Homotopy units in A-infinity algebras

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 . 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