A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categories
aa r X i v : . [ m a t h . K T ] J u l A Quillen model structure on the category ofKontsevich-Soibelman weakly unital dgcategories
Piergiorgio Panero and Boris Shoikhet
Abstract.
In this paper, we study weakly unital dg categories as they weredefined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantlygenerated Quillen model structure on the category C at dgwu ( k ) of small weaklyunital dg categories over a field k . Our model structure can be thought of as anextension of the model structure on the category C at dg ( k ) of (strictly unital) smalldg categories over k , due to Tabuada [Tab]. More precisely, we show that theimbedding of C at dg ( k ) to C at dgwu ( k ) is a right adjoint of a Quillen pair of functors.We prove that this Quillen pair is, in turn, a Quillen equivalence. In course of theproof, we study a non-symmetric dg operad O , governing the weakly unital dgcategories, which is encoded in the Kontsevich-Soibelman definition. We provethat this dg operad is quasi-isomorphic to the operad A ssoc + of unital associativealgebras. Introduction
Weakly unital A ∞ categories firstly appeared in the definition of Fukaya category in Homologicalmirror symmetry [K2]. Since that, weakly unital dg and A ∞ categories have been studied bymany authors, e.g. [LyMa], [Ly], [LH], [KS], [COS] among the others. Currently there areknown three different definitions of a weakly unital A ∞ (or dg) category [LyMa]. These threedefinitions are due to Fukaya, to Lyubashenko, and to Kontsevich-Soibelman, correspondingly.It was proven loc.cit. that the three definitions are equivalent, which means that if a given A ∞ category is weakly unital in one sense it is also weakly unital in another. Nevertheless, thethree categories of weakly unital A ∞ categories are not equivalent. Their homotopy categorieswere supposed to be equivalent, and equivalent to the homotopy category of strictly unital dgcategories. Our Theorem confirms this claim for the Kontsevich-Soibelman definition. . Theorem 2.2 of [COS] confirms this claim for the Lyubashenko definition. The Lyubashenko weakly unitaldg categories seemingly do not admit a closed model structure, and the proof in loc.cit. is direct. C at dgwu ( k ) of small Kontsevich-Soibelman weakly unitaldg categories over a field k admits all small limits and colimits (Theorem ). Our main resultsshow that there is a closed model structure on C at dgwu ( k ), extending the Tabuada closed modelstructure [Tab] on the category C at dg ( k ) of small unital dg categories over k , and that thetwo closed model categories C at dg ( k ) and C at dgwu ( k ) are Quillen equivalent (Theorem andTheorem ). Weakly unital dg categories emerge as well in some elementary algebraic constructions. Thus, let A be a strictly unital dg algebra over k . Then its bar-cobar resolution Cobar(Bar( A )) is a verynice “cofibrant resolution” of A . It is only true if it is considered as a non-unital dg algebra,because Cobar(Bar( A )) lacks a strict unit. In fact, Cobar(Bar( A )) is Kontsevich-Soibelmanweakly unital, see Example .On the other hand, the bar-cobar resolution is a very natural resolution and one likes toconsider it as a cofibrant replacement of A , when one computes Hom sets in the homotopycategory. Certainly, Hom(Cobar(Bar( A )) , B ) in the non-unital setting is the set of all A ∞ maps (or A ∞ functors, for the case of dg categories). However, it is well-known [LH] that thecorrect Hom set in the homotopy category is defined via the unital A ∞ maps (corresp., unital A ∞ functors). The reason is that one has to take Hom(Cobar(Bar( A )) , B ) in the category of(Kontsevich-Soibelman) weakly unital dg categories, see Definition , and it gives rise exactlyto the unital A ∞ functors A → B , see Example .One of our goals is to develop a suitable categorical environment in which the mentionedfacts fit naturally. Some other applications will appear in our next paper. Let us outline in more detail our main results and the organization of the paper.In Section 1, we recall the Kontsevich-Soibelman definition of weakly unital dg categoriesand of their morphisms, which gives rise to a category C at dgwu ( k ). After that, we prove thatthe category C at dgwu ( k ) admits all small limits and colimits. The products, the coproducts,and the equalizers are constructed directly. The coequalizers are less trivial, to define them weuse technique of monads. We adapt some ideas of [Wo] and [Li], where enriched strictly unitalcase is treated. We construct a monad T on the category of dg graphs and prove in Theorem that the categories of T -algebras and of weakly unital dg categories are equivalent. Thecoequalizers are constructed in Proposition . We also construct a non-symmetric dg operad O such that O -algebras in dg graphs are exactly weakly unital dg categories. Recall that an A ∞ map F : A → B is unital if F (1 A ) = 1 B and F k ( . . . , A , . . . ) = 0 for k ≥
2n Section 2, we prove Theorem which says that there is a cofibrantly generated closedmodel structure on C at dgwu ( k ). We construct sets of generating cofibrations I and of generatingacyclic cofibrations J which are paralleled to those in [Tab]. There is a trick, employed in Lemma , with the acyclic cofibration A → K where K is the Kontsevich dg category with two objects.Namely, we notice that, for any closed degree 0 morphism ξ in a weakly unital dg category C ,the replacement of ξ by ξ ′ = 1 · ξ · ξ ] ∈ H ( C ), and, at the sametime, ξ ′ satisfies 1 · ξ ′ = ξ ′ · ξ ′ . It makes us possible to use Tabuada’s acyclic cofibration A → K in the weakly unital case, without any adjustment. Another new and subtle place isLemma , which, even in the unital case, simplifies the argument. In the weakly unital caseit provides, seemingly, the only possible way to prove Theorem .In Section 3, we provide an adjoint pair of functors L : C at dgwu ( k ) ⇄ C at dg ( k ) : R and prove, in Proposition , that it is a Quillen pair. Moreover, we show in Theorem that it is a Quillen equivalence, if the natural projection of dg operads O → A ssoc + is aquasi-isomorphism.Finally, in Section 4 we prove Theorem which states that the natural projection p : O →A ssoc + is a quasi-isomorphism of dg operads. It completes the proof of Theorem . The proofof Theorem goes by a quite tricky computation with spectral sequences.In Appendix A, we provide some detail to the proof of [Dr, Lemma 3.7], which we employin the proof of Lemma . Acknowledgements
We are thankful to Bernhard Keller for his encouragement. The work of both authors waspartially supported by the FWO Research Project Nr. G060118N. The work of B.Sh. waspartially supported by the Russian Academic Excellence Project 5-100.
We adapt the definition of weakly unital dg categories given in [KS, Sect. 4], where a moregeneral context of A ∞ categories is considered. Let A be a (non-unital) dg category. Denote by k A the unital dg category whose objects are Ob ( A ), for any X ∈ Ob( A ) k A ( X, X ) = k , and k A ( X, Y ) = 0 for X = Y . We denote by 1 X the3nit element in k A ( X, X ). By abuse of notations, we denote, for a non-unital dg category A ,by A ⊕ k A the unital dg category having the same objects that A , and( A ⊕ k A )( X, Y ) = ( A ( X, Y ) X = YA ( X, X ) ⊕ k X X = Y One has a natural imbedding i : A → A ⊕ k A sending X to X , and f ∈ A ( X, X ) to the pair( f, ∈ ( A ⊕ k A )( X, X ). Definition 1.1. A weakly unital dg category A over k is a non-unital dg category A over k ,with a distignuished element id X ∈ A ( X, X ) , for any object X in A , such that d (id X ) = 0and id X ◦ id X = id X , subject to the following condition. One requires that there exists an A ∞ functor p : A ⊕ k A → A , which is the identity map on the objects, such that p ◦ i = id A , andwhich fulfils the conditions: p (1 X ) = id X , p n (1 X , . . . , X ) = 0 for n ≥ , for any X ∈ Ob( A ) Example 1.2.
Let A be a strictly unital dg category. Define p : A ⊕ k A → A as p | A ( X,Y ) = id, p (1 X ) = id X , p n = 0 for n ≥
2. Then p is a dg functor, and p ◦ i = id. It makes a strictly unitaldg category a weakly unital dg category. Lemma 1.3.
Let A be a weakly unital dg category. Then the homotopy category H ( A ) is astrictly unital dg category.Proof. The map [ p ] : H ( A ) ⊕ k H ( A ) → H ( A ), induced by the first Taylor component p ofthe A ∞ functor p , is a dg algebra map. One has [ p ](1 X ) = id X and [ p ] ◦ [ i ] = id. It follows thatid X ◦ f = f ◦ id X = f , for any f ∈ H ( A ). Example 1.4.
Let A be an associative dg algebra over k , with a strict unit 1 A . Consider C = Cobar + (Bar + ( A )) where Bar + ( A ) is the bar-complex of A , which is non-counital dgcoalgebra (thus, Bar + ( A ) = T ( A [1]) / k as a graded space), and Cobar + ( B ) is the non-unitaldg algebra (as a graded space, Cobar + ( B ) = T ( B [ − / k ). It is well-known that the naturalprojection Cobar + (Bar + ( A )) → A is a quasi-isomorphism of non-unital dg algebras . We claimthat Cobar + (Bar + ( A )) is (almost) weakly unital, whose weak unit is 1 A ∈ Cobar + (Bar + ( A )).By “almost” we mean that for p n defined below it is not true that p n (1 , , . . . ,
1) = 0 for n ≥
2. (One can easily take a quotient by the corresponding acyclic ideal, or alternatively onecan regard it as an object of the category C at ′ dgwu ( k ) rather than an object of C at dgwu ( k ), seeSection ).We use notations ω = a ⊗ · · · ⊗ a ℓ ∈ Bar + ( A ) for monomial bar-chains, and c = ω ⊠ ω ⊠ · · · ⊠ ω k for monomial elements in Cobar + (Bar + ( A )).Define p n ( x , . . . , x n ), where each x i is either 1 or a monomial c ∈ Cobar + (Bar + ( A )), asfollows. 41): We set p n ( x , . . . , x n ) to be 0 if for some 1 ≤ i ≤ n − x i , x i +1 are elements inCobar + (Bar + ( A )). (2): Otherwise, let x i , . . . , x i + j +1 be a fragment of the sequence x , . . . , x n such that x i = ω ⊠ · · · ⊠ ω a ∈ Cobar + (Bar + ( A )), x i +1 = · · · = x i + j = 1, x i + j +1 = ω ′ ⊠ · · · ⊠ ω ′ b ∈ Cobar + (Bar + ( A )). Then we replace the fragment x i , x i +1 , . . . , x i + j +1 by the following element γ in Cobar + (Bar + ( A )): γ = ω ⊠ · · · ⊠ ω a − ⊠ ( ω a ⊗ id ⊗ · · · ⊗ id j factors id ⊗ ω ′ ) ⊠ · · · ⊠ ω ′ b (3): We perform such replacements succesively for all suitable fragment, and finally we getan element in Cobar + (Bar + ( A )), of degree P deg x i − n + 1. By definition, this element is p n ( x , . . . , x n ). By a suitable fragment we mean either the case considered above, when a groupof succesive 1s is surrounded by elements of Cobar + (Bar + ( A )) from both sides, or one of thetwo extreme case: if x = 1, the leftmost 1 , , . . . , , x i is a suitable fragment, and similarly if x n = 1, the rightmost fragment x s , , . . . , { p n } n ≥ defines an A ∞ morphism p : Cobar + (Bar + ( A )) ⊕ k → Cobar + (Bar + ( A )) such that p ◦ i = id.The construction for the case of Cobar + (Bar + ( C )), for C a dg category, is similar. We endow the weakly unital dg categories with a category structure, as follows.
Definition 1.5.
Let
C, D be weakly unital dg categories, denote by i C : C → C ⊕ k C , i D : D → D ⊕ k D and by p C : C ⊕ k C → C , p D : D ⊕ k D → D the corresponding functors (see Definition ). A weakly unital dg functor F : C → D is defined as a dg functor of non-unital dg categoriessuch that the diagram below commutes: C ⊕ k C F ⊕ id / / p C (cid:15) (cid:15) D ⊕ k Dp D (cid:15) (cid:15) C F / / D ( Note that the upper horizontal map F ⊕ id is automatically a dg functor of unital dg categories,and p , p are A ∞ maps. Note that it follows that F (id X ) = id F ( X ) ( for any X ∈ Ob( C ).Denote by C at dgwu ( k ) the category of small weakly unital dg categories over k .Similarly we define a category C at ′ dgwu ( k ). Its objects are defined as the objects of C at dgwu ( k )but with dropped conditions p n (1 , . . . ,
1) = 0 for n ≥ p (1) · p (1) = p (1). The mor-phisms are defined as for the category C at dgwu ( k ). One sees that the weakly unital dg algebra5obar + (Bar + ( A )), constructed in Example , is an object of C at ′ dgwu ( k ) (but is not an objectof C at dgwu ( k )).Note that the commutativity of diagram ( ) implies F ( p Cn ( f ⊗ · · · ⊗ f n )) = p Dn ( F ( f ) ⊗ · · · ⊗ F ( f n )) ( for any n morphisms f , . . . , f n in C . Lemma 1.6.
Let F : C → D be a weakly unital dg functor between weakly unital dg categories.Then it defines a k -linear functor H ( F ) : H ( C ) → H ( D ) of unital k -linear categories. It is clear.
Example 1.7.
Let A be a strictly unital dg algebra, consider the weakly unital dg algebra C = Cobar + (Bar + ( A )) (which belongs to C at ′ dgwu ( k ))), constructed in Example . Let D be a strictly unital dg algebra. Then the set Hom C at ′ dgwu ( k ) ( C, D ) is identified with the set of unital A ∞ maps A → D . (Recall that for strictly unital dg algebras A, D , an A ∞ morphism f : A → D map is called unital if f (1 A ) = 1 D , and f n ( a , . . . , a n ) = 0 if n ≥ a i = 1 A ).One has a similar description for the case of dg categories. C at dgwu ( k )It is true that the dg category C at dgwu ( k ) is small complete and small cocomplete. One con-structs directly small products and small coproducts. The equalizers are also straightforward,as follows.Let F, G : C → D be two morphisms. Define Eq( F, G ) as the dg category whose objects areOb(Eq(
F, G )) = { X ∈ Ob( C ) | F ( X ) = G ( X ) } Let
X, Y ∈ Ob(Eq(
F, G )). DefineEq(
F, G )( X, Y ) = { f ∈ C ( X, Y ) | F ( f ) = G ( f ) } It is clear that Eq(
F, G ) is a non-unital dg category. For any X ∈ Ob(Eq(
F, G )), F (id X ) =id F ( X ) and G (id X ) = id G ( X ) , therefore id X ∈ Eq(
F, G )( X, X ).One has to construct an A ∞ functor p : Eq( F, G ) ⊕ k Eq(
F,G ) → Eq(
F, G ) such that p (1 X ) =id X , and p ◦ i = id. We define p Eq(
F,G ) n ( f ⊗ · · · ⊗ f n ) = p Cn ( f ⊗ · · · ⊗ f n )One has to check that p Eq(
F,G ) n ( f ⊗ · · · ⊗ f n ) is a morphism in Eq( F, G ), that is, F ( p Cn ( f ⊗ · · · ⊗ f n )) = G ( p Cn ( f ⊗ · · · ⊗ f n )) ( ) one gets F ( p Cn ( f ⊗ · · · ⊗ f n )) = p Dn ( F ( f ) ⊗ . . . F ( f n ))and G ( p Cn ( f ⊗ · · · ⊗ f n )) = p Dn ( G ( f ) ⊗ · · · ⊗ G ( f n ))Now ( ) follows from F ( f i ) = G ( f i ) for all f i , which holds because all f i are morphisms inEq( F, G ). Thus, Eq(
F, G ) is a weakly unital dg category.To construct the coequalizers is a harder task. For the category V − C at of small V -enrichedcategories, the coequalizers were constructed in [Li] and [Wo], assuming V to be a symmetricmonoidal closed and cocomplete, and were constructed in [BCSW] and [KL] in weaker assump-tions on V . All these proofs rely on the theory of monads. We associate a monad which governsthe weakly unital dg categories in Section .We adapt the approach of [Wo] for a proof of existence of the coequalizers in C at dgwu ( k ).We also prove the corresponding monadicity theorem. Here we recall definions and some general facts on monads and algebras over monads. Thereader is referred to [ML], [R] for more detail.Let C be a category. Recall that a monad in C is given by an endofunctor T : C → C and natural transformations η : Id ⇒ T and µ : T ⇒ T so that the following diagrams commute: T T µ + µT (cid:11) (cid:19) T µ (cid:11) (cid:19) T µ + T T ηT + Id (cid:28) $ ❆❆❆❆❆❆❆ ❆❆❆❆❆❆❆ T µ (cid:11) (cid:19) T T η k s Id z (cid:2) ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ T A monad appears from a pair of adjoint functors. Assume we have an adjoint pair F : C ⇄ D : U ( with adjunction unit and counit η : Id C ⇒ U F and ε : F U ⇒ Id D .7t gives rise to a monad in C , defined as: T = U F, η = η : Id C ⇒ T, µ = U ǫF : T ⇒ T An algebra A over a monad T is given by an object A ∈ C equipped with a morphism a : T A → A such that the following diagrams commute: A η A / / Id A ! ! ❈❈❈❈❈❈❈❈ T A a (cid:15) (cid:15) A T A µ A / / T a (cid:15) (cid:15)
T A a (cid:15) (cid:15) T A a / / A The morphisms of algebras over a monad T are defined as morphisms f : A → B in C such thatthe natural diagram commutes.The category of T -algebras is denotes by C T .There is an adjunction F T : C ⇄ C T : U T which by its own gives rise to a monad.There is a functor Φ : D → C T , sending an object Y of D to the T -algebra A = U Y , with a : T A = U F U Y → U Y = A equal to U ε Y . The functor Φ is called the Eilenberg-Moorecomparison functor .An adjunction ( ) is called monadic if the functor Φ : D → C T is an equivalence.There is a criterium when an adjunction is monadic, called the Beck monadicity theorem .We recall its statement below.Recall that a split coequalizer in a category is a diagram A f / / g / / B h / / s (cid:4) (cid:4) C t (cid:4) (cid:4) such that(1) f ◦ s = id B ,(2) g ◦ s = t ◦ h ,(3) h ◦ t = id C ,(4) h ◦ f = h ◦ g Recall 8 emma 1.8.
A split coequalizer is a coequalizer, and is an absolute coequalizer (that is, ispreserved by any functor).
It is enough to prove the first statement, because a split equalizer remains a split equalizerafter application of any functor. See e.g. [R, Lemma 5.4.6] for detail.Given a pair A f ⇒ g B in a category D , and a functor U : D → C , we say that this pair is U -split if the pair U ( A ) f ⇒ g U ( B )in C can be extended to a split coequalizer. Theorem 1.9.
Let F : C ⇄ D : U be a pair of adjoint functors, and let T = U F be the corre-sponding monad. Consider the Eilenberg-MacLane comparison functor
Φ : D → C T . Then:(1) if D has coequalizers of all U -split pairs, the functor Φ has a left adjoint Ψ : C T → D ,(2) if, furthermore, U preserves coequalizers of all U -split pairs, the unit Id C T ⇒ ΦΨ is anisomorphism,(3) if, furthermore, U reflects isomorphisms (that is, U ( f ) an isomorphism implies f anisomorphism), the counit ΨΦ ⇒ Id D is also an isomorphism.Therefore, if (1)-(3) hold, ( U, F ) is monadic. Conversely, if ( U, F ) is monadic, conditions(1)-(3) hold. The reader is referred to [ML] or [R] for a proof.
There is another monadicity theorem, which gives sufficient but not necessary conditions forΦ : D → C T to be monadic.It uses reflexive pairs in D instead of U -split pairs.A pair of morphisms f, g : A → B in D is called reflexive if there is a morphism h : B → A which splits both f and g : f ◦ h = id B = g ◦ h .We refer the reader to [MLM, Ch.IV.4, Th.2] for a proof of the following result, also knownas the crude monadicity Theorem : Theorem 1.10.
Let F : C ⇄ D : U be a pair of adjoint functors, and let T = U F be thecorresponding monad. Consider the Eilenberg-MacLane comparison functor
Φ : D → C T . Then:
1) if D has coequalizers of all reflexive pairs, the functor Φ admits a left adjoint Ψ : C T → D ,(2) if, furthermore, U preserves these coequalizers, the unit of the adjunction Id C T → Φ ◦ Ψ is an isomorphism,(3) if, furthermore, U reflects isomorphisms, the counit of the adjunction Ψ ◦ Φ → Id D is alsoan isomorphism.Therefore, if (1)-(3) hold, ( U, F ) is monadic. Note that, unlike for Theorem , the converse statement is not true . That is, the conditionsfor monadicity, given in Theorem , are sufficient but not necessary.The following construction is of fundamental importance for both monadicity theorems.In the notations as above, let A ∈ D . Consider two morphisms F U F U A f ⇒ g F U A ( where f = F U ε A and g = ε F UA . (Similarly, one defines such two maps for A ∈ C T ).One has two different extensions of this pair of arrows, which form a U -split coequalizer anda reflexive pair, correspondingly.For the first case, consider U F U F U A Uf / / Ug / / U F U A h / / s { { U A t (cid:127) (cid:127) ( with s = η UF UA , t = η UA , h = U ε A .For the second case, consider F U F U A f / / g / / F U A s } } ( with s = F η UA .The following lemma is proven by a direct check: Lemma 1.11.
For any A ∈ D (or A ∈ C T ), ( ) is a split coequalizer in C , whence ( ) is areflexive pair in D (corresp., in C T ). Note that s is not a U -image of a morphism in D , though U f and
U g are. On the otherhand, s is a morphism in D (corresp., in C T ).10 .2.3 The dg operad O and the monad of weakly unital dg categories A dg graph Γ over k is given by a set V Γ of vertices, and a complex Γ( x, y ) for any ordered pair x, y ∈ V Γ . A morphism F : Γ → Γ is given by a map of sets F V : V Γ → V Γ , and by a mapof complexes F E : Γ ( x, y ) → Γ ( F V ( x ) , F V ( y )), for any x, y ∈ V Γ . We denote by G raphs dg ( k )the category whose objects are unital dg graphs over k .There is a natural forgetful functor U : C at dgwu ( k ) → G raphs dg ( k ), where U ( C ) is a graphΓ with V Γ = Ob( C ), and Γ( x, y ) = C ( x, y ). Proposition 1.12.
The functor U admits a left adjoint F : G raphs dg ( k ) → C at dgwu ( k ) .Proof. We provide a construction of the right adjoint to U .Consider the non-Σ the dg operad O define as the quotient-operad of the free operad gen-erated by the composition operations:(a) the composition operation m ∈ O (2) (b) p n ; i , ...,i k ∈ O ( n − k ) − n +1 , 0 ≤ k ≤ n , 1 ≤ i < i < · · · < i k ≤ n , whose meaning isexplained in ( ) below,(c) a 0-ary operation j ∈ O (0) (which generates the morphisms id x , x ∈ Ob C , for a weaklyunital dg category C )by the following relations: ( i ) the associativity of m , and dm = 0( ii ) m ◦ ( j, j ) = j, dj = 0( iii ) p n ; i ,...,i k = 0 if k = 0 or k = n , n ≥ iv ) p = j, p − = id( v ) relation ( ) below ( Note that (ii) formally follows from the part of (iii), saying that p n ;1 , ,...,n = 0, and (v).For a weakly unital dg category C , the operation p n ; i ,...,i k ( f , . . . , f n − k ) is defined as p n (cid:0) f , . . . , f i − , x i , f i , . . . , f i − , x i , f i − , . . . , f i − , x i , . . . . . . , x k i k , f i k − k +1 , . . . , f n − k (cid:1) ( where by 1 x i s are denoted the morphisms 1 x i ∈ k C for the corresponding objects x i ∈ C .The operad O is freely generated by these operations and by m , with the relations being theassociativity of m and the relations on p n ; i ,...,i k which express that ( ) are the summandsof the Taylor components for the A ∞ functor p : C ⊕ k C → C . These relations read:11 p n ; i ,...,i k = X ≤ ℓ ≤ n − ± m ◦ ( p ℓ ; i ,...,i s ( ℓ ) , p n − ℓ ; i s ( ℓ )+1 ,...,i k )+ n − X r =1 ± p n − j ,...,j q ( r ) ◦ (id , . . . , id , m ( a ( r ) , a ( r + 1)) r , id , . . . , id) ( with the notations explained below.We have to explain notations in ( ). By s ( ℓ ) is denoted the maximal s such that i s ≤ ℓ ; a ( r ) is equal to id if r
6∈ { i , . . . , i k } and is equal to j otherwise. Finally, q ( r ) ∈ { k, k − , k − } ; q ( r ) = k if neither r, r + 1 are in { i , . . . , i k } , and in this case j s = i s for i s ≤ r and j s = i s − i s > r ; q ( r ) = k − r or r + 1 are in { i , . . . , i k } but not both, in this case j s = i s for i s < r , and j s = i s +1 − i s +1 > r ; finally, if both r, r + 1 are in { i , . . . , i k } one sets q ( r ) = k − j s = i s for i s < r , and j s = i s +2 − i s +2 > r + 1.The category G raphs dg ( k ) has a natural internal Hom in V ect dg ( k ). We associate with agraph Γ ∈ G raphs dg ( k ) a 1-globular set enriched over V ect dg ( k ), in the sense of Batanin [Ba], ina standard way. Namely, we set X = V Γ , and X = Q x,y ∈ V Γ Γ( x, y ) + , where Γ( x, y ) + = Γ( x, y )for x = y , and Γ + ( x, x ) = Γ( x, x ) ⊕ k id x . There are maps t , t : X → X , mapping an elementin Γ( x, y ) to x and y , correspondingly, and a map s : X → X sending x to id x . It is an(enriched) 1-globular set, meaning that t s = t s = id X . Therefore, one can talk on algebrasin G raphs dg ( k ) over a dg operad.A structure of a weakly unital dg category C on its underlying graph U ( C ) in G raphs dg ( k )is the same that an action of the operad O on U ( C ).Let Γ be a dg graph. Define F (Γ) to be the free O -algebra generated by Γ. Explicitly, F (Γ)is defined as follows.We define a chain of length n in Γ as an ordered set x , x , . . . , x n . Denote by Γ n the set ofall chains of length n in Γ. For c ∈ Γ n , setΓ( c ) := Γ( x , x ) + ⊗ Γ( x , x ) + ⊗ · · · ⊗ Γ( x n − , x n ) + and Γ( n )( x, y ) := X c ∈ Γ n x ( c )= x,x n ( c )= y Γ( c )(for n = 0 we set Γ(0)( x, x ) = k id x and Γ(0)( x, y ) = 0 for x = y ). SetΓ O ( x, y ) := X n ≥ O ( n ) ⊗ Γ( n )( x, y ) ( It gives rise to a graph Γ O ∈ G raphs dg ( k ) with V Γ O = V Γ . Clearly Γ O is an algebra over theoperad O , and therefore it defines a weakly unital dg category F (Γ) such that U F (Γ) = Γ O .12ne has: Hom C at dgwu ( k ) ( F Γ , D ) = Hom G raphs dg ( k ) (Γ , U ( D )) ( which is natural in Γ and D , and gives rise to the required adjunction.The dg operad O plays an important role in our paper. For the proof of Theorem it willbe important to know its cohomology. Despite the answer is easy to state, the computation israther technical. We provide it in Section . Theorem 1.13.
The dg operad O is quasi-isomorphic to the operad A ssoc + of strictly unitalassociative algebras, by the map sending m to m , j to 1, and all p n ; n ,...,n k , k ≥ to 0. G raphs dg ( k )It is standard that coequalizers, and, therefore, all small colimits exist in G raphs dg ( k ).Recall the construction.Let Γ f ⇒ g Γ ( be a pair of morphisms in G raphs dg ( k ).Define its coequalizer Γ f,g as a small graph in G raphs dg ( k ) whose set of objects is thecoequalizer of the corresponding maps of the sets of objectsOb(Γ ) f ⇒ g Ob(Γ )It is the quotient set of Ob(Γ ) by the equivalence relation generated by the binary relation f ( x )R g ( x ), x ∈ Ob(Γ ).Let [ x ] and [ y ] be two equivalence classes. Define a complex Γ f,g ([ x ] , [ y ]) as the coequalizerin V ect dg ( k ) of M w,z ∈ Ob(Γ )[ f ( w )]=[ g ( w )]=[ x ][ f ( z )]=[ g ( z )]=[ y ] Γ ( w, z ) f ∗ ⇒ g ∗ M a,b ∈ Ob(Γ )[ a ]=[ x ] , [ b ]=[ y ] Γ ( a, b ) ( where f ∗ maps φ ∈ Γ ( w, z ) to f ( φ ), and g ∗ maps it to g ( φ ). If at least one class of [ x ] , [ y ] is notin the image of f (which is the same that the image of g ), we define source complex in ( )as 0.It is easy to check that the constructed dg graph Γ f,g is a coequalizer of ( ).13 .2.5 The coequalizers in C at dgwu ( k ) , I Consider a pair of maps of weakly unital dg categories A F ⇒ G B ( It is not straightforward to find (or to prove existence of) its coequalizer.However, one always can find the coequalizer of the maps of graphs U ( A ) U ( F ) ⇒ U ( G ) U ( B ) ℓ −→ Coeq( U ( F ) , U ( G )) ( as in Section . For some special diagrams ( ), the functor U creates coequalizers, seebelow. Afterwards, we reduce the general coequalizers ( ) to these special ones, in Section . Definition 1.14.
We say that the diagram ( ) is good if Ob( A ) = Ob( B ), and both F and G are identity maps on the sets objects.Assume that ( ) is good. Then the graph Coeq( U ( F ) , U ( G )), which is a particular caseof general coequalizers ( ) in G raphs dg ( k ), is especially simple. It has the set of verticesequal to Ob( A ) = Ob( B ), and its morphisms are the quotient-complexesCoeq( U ( F ) , U ( G ))( X, Y ) = B ( X, Y ) / ( F ( f ) − G ( f )) f ∈ A ( X,Y ) Lemma 1.15.
Suppose we are given a diagram ( ) which is good. Then a weakly unital dgcategory structure Q and a map of weakly unital dg categories L : B → Q such that A F ⇒ G B L −→ Q is a coequalizer, and U ( Q ) = Coeq( U ( F ) , U ( G )) , U ( L ) = ℓ , exist if and only if the followingtwo conditions hold:(1) the subcomplexes ( F ( f ) − G ( f )) f ∈ A ( X,Y ) , X, Y ∈ Ob( A ) , form a two-sided ideal in B : ℓ ( g ◦ ( F ( f ) − G ( f )) ◦ g ′ ) = 0 ( for any morphism f in A and any morphisms g, g ′ in B (such that the compositions aredefined),(2) ℓ ( p Bn ( g ⊗ . . . g k ⊗ ( g ◦ ( F ( f ) − G ( f )) ◦ g ′ ) ⊗ g k +1 ⊗ · · · ⊗ g n − )) = 0 ( for n ≥ , and any morphism f in A (some of g i are elements of k B ). n particular, the weakly unital dg category Q , if it exists, is uniquely defined (which means thatin this case U strictly creates the coequalizer). It is clear.Recall that diagram ( ) is called reflexive if there exists H : B → A such that F H = GH = id B . Proposition 1.16.
Assume we are given a good and reflexive diagram ( ) . Then condi-tions (1) and (2) of Lemma are fulfilled. Consequently, the functor U strictly creates thecoequalizer.Proof. Prove that (1) holds. One has: ℓ ( g ◦ ( F ( f ) − G ( f )) ◦ g ′ ) = ℓ ( g ◦ F ( f ) ◦ g ′ ) − ℓ ( g ◦ G ( f ) ◦ g ′ ) = ℓ ( F H ( g ) ◦ F ( f ) ◦ F H ( g ′ )) − ℓ ( GH ( g ) ◦ G ( f ) ◦ GH ( g ′ )) = ℓ ( F ( H ( g ) ◦ f ◦ H ( g ′ )) − ℓ ( G ( H ( g ) ◦ f ◦ H ( g ′ )) = 0 ( Prove that (2) holds. One has: ℓ ( p Bn ( g ⊗ · · · ⊗ ( g ◦ ( F ( f ) − G ( f )) ◦ g ′ ) ⊗ · · · ⊗ g n − )) = ℓ ( p Bn ( g ⊗ · · · ⊗ ( g ◦ F ( f ) ◦ g ′ ) ⊗ · · · ⊗ g n − )) − ℓ ( p Bn ( g ⊗ · · · ⊗ ( g ◦ G ( f ) ◦ g ′ ) ⊗ · · · ⊗ g n − )) = ℓ ( p Bn ( F H ( g ) ⊗ · · · ⊗ ( F H ( g ) ◦ F ( f ) ◦ F H ( g ′ )) ⊗ · · · ⊗ F H ( g n − )) − ℓ ( p Bn ( GH ( g ) ⊗ · · · ⊗ ( GH ( g ) ◦ G ( f ) ◦ GH ( g ′ )) ⊗ · · · ⊗ GH ( g n − ))) = ℓ ( p Bn ( F H ( g ) ⊗ · · · ⊗ ( F ( H ( g ) ◦ f ◦ H ( g ′ )) ⊗ · · · ⊗ F H ( g n − )) − ℓ ( p Bn ( GH ( g ) ⊗ · · · ⊗ ( G ( H ( g ) ◦ f ◦ H ( g ′ ))) ⊗ · · · ⊗ GH ( g n − ))) ∗ = ℓ ( F p An ( H ( g ) ⊗ · · · ⊗ ( H ( g ) ◦ f ◦ H ( g ′ )) ⊗ · · · ⊗ H ( g n − ))) − ℓ ( Gp An ( H ( g ) ⊗ · · · ⊗ ( H ( g ) ◦ f ◦ H ( g ′ )) ⊗ · · · ⊗ H ( g n − ))) = 0 ( where the equality marked by * follows from the fact that F, G are functors of weakly unital dgcategories and ( ). C at dgwu ( k ) , II In this Section, we closely follow the arguments in [Wo, Prop. 2.11]. We reproduce them herefor completeness.We make use of the following lemma, due to [Li, pp. 77-78], and known as the 3x3-lemma.15 emma 1.17.
Consider the following diagram in a category A h / / h / / α (cid:15) (cid:15) α (cid:15) (cid:15) B h / / β (cid:15) (cid:15) β (cid:15) (cid:15) C γ (cid:15) (cid:15) γ (cid:15) (cid:15) A g / / g / / α (cid:15) (cid:15) B ∗ g / / β (cid:15) (cid:15) C γ (cid:15) (cid:15) A f / / f / / B f / / C ( in which the top and the middle rows are coequalizers, the leftmost and the middle columns arecoequalizers, and all squares commute: g i α i = β i h i , f i α = β g i , g β i = γ i h , f β = γ g , i = 1 , . Then the following statements are equivalent:(1) the bottom row is a coequalizer,(2) the rightmost column is a coequalizer,(3) the square in the lower right corner (marked by ∗ ) is a push-out. Proposition 1.18.
The category C at dgwu ( k ) has all coequalizers.Proof. Let A H / / H / / B ( be two arrows in C at dgwu ( k ) coequalizer of which we’d like to compute. Embed it to the followingsolid arrow diagram, where ( F, U ) is the adjoint pair of functors from Proposition . F U F U A
F UF U ( H ) / / F UF U ( H ) / / ǫ F UA (cid:15) (cid:15)
F Uǫ A (cid:15) (cid:15) F U F U B F ( L ′ ) / / ǫ F UB (cid:15) (cid:15)
F Uǫ B (cid:15) (cid:15) F E ′ α (cid:15) (cid:15) α (cid:15) (cid:15) F U A
F U ( H ) / / F U ( H ) / / ǫ A (cid:15) (cid:15) F U B F ( L ) / / ǫ B (cid:15) (cid:15) F E p (cid:15) (cid:15) A H / / H / / B q / / X ( ) by application of F U F U and
F U ,correspondingly. Denote by E the coequalizer of ( U H , U H ) in G raphs dg ( k ), and by E ′ thecoequalizer of ( U F U H , U F U H ) in G raphs dg ( k ). As F is left adjoint, F E and
F E ′ arethe coequalizers of ( F U H , F U H ) and ( F U F U H , F U F U H ) in C at dgwu ( k ), correspondingly.Therefore, the upper and the middle rows of ( ) are coequalizers.The leftmost and the middle columns fulfil the assumptions of Proposition . Indeed,the upper pairs of arrows are reflexive, by the second case of Lemma , see ( ). Therefore,these columns are coequalizers, by Proposition .The dotted arrows α , α are constructed as follows. For α , consider the map F ( L ) ◦ ǫ F UB : F U F U B → F E
The two compositions
F U F U A
F UF UH ⇒ F UF UH F U F U B F ( L ) ◦ ǫ F UB −−−−−−−→
F E are equal, which gives rise to a unique map α : F E ′ → F E .Similarly, taking
F U ǫ B instead of ǫ F UB , one gets a unique map α : F E ′ → F E , whichcoequalizes the corresponding two arrows.We claim that the pair ( α , α ) is reflexive. We construct κ E : F E → F E ′ such that α ◦ κ E = α ◦ κ E = id F E .Recall κ A : F U A → F U F U A and κ B : F U B → F U F U B given as in ( ): κ A = F η UA , κ B = F η UB These maps are sections of the corresponding pairs of maps, which make them reflexive pairs,see Lemma . Consider F ( L ′ ) ◦ κ B : F U B → F E ′ The two maps
F U A ⇒ F U B F ( L ′ ) ◦ κ B −−−−−−→ F E ′ are equal, which gives rise to a unique map κ E : F E → F E ′ A simple diagram chasing shows that α ◦ κ E = α ◦ κ E = id F E .One has Ob(
F E ) = Ob(
F E ′ ), and Proposition is applied. We get an arrow p : F E → X which is a coequalizer of ( α , α ).Finally, we have to construct an arrow q : B → X making the square in the lower rightcorner commutative. To this end, consider p ◦ F ( L ) : F U B → X . The two compositions F U F U B ⇒ F U B p ◦ F ( L ) −−−−→ X q : B → X . One checks that the lower right squarecommutes.One makes use of Lemma to conclude that the bottom row is a coequalizer.We have already seen in Section that the products, the coproducts, and the equalizersin C at dgwu ( k ) are constructed straightforwardly. Then Proposition , and the classic result[R, Th. 3.4.11] give: Theorem 1.19.
The category C at dgwu ( k ) is small complete and small cocomplete. Although we will not be using the following result in this paper, it may have an independentinterest. The argument is close to [Wo, Th. 2.13].
Theorem 1.20.
The adjunction F : G raphs dg ( k ) ⇄ C at dgwu ( k ) : U is monadic.Proof. We deduce the statement from the Beck Monadicity Theorem , for which we have toprove that the assumptions in (1)-(3) in Theorem hold.(1) has been proven in Proposition , by which C at dgwu ( k ) has all coequalizers, and (3)is clear. One has to prove (2), that is, that the functor U : C at dgwu ( k ) → G raphs dg ( k ) preservesall U -split coequalizers. We make use of Lemma , once again.Let a pair of arrows in C at dgwu ( k ) A H ⇒ H B ( be U -split. Then U A UH ⇒ UH U B L −→ E ( is a split coequalizer, for some L and E . The upper and the middle rows in ( ) are definednow as the result of application of F U F and F , correspondingly, to ( ). (In particular, now E ′ = U F ( E ), L ′ = U F ( L )). Therefore, the upper and the middle rows are split, and, therefore, absolute coequalizers, by Lemma .Then we get the dotted arrows in ( ), and construct X , as in the proof of Proposition . In particular, we get a coequalizer A H ⇒ H B q −→ X (
18t the bottom row of ( ). One has to prove that
U X ≃ E .In the obtained diagram all columns and two upper rows are split coequalizers, but thebottom row is also a coequalizer but possibly not split. Now apply the functor U to thewhole diagram. As split coequalizers are absolute, by Lemma , the upper two rows and allthree columns remain coequalizers. Therefore, by Lemma , the bottom row also remains acoequalizer, after application of the functor U . C at dgwu ( k ) Here we construct a cofibrantly-generated closed model structure on the category C at dgwu ( k ).The construction generalises the Tabuada construction [Tab] of a cofibrantly-generated closedmodel structure on C at dg ( k ). Some arguments are new, such as Lemma and Lemma .We assume some familiarity with closed model categories, in particular with [Ho, Ch.2]. C at dgwu ( k ) Denote by A ssoc + the k -linear operad of unital associative algebras, A ssoc + ( n ) = k for any n ≥
0, with standard operadic compositions.
Define weak equivalences W in C at dgwu ( k ) as the weakly unital dg functors F : C → D suchthat the following two conditions hold:(W1) for any two objects x, y ∈ C , the map of complexes C ( x, y ) → D ( F x, F y ) is a quasi-isomorphism of complexes,(W2) the functor H ( F ) : H ( C ) → H ( D ) is an equivalence of k -linear categories.Note that for a weakly unital dg category C , the category H ( C ) is strictly unital, and thefunctor H ( F ) is well-defined, see Lemmas and .Define fibrations in C at dgwu ( k ) as the weakly unital dg functors F : C → D such that thefollowing two conditions hold:(F1) for any two objects x, y ∈ C , the map of complexes C ( x, y ) → D ( F x, F y ) is component-wise surjective,(F2) for any x ∈ C and a closed of degree 0 arrow g : F x → z in D (where z is an object of D , apriori not necessarily in the image of F ), such that g becomes an isomorphism in H ( C ),there is an object y ∈ C , and a closed degree 0 map f : x → y inducing an isomorphismin H ( D ), and such that F ( f ) = g (in particular, F ( y ) = z ).19e denote the class of all fibrations by Fib.Define a class of weakly unital dg functors Surj. A weakly unital dg functor F : C → D belongs to Surj if F is surjective on objects, and if (F1) holds. The lemma below is standard:
Lemma 2.1.
A weakly unital dg functor F : C → D belongs to Fib ∩ W if and only if it belongsto Surj ∩ ( W .Proof. It is clear that Surj ∩ ( W
1) implies Fib ∩ W . Conversely, assume F obeys Fib ∩ W . Onehas to prove that F is surjective on objects. From ( W
2) we know that H ( F ) is essentiallysurjective, that is, for any object z in D there is a homotopy equivalence g : F x → z . By (F2),there is a homotopy equivalence f : x → y such that F ( f ) = g . In particular, F ( y ) = z .One of our main results is: Theorem 2.2.
The category C at dgwu ( k ) admits a cofibrantly generated closed model structurewhose weak equivalences and fibrations are as above, and whose sets of generating cofibrationsand generating acyclic cofibrations are as it is defined in Section below. I and J Here we define sets I and J of morphisms in C at dgwu ( k ) which later are proven to be the setsof generating cofibrations and of generating acyclic cofibrations for the closed model structure,whose existence is stated in Theorem . The Kontsevich dg category
Denote by K the strictly unital dg category with two objects 0 and 1, whose morphisms aredescribed by generators and relations, as follows: • a closed degree 0 morphism f ∈ K (0 ,
1) and a closed degree 0 morphism g ∈ K (1 , • degree -1 morphisms h ∈ K (0 ,
0) and h ∈ K (1 ,
1) such that gf = id + dh . f g = id + dh ( • degree -2 morphism r ∈ K (0 ,
1) such that dr = h f − f h ( This category was introduced by Kontsevich in [K, Lecture 6]. Note the our use of notation Surj does not coincide with the one in [Tab].
20t was proven in [Dr, 3.7] that K is a (semi-free) resolution of the dg category which isthe k -linear envelope of the ordinary category with two objects 0 and 1, and two morphisms f ′ : 0 → g ′ : 1 → gf = id , f g = id . On the other hand, Kontsevich proved in [K1, Lecture 6] the following fact:Assume we are given a dg category C , and a closed degree 0 morphism ξ ∈ C ( x, y ), whichis a homotopy equivalence (that is, which descends to an isomorphism in H ( C )). Then thereis a (not unique) dg functor F : K → C such that F ( f ) = ξ .Lemma below shows that this property still holds, with minor changes, when C is a weaklyunital dg category: Lemma 2.3.
Let C be a weakly unital dg category, and ξ ∈ C ( x, y ) be a closed degree 0morphism, such that [ ξ ] ∈ H ( C ) is a homotopy equivalence. Then there is a weakly unital dgfunctor F : K → C such that F ( f ) = ξ ′ , where ξ ′ ∈ C ( x, y ) is a closed degree 0 morphism suchthat [ ξ ] = [ ξ ′ ] in H ( C ) .Proof. The proof uses basically the same computation as in Kontsevich’s proof for strictly unitalcase, with some adjustments.The problem is that 1 y · ξ and ξ · x may be distinct from ξ . Consider ξ ′ := 1 y · ξ · x . Then1 y · ξ ′ = ξ ′ · x = ξ ′ (because 1 z z = 1 z for any z , see Definition ). By assumption, there isdegree 0 morphism η ∈ C ( y, x ) which is inverse to ξ (and, therefore, inverse to ξ ′ as well) in H ( C ). We get: η · ξ ′ = 1 x + dh x , ξ ′ · η = 1 y + dh y ( Set η ′ = 1 x · η · y , h ′ x = 1 x · h x · x , h ′ y = 1 y · h y · y From ( ) we find η ′ · ξ ′ = 1 x + dh ′ x , ξ ′ · η ′ = 1 y + dh ′ y ( However, ( ) (for the corresponding morphisms) may fail.The rest of the proof is as in [K1, Lecture 6]. Maintain ξ ′ , η ′ , h ′ x , and set h ′′ y := h ′ y − ξ ′ · h ′ x · η ′ − h ′ y · ξ ′ · η ′ ( It is checked directly that ( ξ ′ , η ′ , h ′ x , h ′′ y ) satisfy ( ) and ( ), with r = − h ′′ y · ξ ′ · h ′ x + ξ ′ · h ′ x · h ′ x ( It follows from this result that any cocycle of negative degree is a coboundary in the complexes of morphismsof K . Clearly h g − gh is a cycle of degree -1 in K (1 , K (1 ,
0) whose boundary is h g − gh . he sets I and J Define, for any integral number n , the complex D ( n ) := Cone( k [ n ] id −→ k [ n ]). It is thecomplex k [ n ] id −→ k [ n − S ( n −
1) = k [ n − i : S ( n − → D ( n ) the natural imbedding ofcomplexes.Denote by A the dg category with a single object 0, and with A (0 ,
0) = k . Denote by κ thestrictly unital dg functor κ : A → K , sending 0 to 0.Denote by B the (strictly unital) dg category with two objects 0 and 1, such that B (0 ,
0) = k , B (1 ,
1) = k , B (0 ,
1) = 0, B (1 ,
0) = 0.Denote by P ( n ) the dg category with two objects 0 and 1, and P ( n )(0 ,
1) = D ( n ), P ( n )(0 ,
0) =0, P ( n )(1 ,
1) = 0, P ( n )(1 ,
0) = 0. Denote by P ( n ) the weakly unital dg category P ( n ) = F U ( P ( n ))(the functors F : G raphs dg ( k ) → C at dgwu ( k ), U : C at dgwu ( k ) → G raphs dg ( k ) are defined in ).Denote by α ( n ) the (weakly unital) dg functor α ( n ) : B → P ( n ) sending 0 to 0 and 1 to 1.Denote by C ( n ) the dg category with two objects 0 and 1, and with morphisms C ( n )(0 ,
1) = S ( n − C ( n )(0 ,
0) = 0, C ( n )(1 ,
1) = 0, C ( n )(1 ,
0) = 0. Denote C ( n ) = F U ( C ( n ))the corresponding weakly unital dg category.Consider the morphism b ( n ) : C ( n ) → P ( n ) the map of dg categories, sending 0 to 0, 1 to1, and such that S ( n −
1) = C ( n )(0 , → P ( n )(0 ,
1) = D ( n ) is the imbedding i . Define β ( n ) = F U ( b ( n )) : C ( n ) → P ( n )It is a weakly unital dg functor.Let Q : ∅ → A be the natural dg functor.Let I be a set of morphisms in C at dgwu ( k ) which comprises the dg functor Q and the weaklyunital dg functors β ( n ), n ∈ Z .Let J be a set of morphisms in C at dgwu ( k ) which comprises κ and α ( n ), n ∈ Z .The sets I and J are referred to as the sets of generating cofibrations and of generatingacyclic cofibrations , correspondingly. I and J The morphisms with
RLP with repsect to a set S of morphisms is denoted by S − inj.22 weakly unital dg functor P ( n ) → D , for D in C at dgwu ( k ), is 1-to-1 corresponded toa morphism in D of degree − n . Similarly, a weakly unital dg functor C ( n ) → D is 1-to-1corresponded to a closed degree − n + 1 morphism in D . It is straightforward.Assume a weakly unital dg functor f : C → D has RLP with respect to all α ( n ), n ∈ Z : B t / / α ( n ) (cid:15) (cid:15) C φ (cid:15) (cid:15) P ( n ) t / / = = D ( For the functor φ it means that any morphism in D ( φx, φy ) is φ ( q ), for some q ∈ C ( x, y ). Thatis, φ is surjective on morphisms.Assume that a weakly unital dg functor φ : C → D has RLP with respect to all β ( n ), n ∈ Z : C ( n ) t / / β ( n ) (cid:15) (cid:15) C φ (cid:15) (cid:15) P ( n ) t / / = = D ( One deduces from this property that for any x, y ∈ C , the map of complexes C ( x, y ) → D ( φx, φy ) is component-wise surjective, and is a quasi-isomorphism.We summarize: Lemma 2.4.
A weakly unital dg functor φ : C → D has RLP with respect to all α ( n ) , n ∈ Z ifand only if φ obeys ( F . A weakly unital dg functor φ : C → D has RLP with respect to all β ( n ) , n ∈ Z if and only if φ obeys ( F ∩ ( W . In fact, we have proved the “only if” parts of both statements. The proofs of the “if” partsare standard and are left to the reader.
Proposition 2.5.
One has: I − inj = Surj ∩ ( W
1) = J − inj ∩ W ( Proof.
In virtue of Lemma , for the first identity it is enough to show that any φ havingRLP with respect to Q is surjective on objects, which is trivial.For the second identity, we prove a statement which also will be used later. Lemma 2.6.
One has
Fib = J − inj . roof of J − inj ⊂ Fib:(F1) follows from RLP with respect to α ( n ), n ∈ Z , see Lemma . Prove (F2). Let φ : C → D be in J − inj. Let x be an object in C , and ξ : f ( x ) → z a homotopy equivalence.Consider ξ ′ = 1 z · ξ · f ( x ) . By Lemma , there is a weakly unital dg functor F : K → D such that F ( f ) = ξ ′ . Then the RLP gives a weakly unital dg functor ˆ F : K → C such that φ ◦ ˆ F = F . In particular, η ′ = ˆ F ( f ) ∈ C ( x, ?) is a homotopy equivalence, such that φ ( η ′ ) = ξ ′ .Now ξ ′ − ξ = dt , by (F1) there exists t ′ such that φ ( t ′ ) = t . Finally, set η := η ′ − dt ′ . Then[ η ] = [ η ′ ], and φ ( η ) = ξ . It completes the proof of (F2). Proof of
Fib ⊂ J − inj:Let φ : C → D in Fib. (F1) is equivalent to the RLP with respect to α ( n ), n ∈ Z . It remainsto prove the RLP with respect to κ for φ . The proof is quite involved.We are given a weakly unital dg functor F : K → D . Apply (F2) to ξ = F ( f ) ∈ D ( φ ( x ) , z ),it gives η ′ ∈ C ( x, y ) of degree 0, which is homotopy equivalence, φ ( y ) = z . Set η = 1 y · η ′ · x .We will construct ˆ F : K → C such that φ : ˆ F = F and ˆ F ( f ) = η .To this end, we make use of a construction from [Dr, 3.7], which links the Kontsevich dgcategory K with the Drinfeld dg quotient (loc.cit.). Let I be the (strictly unital) dg categorywith two objects 0 and 1 and generated by a single morphism f ∈ I (0 ,
1) of degree 0, df = 0.Denote I := I pre − tr0 the pre-triangulated hull of I (see [Dr, 2.4]). Consider the object Cone( f ) ∈ I , and define J as the full dg sub-category in I with a single object Cone( f ). Consider theDrinfeld dg quotient D := I / J , and denote by D the full dg subcategory in D with objects 0and 1. The following result is due to Drinfeld, loc.cit.: Lemma 2.7.
One has D = K . We reconstruct the argument in Appendix A.It gives rise to the following construction. Let E be a (strictly unital) dg category, ξ ∈ E ( x, y )a closed degree 0 morphism which is a homotopy equivalence. One has a dg functor F : I → E , F ( f ) = ξ . It gives rise to F pre − tr : I → E pre − tr . Denote by X ⊂ E pre − tr the full dg subcategorywhich has a single object Cone( ξ ). One gets D = I / J → E pre − tr / X The fact the ξ is a homotopy equivalence implies that one has a dg functor E pre − tr / X → E pre − tr ,depending on a contraction of Cone( ξ ).We get a dg functor D → E K → E . Conversely, any dg functor K → E is obtained in thisway. If all our categories were strictly unital, we would make use of this construction, to provethat Fib ⇒ κ − inj, as follows.One has: Lemma 2.8.
Let X be a dg category, x ∈ X an object. Assume there are two degree -1 maps h , h ∈ X − ( x, x ) such that dh i = id x , i = 1 , . Then there is t ∈ X − ( x, x ) such that dt = h − h It is true for t = h h .A dg functor F : K → D , F ( f ) = ξ , amounts to the same that a contraction of Cone( ξ ) in D pre − tr . That is, we get h ∈ D pre − tr (Cone( ξ ) , Cone( ξ )) such that dh = id Cone( ξ ) . We know from(F2) that Cone( η ) is contractible. It gives rise to ˜ h ∈ C pre − tr (Cone( η ) , Cone( η )) such that d ˜ h = id Cone( η ) . We may have not φ (˜ h ) = h . In any case, d ( φ (˜ h )) = id Cone( ξ ) . By Lemma one has φ (˜ h ) − h = dt . By (F1), we lift t to ˜ t , φ (˜ t ) = t . Set ˜ h := ˜ h − d ˜ t . One has d ˜ h = id Cone( η ) and φ (˜ h ) = h . It gives a lift of the dg functor ˆ F : K → C such that φ ◦ ˆ F = F .In the weakly unital case, this speculation should be adjusted.The main point is that, for a weakly unital dg category C and for a morphism ξ : x → y in C , we can not define Cone( ξ ). Indeed, we want any object to have a weak unit. One checksthat 1 Cone( ξ ) := (1 x , y [ − ) satisfies d Cone( ξ ) = 0 if and only if one has f · x = 1 y · f . It meansthat we can define C pre − tr but it fails to be weakly unital, even if C is.For a weakly unital dg category C , denote by C u the dg subcategory of C , whose objects areOb( C ), and whose morphisms are those morphisms f in C for which 1 · f = f ·
1. We consider C u as a unital dg category.If φ : C → D is in Fib, then φ u : C u → D u is also in Fib, as follows from the argument above,with replacement of f by 1 · f · K is strictly unital, a weakly unital dg functor F : K → D defines a dg functor F u : K → D u . Then we construct ˆ F : K → C u , as in the strictly unital case. It completes the proof.Now the second identity is proved as follows. One has J − inj ∩ W = Fib ∩ W = Surj ∩ ( W , and the second one follows from Lemma . It gives, in particular, a more conceptual replacement for the Kontsevich computation reproduced in Lemma (for its strictly unital case). .1.4 The proof of Theorem 2.2 The proof relies on [Ho, Th. 2.1.19]. Recall this theorem in a slightly different form, adaptedfor our needs:
Theorem 2.9.
Let C be a category with all small limits and colimits. Suppose W is a subcategoryof C , and I and J are sets of maps. Assume the following conditions hold:1. the subacategory W has two out of three property and is closed under retracts,2. the domains of I are small relative to I − cell ,3. the domains of J are small relative to J − cell ,4. J − cell ⊂ W ∩ I − cof ,5. I − inj = W ∩ J − inj .Then there is a cofibrantly generated closed model structure on C , for which the morphisms W of W are weak equivalences, I are generating cofibrations, J are acyclic generating cofibrations.Its fibrations are defined as J − inj . The reader is referred to [Ho, Sect.2.1] for notations S − cof and S − cell.Prove Theorem .We check conditions (1)-(5) of Theorem . (1)-(3) are clear. We proved (5) in Proposition . It follows from (5) that I − inj ⊂ J − inj, therefore, I − cof ⊃ J − cof. Therefore, it remainsto prove the part J − cell ⊂ W of (4), which we do below. The fact that J − inj coincides withthe class Fib defined in Section is proven in Lemma . Proof of J − cell ⊂ W : We have to prove that in the following push-outs squares in C at dgwu ( k )the weakly unital dg functor f : X → Y is a weak equivalence:( a ) B g / / α ( n ) (cid:15) (cid:15) X f (cid:15) (cid:15) P ( n ) / / Y ( b ) A h / / κ (cid:15) (cid:15) X f (cid:15) (cid:15) K / / Y ( where the (weak unital) dg functors g and h are arbitrary. We consider the cases (a) and (b)separately. The case (a):
It is clear that Ob( X ) = Ob( Y ), and f acts by the identity map on theobjects. Therefore, we have to show that, for any objects a, b ∈ X , the map of complexes26 ( a, b ) : X ( a, b ) → Y ( a, b ) is a quasi-isomorphism. For objects 0 and 1 in B , denote u = g (0) , v = g (1). Then Y ( a, b ) = X ( a, b ) M O (3) ⊗ X ( a, u ) ⊗ D ( n ) ⊗ X ( v, b ) M O (5) ⊗ X ( a, u ) ⊗ D ( n ) ⊗ X ( v, u ) ⊗ D ( n ) ⊗ X ( v, b ) M . . . ( where O is the operad introduced in . The map f ( a, b ) sends X ( a, b ) to the first summand.All other summands have 0 cohomology by the K¨unneth formula, because D ( n ) is acyclic. The case (b):
In this case, Ob( Y ) = Ob( X ) ⊔ K . It is clear that H ( f ) is essentiallysurjective. One has to prove that f is locally quasi-isomorphism: X ( a, b ) quis −−→ Y ( a, b ), a, b = 1 K .Denote h (0 A ) = u .By [Dr, 3.7], one knows that K is a resolution of the k -linear envelope of the ordinarycategory with two objects 0 and 1, and with only morphism between any two objects. Inparticular, K (0 ,
0) is quasi-isomorphic to k [0]. Therefore, one can decompose (as a complex): K (0 ,
0) = ¯ K ⊕ k [0] ( where ¯ K is a complex acyclic in all degrees. At the same time, k [0] is corresponded to amorphism in h ( A (0 , ∈ X ( u, u ); thus it is not a “new morphism”.One has: Y ( a, b ) = X ( a, b ) M O (3) ⊗ X ( a, u ) ⊗ ¯ K ⊗ X ( u, b ) M O (5) ⊗ X ( a, u ) ⊗ ¯ K ⊗ X ( u, u ) ⊗ ¯ K ⊗ X ( u, b ) M . . . ( Note that ( ) is a direct sum of complexes .The map of complexes f ( a, b ) maps X ( a, b ) to the first summand. All other summands have0 cohomology, because ¯ K is acyclic by [Dr, 3.7], and by the K¨unneth formula.Note that we did not use Theorem here, the proof does not rely on a computation ofthe cohomology of the dg operad O .Theorem is proven. C at dg ( k ) and C at dgwu ( k ) Let C , C be closed model categories. Recall that a Quillen pair of functors L : C ⇄ C : R is an adjoint pair of functors with an extra condition saying that L preserves cofibrations and27rivial cofibrations, or, equivalently, R preserves fibrations and trivial fibrations. Either of theseconditions guarantee that a Quillen pair of functors descends to a pair of adjoint functors L : Ho( C ) ⇄ Ho( C ) : R ( between the homotopy categories, see e.g. [Hi, Sect. 8.5] or [Ho, Sect. 1.3].In the case when C is cofibrantly generated, there is a simpler criterium [Ho, Lemma 2.1.20]for a pair of adjoint functors to be a Quillen pair. We reproduce it here for reader’s convenience. Proposition 3.1.
Let C , C be closed model categories, with C cofibrantly generated with gen-erating cofibrations I and generating acyclic cofibrations J . Let L : C ⇄ C : R be an adjointpair of functors. Assume that L ( f ) is a cofibration for all f ∈ I , and L ( f ) is a trivial cofibrationfor all f ∈ J . Then the pair ( L, R ) is a Quillen pair. See [Ho, Lemma 2.1.20] for a proof.Let C be a weakly unital dg category. Define L ( C ) = C/I where I is the dg category-ideal generated by p n ( x , . . . , x n ), x i ∈ C ⊕ k C , n ≥
2. (Recall that p n ( x , . . . , x n ) = 0 if n ≥ x i belong to C ⊂ C ⊕ k C ). Clearly L ( C ) is a unital dgcategory.The assignment C L ( C ) gives rise to a functor L : C at dgwu ( k ) → C at dg ( k ).Let A be a unital dg category. Define R ( A ) = ( A ⊕ k A , p dg )where p dg : A ⊕ k A → A is the dg functor constructed in Example . Recall that p dg (1 x ) = id x , x ∈ A . It gives rise to a functor R : C at dg ( k ) → C at dgwu ( k ). Proposition 3.2.
The following statements are true:(1) there is an adjunction
Hom C at dg ( k ) ( L ( C ) , A ) ≃ Hom C at dgwu ( k ) ( C, R ( A )) (2) the functors L : C at dgwu ( k ) ⇄ C at dg ( k ) : R form a Quillen pair of functors.Proof. (1): any map F : C → R ( A ) in C at dgwu ( k ) sends p Cn ( − , . . . , − ) , n ≥ C is strictly unital, see ( ). Therefore, this map is the same that a map L ( C ) → A in C at dg ( k ).(2): Clearly { L ( β ( n )) , L ( Q ) } form the set I T of generating cofibrations for the Tabuadaclosed model structure [Tab], and { L ( α ( n )) , L ( κ ) } for the set J T of generating trivial cofibrationsfor this model structure. The statement follows from Proposition .28 .2 Recall that a Quillen pair L : C ⇄ C : R is called a Quillen equivalence if the following conditionholds:For all cofibrant X ∈ C and all fibrant Y ∈ C a morphism f : LX → Y is a weak equivalencein C if and only if the corresponding morphism g : X → RY is a weak equivalence in C , seee.g. [Hi, Sect. 8.5.19], [Ho, Sect. 1.3.3].Recall that this condition implies that the corresponding adjoint pair between the homotopycategories ( ) is an adjoint equivalence of categories. Theorem 3.3.
The Quillen pair of functors L : C at dgwu ( k ) ⇄ C at dg ( k ) : R is a Quillen equivalence.Proof. Let X ∈ C at dgwu ( k ) be cofibrant, and Y ∈ C at dg ( k ) fibrant (therefore, Y is an arbitraryobject). On has to prove that f : LX → Y is a weak equivalence iff the adjoint map f ∗ : X → RY also is.It is enough to prove the statement for the case when X is an I -cell. Indeed, by thesmall object argument, for any X there exist an I -cell X ′ such that p : X ′ → X is an acyclicfibration. The Quillen left adjoint L maps the weak equivalences between cofibrant object toweak equivalences, by [Hi, Prop. 8.5.7]. Therefore, L ( p ) : L ( X ′ ) → L ( X ) is a weak equivalence.There is a map i : X → X ′ such that p ◦ i = id, given by the RLP. By 2-of-3 axiom, i is a weakequivalence, and L ( i ) also is.Assume L ( X ) f −→ Y is a weak equivalence, then L ( X ′ ) L ( p ) ◦ f −−−−→ Y is also a weak equivalence.If we know that the adjoint map ( f ◦ L ( p )) ∗ : X ′ → R ( Y ) is a weak equivalence, then the adjointmap f ∗ = ( f ◦ L ( p )) ∗ ◦ i is also a weak equivalence. The converse statement is proven similarly.Consider the case when X is an I -cell for C at dgwu ( k ). We reduce this case of the statementto Theorem .Denote by V the graded graph of generators of X . Prove that for any objects x, x ′ ∈ X , y ∈ Y , the cone L = Cone( LX ( x, x ′ )) f −→ Y ( f x, f x ′ )) is acyclic iff the cone L = Cone( X ( x, x ′ ) f ∗ −→ RY ( f ∗ x, f ∗ x ′ )) is acyclic. Denote ¯ O = Ker( P : O → A ssoc + ), where P is the dg operad mapsending all p n ; − to 0. There is a canonical map ω : L → L , and Cone( ω ) is quasi-isomorphic F ¯ O ( V )( x, x ′ ), where F ¯ O ( V ) is the free algebra over ¯ O generated by V , with an extra differentialcoming from the differential in the I -cell X . By Theorem , ¯ O is acyclic. Therefore, F ¯ O ( V )is acyclic by the K¨unneth formula. Therefore, Cone( ω ) is acyclic, and L is quasi-isomorphicto L . Therefore, L is acyclic iff L is. 29 A proof of Theorem 1.13 O ′ and its cohomology Recall that the dg operad O is generated by an n -ary operations p n ; n ,...,n k , acting as p n ( f , . . . , f n k − , n k , f n k +1 , . . . ), a binary operation m , with the relations and the differential asin ( ).Define a dg operad O ′ , for which the dg operad O is a quotient-operad, as follows. Thedefinition of O ′ is similar to O , but for the case of O ′ we drop the relation p n (1 , , . . . ,
1) = 0 for n ≥
2, which holds in O . We set j = p (1), and thus dp (1 ,
1) = m ( j, j ) − j = 0, dp (1 , ,
1) = m ( p (1) , p (1 , − m ( p (1 , , p (1)), and so on. The other relations and identities from ( )remain the same.There is a natural map of dg operads P : O ′ → A ssoc + , sending all p n ; ... , n ≥
2, to 0.
Theorem 4.1.
The map of dg operads P : O ′ → A ssoc + is a quasi-isomorphism.Proof. Let ω ∈ O ′ . Then ω is a linear combination of labelled “trees”, where each vertex(excluding the leaves) is labelled either by p n ; n ,...,n k or by m . We say that p n ; n ,...,n k has n − k operadic arguments (the remaining k arguments are 1’s). We use notation ♯ ( p n ; n ,...,n k ) = n − k .Given a tree T in which a vertex v is labelled by p n ; n ,...,n k , we write ♯ ( v ) = n − k . We extend ♯ ( − ) to all vertices of T , by setting ♯ ( v ) = 0 if v is labelled by m . Denote by V T the set of allvertices of T excluding the leaves.For a given tree T , denote ♯ ( T ) = X v ∈ V T ♯ ( v )We also denote by ♯ p ( T ) the total number of vertices with p ... , excluding p (1) , p (1 , , . . . .Define a descending filtration F q on O ′ , as follows. Its ( − ℓ )-th term F − ℓ is formed by linearcombinations of labelled trees T for which ♯ ( T ) − ♯ p ( T ) ≤ ℓ Note that for any tree T one has ♯ ( T ) − ♯ p ( T ) ≥ · · · ⊃ F − ⊃ F − ⊃ F − ⊃ F ⊃ dF − ℓ ⊂ F − ℓ , and any component of the differential on O ′ either preserves ♯ ( T ) − ♯ p ( T ) or decreases it by 1.We get a similar filtration F q on the component O ′ ( N ) of the airity N operations.We compute cohomology of O ′ ( N ) using the spectral sequence associated with filtration F q on O ′ ( N ). The spectral sequence lives in the quadrant { x ≤ , y ≤ } , the differential d ishorizontal. One easily sees that the spectral sequence converges. In fact, we show the spectralsequence collapses at the term E . 30 emma 4.2. Consider the filtration F q on O ′ ( N ) . One has: E − ℓ,m = ( A ssoc + ( N ) ℓ = 0 , m = 00 otherwiseIn particular, the spectral sequence collapses at the term E .Proof. We write p n ; n ,...,n k as p n ( f , f , . . . , , . . . , f n − k ) where f , . . . , f n − k are operadic argu-ments, and 1s stand on the places n , n , . . . , n k . In these notations, describe the differential in E − ℓ, q = F − ℓ /F − ℓ +1 .It has components of the following three types, which we refer to as Type I, Type II andType III components. Type I components: a component of Type I acts on a group of consequtive 1s, surroundedby operadic arguments from both sides, such as p n ( . . . , f s , , , . . . , | {z } a group of i consequtive 1s , f s +1 , . . . )For such a group, the component of d is a sum of expressions, each summand of which iscorresponded to either a product 1 · f s · · f s +1 , taken with alternated signs. It is clear that totally the component d S corresponded tosuch a group S is equal to d S ( p n ( . . . , f s , , . . . , | {z } i of 1s in the group S , f s +1 , . . . )) = ± p n ( . . . , f s , , , . . . , | {z } i − , f s +1 , . . . ) if i is even0 if i is odd Type II components: a component of Type II acts on the groups of leftmost (corresp., right-most) 1s, such as p n (1 , , . . . , , f , . . . ) or p n ( . . . , f n − k , , , . . . , ≥ p n ( . . . ) contains at least one operadic argument.The corresponding component d S of the differential is a sum of two subcomponents: d S = d S, + d S, .The first subcomponent d S, = d S, , − ± d S, , +0 , where d S, , − ( p n (1 , . . . , | {z } i of 1s , f , . . . )) = p n (1 · , , . . . , , f , . . . ) − p n (1 , · , . . . , f , . . . ) + · · · + ( − i − p n (1 , . . . , , · f , . . . )and similarly for d S, , +0 for the group of rightmost 1s.31ne has d S, , − ( p n (1 , . . . , | {z } i of 1s , f , . . . )) = p n ( 1 , . . . , | {z } i − , f , . . . ) if i is odd0 if i is evenand similarly for d S, , +0 .The second subcomponent d S, = d S, , − ± d S, , +0 , where d S, , − ( p n (1 , . . . , | {z } i of 1s , f , . . . )) = p (1) · p n − (1 , . . . , i − , f , . . . ) − p (1 , · p n − (1 , . . . , i − , f , . . . ) + · · · + ( − i − p i (1 , , . . . , · p n − i ( f , . . . )and similarly for d S, , +0 for the rightmost group of 1s.One checks that all other components of the differential d on O ′ decrease ♯ ( T ) − ♯ p ( T ) by 1. Type III components:
Here we have d acting on p n (1 , , . . . , n of 1s ).One has: d ( p n (1 , , . . . , p n − (1 · , , . . . , − p n − (1 , · , , . . . ,
1) + · · · + ( − i − p n − (1 , , . . . , · ± X ≤ i ≤ n − ( − i − p i (1 , , . . . , · p n − i (1 , , . . . , Denote the first summand by d S, and the second summand by d S, One sees that d S, ( p n (1 , , . . . , ( p n − (1 , , . . . ,
1) if n is even0 if n is oddThe computation of cohomology of the complex ( E − ℓ, q , d ) is reduced to the computationof the cohomology of a tensor product of complexes (the factors are labelled by combinatorialdata of the labelled tree T ), corresponded to different components S as listed above: E − ℓ, q = O S,T K q S ( The complexes K S corresponded to Type I components are isomorphic to K q = { . . . −→ k i =4 id −→ k i =3 0 −→ k i =2 id −→ k i =1deg= − → } ( K q is acyclic in all degrees. It implies that the complex ( E − ℓ, q , d ) is quasi-isomorphic to its subcomplex which is formed by the trees in which any p is of the type p n (1 , , . . . , , f , . . . , f n − k , , . . . , n − k operadic arguments stand in turn, without1s between them.It remains to treat the Type II and Type III cases.The complexes whose cohomology we need to compute are of two types. They are formedeither by linear combinations of p n (1 , , . . . , · p n (1 , , . . . , . . . p n k (1 , , . . . , · p n (1 , , . . . , , f , . . . )or by all linear combinations of p n (1 , , . . . , · p n (1 , , . . . , . . .p n k (1 , , . . . , K q and K q .Their cohomology are computed similarly, we consider the case of K q , leaving the case of K q to the reader.Denote p ℓ = p ℓ (1 , , . . . ,
1) and by P ℓ the 1-dimensional vector space k p ℓ (1 , , . . . ,
1) = k p ℓ , ℓ ≥ K − n = M k ≥ , n + ··· + n k − k = n P n ⊗ P n ⊗ · · · ⊗ P n k We denote the differential d on K q , see ( ), by d . Lemma 4.3.
The complex ( K q , d ) is quasi-isomorphic to P [0] .Proof. Consider on K q the following descending filtration Φ q , whereΦ − ℓ = M n + n + ··· + n k ≤ ℓ P n ⊗ P n ⊗ · · · ⊗ P n k One has · · · ⊃ Φ − ⊃ Φ − ⊃ Φ − ⊃ Φ = 0 d Φ − ℓ ⊂ Φ − ℓ Denote by d , Φ the differential in E − ℓ, q , Φ = Φ − ℓ / Φ − ℓ +1 . It is given by d , Φ ( p n ⊗ p n ⊗ · · · ⊗ p n k ) = k X i =1 ( − n + ··· + n i − − i +1 p n ⊗ · · · ⊗ d , Φ ( p n i ) ⊗ · · · ⊗ p n k ( where d ( p n ) = X ≤ i ≤ n − ( − i − p i ⊗ p n − i (
33t is well-known that the complex E − ℓ, q , Φ is acyclic when ℓ ≥
2, and is quasi-isomorphic to P [0]when ℓ = 1.We can identify P n ≃ ( k [1]) ⊗ n , then ⊕ n ≥ k [1] ⊗ n = P becomes the (non-unital) cofreecoalgebra cogenerated by k [1]. The complex ( ), ( ) is identified with the cobar-complex ofthe cofree coalgebra P . It is standard that its cohomology is equal to k [1][ − ≃ k .Therefore, the spectral sequence collapses at the term E by dimensional reasons.It completes the proof of Lemma .Similarly we prove that K q is acyclic in all degrees.In this way we see that any cohomology class in E − ℓ, q can be represented by a linearcombination of trees which do not contain p n s with n ≥ m and p (1), and all such trees have cohomological degree 0.It completes the proof.Theorem immediately follows from Lemma . O We are to prove Theorem . Proof.
The dg operad O is the quotient-operad of O ′ by the dg operadic ideal I generated by p n (1 , . . . , n ≥
2. It is enough to prove that I is acyclic. It would be natural to deducethe acyclicity of I from the acyclicity of the complex K q = K q / ( k p (1)), established above, byapplication of the K¨unneth formula. However, the K¨unneth formula is not applicable, becausewe do not have a decomposition such as I = O ′ ◦ K q ◦ O ′ , compatible with the differential.Alternatively, we repeat the arguments in the proof of Theorem . The main point is thatthe filtration F q on O ′ , defined in the course of the proof of Theorem , descends to O ′ /I .Indeed, both numbers ♯ ( T ) and ♯ p ( T ) are well-defined on the quotient O ′ /I . The statement ofLemma holds in this case, and its proof follows the same line. It becomes even simpler,because for Type II and Type III summands we make use that p n (1 , . . . ,
1) = 0 for n ≥
2, whichsubstantially simplifies the computation.
A The Drinfeld dg quotient and the Kontsevich dg cate-gory K Here we reconstruct the proof of Lemma sketched in [Dr, 3.7].34n this Appendix, we denote by X , X the objects of the dg category I , generated by aclosed degree 0 morphism f ∈ I ( X , X ) (our former notations for these objects were 0 and1). Then define I := I pre − tr0 , and D := I / J where J is the full dg subcategory with a singleobject Cone( f ). Finally, consider the full dg subcategory D of D , whose objects are X and X . Lemma states that D is isomorphic to K , the Kontsevich dg category, introduced inSection . Cone ( f ) fg X X " i j j i h h Figure 1: The derivation of the Kontsevich dg category K from the Drinfeld dg quotient.To describe D explicitly, consider the fragment of the dg category D , drawn in Figure 1.We start with the morphism f of degree 0, df = 0.Then there are morphisms (in notations of Figure 1): • i of degree 1, j of degree 0, • i of degree 0, j of degree -1, • ε of degree -1 (it is the morphism which was added in passage to the Drinfeld dg quotient).One has: j i = id , j i = id , j i = 0 , j i = 0 , i j + i j = id Cone( f ) (A.1)and di = 0 , dj = 0 , di = i f, dj = f j , dε = id Cone( f ) (A.2)35n the basis of these morphisms we define g := j εi , h := j εi , h := j εi , r := j εi (A.3)One checks directly from (A.1) and (A.2) that the relations ( ),( ) hold for these morphisms.One can show that the full dg subcategory D of D , whose objects are X and X , is generatedby f, g, h , h , r , and the relations as above.It identifies the Kontsevich dg category K with a full subcategory in the dg quotient. Then,the standard results such as [Dr, 3.4] are applied to compute the cohomology of all Hom com-plexes in K . See [Dr, 3.7.2-3.7.4]. Bibliography [Ba] M.A.Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories,
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