aa r X i v : . [ m a t h . QA ] M a y A CLOSER LOOK AT KADEISHVILI’S THEOREM
DAN PETERSENA
BSTRACT . We give a proof of the Homotopy Transfer Theorem following Kadeishvili’s original strategy. Al-though Kadeishvili originally restricted himself to transferring a dg algebra structure to an A ∞ -structure onhomology, we will see that a small modification of his argument proves the general case of transferring any kindof ∞ -algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.
1. I
NTRODUCTION
In 1980, Tornike Kadeishvili published the following celebrated theorem [Kad80]:
Theorem 1 (Homotopy Transfer Theorem) . Let A be a dg algebra over a commutative ring R . Assume thatthe homology H ( A ) is a projective R -module, so that there exists a quasi-isomorphism f : H ( A ) → A . There ex-ists noncanonically an A ∞ -algebra structure on H ( A ) with vanishing differential, and an A ∞ -quasi-isomorphism H ( A ) → A whose arity term is given by f . A very large number of different proofs of the Homotopy Transfer Theorem have been given since, invarious strengthened forms. Most of these arguments use either sums over trees (e.g. [LV12, Section 10.3])or the Homological Perturbation Lemma (e.g. [Ber14]), but other completely different methods of proof arealso possible [BM03, Mar04, Rog18, DSV16]. In more general versions of this theorem one can allow A itselfto be an A ∞ -algebra to begin with, the map H ( A ) → A can be replaced by a quasi-isomorphism from somechain complex to A , and rather than A ∞ -algebras one may consider ∞ -algebras over other operads. On theother hand all these other versions require stronger hypotheses on R or the complexes involved.Each of the proofs mentioned above requires setting up some amount of general machinery. By contrast,Kadeishvili’s argument is as direct as it could be: he writes down the infinite list of equations describing an A ∞ -structure on H ( A ) and an A ∞ -morphism H ( A ) → A , and argues inductively that each of the equationscan be solved in turn. In this note we will give a direct proof of the Homotopy Transfer Theorem followingKadeishvili’s original approach. Somewhat surprisingly, we will see that once Kadeishvili’s argument hasbeen written down in the right way it works for all sorts of ∞ -algebras, and we can transfer the algebraicstructure over a general quasi-isomorphism. In fact the resulting version of the Homotopy Transfer The-orem requires weaker assumptions than any statement that I am aware of in the literature. (Its drawbackcompared to arguments by sums over trees or homological perturbation theory is of course the noncon-structive nature: one does not get an explicit formula for the transferred structure.) That being said, wecertainly do not claim any great originality in this result.Before stating the theorem we will need some notation. Let R be any commutative ring, and let C bea conilpotent cooperad in graded R -modules satisfying C (0) = 0 and C (1) ∼ = R . We denote the cofreeconilpotent C -coalgebra cogenerated by a graded R -module V by C ( V ) = L n ≥ ( C ( n ) ⊗ V ⊗ n ) S n . Anycoderivation of C ( V ) is uniquely determined by a linear map C ( V ) → V , and by the arity term of acoderivation we mean the component V = C (1) ⊗ V → V . Similarly a C -coalgebra morphism C ( V ) → C ( W ) is described by a linear map C ( V ) → W , and by the arity term of such a morphism we mean the The author gratefully acknowledges support by ERC-2017-STG 759082 and a Wallenberg Academy Fellowship. omponent V → W . If M and N are dg R -modules, then we let Hom R ( M, N ) denote the “internal Hom”dg R -module of homomorphisms M → N . Theorem 2.
Let ( V, d V ) and ( W, d W ) be dg R -modules, and f : V → W a morphism. Let ν be a square-zerocoderivation of C ( W ) of degree − whose arity term equals the given differential d W . Assume that f induces a quasi-isomorphism Hom R ( C ( V ) , V ) → Hom R ( C ( V ) , W ) . Then there exists noncanonically a square-zero coderivation µ of C ( V ) whose arity term is d V , and a morphism of C -coalgebras F : C ( V ) → C ( W ) whose arity term is f andwhich is a chain map with respect to the differentials defined by µ and ν . In particular, we get a Homotopy Transfer Theorem if we assume that f is a quasi-isomorphism and that C ( V ) is K -projective in the sense of Spaltenstein [Spa88]. For R = Z this holds e.g. if each C ( n ) is a freeabelian group, and V is a bounded below chain complex of free abelian groups. Note that we do not needto assume e.g. that C is Σ -cofibrant, or that f has a quasi-inverse. For example, V might be a projectiveresolution of W ; this is a naturally occuring situation not covered by the usual forms of the HomotopyTransfer Theorem.Theorem 2 specializes to Kadeishvili’s original result since the structure of an A ∞ -algebra on a graded R -module V is equivalent to a square zero coderivation of the reduced tensor coalgebra of the suspension of V .More generally if P is a Koszul operad and C is its Koszul dual cooperad, then a square zero coderivationof C ( V ) is the same thing as a P ∞ -algebra structure on V , and a P ∞ -morphism V → W is a morphism C ( V ) → C ( W ) which is a chain map with respect to the differentials defined by the coderivations. Sowe recover the Homotopy Transfer Theorem for ∞ -algebras over a Koszul operad. For example, L ∞ -algebras and C ∞ -algebras are described as coderivations of cofree cocommutative coalgebras and cofreeLie coalgebras, respectively. See [LV12, Chapter 10].By minor modifications of the argument one also obtains a Homotopy Transfer Theorem in the “otherdirection”, as well as proofs of uniqueness of the transferred structures. More precisely we have: Theorem 3.
Let ( V, d V ) and ( W, d W ) be dg R -modules, and f : V → W a morphism. Let µ be a square-zerocoderivation of C ( V ) of degree − whose arity term equals the given differential d V . Assume that f induces a quasi-isomorphism Hom R ( C ( W ) , W ) → Hom R ( C ( V ) , W ) . Then there exists noncanonically a square-zero coderivation ν of C ( W ) whose arity term is d W , and a morphism of C -coalgebras F : C ( V ) → C ( W ) whose arity term is f and which is a chain map with respect to the differentials defined by µ and ν . Theorem 4.
Keep the hypotheses and notation of Theorem 2. Let µ and µ ′ be square zero coderivations of C ( V ) obtained as in Theorem 2, and let F, F ′ : C ( V ) → C ( W ) be the corresponding morphisms. There exists a noncanon-ical isomorphism of C -coalgebras Φ : C ( V ) → C ( V ) whose arity term is the identity and which is a chain mapwith respect to the differentials defined by µ and µ ′ . Moreover, if the cooperad C is nonsymmetric then there exists acoderivation homotopy H between F ′ ◦ Φ and F . Theorem 5.
Keep the hypotheses and notation of Theorem 3. Let ν and ν ′ be square zero coderivations of C ( W ) obtained as in Theorem 3, and let F, F ′ : C ( V ) → C ( W ) be the corresponding morphisms. There exists a noncanon-ical isomorphism of C -coalgebras Ψ : C ( W ) → C ( W ) whose arity term is the identity and which is a chain mapwith respect to the differentials defined by ν and ν ′ . Moreover, if the cooperad C is nonsymmetric then there exists acoderivation homotopy H between Ψ ◦ F ′ and F .
2. R
ECOLLECTIONS ON COALGEBRAS AND CODERIVATIONS
Let C be a conilpotent cooperad in dg R -modules. For any dg R -module V and any n ≥ we set C n ( V ) =( C ( n ) ⊗ V ⊗ n ) S n , and denote by C ( V ) = M n ≥ C n ( V ) the cofree conilpotent C -coalgebra generated by V . We also write C ≤ N ( V ) = L Nn =0 C n ( V ) , so that C ( V ) is the increasing union of subcoalgebras C ≤ n ( V ) . The reader who does not like cooperads and is only nterested in the A ∞ -case of the main theorem may restrict their attention to the case that C ( V ) is thereduced tensor coalgebra on V .Let E be a conilpotent dg C -coalgebra. The fact that C ( V ) is cofree means precisely that coalgebra homo-morphisms E → C ( V ) are in natural bijection with R -linear maps E → V . In particular, this means thatcoalgebra homomorphisms between cofree coalgebras C ( V ) → C ( W ) are given by elements of Hom R ( C ( V ) , W ) ∼ = Y n ≥ Hom R ( C n ( V ) , W ) . If F : C ( V ) → C ( W ) is a map of cofree conilpotent coalgebras then we denote by F ( n ) : C n ( V ) → W thecorresponding factor in the above decomposition, and we call it the arity n component of F .Let E be a conilpotent C -coalgebra and M an E -comodule. A map of dg R -modules M → E is called a coderivation if it satisfies the co-Leibniz rule, meaning that the diagram M E L n − k =0 C ( n ) ⊗ E ⊗ k ⊗ M ⊗ E ⊗ ( n − k − C ( n ) ⊗ E ⊗ n commutes for all n .If M → E and E → F are coderivations, then their composition M → F is in general not a coderivation.However, the composition of two coderivations of odd homological degree is always a coderivation: theterms in the composition which violate the co-Leibniz rule will cancel pairwise because of Koszul signs. Arelated fact is that the commutator of two coderivations E → E is a coderivation: Coder R ( E, E ) is a Liesubalgebra of Hom R ( E, E ) .The pre- or postcomposition of a morphism of coalgebras with a coderivation is again a coderivation.Coderivations into the cofree coalgebra C ( V ) are in natural bijections with linear maps into V , in the sameway as we could describe coalgebra homomorphisms into cofree coalgebras. Specifically, we have for any C ( V ) -comodule M a natural bijection Coder R ( M, C ( V )) ∼ = Hom R ( M, V ) . In particular, if we are given a coalgebra morphism C ( V ) → C ( W ) , so that C ( V ) is a C ( W ) -comodule, thencoderivations from C ( V ) to C ( W ) are in bijection with elements of Y n ≥ Hom R ( C n ( V ) , W ) . If η : C ( V ) → C ( W ) is a coderivation, then we denote by η ( n ) : C n ( V ) → W the n th factor of the abovedecomposition, and we call it the arity n component of η .Any coderivation η : C ( V ) → C ( W ) can be uniquely decomposed into homogeneous components η = P n ≥ η n , where η n has η ( n ) as its arity n component and all other components vanish. Then η n maps C k ( V ) into C k − n +1 ( W ) for all k , so η n is “homogeneous of weight n − ” with respect to the arity decompositionsof C ( V ) and C ( W ) . The commutator (and composition) of coderivations respects this decomposition intohomogeneous components: if η is nonzero only in arity n and θ is nonzero only in arity m , then [ η, θ ] isnonzero only in arity n + m − .We caution the reader that although we speak of the “arity n component F ( n ) ” of a morphism F : C ( V ) → C ( W ) , none of the statements in the preceding paragraph are valid for morphisms. The reason is thata coderivation C ( V ) → C ( W ) , considered as an element of Hom R ( C ( V ) , C ( W )) , depends linearly on itsvarious arity components C n ( V ) → W ; a morphism F : C ( V ) → C ( W ) depends nonlinearly on its aritycomponents. . P ROOF OF THE H OMOTOPY T RANSFER T HEOREM
We denote by
Hom R the inner Hom of dg R -modules: if ( M, d M ) and ( N, d N ) are dg R -modules, then Hom R ( M, N ) n = Y k ∈ Z Hom R ( M k , N n + k ) , with differential defined by ∂f = d N ◦ f − ( − n f ◦ d M for f ∈ Hom R ( M, N ) n .We fix a cooperad C in graded R -modules satisfying C (0) = 0 and C (1) ∼ = R . If V is a dg R -module,then C ( V ) is naturally differential graded; the induced differential of C ( V ) is a coderivation of arity . Inthe following proof we will repeatedly consider the Hom-complexes Hom R ( C n ( V ) , W ) , where ( V, d V ) and ( W, d W ) are dg R -modules, and we always use ∂ to denote the above differential on Hom R ( C n ( V ) , W ) . Theorem 2.
Let ( V, d V ) and ( W, d W ) be dg R -modules, and f : V → W a quasi-isomorphism. Let ν be a square-zero coderivation of C ( W ) of degree − whose arity term equals the given differential d W . Assume either that C n ( V ) is a K -projective complex of R -modules for all n , or that f is a chain homotopy equivalence. Then there existsnoncanonically a square-zero coderivation µ of C ( V ) whose arity term is d V , and a morphism of C -coalgebras F : C ( V ) → C ( W ) whose arity term is f and which is a chain map with respect to the differentials defined by µ and ν .Proof. Let n ≥ , and suppose we are given a degree − coderivation µ : C ( V ) → C ( V ) with µ (1) = d V , anda morphism F : C ( V ) → C ( W ) with F (1) = f , such that the restrictions of µ and F to C ≤ ( n − ( V ) satisfy ( µ ◦ µ = 0 F ◦ µ − ν ◦ F = 0 . We will prove that by modifying the arity n components of µ and F we can arrange so that these equationsare satisfied also on C ≤ n ( V ) . This will finish the proof by induction on n , since both equations are clearlysatisfied on C ≤ ( V ) = V .Note that µ ◦ µ is a coderivation into C ( V ) , being the composition of two odd coderivations, and that F ◦ µ − ν ◦ F is a coderivation into C ( W ) . The following equations are obviously satisfied:(1) µ ◦ ( µ ◦ µ ) − ( µ ◦ µ ) ◦ µ = 0 and(2) F ◦ ( µ ◦ µ ) = ( F ◦ µ − ν ◦ F ) ◦ µ + ν ◦ ( F ◦ µ − ν ◦ F ) . Let us compute the arity n component of both these equations. For (1), we use that components of µ ◦ µ ofarity less than n vanish. Hence the arity n component of the left hand side of (1) is given by a sum over allways of precomposing and postcomposing ( µ ◦ µ ) ( n ) with µ (1) , as all other terms in this arity vanish, so thatwe obtain the identity(3) ∂ ( µ ◦ µ ) ( n ) = 0 in Hom R ( C n ( V ) , V ) . Similarly we may consider the arity n component of (2), and use that all components of both µ ◦ µ and ( F ◦ µ − ν ◦ F ) of arity below n vanish. We obtain from (2) the identity(4) f ◦ ( µ ◦ µ ) ( n ) = ∂ ( F ◦ µ − ν ◦ F ) ( n ) in Hom R ( C n ( V ) , W ) , since the differential in Hom R ( C n ( V ) , W ) is the sum over all ways of precomposing and postcomposingwith µ (1) and ν (1) , respectively.Recall that f induces a quasi-isomorphism of chain complexes Hom R ( C n ( V ) , V ) → Hom R ( C n ( V ) , W ) . Now ( µ ◦ µ ) n is a cycle by (3), and it is mapped under f to a boundary by (4). It follows that ( µ ◦ µ ) n must itself be a boundary in Hom R ( C n ( V ) , V ) ; say that there exists e : C n ( V ) → V of degree − such that e = ( µ ◦ µ ) ( n ) . Let µ ′ be the coderivation of C ( V ) which has the same homogeneous arity components as µ except µ ′ ( n ) = µ ( n ) − e . Then(5) ( µ ′ ◦ µ ′ ) ( n ) = 0 , i.e. µ ′ ◦ µ ′ = 0 in C ≤ n ( V ) . Moreover we then have by (4) that ( F ◦ µ ′ − ν ◦ F ) is a cycle in Hom R ( C n ( V ) , W ) ,which (again since Hom R ( C ( V ) , V ) → Hom R ( C ( V ) , W ) is a quasi-isomorphism) means that it can be writ-ten as the sum of the image of a cycle under f , and a boundary. Thus we choose e ′ ∈ Hom R ( C n ( V ) , V ) with ∂e ′ = 0 , and e ′′ ∈ Hom R ( C n ( V ) , W ) , such that(6) ( F ◦ µ ′ − ν ◦ F ) ( n ) = f ◦ e ′ + ∂e ′′ . Let µ ′′ be the coderivation of C ( V ) which has the same arity components as µ ′ except µ ′′ ( n ) = µ ′ ( n ) − e ′ , anddefine similary a new morphism F ′ with F ′ ( n ) = F ( n ) − e ′′ . Since we only modified µ ′ by adding a cycle, wecan still argue as in (5) to see that µ ′′ ◦ µ ′′ = 0 in C ≤ n ( V ) . Moreover, it is straightforward to check that (6)says exactly that ( F ′ ◦ µ ′′ − ν ◦ F ′ ) ( n ) = 0 . The theorem is proven. (cid:3) The preceding proof is essentially Kadeishvili’s. Let us make the comparison explicit. At one point ofthe argument Kadeishvili writes “Direct calculations show that ∂U n = 0 ”, and later that “The remainingcondition (1) can be proved by a straightforward check”. Kadeishvili’s ∂U n is our ∂ ( F ◦ µ − ν ◦ F ) ( n ) ,and his condition (1) is our condition ( µ ◦ µ ) ( n ) = 0 . As in the above argument a calculation shows that f ◦ ( µ ◦ µ ) ( n ) = ∂ ( F ◦ µ − ν ◦ F ) ( n ) . For Kadeishvili the map f is a cycle-choosing homomorphism, so thefact that the value of f is a boundary implies both that ( µ ◦ µ ) ( n ) = 0 and ( F ◦ µ − ν ◦ F ) ( n ) = 0 .4. V ARIATIONS
Let us briefly explain the necessary modifications of the argument to obtain the transfer in the other direc-tion, and the uniqueness of the transferred structure.
Theorem 3.
Let ( V, d V ) and ( W, d W ) be dg R -modules, and f : V → W a morphism. Let µ be a square-zerocoderivation of C ( V ) of degree − whose arity term equals the given differential d V . Assume that f induces a quasi-isomorphism Hom R ( C ( W ) , W ) → Hom R ( C ( V ) , W ) . Then there exists noncanonically a square-zero coderivation ν of C ( W ) whose arity term is d W , and a morphism of C -coalgebras F : C ( V ) → C ( W ) whose arity term is f and which is a chain map with respect to the differentials defined by µ and ν .Proof. The structure of the argument is the same as in the previous proof. We suppose instead that we havea degree − coderivation ν : C ( W ) → C ( W ) with ν (1) = d W , and a morphism F : C ( V ) → C ( W ) with F (1) = f , such that the restrictions of ν and F to C ≤ ( n − ( W ) resp. C ≤ ( n − ( V ) satisfy ( ν ◦ ν = 0 F ◦ µ − ν ◦ F = 0 . We now consider the two equations ν ◦ ( ν ◦ ν ) − ( ν ◦ ν ) ◦ ν = 0 and ( ν ◦ ν ) ◦ F = ( F ◦ µ − ν ◦ F ) ◦ µ + ν ◦ ( F ◦ µ − ν ◦ F ) and compute the arity n component of both these equations. By the same argument as before we obtain ∂ ( ν ◦ ν ) ( n ) = 0 in Hom R ( C n ( W ) , W ) , and the identity ( ν ◦ ν ) ( n ) ◦ f = ∂ ( F ◦ µ − ν ◦ F ) ( n ) in Hom R ( C n ( V ) , W ) . y the same argument as before it follows that ( ν ◦ ν ) n is a boundary in Hom R ( C n ( W ) , W ) ; say that thereexists e : C n ( W ) → W such that ∂e = ( ν ◦ ν ) ( n ) . Let ν ′ be the coderivation of C ( W ) for which ν ′ ( n ) = ν ( n ) − e .Then ( ν ′ ◦ ν ′ ) ( n ) = 0 , and it follows that ( F ◦ µ − ν ′ ◦ F ) is a cycle in Hom R ( C n ( V ) , W ) , which again means that it can be writtenas the sum of the image of a cycle under f , and a boundary. Thus we choose e ′ ∈ Hom R ( C n ( W ) , W ) with ∂e ′ = 0 , and e ′′ ∈ Hom R ( C n ( V ) , W ) , such that ( F ◦ µ − ν ′ ◦ F ) ( n ) = e ′ ◦ f + ∂e ′′ . Let ν ′′ be the coderivation of C ( W ) with ν ′′ ( n ) = ν ′ ( n ) − e ′ , and define similary a new morphism F ′ with F ′ ( n ) = F ( n ) − e ′′ . By the same argument as before we see that ( F ′ ◦ µ − ν ′′ ◦ F ′ ) ( n ) = 0 as claimed. (cid:3) Suppose that C is a nonsymmetric cooperad, and let F, F ′ : E → D be morphisms of C -coalgebras. An ( F, F ′ ) -coderivation is a map H : E → D making the following diagram commute for all n : E DC ( n ) ⊗ E ⊗ n C ( n ) ⊗ D ⊗ n . H P n − k =0 id C ( n ) ⊗ F ⊗ k ⊗ H ⊗ ( F ′ ) ⊗ ( n − k +1) There is a natural bijection between ( F, F ′ ) -coderivations E → C ( W ) and R -linear maps E → W , justas for usual coderivations. If H : C ( V ) → C ( W ) is an ( F, F ′ ) -coderivation then we write H ( n ) for thecorresponding map C n ( V ) → W , the “arity n component” of H . We say that an ( F, F ′ ) -coderivation H is a coderivation homotopy between F and F ′ if ∂H = F − F ′ in Hom R ( E, D ) .If C is a symmetric cooperad the above definition of ( F, F ′ ) -derivation still makes sense but is not useful;the image of the left vertical arrow in the diagram lands in the S n -invariants and the lower horizontal arrowis very much not S n -invariant, so nontrivial ( F, F ′ ) -coderivations will generally not exist. Theorem 4.
Keep the hypotheses and notation of Theorem 2. Let µ and µ ′ be square zero coderivations of C ( V ) obtained as in Theorem 2, and let F, F ′ : C ( V ) → C ( W ) be the corresponding morphisms. There exists a noncanon-ical isomorphism of C -coalgebras Φ : C ( V ) → C ( V ) whose arity term is the identity and which is a chain mapwith respect to the differentials defined by µ and µ ′ . Moreover, if the cooperad C is nonsymmetric then there exists acoderivation homotopy between F ′ ◦ Φ and F .Proof. Let us focus on the case that C is nonsymmetric. Let n ≥ , and suppose we are given a morphism Φ : C ( V ) → C ( V ) with Φ (1) = id V , and a degree F ′ ◦ Φ , F ) -coderivation H with H (1) = 0 , such that therestrictions of Φ and H to C ≤ ( n − ( V ) satisfy the equations ( µ ′ ◦ Φ − Φ ◦ µ = 0 F ′ ◦ Φ − F = ν ◦ H + H ◦ µ. One easily checks that the following two equations are satisfied: µ ′ ◦ ( µ ′ ◦ Φ − Φ ◦ µ ) + ( µ ′ ◦ Φ − Φ ◦ µ ) ◦ µ = 0 and F ◦ ( µ ′ ◦ Φ − Φ ◦ µ ) = ν ◦ ( F ′ ◦ Φ − F − ν ◦ H − H ◦ µ ) − ( F ′ ◦ Φ − F − ν ◦ H − H ◦ µ ) ◦ µ. Considering the arity n component of these equations and using the fact that both µ ′ ◦ Φ − Φ ◦ µ = 0 and F ′ ◦ Φ − F − ν ◦ H − H ◦ µ = 0 in arities below n , we deduce that ∂ ( µ ′ ◦ Φ − Φ ◦ µ ) ( n ) = 0 and f ◦ ( µ ′ ◦ Φ − Φ ◦ µ ) ( n ) = ∂ (cid:16) ( F ′ ◦ Φ) ( n ) − F ( n ) − ( ν ◦ H + H ◦ µ ) ( n ) (cid:17) . s in the previous proofs it follows that ( µ ′ ◦ Φ − Φ ◦ µ ) ( n ) is itself a boundary, say ∂e . Then if we let Φ ′ denote the morphism which has Φ ′ ( n ) = Φ ( n ) − e and agrees with Φ in all other arities, we will have ( µ ′ ◦ Φ ′ − Φ ′ ◦ µ ) ( n ) = 0 . Hence also ∂ (cid:16) ( F ′ ◦ Φ ′ ) ( n ) − F ( n ) − ( ν ◦ H + H ◦ µ ) ( n ) (cid:17) = 0 , and so (again using that f is a quasi-isomorphism) we see that ( F ′ ◦ Φ ′ ) ( n ) − F ( n ) − ( ν ◦ H + H ◦ µ ) ( n ) is thesum of a boundary and the image of a cycle under f , say f ◦ e ′ + ∂e ′′ . We now modify Φ ′ and H in arity n ,setting Φ ′′ ( n ) = Φ ′ ( n ) − e ′ and H ′ ( n ) = H ( n ) − e ′′ . Now ( F ′ ◦ Φ ′′ ) ( n ) − F ( n ) − ( ν ◦ H ′ + H ′ ◦ µ ) ( n ) = 0 and theproof is done by induction.If C is a symmetric operad, a nearly identical argument would show that there exists a coderivation H inthe usual sense for which ( µ ′ ◦ Φ − Φ ◦ µ = 0 F ′ ◦ Φ − F = ν ◦ H + H ◦ µ. In particular, Φ still provides an isomorphism between the two transferred structures µ and µ ′ , but H canno longer be interpreted as a homotopy between ∞ -morphisms. (cid:3) The proof of Theorem 5 very similar to the three preceding proofs, and it is obtained by modifying the proofof Theorem 4 in exactly the same way as Theorem 3 is obtained by modifying the proof of Theorem 2. Weomit the argument.The notion of homotopy between morphisms of coalgebras over a symmetric cooperad is a bit more com-plicated to define. There are three possible definitions, all of which give rise to equivalent notions, but notobviously so; see [DP16]. None of the possible definitions make sense in general unless our ground ring R contains Q , and it is not clear to me whether the above argument could be modified to produce a homotopyin this symmetric sense, for any of the definitions.R EFERENCES[Ber14] Alexander Berglund. Homological perturbation theory for algebras over operads.
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E-mail address : [email protected]@math.su.se