Koszul duality of operads and homology of partition posets

Koszul duality of operads and homology of partition posets

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Koszul duality of operads
and
homology of partition posets
Benoit Fresse
Abstract. Weconsiderpartitionsofasetwith r elementsorderedbyrefine-
¯ment. Weconsiderthesimplicialcomplex K(r)formedbychainsofpartitions
which starts at the smallest element and ends at the largest element of the
¯partitionposet. Aclassicaltheoremassertsthat K(r)isequivalenttoawedge
of r−1-dimensional spheres. In addition, the poset of partitions is equipped
with a natural action of the symmetric group in r letters. Consequently, the
associated homology modules are representations of the symmetric groups.
¯One observes that the r−1th homology modules of K(r), where r = 1,2,...,
are dualto the Lie representationofthe symmetricgroups. In thisarticle, we
would like to point out that this theorem occurs a by-product of the theory
of Koszul operads. For that purpose, we improve results of V. Ginzburg and
M. Kapranov in several directions. More particularly, we extend the Koszul
duality of operads to operads defined over a field of positive characteristic (or
over a ring). In addition, we obtain more conceptual proofs of theorems of
V. Ginzburg and M. Kapranov.
Contents
Prologue and introduction
0. Conventions
1. Composition products and operad structures
2. Chain complexes of modules over an operad
3. The reduced bar construction
4. Bar constructions with coefficients
5. Koszul duality for operads
6. Epilogue: partition posets
References
Glossary and notation index
Prologue
We consider the set of partitions ...

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Koszul duality of operads and homology of partition posets Benoit Fresse Abstract. Weconsiderpartitionsofasetwith r elementsorderedbyrefine- ¯ment. Weconsiderthesimplicialcomplex K(r)formedbychainsofpartitions which starts at the smallest element and ends at the largest element of the ¯partitionposet. Aclassicaltheoremassertsthat K(r)isequivalenttoawedge of r−1-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in r letters. Consequently, the associated homology modules are representations of the symmetric groups. ¯One observes that the r−1th homology modules of K(r), where r = 1,2,..., are dualto the Lie representationofthe symmetricgroups. In thisarticle, we would like to point out that this theorem occurs a by-product of the theory of Koszul operads. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov. Contents Prologue and introduction 0. Conventions 1. Composition products and operad structures 2. Chain complexes of modules over an operad 3. The reduced bar construction 4. Bar constructions with coefficients 5. Koszul duality for operads 6. Epilogue: partition posets References Glossary and notation index Prologue We consider the set of partitions of{1,...,r} equipped with the partial or- der defined by the refinement of partitions (for instance, we have{1,3},{2,4}≤ {1},{3},{2,4}). Thesingleset{1,...,r}formsthesmallestpartitionof{1,...,r} and the collection{1},...,{r} forms the largest partition. We mention that the 1991 Mathematics Subject Classification. 18D50; 18C15; 17B01; 05E25; 05C05. This work was carried out at the Laboratoire Jean-Alexandre Dieudonn´e, Universit´e de Nice-Sophia-Antipolis et CNRS. 1 6 2 BENOIT FRESSE components of a partition are not assumed to be ordered (for instance, the collec- tions{1,3},{2,4} and{2,4},{1,3} define the same partition). We observe also that the set of partitions is equipped with an action of the symmetric group Σ ,r since a permutation w :{1,...,r}→{1,...,r} maps a partition of{1,...,r} to another partition. ¯We consider the simplicial set K(r) which consists of sequences of partitions λ ≤λ ≤···≤λ such that λ ={1,...,r} and λ ={1},...,{r}. The face d0 1 n 0 n i is given by the omission ofλ . The degeneracys is given by the repetition ofλ .i j j We have the following result: ¯Theorem. We consider the reduced homology of the simplicial setK(r) with coefficients inK=Z,Q orF . This homology is a graded Σ -module since the setp r of partitions is equipped with an action of the symmetric group. We have ( ∨L(r) ⊗sgn , if∗=r−1,r˜ ¯H (K(r),K)=∗ 0, otherwise, whereL(r) denotes the rth component of the Lie operadL. The Lie operadL consists precisely of the sequence of representationsL(r) formed by the “multilinear” components of free Lie algebrasL(Kx ⊕···⊕Kx )1 r (more explicitly, the moduleL(r) is spanned by the Lie monomials of degree 1 in eachvariable). Inthetheorem,weconsidertensorproductsofdualrepresentations ∨L(r) with the classical signature representations sgn .r Here are bibliographical references about this theorem. The character of the ˜ ¯representationH (K(r),K)isdeterminedbyR.Stanleyin[81]andbyP.Hanlonr−1 in [35]. The relationship with the character of the Lie operad is pointed out by A. Joyal in [41]. These results about characters are refined by A. Robinson and S. Whitehouse in [74] (see also S. Whitehouse [86] for the integral version). One can deduce from results of A. Bjorn¨ er that the (reduced) homology of partition posets vanishes in degree∗ = r−1 (cf. [14]). Then, in [5], H. Barcelo ∨˜ ¯defines an isomorphism of representations H (K(r),K)’L(r) ⊗sgn , basedr−1 r on the Lyndon basis of the Lie operad (cf. M. Lothaire [51], C. Reutenauer [71]). This result is improved by P.Hanlonand M. Wachs in [36]. Namely, these authors define a natural morphism (which does not involve the choice of a basis of the Lie operad) from the dual of the Lie operad to the chain complex of the partition poset ∨ ¯L(r) ⊗sgn → C (K(r)).r−1r This morphism fixes a representative of the homology class associated to a given element of the Lie operad. In addition, P. Hanlon and M. Wachs generalize the theorem above and give a relationship between partition posets and structures of Lie algebras with k-ary brackets. On the other hand, a topological proof of the theorem above, based on cal- culations of F. Cohen (cf. [20]), is given by G. Arone and M. Kankaanrinta in [2]. In connection with this result, we should mention that an article of G. Arone and M. Mahowald (cf. [3]) sheds light on the importance of partition posets in homotopy theory. Namely, these authors prove that the Goodwillie tower of the identity functor on topological spaces is precisely determined by partition posets. Introduction In this article, we would like to point out that the theorem of the prologue occurs as a by-product of the theory of Koszul operads. For that purpose, we KOSZUL DUALITY OF OPERADS 3 improve results of V. Ginzburg and M. Kapranov (cf. [31]) in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov. We recall that an operad is an algebraic structure which parametrizes a set of operations. Originally, the notion of an operad is introduced by P. May in a topological context in order to model the stucture of an iterated loop space (cf. [61]). To be precise, the definition of an operad arises on one hand from the notion of an algebraic theory (cf. J. Boardman and R. Vogt [15], F. Lawvere [48], S. Mac Lane [53]) and on the other hand from the structure of a monad (also called triple, cf. J. Beck [10], S. Eilenberg and C. Moore [23]). In particular, we should mention that J. Boardman and R. Vogt consider certain prop structures, which are equivalent to operads, in their work on iterated loop spaces (cf. [15]). We refer to Adams’s book [1] for a nice survey of this subject. In this article, we are concerned with operads in a category of modules over a ground ringK. In this context, an operad P consists of a sequence of representa- tions P(r) of the symmetric groups Σ together with composition productsr P(r)⊗P(n )⊗···⊗P(n )→ P(n +···+n ).1 r 1 r In general, an operadP is associated to a category of algebras, calledP-algebras, forwhichtheelementsofP(r)representmultilinearoperationsonr variables. The composition products of P determine the composites of these operations. For in- stance,thereareoperadsdenotedbyC,respectivelyL,associatedtothecategoryof commutativeandassociativealgebras, respectivelytothecategoryofLiealgebras. (To be precise, in this article, we consider commutative algebras without units.) TheK-moduleC(r)isgeneratedbythesinglemonomialp(x ,...,x )=x ·····x1 r 1 r inr commutative variable which has degree 1 in each variable. Consequently, this K-moduleC(r) is the trivial representation of the symmetric group Σ . The K-r moduleL(r) is generated by the Lie monomials in r variables which have degree 1 in each variable. Classically, a sequence of representations of the symmetric groups M(r) is equivalent to a functor S(M):KMod→ KMod whose expansion ∞M ⊗rS(M)(V)= (M(r)⊗V )Σr r=0 ⊗rconsists of coinvariant modules S (M)(V) = (M(r)⊗ V ) . For instance,r Σr the functor associated to the commutative operad S(C) : KMod → KMod can be identified with the augmentation ideal of the symmetric algebra S(C)(V) = L∞ ⊗r(V ) . Similarly,thefunctorassociatedtotheLieoperadS(L):KMod→Σr=1 r KMod is given by the structure of a free Lie algebra. In general, the structure formed by a sequence of representations of the symmetric groupsM(r) is called a Σ -module (or a symmetric module in plain English).∗ Oneobservesthatclassicalcomplexesassociatedtocommutativealgebrasand Lie algebras are functors determined by Σ -modules. On one hand, we consider∗ the cotriple construction of a commutative algebra A which is a chain complex cotriple C (A) such that∗ cotripleC (A)=S(C)◦···◦S(C)(A)∗ | {z } ∗−1 4 BENOIT FRESSE (cf. M. Barr [6], J. Beck [10]). We prove precisely that the cotriple construction ¯is the functor associated to the chain complex of the partition posets C (K(r)).∗ Lcotriple ∞ ⊗r¯We have explicitly C (A)= (C (K(r))⊗A ) .∗ ∗ Σrr=0 HarrOn the other hand, we consider the Harrison complex C (A) (cf. D. Har-∗ rison, [37]) which has the coinvariant module Harr ∨ ⊗rC (A)=(L(r) ⊗sgn ⊗A )Σr−1 r r in degree r−1 (see paragraph 6.6 for more precisions). We define a comparison morphism Harr cotriple C (A)→ C (A).∗ ∗ In positive characteristic, the cotriple homology differs from the Harrison homol- cotriple ogy. Inparticular,thecotriplehomologyofasymmetricalgebraH (S(C)(V))∗ Harrvanishes for general reasons while the Harrison homology H (S(C)(V)) does∗ not(cf. M.Barr[6],D.Harrison[37],S.Whitehouse[86]). Consequently,thecom- parison morphism above does not define a quasi-isomorphism of chain complexes. Nevertheless, we deduce from the Koszul duality of operads that this compari- ∼∨son morphism is induced by a quasi-isomorphism of Σ -modulesL(r) ⊗sgn −→∗ r ¯C (K(r)). (In fact, we recover exactly the chain equivalence of M. Wachs and P.∗ Hanlon.) The theorem of the prologue follows from this assertion. Similarly, for Lie algebras, we have, on one hand, the cotriple complex cotripleC (G)=S(L)◦···◦S(L)(G)∗ | {z } ∗−1 CEand, on the other hand, the Chevalley-Eilenberg complex C (G), which can be∗ defined by the formula CE ∨ ⊗rC (G)=(C(r) ⊗sgn ⊗G )r−1 r Σr (cf. M. Barr [7], J.-L. Koszul [45]). One can deduce from classical arguments that the cotriple homology agrees with the Chevalley-Eilenberg homology (cf. M. Barr [7]). We make this assertion more precise. Namely, in the case of Lie algebras, we have a comparison morphism CE cotripleC (G)→ C (G)∗ ∗ which gives rise to an isomorphism from the Chevalley-Eilenberg homology to the cotriple homology for all Lie algebrasG which are projective over the ground ring K. In fact, one purpose of this article is to point out that the comparison morphisms Harr cotriple CE cotripleC (A)→ C (A) and C (G)→ C (G)∗ ∗ ∗ ∗ are determined by quasi-isomorphisms of Σ -modules in both cases, although the∗ firstonedoesnotgiveaquasi-isomorphismofchaincomplexesforallcommutative algebras A. More generally, we observe that Σ -modules have better homological prop-∗ erties than associated functors and this makes results about coefficients easier. Therefore, we introduce an operation on Σ -modules M,N7→ M◦N which cor-∗ responds to the composition product of functors. We have explicitly S(M◦N)= S(M)◦S(N). The cotriple constructions above are determined by composites Σ -∗ modules, sinceS(C)◦···◦S(C)(V)=S(C◦···◦C)(V) andS(L)◦···◦S(L)(V)= KOSZUL DUALITY OF OPERADS 5 S(L◦···◦L)(V). Similarly,thecomparisonmorphismsareinducedbyembeddings of Σ -modules∗ ∨ ∨L(r) ⊗sgn →C◦···◦C(r) and C(r) ⊗sgn →L◦···◦L(r)r r| {z } | {z } r−1 r−1 respectively. Let us introduce the general constructions for operads which give rise to these comparisonmorphisms. Theideaistogeneralizeclassicalstructuresfromalgebras to operad. The tensor product ofK-modules is replaced by the composition prod- uct of Σ -modules. The bar construction of algebras is replaced by the cotriple∗ construction of operads (which corresponds to the cotriple construction of monads considered by J. Beck in [10] and by P. May in [61]). Explicitly, for a given operad ¯P, we let C (P) denote the chain complex such that∗ ¯C (P)=P◦···◦P.∗ | {z } ∗−1 Weanalyzethestructureofthischaincomplex. First,weobservethattheoperadic ¯bar construction B (P) introduced by E. Getzler and J. Jones in [30] and by V.∗ ¯GinzburgandM.Kapranovin[31]isidentifiedwithasubcomplexofC (P). Then,∗L∞ we assume that P is equipped with a weight grading P = P . In this(s)s=0 ¯ ¯case, the bar construction B (P) is equipped with an induced grading B (P) =∗ ∗L∞ ¯ ¯B (P) . Furthermore, we have B (P) = 0 if∗ > s. Therefore, the∗ (s) ∗ (s)s=0 ¯ ¯ ¯homology modules K (P) =H (B (P) ) form a subcomplex of B (P). Finally,s s ∗ ∗(s) we obtain embeddings of Σ -modules∗ ¯ ¯ ¯K (P) → B (P) → C (P).∗ ∗ ∗ ¯The definition ofK (P) generalizes a construction of V. Ginzburg and M. Kapra-∗ nov. Tobeprecise,theseauthorsconsideroperadsforwhichtheweightgradingco- incides with the operadic grading. More explicitly, any operadP can be equipped with a weight grading such that P (r) =P(r) if s =r+1 and P (r) = 0 oth-(s) (s) erwise. In this context, V. Ginzburg and M. Kapranov determine the Σ -modules∗ ¯K (P) associated to the commutative operadP =C and to the Lie operadP =L.∗ ∨ ∨¯ ¯WeobtainpreciselyK (C)(r)=L(r) ⊗sgn andK (L)(r)=C(r) ⊗sgn . Hence,∗ r ∗ r the embeddings above provide the required comparison morphisms. Here is how we deduce the theorem of the prologue from Koszul duality argu- ments. We introduce constructions with coefficients C (L,P,R) and K (L,P,R)∗ ∗ such that ¯ ¯C (I,P,P)=C (P)◦P and K (I,P,P)=K (P)◦P.∗ ∗ ∗ ∗ Then, we generalize a classical comparison argument from algebras to operads. ¯Namely, we prove that amorphism ofΣ -modulesφ:M → N, whereM =M◦P∗ ¯ ¯ ¯ ¯and N =N◦P, induces a quasi-isomorphism φ :M → N if the Σ -modules M∗ andN are both acyclic chain complexes. (Let us mention that, in positive charac- teristic, this property holds for Σ -modules but not for functors.) Consequently,∗ ¯ ¯we conclude that the embedding K (P) → C (P) is a quasi-isomorphism if and∗ ∗ ¯only if the complex K (I,P,P)=K (P)◦P is acyclic.∗ ∗ ¯We have in general S(K (I,P,P)) = S(K (P))◦ S(P). Accordingly, the∗ ∗ complex of Σ -modules K (I,P,P) is associated to the Harrison complex of the∗ ∗ Harrsymmetric algebra C (S(C)(V)) in the case P = C and to the Chevalley-∗ CEEilenberg complex of the free Lie algebra C (S(L)(V)) in the case P =L. We∗ mention that the latter complex is acyclic while the former is not. Nevertheless, 6 BENOIT FRESSE we prove that the associated complexes of Σ -modulesK (I,C,C) andK (I,L,L)∗ ∗ ∗ are both acyclic. To be precise, the complex K (I,L,L) is acyclic because this∗ CEproperty holds for the associated functor S(K (I,L,L))(V) = C (S(L)(V)).∗ ∗ Then, wededucethatthecomplexK (I,C,C)isacyclicbyKoszuldualitybetween∗ the commutative operad and the Lie operad. ¯As mentioned above, our definition of the Koszul construction P 7→ K(P) generalizes the definition of V. Ginzburg and M. Kapranov. In fact, there are Koszul duality results for generalizations of the Lie operad in which one considers Lie brackets in more than 2 variables (cf. E. Getzler [29], A.V. Gnedbaye [32], A.V.GnedbayeandM.Wambst[33],P.HanlonandM.Wachs[36],Y.Manin[57]). The theory of V. Ginzburg and M. Kapranov does not work for these examples, but the generalized construction gives the right results. Finally, we would like to mention that similar Koszul duality results seem to occur for the derived functors ∞M Σ ⊗rrTor (C)(V)= Tor (C(r),V )∗ ∗ r=0 ∞M Σ ⊗rrand Tor (L)(V)= Tor (L(r),V )∗ ∗ r=0 associated to the commutative and Lie operads. Hints are given, on one hand, by the calculations of G. Arone and M. Mahowald (cf. [3], see also G. Arone and M. Kankaanrinta[2])and,ontheotherhand,bytheoremsobtainedbyA.K.Bousfield and E.B. Curtis in the context of unstable algebras (cf. [18]). Hereareafewindicationsaboutthecontentsofthisarticle. Werecallclassical definitionsandresultsaboutoperadstructuresinchapter1. Moreparticularly, we point out special phenomena which do not occur in characteristic zero and which motivate the definitions of this article. We generalize results of classical homology theory from modules over an algebra to modules over an operad in chapter 2. ¯We recall the main properties of the bar construction B (P) in chapter 3. We∗ give a precise account of this construction to fix conventions and for the sake of completeness, but this section does not contain any original result. We introduce ¯ ¯ ¯the cotriple construction C (P) and the comparison morphism B (P) → C (P)∗ ∗ ∗ ¯in chapter 4. We devote chapter 5 to the Koszul constructionK (P). We go back∗ to partition posets and applications in the last section of this article. Thechapters2,3and4haveindependentintroductionswheremainresultsare stated. Proofs and technical constructions are postponed (and could be skipped in a first reading). Part 0. Conventions 0.1. Notation. We fix a commutative ground ring K. We work within the category of K-modules, denoted by KMod. To be precise, if the ground ring is ∨not a field, then we consider tacitely only projective K-modules. We let V = Hom (V,K) denote the dualK-module of any V ∈KMod. K The group of permutations of{1,...,r} is denoted by Σ . If M is a Σ -r r ∨module, then M is the dual K-module of M and is equipped with an unsigned −1action of Σ . Explicitly, if f :M → K, then we set w·f(x) =f(w ·x), for allr w∈Σ .r KOSZUL DUALITY OF OPERADS 7 0.2. Symmetric monoidal categories. We recall that a symmetric mo- noidal category consists of a categoryM together with an associative bifunctor ⊗:M×M→M which possesses a symmetry isomorphism c(X,Y):X⊗Y → Y⊗X and a unit 1∈M. In this article, we are concerned with the following basic symmetric monoidal categories: thecategoryofK-modules, denotedbyKMod; thecategoryofdifferen- tialgradedK-modules,denotedbydgKMod;thecategoryofsimplicialK-modules, denoted by sKMod. In the next paragraphs, we recall the definition of these cat- egories and some related conventions. 0.3. Differential graded modules. A differential gradedK-module (a dg- L module, for short) denotes a lower N-graded K-module V = V equipped∗∗∈N with a differential δ : V → V which decreases degrees by 1. A dg-module is∗ ∗−1 equivalent to a chain complex δ δ δ δ ···−→ V −→···−→ V −→ V .d 1 0 In general, the homology of this chain complex is denoted by H (V). But, in the∗ caseofanoncanonicaldifferentialδ :V → V , weadoptthenotationH (V,δ).∗ ∗−1 ∗ The notation|v|=d indicates the degree of a homogeneous element v∈V .d 0A morphism f : V → V is homogeneous of lower degree|f| = d if we have 0 ∨f(V )⊂ V . The dual of a dg-module V, denoted by V , is the dg-module∗ ∗+d generated by homogeneous morphisms f : V → K. More precisely, we assume that the ground ring K is equivalent to a dg-module concentrated in degree 0. Accordingly, a homogeneous morphism of lower degree|f| = d is equivalent to a ∨ ∨ ∨map f :V → K and V is the dg-module such that (V ) =(V ) .−d d −d 0A morphism of dg-modulesf :V → V is a homogeneous morphism of degree ∼ 00 such thatδ(f(v))=f(δ(v)), for allv∈V. A quasi-isomorphismf :V −→ V is ’ a morphism of dg-modules which induces a homology isomorphismf :H (V)−→∗ ∗ 0H (V ).∗ 0.4. The tensor product of dg-modules. The category of dg-modules, which we denote by dgKMod, is equipped with the structure of a symmetric monoidal category. We just consider the classical tensor product of dg-modules. L By definition, we have (V ⊗W) = V ⊗W and the differential of ad i ji+j=d homogeneous tensorv⊗w∈V⊗W is given by the formulaδ(v⊗w)=δ(v)⊗w+ |v|(−1) v⊗δ(w). We recall that the symmetry isomorphism c(V,W) : V⊗W → W⊗V involves a sign. Precisely, for v⊗w ∈ V ⊗W, we have the formula |v||w|c(V,W)(v⊗w) = (−1) w⊗v. In general, a commutation of homogeneous deelements of respective degrees d and e is supposed to produce a sign± = (−1) as in the definition of the symmetry isomorphism. In this article, we may denote this sign by± without more specification. 0For instance, suppose given homogeneous morphisms f : V → V and g : 0W → W . According to Koszul, we have a homogeneous morphism f⊗g : 0 0 |v||g|V⊗W → V ⊗W defined by the formula (f⊗g)(v⊗w)=(−1) f(v)⊗g(w), for all v⊗w∈V⊗W. 0.5. Graded modules. We have an inclusion of symmetric monoidal cat- egories (KMod,⊗) ⊂ (dgKMod,⊗). Precisely, a K-module is equivalent to a dg-module which has only one component in degree 0. We have a similar inclusion for the category of graded modules (grKMod,⊗)⊂ (dgKMod,⊗). By definition, 8 BENOIT FRESSE a gradedK-module is a dg-module equipped with a trivial differential δ = 0. Be- cause of this definition, we assume the same commutation rule for graded modules as for dg-modules. In section 5, we introduce the notion of a weight in order to make distinct a grading which does not contribute to the sign of a symmetry isomorphism. 0.6. Simplicial modules. We assume the classical definition of a simplicial K-module,asasequenceofK-modulesV equippedwithfacesd :V → V andn i n n−1 degeneraciess :V → V . ThetensorproductofsimplicialK-modulesV⊗W isj n n+1 definedby(V⊗W) =V ⊗W foralldimensionsn∈N. Thefaced :(V⊗W) →n n n i n (W⊗V) (respectively, the degeneracys :(V⊗W) → (W⊗V) ) satisfiesn−1 j n n+1 the identity d (v⊗w) = d (v)⊗d (w) (respectively, s (v⊗w) = s (v)⊗s (w)),i i i j j j for allv⊗w∈V ⊗W . The symmetry isomorphismc(V,W):V⊗W → W⊗Vn n verifies c(V,W)(v⊗w)=w⊗v, for all v⊗w∈V ⊗W .n n We define the normalized chain complex of a simplicialK-module by the for- mula N (V)=V /s (V )+···+s (V ).n n 0 n−1 n−1 n−1 The differential of N (V) is given by the alternate sum∗ nX iδ(v)= (−1) d (v),i i=0 for all v∈V . One observes that the normalization functorn N :sKMod→ dgKMod∗ isweak-monoidal, sincetheclassicalEilenberg-MacLanemorphismdefinesafunc- torial and associative quasi-isomorphism ∼ N (V)⊗N (W)−→ N (V⊗W)∗ ∗ ∗ which commutes with symmetry isomorphisms (cf. S. Mac Lane [54, Chapter 8]). 0.7. Multipledg-modules. Abi-gradedmoduleV isequippedwithasplit- L ting V = V . We refer to the module V as the homogeneous componentst sts,t∈N of V of horizontal degree s and vertical degree t. In the context of a bi-graded module, the notation|v| refers to the total degree|v| = s+t of a homogeneous element v∈ V . A bi-dg-module V is a bi-graded module equipped with com-st muting differentials such that δ : V → V and δ : V → V . To beh st s−1t v st st−1 precise, the commutation relation reads δ δ +δ δ = 0, because δ and δ haveh v v h h v total degree−1. We generalize the commutation rule of tensors in the context of bi-dg-modules. We assume precisely that signs are determined by total degrees. 0.8. The spectral sequence of a bicomplex. We consider the total com- plex (V,δ) of a bi-dg-module (V,δ ,δ ). The homogeneous component of degreedh vL of this dg-module is the sum V . The differential of the total complex issts+t=d rthe sum δ =δ +δ . We have a spectral sequence I (V)⇒ H (V,δ) such thath v ∗ 0 0 1 1(I ,d )=(V ,δ ), (I ,d )=(H (V ,δ ),δ )st v t s∗ v hst st 2and I =H (H (V ,δ ),δ ).s t ∗∗ v hst rWe consider also the spectral sequence II (V)⇒ H (V,δ) such that∗ 0 0 1 1(II ,d )=(V ,δ ), (II ,d )=(H (V ,δ ),δ )st h s ∗t h vst st 2and II =H (H (V ,δ ),δ ).t s ∗∗ h vst $ $ o / z / o / z / / / KOSZUL DUALITY OF OPERADS 9 0.9. Chain complexes of dg-modules. In chapters 3 and 4, we consider chain complexes ∂ ∂ ∂ ∂ ···−→ C (X)−→ C (X)−→···−→ C (X),d d−1 0 where each component C (X) is a dg-module C (X) = (C (X) ,δ). We refer tod d d ∗ δ :C (X) → C (X) astheinternaldifferentialofC (X). Weassumethatthed ∗ d ∗−1 d external differential∂ :C (X)→ C (X) is a homogeneous morphism of degreed d−1 −1 and commutes with internal differentials. We have explicitly ∂(C (X) )⊂d ∗ C (X) and δ∂ +∂δ = 0. In this situation, the chain complex (C (X),∂) isd−1 ∗−1 d equivalent to a bi-dg-module which has theK-moduleC (X) in bidegree (s,t).s s+t In general, such chain complexes are associated to a dg-object X equipped with extra algebraic structures. The map X 7→ C (X) defines a functor in X.d The internal differential δ : C (X) → C (X) is induced by the differentiald ∗ d ∗−1 of X. The external differential ∂ : C (X) → C (X) is determined byd ∗ d−1 ∗−1 the algebraic structure of X. The external subscript refers always to the internal degree of the dg-module C (X) (and is equivalent to the total degree of thed ∗ associated bi-dg-module). Part 1. Composition products and operad structures 1.1. Introduction: monads and operads The purpose of this section is to recall the definition of an operad and to survey some classical constructions related to this structure. More specifically, we give in paragraph 1.1.11 the definition of the classical operadsC,A andL, which are associated to commutative and associative algebras, associative algebras and Lie algebras respectively. 1.1.1. The category of functors. We letF =F(KMod) denote the cate- gory of functors on K-modules S : KMod → KMod. We equip this categoryF with the composition product of functors◦ :F×F → F. We observe that the composition product of functors is associative and has a unit I∈F which is sup- pliedbytheidentityfunctorI(V)=V. Oneconcludesthatthecompositionprod- uct provides the category of functorsF with the structure of a (non-symmetric) monoidal category (cf. S. Mac Lane [55]). 1.1.2. Monads. According to S. Mac Lane [55], a monad is a monoid in the monoidal category of functors (F,◦). Consequently, a monad consists of a functor S∈F equipped with a product μ : S◦S → S and a unit η : I → S that verify the classical associativity and unit relations. Explicitly, the following classical diagrams are supposed to commute: a) Associativity relation: b) Unit relations: μ◦S η◦S S◦η S◦S◦S S◦S I◦S S◦S S◦I H H v H v H v H vμ H μ vS◦μ H v= H = v H v H vμ v S◦S S S 0A monad morphism φ : S → S is a functor morphism which preserves monad products. / / / ! / / ! / / < / < ! ! 10 BENOIT FRESSE 1.1.3. Algebras over a monad. An algebra over a monad S is aK-module A equipped with a left monad action ρ : S(A) → A such that the following diagrams commute: S(ρ) η(A) S(S(A)) S(A) S(A)A D D D D Dandρμ(A) D ρ D= D D ρ S(A) A A 0A morphism of S-algebras φ:A→ A is aK-module morphism which commutes with monad actions. The K-module S(V), where V ∈KMod, represents the free S-algebra gener- ated byV. More precisely, the monad productμ(V):S(S(V))→ S(V) defines a canonicalmonadactiononS(V). Inaddition, themonadunitinducesaK-module morphismη(V):V → S(V) that satisfies the classical universality property. Ex- 0 0plicitly,aK-modulemorphismφ:V → A ,whereA isanS-algebra,hasaunique extension φ 0V A D D z D D z D D z Dη(V) D ˜φ z S(V) 0˜such that φ :S(V)→ A is a morphism of S-algebras. Equivalently, we have an adjunction relation 0 0 Hom (S(V),A )=Hom (V,A ).SAlg KMod One deduces from this relation that classical algebra categories, which possess free objects, are determined by monads (cf. S. Eilenberg and C. Moore [23], S. Mac Lane [55]). For instance, the category of associative algebras, the category of associative and commutative algebras and the category of Lie algebras are as- sociated to monad structures. In paragraph 1.1.11, we deduce the construction of such monads from operad structures. 1.1.4. The classical definition of an operad. A Σ -module M is a se-∗ quence M(r), r∈N, such that M(r) is a representation of the symmetric group 0Σ . A morphism of Σ -modules f :M → M consists of a sequence of represen-r ∗ 0tation morphisms f :M(r)→ M (r). A Σ -module M has an associated functor∗ S(M):KMod→ KMod defined by the formula ∞M ⊗rS(M)(V)= (M(r)⊗V ) .Σr r=0 According to P. May (cf. [61]), an operad is a Σ -module P such that the∗ functor S(P) :KMod → KMod is equipped with the structure of a monad. An 0operad morphismφ:P → P is a morphism of Σ -modules such that the functor∗ 0morphism S(φ):S(P)→ S(P ) defines a morphism of monads. In the next paragraph, we interpret an operad element p∈ P(r) as a mul- tilinear operation in r variables p = p(x ,...,x ). The action of a permutation1 r w :P(r)→ P(r) is equivalent to a permutation of variables. We have explicitly∗ w (p)(x ,...,x )=p(x ,...,x ).∗ 1 r w(1) w(r)