o
he quasi-uni . Ne e heless, we ha e been unable o each a con adic ion.
P oposi ion 5.12. Any endomo phism o uACh(k)
∞is a homo opy au omo phism.
P oo . The monoid o homo opy classes o maps om uACh(k)
∞ o i sel coincides
wi h he endomo phism monoid o uAssCh(k), since he homology ope ad o uACh(k)
∞
is uAssCh(k), which is concen a ed in deg ee 0. Recall he p esen a ion o uAssCh(k)
in Example 3.10. Any mo phism ϕ:uAssCh(k)→uAssCh(k)is de e mined by he
image o he gene a o s, which by a i y and deg ee easons mus be o he o m
ϕ(µ) = α·µ, ϕ(u) = β·u, α, β ∈k.
36 FERNANDO MURO
The ela ion id = µ◦2uimplies
id = ϕ(id)
=ϕ(µ◦2u)
=ϕ(µ)◦2ϕ(u)
= (α·µ)◦2(β·u)
=α·β·(µ◦2u)
=α·β·id.
The e o e α·β= 1, i.e. α∈k×and β=α−1. One can con e sely check ha o
any α∈k×,µ7→ αµ and u7→ α−1ude ine an au omo phism o uAssCh(k). Hence,
π0MapOp(Ch(k))(uACh(k)
∞,uACh(k)
∞) = Au Op(Ch(k))(uAssCh(k))∼
=k×.

6. T ans e ence
In his sec ion we p o e he main esul o his pape , Theo em 1.2. The ollowing
p oposi ion is a di ec consequence o [Mu 14, P oposi ion 4.1, Co olla y C.5, and
Theo em E.2] and Co olla y 2.9. I allows o ans e ou p e ious main esul s o
a wide class o symme ic monoidal model ca ego ies.
P oposi ion 6.1. Le F:V⇄W:Gbe a weak symme ic monoidal Quillen ad-
junc ion be ween symme ic monoidal model ca ego ies as in Theo em 3.13. Sup-
pose ha Vand Wsa is y he s ong uni axiom and F⊣Gsa is ies he pseudo-
co ib an axiom and he I-co ib an axiom. The ollowing s a emen s hold:
(1) I φV:AssV→uAssVis a homo opy epimo phism in Op(V) hen he map
φW:AssW→uAssWis a homo opy epimo phism in Op(W).
(2) Suppose in addi ion ha LF: Ho V→Ho W e lec s isomo phisms, e.g. i
F⊣Gis a Quillen equi alence. In his case he con e se o (1) also holds.
The s ong uni axiom, he pseudo-co ib an axiom, and he I-co ib an axiom
we e in oduced in [Mu 14, De ini ions A.9 and B.6]. The s ong uni axiom holds in
all symme ic monoidal model ca ego ies wi h co ib an uni . The pseudo-co ib an
axiom and he I-co ib an axiom hold in all Quillen pai s F⊣Gwhe e he sou ce
o Fhas a co ib an enso uni .
We now p o e he main heo em o he ca ego y Ch(k)≥0o non-nega i e chain
complexes. Weak equi alences and he monoidal s uc u e a e de ined as in Ch(k),
ib a ions a e chain maps which a e su jec i e in posi i e deg ees, compa e [SS03,
§4.1].
Theo em 6.2. The mo phism φCh(k)≥0:AssCh(k)≥0→uAssCh(k)≥0in Example 3.5
is a homo opy epimo phism in Op(Ch(k)≥0).
P oo . Conside he symme ic monoidal adjoin pai
Ch(k)≥0
inclusion //Ch(k).
≥0
oo
The igh adjoin ≥0is he unca ion unc o . I is de ined by he ac ha
he couni ≥0(X)→Xis he iden i y in posi i e deg ees and he inclusion
Ke [d:X0→X−1]⊂X0in deg ee 0. Clea ly, ≥0p ese es weak equi alences
HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 37
and ib a ions, so his is a Quillen pai . The inclusion unc o e lec s weak equi a-
lences, hence i s le de i ed unc o Ho Ch(k)≥0→Ho Ch(k) e lec s isomo phisms.
The ca ego ies Ch(k)≥0and Ch(k) ha e co ib an enso uni s. These obse a ions
show ha ou adjunc ion sa is ies he assump ions o P oposi ion 6.1 (2). The e o e
he esul ollows om Theo em 5.1. 
The ollowing heo em is he main esul o simplicial k-modules wi h he sym-
me ic monoidal model s uc u e conside ed in [SS03, §4.1].
Theo em 6.3. The mo phism φMod(k)∆op :AssMod(k)∆op →uAssMod(k)∆op in Ex-
ample 3.5 is a homo opy epimo phism in Op(Mod(k)∆op ).
P oo . This ollows om Theo em 6.2 and P oposi ion 6.1 (1) applied o he Dold–
Kan equi alence Ch(k)≥0⇄Mod(k)∆op , which is a weak symme ic monoidal
Quillen equi alence, see [SS03, §4.2]. He e, bo h ca ego ies ha e co ib an enso
uni . 
We can now p o e he main esul o simplicial se s, wi h he usual ca esian
symme ic monoidal model s uc u e, see [Ho 99, P oposi ion 4.2.8].
Theo em 6.4. The mo phism φSe ∆op :AssSe ∆op →uAssSe ∆op in Example 3.5 is
a homo opy epimo phism in Op(Se ∆op ).
P oo . We ha e o show ha ϕSe ∆op in De ini ion 3.17 is a weak equi alence, see
Lemma 3.18, o equi alen ly, ha u∞uASe ∆op (n) is con ac ible o all n≥0.
Conside he ollowing wo Quillen pai s,
Se ∆op Π1//G d,
Ne
ooSe ∆op Z·− //Mod(Z)∆op .
o ge
oo
The i s Quillen pai was al eady conside ed in he p oo o P oposi ion 4.2. I is a
symme ic monoidal Quillen pai in he sense o [Ho 99]. The second Quillen pai ,
induced by he ee abelian g oup unc o , is also symme ic monoidal. These ou
unc o s happen o p ese e weak equi alences, so hey coincide wi h hei de i ed
unc o s. These adjoin pai s induce Quillen pai s be ween ope ad ca ego ies, see
[Mu 14, P oposi ion 4.1]. Applying Π1and Z·− o he push-ou squa e in De ini ion
3.17 o V= Se ∆op , we ob ain push-ou diag ams in Op(G d) and Op(Mod(Z)∆op ),
espec i ely,
AssG d
φG d

Π1AssSe ∆op
//
Π1¯
φSe ∆op
∞//
Π1φSe ∆op

push
Π1u∞ASe ∆op
Π1ψSe ∆op

uAssG d Π1uAssSe ∆op
//
Π1ϕSe ∆op //Π1u∞uASe ∆op
AssMod(Z)∆op
φMod(Z)∆op

Z·AssSe ∆op
//
Z·¯
φSe ∆op
∞//
Z·φSe ∆op

push
Z·u∞ASe ∆op
Z·ψSe ∆op

uAssMod(Z)∆op
Z·uAssSe ∆op
//
Z·ϕSe ∆op //Z·u∞uASe ∆op

38 FERNANDO MURO
He e, Π1¯
φSe ∆op
∞and Z·¯
φSe ∆op
∞a e models o ¯
φG d
∞and ¯
φMod(Z)∆op
∞, espec i ely,
see [Mu 14, Theo em 1.7]. Hence, Π1ϕSe ∆op and Z·ϕSe ∆op a e weak equi alences
by Theo ems 4.3 and 6.3 and Lemma 3.18. In pa icula , u∞uASe ∆op (n) is simply
connec ed and has he homology o a poin o all n≥0, he e o e i is con ac ible.

Le us inally p o e ou main heo em.
P oo o Theo em 1.2. I is enough o check ha we can apply P oposi ion 6.1
(1) o he s uc u e symme ic monoidal Quillen pai F⊣Go he simplicial o
complicial monoidal model ca ego y V. We a e assuming ha Vsa is ies he s ong
uni axiom. The ca ego ies Se ∆op and Ch(k) ha e co ib an enso uni s. Hence,
hey sa is y he s ong uni axiom and F⊣Gsa is ies he pseudo-co ib an axiom
and he I-co ib an axiom. 
Rema k 6.5.Once Theo em 1.2 is p o ed, i is easonable o wonde whe he he e
is a iendly iden i ica ion o he image o π0(φV)∗as in Rema k 5.11 o any V
sa is ying he hypo heses o ha heo em and O=EndV(X) he endomo phism
ope ad o a ib an -co ib an objec X.
Suppose o simplici y ha he enso uni Iis co ib an . In his case, i is
possible o de ine quasi-uni al A-in ini y algeb as as ollows. Le us conside a
co ib an esolu ion AV
∞
∼
։AssVwhich is a i ial ib a ion. Since Iis co ib an ,
he i ial ib a ion AV
∞(2) ∼
։AssV(2) = Iis a e ac ion which admi s a sec ion
˜g:I→AV
∞(2). Gi en an A-in ini y s uc u e on X, we de ine m2:X⊗X→Xas
he composi e
X⊗X∼
=I⊗X⊗X˜g⊗id
−→ AV
∞(2) ⊗X⊗X−→ X,
whe e he las mo phism is pa o he A-in ini y s uc u e. We say ha an A-
in ini y s uc u e is quasi-uni al i he e exis s a mo phism :I→Xsuch ha he
maps m2( ⊗X), m2(X⊗ ): X→Xa e homo opic o he iden i y.
I looks like i [Lu 12, Theo em 5.2.3.5] implied a posi i e answe o all V
sa is ying also he hypo heses in [Lu 12, Theo em 4.1.4.4], e.g. chain complexes and
simplicial se s, bu no opological spaces. Howe e , [Lu 12, Theo em 5.2.3.5] is no
abou moduli spaces o algeb a s uc u es, bu abou (gene aliza ions o ) Dwye –
Kan simplicial localiza ions o ca ego ies o algeb as. The connec ion be ween hese
spaces was es ablished by Rezk [Rez96] o symme ic ope ads and V= Se ∆op
o Mod(k)∆op . In [Mu 11b] we p o e he analogous esul in he non-symme ic
con ex o any Vas in Theo em 3.13. Wi h ha esul a hand, we will be able
o answe posi i ely he ques ion aised he e [Mu 11b, Rema k 6.8].
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Uni e sidad de Se illa, Facul ad de Ma em´
a icas, Depa amen o de ´
Algeb a, A da.
Reina Me cedes s/n, 41012 Se illa, Spain
E-mail add ess: mu[email p o ec ed]s
URL:h p://pe sonal.us.es/ mu o