BAR CONSTRUCTIONS FOR TOPOLOGICAL OPERADS AND THE GOODWILLIE DERIVATIVES OF THE IDENTITY MICHAEL CHING Abstract.We describe a cooperad structure on the simplicial bar construc* *tion on a re- duced operad of based spaces or spectra and, dually, an operad structure * *on the cobar construction on a cooperad. We also show that if the homology of the orig* *inal operad (re- spectively, cooperad) is Koszul, then the homology of the bar (respective* *ly, cobar) construc- tion is the Koszul dual. We use our results to construct an operad struct* *ure on the partition poset models for the Goodwillie derivatives @*I of the identity functor o* *n based spaces and show that this induces the `Lie' operad structure on the homology groups * *H*(@*I). We also extend the bar construction to modules over operads (and, dually, to* * comodules over cooperads) and show that a based space naturally gives rise to a right mo* *dule over the operad @*I. Introduction The motivation for this paper was an effort to construct an operad structure * *on the derivatives (in the sense of Tom Goodwillie's homotopy calculus [9, 10, 11]) of* * the identity functor I on the category of based spaces. Such an operad structure has been `* *known' intuitively by experts for some time but, as far as the author knows, no explic* *it construction has previously been given. One piece of evidence for such a structure is the ca* *lculation, due to various people, of the homology of these derivatives. This homology is the * *suspension of the standard Lie operad and so is itself an operad. It is reasonable to ask,* * therefore, if there is an operad structure on the derivatives themselves1 that induces this s* *tructure on the homology. Our construction is based on the partition poset model for the derivatives @** *I described by Arone and Mahowald in [1]. They show that the derivatives are the dual spectra * *associated to certain finite complexes known as the partition poset complexes. In the pres* *ent work we notice that these complexes are precisely the simplicial bar construction2 on t* *he operad P in based spaces with P (n) = S0 for all n. Most of the paper is concerned with* * showing that such a bar construction has a natural cooperad structure. We do this by re* *interpreting the bar construction in terms of spaces of trees. The cooperad structure then c* *omes from a natural way to break trees apart. Taking duals, we get the required operad st* *ructure on the derivatives of the identity. In fact, we can view the derivatives of the id* *entity as a cobar construction on the cooperad Q in spectra with Q(n) = S, the sphere spectrum, f* *or all n. ___________ Date: January 25, 2005. 2000 Mathematics Subject Classification. Primary 55P48, Secondary 18D50. 1The Goodwillie derivatives of a homotopy functor are a sequence of spectra w* *ith actions by the symmetric groups, but are only defined up to homotopy. By an operad structure on these de* *rivatives, we mean choices of models for these spectra in a suitable symmetric monoidal category, such as * *the category of S-modules of EKMM [5], together with an operad structure on those models. 2See, for example, [15, II.2.3] for the general form of the two-sided simplic* *ial bar construction. 1 2 MICHAEL CHING In the final part of the paper (x9) we show that by taking homology we do ind* *eed recover the `Lie' operad structure on H*(@*I). We do this by introducing spectral sequ* *ences for calculating the homology of the topological bar and cobar constructions. The E* *1 terms of these spectral sequences can be identified with algebraic versions of the ba* *r and cobar constructions, which in turn are related to the theory of Koszul duality for op* *erads introduced by Ginzburg and Kapranov in [8]. Our main result on this connection is that if * *the homology of a topological operad P is Koszul, then the homology of the bar construction * *B(P ) is its Koszul dual cooperad. In our case of interest, we deduce that the induced opera* *d structure on the homology of the derivatives of the identity is that of the Koszul dual of t* *he cocommutative cooperad. This is precisely the `Lie' operad structure referred to above. Outline of the paper. We now give a more detailed description of the paper. The* * first two sections are concerned with preliminaries. In x1 we recall the notions of symme* *tric monoidal and enriched categories and specify the categories we will be working with in t* *his paper. These are symmetric monoidal categories that are enriched, tensored and cotenso* *red over the category T of based compactly-generated spaces. It is to operads in these c* *ategories that we refer in the title when we say `topological operads'. We also require an ext* *ra condition that relates the symmetric monoidal structure to the tensoring over T. This con* *dition (see Definition 1.10) is crucial to our later constructions. The two main examples o* *f such cate- gories are based spaces themselves and a suitable symmetric monoidal category o* *f spectra, such as that of EKMM [5]. In x2 we recall the definitions of operads and cooperads. We should stress th* *at the con- structions of this paper only apply to what we call reduced operads and coopera* *ds. These are P with P (0) = * and P (1) = S the unit of the symmetric monoidal structure* *. The bar construction can still be defined for more general operads, but the coopera* *d structure described here does not seem to extend to such cases. In this section we also d* *efine modules and comodules over operads and cooperads respectively. The real substance of the paper starts in x3. Here we define the trees that * *will form the combinatorial heart of our description of the bar and cobar constructions. * * It is not a coincidence that these trees are the same species used by, for example, Getzler* * and Jones in their work [7] on the bar constructions for algebraic operads and Koszul dualit* *y. We also describe what we call a weighting on a tree (Definition 3.7), that is, a suitab* *le assignment of lengths to the edges of the tree. The spaces w(T ) of weightings are at the hea* *rt of everything we do in this paper. In x4.1 we give our description of the bar construction on an operad in terms* * of such trees. If P is an operad of based spaces, we can think of a point in the bar c* *onstruction B(P ) as a weighted tree (that is, a tree with lengths assigned to the edges) w* *ith vertices labelled by points coming from the spaces P (n). See Definition 4.1 for a preci* *se statement and Definition 4.4 for a more formal approach. In 4.2 we show that what we have* * defined is isomorphic to the standard simplicial bar construction on an operad. In x4.3 we concern ourselves with the cooperad structure on B(P ). This is g* *iven by the process of `ungrafting' trees (see Definition 4.11 and beyond). This invol* *ves taking a weighted, labelled tree and breaking it up into smaller trees. Finding the righ* *t way to weight and label these smaller trees gives us the required cooperad structure maps. One of the advantages of the way we have set up the theory is that the cobar * *construction on a cooperad is strictly dual to the bar construction on an operad. In x5 we g* *o through the definitions and results dual to those of x4. OPERADIC BAR CONSTRUCTIONS 3 The short section x6 is devoted to a simple but key result (Proposition 6.4) * *that relates the bar and cobar constructions via a duality functor that reduces to Spanier-White* *head duality in the case of spectra. This result says that, under the right circumstances, t* *he dual of the bar construction on an operad P is isomorphic to the cobar construction on the * *dual of P . This allows us, later on, to identify the derivatives of the identity as the co* *bar construction on a cooperad of spectra. Before turning to our main example and application, we deal in x7 with the tw* *o-sided bar and cobar constructions. This includes the bar construction for a module over * *an operad and, dually, for a comodule over a cooperad. This requires a fairly simple gene* *ralization of much of the work we did in xx3-4, in particular, a more general notion of tree * *(see Definition 7.1). Finally, in x8 we are able to complete the main aim of this paper. We identif* *y the partition poset complexes with a bar construction and deduce the existence of an operad s* *tructure on the derivatives of the identity functor (Corollary 8.7). We also give exampl* *es of modules over the resulting operad, including, in particular, a module MX naturally asso* *ciated to a based space X. The last section of the paper x9 is concerned with the relationship of our wo* *rk to the alge- braic bar construction and Koszul operads. As promised, we construct a spectral* * sequence (Proposition 9.38) relating the two and deduce the result on Koszul duality (Pr* *oposition 9.47). Future Work. The work of this paper raises various questions that seem to the a* *uthor to warrant further attention: o What is the homotopy theory of the topological bar and cobar constructio* *ns of this paper? In particular, how do they relate to known model structures on th* *e categories of operads and cooperads (see, for example, Berger-Moerdijk [2])? o Is there a deeper relationship between Goodwillie's homotopy calculus an* *d the theory of operads? The present paper does not do any calculus, the only connect* *ion being via the partition poset complexes. One might ask, for example, if the de* *rivatives of other functors can be described and/or treated using these ideas. o What object is described by an algebra or module over the operad formed * *by the derivatives of the identity? In Remark 8.9 we show that a based space X * *gives rise to such a module. How much of the (homotopy theory of) the space X is re* *tained by this module? Acknowledgements. The idea that the derivatives of the identity might be relate* *d to a cobar construction was suggested by work in progress of Kristine Bauer, Brenda * *Johnson and Jack Morava. The observation that the partition poset complexes (and hence the * *derivatives of the identity) can be described in terms of spaces of trees was mentioned to * *the author by Tom Goodwillie. The work of Benoit Fresse [6] on the algebraic side of the t* *heory was invaluable to the present paper. The author also benefitted greatly from conver* *sations with Mark Behrens and Andrew Mauer-Oats while he was writing this paper and wishes t* *o thank, in particular, his advisor Haynes Miller for his constant support, encouragemen* *t and advice. 1. Symmetric monoidal and enriched categories On the one hand, the bar and cobar constructions are most easily defined (and* * understood) in the category of based spaces. On the other hand, our main application is in * *a category of 4 MICHAEL CHING spectra. We will develop the theory in a general setting that encompasses both * *cases. This approach will also allow us to appreciate more readily the duality between the * *bar and cobar constructions. In this section we recall the basic theory of symmetric monoidal and enriched* * categories (see [4, x6] for a detailed account). We state precisely (Definition 1.10) the * *structure we will require of a category to make the bar and cobar constructions in it. Definition 1.1 (Symmetric monoidal categories). A monoidal category consists of: o a (locally small) category V; o a functor - ^ - : V x V ! V; o a unit object I; o a natural isomorphism X ^ (Y ^ Z) ~=(X ^ Y ) ^ Z; o natural isomorphisms X ^ I ~=X ~=I ^ X; such that the appropriate three coherence diagrams commute [14, xVII]. A symmetric monoidal category is a monoidal category together with: o a natural isomorphism X ^ Y ~=Y ^ X; such that four additional coherence diagrams also commute. Our notation for a s* *ymmetric monoidal category will be (V, ^, I) or just V with the rest of the structure un* *derstood. Remark 1.2. We will not give names to the associativity and commutativity isomo* *rphisms in a symmetric monoidal category. When we write unbracketed expressions such as X ^ Y ^ Z or unordered expressions such as ^ Xa a2A we strictly mean the colimit of the diagram formed by all possible ways to brac* *ket and order the expression together with the associativity and commutativity isomorphisms b* *etween them. This colimit is isomorphic to any one particular way to bracket and order* * the given expression. Definition 1.3 (Closed symmetric monoidal categories). A closed symmetric monoi* *dal cat- egory is a symmetric monoidal category (V, ^, I) together with a functor Vopx V ! V; (X, Y ) 7! Map (X, Y ) and a natural isomorphisms of sets Hom V (X ^ Y, Z) ~=Hom V(X, Map (Y, Z)) (Hom V (X, Y ) denotes the set of morphisms from X to Y in the category V). Remark 1.4. The natural isomorphism of sets in Definition 1.3 can be made into * *an iso- morphism within V. That is, in any closed symmetric monoidal category there is * *a natural isomorphism: Map (X ^ Y, Z) ~=Map (X, Map (Y, Z)). See [4, 6.5.3] for details. Definition 1.5 (Enriched categories). Let (V, ^, S) be a closed symmetric monoi* *dal cate- gory. A V-category or category enriched over V consists of: o a class of objects C OPERADIC BAR CONSTRUCTIONS 5 o for each pair of objects C, D 2 C, an object Map V(C, D) of V; o composition morphisms MapV (C, D) ^ Map V(D, E) ! Map V(C, E) o identity morphisms I ! Map V(C, C) that satisfy the appropriate conditions [4, 6.2.1]. Remark 1.6. Some basic observations about enriched categories from [4, 6.2]: (1) A Set-category is precisely a (locally small) category. (2) A V-category C has an underlying category whose objects are the objects * *of C and whose morphisms C ! D are the elements of the set Hom V(I, Map V(C, D)),* * where I is the unit object of V. We shall abuse notation and also write C for * *this category. (3) A closed symmetric monoidal category V is enriched over itself with Map * *V(X, Y ) := Map (X, Y ). Definition 1.7 (Tensoring and cotensoring). Let C be a V-category. A tensoring * *of C over V is a functor V x C ! C; (X, C) 7! X C together with a natural isomorphism Map V(X C, D) ~=Map (X, Map V(C, D)). A cotensoring of C over V is a functor Vopx C ! C; (X, D) 7! Map C(X, D) together with a natural isomorphism Map V(C, Map C(X, D)) ~=Map (X, Map V(C, D)). Remark 1.8. Some basic observations about tensorings and cotensorings: (1) A closed symmetry monoidal category (V, ^, I) is tensored and cotensored* * over itself with X Y := X ^ Y and Map V(X, Y ) := Map (X, Y ). (2) If C is tensored over V we have (X ^ Y ) C ~=X (Y C). If C is cotensored over V we have Map C(X ^ Y, C) ~=Map C(X, Map C(Y, C)). Proposition 1.9 (Duality). Let C be a V-category. Then Cop has a natural enrich* *ment over V. If C is tensored, then Cop is cotensored and vice versa. Proof.We define an enrichment on Cop by Map V(Cop, Dop) := Map V(D, C) where Cop is the object in Cop corresponding to C 2 C. If - - is a tensoring * *for C then we get a cotensoring for Cop by setting Map Cop(X, Dop) := (X D)op. 6 MICHAEL CHING The required natural isomorphism comes from Map V(Cop, Map Cop(X, Dop))= Map V(X D, C) ~=Map (X, Map V(D, C)) = Map (X, Map V(Cop, Dop)). The vice versa part is similar. We are interested in enriched categories that are also themselves symmetric m* *onoidal categories. As far as the author knows there are no standard definitions for t* *hese. The following definition contains the properties we require in this paper. Definition 1.10. Let (V, ^, I) be a closed symmetric monoidal category. A symm* *etric monoidal V-category consists of: o a symmetric monoidal category (C, Z, S) with C enriched, tensored and co* *tensored over V; o a natural transformation d : (X ^ Y ) (C Z D) ! (X C) Z (Y D); satisfying axioms o (associativity) (X ^ Y ^ Z) (C Z D Z E)______/((X/^_Yd) (C Z D)) Z (Z E) | | | | d| |idZd | | fflffl| idZd fflffl| (X C) Z ((Y ^ Z) (D Z E))_____/(X/ C) Z (Y D) Z (Z E) commutes for all X, Y, Z 2 V and C, D, E 2 C; o (unit) The following composite is the identity: X C ~=(X ^ I) (C Z S) __//_d(X C) Z (I S) ~=X C for any X 2 V and C 2 C. The transformation d (for `distribute') is our way of relating the symmetric * *monoidal structures in the two categories. It will be essential in making the bar const* *ruction of an operad into a cooperad (see Definition 4.23). Remark 1.11. A closed symmetric monoidal category V is itself a symmetric monoi* *dal V-category by the symmetry isomorphism: (X ^ Y ) ^ (C ^ D) ~=(X ^ C) ^ (Y ^ D). A crucial feature Definition 1.10 is that it is self-dual. Proposition 1.12. Let C be a symmetric monoidal V-category. Then Cop is natural* *ly also a symmetric monoidal V-category. Proof.We already know from Proposition 1.9 that Cop is enriched, tensored and c* *otensored over V and there is a canonical symmetric monoidal structure on Cop given by th* *at on C. It therefore only remains to construct the map d. The tensoring in Cop is give* *n by the cotensoring in C. Therefore d for Cop corresponds to the following map in C: Map C(X, C) Z Map C(Y, D) ! Map C(X ^ Y, C Z D) OPERADIC BAR CONSTRUCTIONS 7 This would be adjoint to a map (X ^ Y ) (Map C(X, C) Z Map C(Y, D)) ! C Z D. But we can construct such a map by first using d for C to get to (X Map C(X, C)) Z (Y Map C(Y, D)) and then using the evaluation maps X Map C(X, C) ! C, Y Map C(Y, D) ! D and the naturality of Z, we get the required map to C Z D. An important property of the categories that we work with in this paper is th* *at they are pointed (that is, they have a null object * that is both initial and terminal).* * The following proposition describes how null objects interact with symmetric monoidal structu* *res and enrichments. Proposition 1.13. Let (V, ^, I) be a closed symmetric monoidal category that is* * pointed with null object *. Then * ^ X ~=* ~=Map (*, X) ~=Map (X, *) for all X 2 V. Let C be a category enriched over V. If C is tensored then * C is an initial object in C for all C 2 C. If C is cotensored then Map C(*, D) is a terminal object in C for all D 2 C. If C is both tensored and cotensored o* *ver V then the initial and terminal objects are isomorphic and so C is itself pointed. Proof.We have Hom V(* ^ X, Y ) ~=Hom V(*, Map (X, Y )) which has one element for any X, Y . This tells us that * ^ X is initial and he* *nce isomorphic to *. The other isomorphisms in the first part of the proposition are similar. The tensoring functor - C : V ! C is a left adjoint so preserves an initial o* *bject. Dually, the cotensoring functor Map C(-, D) : Vop ! C is a right adjoint so preserves t* *he terminal object. If C is both tensored and cotensored, we get a map from the terminal ob* *ject to the initial object by Map C(*, D) ! I Map C(*, D) ! * Map C(*, D). The first map here is an example of a general isomorphism C ! I C where I is * *the unit object of V. The second map comes from I ! *. A map from a terminal object to a* *n initial object must be an isomorphism. Therefore C is pointed. Examples 1.14. The categories we will mainly be concerned with in this paper ar* *e the following. (1) Let T be the category of compactly generated based spaces and basepoint-* *preserving continuous maps of [13]. Then T is a pointed closed symmetric monoidal * *category with ^ the usual smash product, unit S0 and Map (X, Y ) the space of bas* *epoint- preserving maps X ! Y . 8 MICHAEL CHING (2) Let Sp be the category of S-modules of EKMM [5]. Then (Sp, ^S, S) is a s* *ymmetric monoidal T-category, where S is the sphere spectrum and ^S is the smash * *product of S-modules [5, II.1.1]. The enrichment, tensoring and cotensoring are * *described in [5, VII.2.8]. For d we have a natural isomorphism ~= d : (X ^ Y ) ^ (E ^S F ) __//_(X ^ E) ^S (Y ^ F ) given by the fact that X ^ E ~=(X ^ S) ^S E [5, II.1.4]. We will usually work with a general symmetric monoidal T-category denoted (C, Z* *, S), but these examples will be foremost in our minds. 2.Operads and cooperads In this section (C, Z, S) is a pointed symmetric monoidal category with null * *object *. We will assume that C has all necessary limits and colimits. We will write the cop* *roduct in C as a wedge product using _. Definition 2.1 (Symmetric sequences). A symmetric sequence in C is a functor F * *from the category of nonempty finite sets and bijections to C. For each nonempty finite * *set A, F (A) has an action by the symmetric group A. We will write F (n) for F ({1, . .,.n}* *). Note that our symmetric sequences (and hence our operads) do not have an F (0) term becau* *se our indexing sets are nonempty. We will often write `finite set' when we mean `none* *mpty finite set' and these will usually be labelled A, B, . ...We write C for the category* * of symmetric sequences in C (whose morphisms are the natural transformations). There are several different but equivalent ways to define operads (see Markl-* *Shnider- Stasheff [15] for a comprehensive guide). We will use the following definition. Definition 2.2 (Operads). An operad in (C, Z, S) is a symmetric sequence P toge* *ther with partial composition maps: - Oa - : P (A) Z P (B) ! P (A [a B) for each pair of finite sets A, B and each a 2 A (where A [a B denotes (A \ {a}* *) q B), and a unit map j : S ! P (1). The composition maps are natural in A and B and satisfy the following four axio* *ms: (1) P (A) Z P (B) Z P (C)___/P/(A) Z P (B [bC) | | | | | | | | fflffl| fflffl| P (A [a B) Z P (C)______/P/(A_[a B [bC) for all a 2 A and b 2 B. (Notice that (A [a B) [bC = A [a (B [bC).) OPERADIC BAR CONSTRUCTIONS 9 (2) ~= P (A) Z P (B) Z P (C)___/P/(A)_Z P (C) Z P (B)_____/P/(A [a0C) Z P (B) | | | | | | | | fflffl| fflffl| P (A [a B) Z P (C)_________________________________/P/(A [a B [a0C) for all a 6= a02 A. (Notice that (A [a B) [a0C = (A [a0C) [a B.) (3) idZj P (A) _____/P/(A) Z P (1) JJ | JJJ | JJJ | ~= JJJ | J$$Jfflffl| P (A [a {1}) for all a 2 A (the diagonal map is induced by the obvious bijection A ! * *A [a {1}); (4) jZid P (A) _____/P/(1) Z P (A) JJ | JJJ | JJJ | id JJJ | J$$Jfflffl| P ({1} [1 A) A morphism of operads P ! P 0is a morphism of symmetric sequences that commutes* * with the composition and unit maps in the obvious way. We write Op (C) for the cate* *gory of operads in C and their morphisms. Definition 2.3 (Augmented and reduced operads). An augmentation of an operad P * *is a map " : P (1) ! S such that the composite j " S _____/P/(1)_____/S/_ is the identity on S. An augmented operad is an operad together with an augment* *ation. An operad P is reduced if the unit map j : S ! P (1) is an isomorphism. A reduced* * operad has a unique augmentation given by the inverse of the unit map. A morphism of a* *ugmented operads is a morphism of operads that commutes with the augmentation. Remark 2.4. Operads are a generalization of monoids for the symmetric monoidal * *category (C, Z, S). A monoid A gives rise to an operad PA with PA(1) = A and PA(n) = * f* *or n > 1. Conversely, given an operad P in the symmetric monoidal category C, P (1) forms* * a monoid in C. An alternative definition of an operad is based on the following monoidal str* *ucture on the category of symmetric sequences. Definition 2.5 (Composition product of symmetric sequences). We define the comp* *osition product of the two symmetric sequences M, N to be the symmetric sequence M O N * *with ` ^~ M O N(A) := M(~) Z N(B). partitions ~ of A B2~ 10 MICHAEL CHING By a partition of A we mean a set of disjoint nonempty subsets whose union is A* *. (Note that our partitions are unordered.) A bijection A ! A0determines a bijection between* * partitions of A and partitions of A0in an obvious way. Thus we match up the terms in the c* *oproducts for M O N(A) and M O N(A0). If ~ corresponds to ~0, then in fact we have a bije* *ction ~ ! ~0 and if B 2 ~ and B02 ~ correspond under this bijection then we in turn have B !* * B0. We use the effect of M and N on these bijections to get an isomorphism M O N(A) ! M O N(A0). Thus M O N becomes a symmetric sequence in C. Proposition 2.6 ([15], x1.8). The composition product gives us a monoidal struc* *ture on the category of symmetric sequences in C with unit object I given by ( S if |A| = 1; I(A) := * otherwise. An operad in C is precisely a monoid for this monoidal structure. Thus an operad P consists of a map of symmetric sequences P O P ! P that encodes all the partial compositions, and a map I ! P that is the unit. The four axioms of Definition 2.2 together make up the associ* *ativity and unit axioms for P to be a monoid. An augmentation for P is precisely a map P !* * I of symmetric sequences, left inverse to the unit I ! P . This definition of an operad allows us to succintly say what we mean by a mod* *ule over an operad. Definition 2.7 (Modules over operads). A left module over the operad P is a sym* *metric sequence M together with a left action of the monoid P , that is, a map P O M ! M such that (P O P ) O M_____/P/O_M | RRRRRR | RRRR | R))R || ll5M5l | lllll fflffl| llll P O (P O M) _____/P/O_M commutes and M ~=I O M ! P O M ! M is the identity on M. A right module over P is a symmetric sequence M with a right action of P , th* *at is a map M O P ! M OPERADIC BAR CONSTRUCTIONS 11 satisfying corresponding axioms. A P -bimodule is a symmetric sequence M that i* *s both a right and a left module over P such that (P O M) O P _____/M/O_P | RRRRRR | RRRR | R))R || ll5M5l | lllll fflffl| llll P O (M O P )_____/P/O_M commutes. Clearly, P itself is a P -bimodule. Remark 2.8. It's useful to have a slightly more explicit description of a modul* *e over an operad. The action map for a left P -module M consists of maps P (r) Z M(A1) Z . .Z.M(Ar) ! M(A) ` r for every partition A = i=1Ai of a finite set A into nonempty subsets. Simila* *rly, a right module structure consists of maps M(r) Z P (A1) Z . .Z.P (Ar) ! M(A). Remark 2.9. In the same way that operads are a generalization of monoids in C, * *modules over those operads are generalization of modules over the monoids. A module M o* *ver the monoid A gives rise to a module PM over the operad PA described in Remark 2.4 * *with PM (n) = * if n > 1 and PM (1) = M. Remark 2.10. An augmentation for the operad P is equivalent to either a left or* * right module structure on the unit symmetric sequence I. The standard notion of an algebra over an operad is closely related to that o* *f a module. We briefly describe how this works. Definition 2.11 (Algebras over an operad). An algebra over the operad P is an o* *bject C 2 C together with maps ^~ P (A) Z C ! C a2A that satisfy appropriate naturality, associativity and unit axioms. We can think of a P -algebra as a left P -module concentrated in the M(0) ter* *m. Unfortu- nately, in order for the bar and cobar constructions to work out, we are forced* * in this paper to rule out modules with nontrivial M(0). However, the following result allows * *us to treat algebras indirectly. Lemma 2.12. Let C be an algebra over the operad P . Then there is a natural lef* *t P -module structure on the symmetric sequence C_with C_(A) = C (with trivial A-action) f* *or all finite sets A. Proof.The module structure map P OC_ ! C_is easily constructed from the algebra* * structure maps ^~ P (A) Z C ! C. a2A 12 MICHAEL CHING Definition 2.13 (Cooperads). The notion of cooperad is dual to that of operad, * *that is, a cooperad in C is an operad in the opposite category Cop with the canonical sy* *mmetric monoidal structure given by that in C. More explicitly, a cooperad consists of * *a symmetric sequence Q in C together with cocomposition maps Q(A [a B) ! Q(A) Z Q(B) and a counit map Q(1) ! S satisfying axioms dual to (1)-(4) of Definition 2.2. A morphism of cooperads is* * a morphism of symmetric sequences that commutes with the cocomposition and counit maps. Coope* *rads in C and their morphisms form a category which we will denote by Coop(C). A coaugm* *entation for a cooperad is a map S ! Q(1) left inverse to the counit map. A cooperad Q i* *s reduced if the counit map is an isomorphism. Remark 2.14. A cooperad is a comonoid for the monoidal product "Oon symmetric s* *equences obtained by replacing the coproduct in Definition 2.5 with a product. That is: Y ^~ M "ON(A) := M(~) Z N(B). partitions ~ of A B2~ This is again a monoidal product on the category of symmetric sequences with th* *e same unit object I. A cooperad then consists of a symmetric sequence Q with maps Q ! Q "O* *Q and Q ! I. An augmentation for Q is a map I ! Q such that I ! Q ! I is the identity. In [7] Getzler and Jones define a cooperad to be a comonoid for the compositi* *on product O. In their case, O and "Oare equal because finite products are isomorphic to f* *inite coproducts in the category of chain complexes. Definition 2.15 (Comodules over a cooperad). A left comodule C over the coopera* *d Q is a left module over Q considered as an operad in Cop. More explicitly, C is a symm* *etric sequence together with a left coaction of the comonoid Q, that is, a map C ! Q "OC. Equi* *valently, we have a collection of cocomposition maps C(A) ! Q(r) Z C(A1) Z . .Z.C(Ar) ` for partitions A = i2IAi. Similarly a right comodule is a symmetric sequence* * C with a right coaction C ! C "OQ, or equivalently, cocomposition maps C(A) ! C(r) Z Q(A1) Z . .Z.Q(Ar). A bicomodule is a symmetric sequence with compatible left and right comodule st* *ructures. The cooperad Q is itself a Q-bicomodule. A coalgebra over a cooperad is the dual concept of an algebra over an operad * *and the constant symmetric sequence with value equal to a Q-coalgebra is a left Q-comod* *ule. 3.Spaces of trees As mentioned in the introduction to the paper, the key to finding a cooperad * *structure on the bar construction on an operad is its reinterpretation in terms of trees.* * These are the same sorts of trees used in many other places to work with operads. See Getzler* *-Jones [7], Ginzburg-Kapranov [8] and Markl-Shnider-Stasheff [15] for many examples. OPERADIC BAR CONSTRUCTIONS 13 Definition 3.1 (Trees). A typical tree of the sort we want is shown in Figure 1* *. It has a root element at the base, a single edge attached to the root, and no other vert* *ices with only one incoming edge. We encode these geometric requirements in the following comb* *inatorial definition. A tree T is a finite poset satisfying the following conditions: (1) T has at least two elements: an initial (or minimal) element r, the root* *, and another element b such that b t for all t 2 T except r; (2) for any elements t, u, v 2 T , if u t and v t, then either u v or * *v u; (3) for any t < u in T with t not equal to r, there is some v 2 T such that * *t < v but u v. We picture a tree by its graph, whose vertices are the elements of T with an ed* *ge between t and u if t < u and there is no v with t < v < u. An incoming edge to a vertex t* * is an edge corresponding to t < u. Condition (1) above ensures that the tree has a root r * *with exactly one incoming edge (that connects it to b). The second condition ensures that t* *his graph is indeed a tree in the usual sense.3 The third condition ensures that there ar* *e no vertices except the root that have exactly one incoming edge. More terminology: the maximal elements of the tree T will be called leaves. F* *rom now on, by a vertex, we mean an element other than the root or a leaf (see Figure 1* *). A tree is binary if each vertex has precisely two incoming edges. The root edge is the ed* *ge connected to the root element. The leaf edges are the edges connected to the leaves. The * *other edges in the tree are internal edges. Given a vertex v of a tree, we write i(v) for the * *set of incoming edges of the vertex v. We generally denote trees with the letters T, U, . ... Remark 3.2. We stress that our trees are not allowed to have vertices with only* * one incoming edge, as guaranteed by condition (3) of the definition. This reflects * *the fact that we will deal only with reduced operads in this paper. See the beginning of sec* *tion x4 for more discussion of this. leaves vertices root Figure 1. Terminology for trees Definition 3.3 (Labellings). A labelling of the tree T by a finite set A is a b* *ijection between A and the set of leaves of T . An isomorphism of A-labelled trees is an isomorp* *hism of the underlying trees that preserves the labelling. We denote the set of isomorphism* * classes of A-labelled trees by T(A). For a finite set A, T(A) is also finite. For a positi* *ve integer n, we write T(n) for the set T({1, . .,.n}). ___________ 3A graph is usually said to be a tree if it is connected and contains no cycl* *es. 14 MICHAEL CHING Example 3.4. There is up to isomorphism only one tree with one leaf. It has a s* *ingle edge whose endpoints are the root and the leaf. Thus T(1) has one element. It is eas* *y to see that T(2) also only has one element: the tree with one vertex that has two input edg* *es. Figure 2 shows T(1), T(2), T(3). 1 1 2 T(1) T(2) 1 2 3 1 2 3 2 3 1 3 1 2 T(3) Figure 2. Labelled trees with three or less leaves Definition 3.5 (Edge collapse). Given a tree T and an internal edge e, denote b* *y T=e the tree obtained by colllapsing the edge e, identifying its endpoints. (In poset * *terms, this is equivalent to removing from the poset the element corresponding to the upper en* *dpoint of the edge.) If u and v are those endpoints, write u O v for the resulting vertex* * of T=e. Note that T=e has the same leaves as T so retains any labelling. See Figure 3 for an* * example. a b c a b c v e u _ u O v T T=e Figure 3. Edge collapse of labelled trees OPERADIC BAR CONSTRUCTIONS 15 Definition 3.6 (The categories T(A)). The process of collapsing edges gives us * *a partial order on the set T(A) of isomorphism classes of A-labelled trees. We say that T* * T 0if T can be obtained from T 0be collapsing a sequence of edges. We think of the resu* *lting poset as a category. We now give our trees topological significance by introducing `weightings' on* * them. A weighting on a tree T assigns lengths to the edges of T . The operadic W -cons* *truction of Boardman and Vogt [3, 16] is based on the similar idea of a length function on * *the edges of trees. Their length functions satisfy slightly different conditions to our weig* *htings. Definition 3.7 (Weightings). A weighting of a tree T is an assignment of nonneg* *ative `lengths' to the edges of T in such a way that the `distance' from the root to * *each leaf is exactly 1. The set of weightings on a tree T is a subset of the space of functi* *ons from the set of edges of T to the unit interval [0, 1] and we give it the subspace topol* *ogy. We denote the resulting space by w(T ). A tree together with a weighting is a weighted tr* *ee. Example 3.8. There is only one way to weight the unique tree T 2 T(1) (the sing* *le edge must have length 1), so w(T ) = *. For any n, T(n) contains a tree Tn with a si* *ngle vertex that has n incoming edges. For this tree we have w(Tn) = 1 the topological 1-s* *implex or unit interval. Figure 2 displays another shape of tree with three leaves, one * *that has two vertices. For such a tree U, we have w(U) = 2, the topological 2-simplex. Not * *all spaces of weightings are simplices, but we do have the following result. Lemma 3.9. Let T be a tree with n vertices. Then w(T ) is homeomorphic to the* * n- dimensional disc Dn. If n 1, the boundary @w(T ) is the subspace of weighting* *s for which at least one edge has length zero. Proof.Suppose T has l leaves. Then it has n + l total edges and using the lengt* *hs of the edges as coordinates we can think of w(T ) as a subset of Rn+l. For each leaf l* *iof T there is a condition on the lengths of the edges in a weighting that translates into an af* *fine hyperplane Hiin Rn+l. Then w(T ) is the intersection of all these hyperplanes with [0, 1]n* *+l. Now these hyperplanes all pass through the point that corresponds to the root* * edge hav- ing length 1 and all other edges length zero. Therefore their intersection is * *another affine subspace of Rn+l. To see that they intersect transversely, we check that each H* *i does not contain the intersection of the Hj for j 6= i. Consider the point pi in Rn+l t* *hat assigns length 1 to each leaf edge except that corresponding to leaf li, and length 0 t* *o all other edges (including the leaf edge for li). Since the equation for the hyperplane Hj cont* *ains the length of exactly one leaf edge, this point piis in " Hj j6=i but not in Hi. This shows that the Hi do indeed intersect transversely. There* *fore their intersection is an n-dimensional affine subspace V of Rn+l. Finally, notice that, as long as n > 0, V passes through an interior point of* * [0, 1]n+l, for example, the point where all edges except the leaf edges have length " for some* * small " > 0 and the leaf edges then have whatever lengths they must have to obtain a weight* *ing. It then follows that w(T ) = V \ [0, 1]n+l is homeomorphic to Dn. If n = 0, there is on* *ly one tree and its space of weightings is a single point, that is, D0. For the second statement, notice that the boundary of w(T ) is the intersecti* *on of V with the boundary of the cube [0, 1]n+l. If a weighting includes an edge of length z* *ero, it lies in 16 MICHAEL CHING this boundary. Conversely, a weighting in this boundary must have some edge wit* *h length either 0 or 1. If the root edge has length 1, all other edges must have length* * 0. If some other edge has length 1, the root edge must have length 0. In any case, some ed* *ge has length 0. Definition 3.10 (The functors w(-) and w~(-)). For each finite set A, the assig* *nment T 7! w(T ) determines a functor w(-) : T(A) ! U where U is the category of unbased spaces. To see this we must define maps w(T=e) ! w(T ) whenever e is an internal edge in the A-labelled tree T . Given a weighting on * *T=e we define a weighting on T by giving edges in T their lengths in T=e with the edge e havi* *ng length zero. This is an embedding of w(T=e) as a `face' of the `simplex' w(T ). It's e* *asy to check that this defines a functor as claimed. Let w0(T ) be the subspace of w(T ) containing weightings for which either th* *e root edge or some leaf edge has length zero. We set ~w(T ) := w(T )=w0(T ). This is a based space with basepoint given by the point to which w0(T ) has bee* *n identified. If T is the tree with only one edge then w0(T ) is empty. We use the convention th* *at taking the quotient by the empty set is given by adjoining a disjoint basepoint. Therefore* *, ~w(T ) = S0. The maps w(T=e) ! w(T ) clearly map w0(T=e) to w0(T ) and so give us maps ~w(T=e) ! ~w(T ). For each finite set A, these form a functor ~w(-) : T(A) ! T where T is the category of based spaces. Example 3.11. Figure 4 displays the spaces w(T ) for T 2 T(3) and how the funct* *or w(-) fits them together. Recall that the poset T(3) has four elements: one minimal e* *lement (the tree with one vertex and three incoming edges) and three maximal elements (thre* *e binary trees with two vertices). As the picture shows, the functor w(-) embeds a 1-sim* *plex for the minimal element as one of the 1-dimensional faces of a 2-simplex for each of th* *e maximal elements. The subspaces w0(T ) are outlined in bold. Collapsing these we get * *the functor w~(-) which embeds S1 (for the minimal element) as the boundary of D2 (for each* * maximal element). 4.Bar constructions for reduced operads This section forms the heart of this paper. We show that by giving an explici* *t description of the simplicial bar construction in terms of trees, we can construct a cooper* *ad structure on it. In x4.1 we give our definition of the bar construction B(P ) for an operad * *P in C. In x4.2 we show that this is isomorphic to the standard simplicial reduced bar construc* *tion on P . Then in x4.3 we prove the main result of this paper: that B(P ) admits a natura* *l cooperad structure. We will work in a fixed symmetric monoidal T-category (C, Z, S) where T is th* *e category of based compactly-generated spaces and basepoint preserving maps. Since T is * *pointed, OPERADIC BAR CONSTRUCTIONS 17 1 2 3 w( ) 1 2 3 2 3 1 w( ) w( ) 3 1 2 w( ) Figure 4. Spaces of weightings of trees with three leaves Proposition 1.13 implies that C too is pointed. We denote the null object in C * *also by *. We assume that C has all limits and colimits. The examples to bear in mind are C =* * T itself and C = Sp. We will use the notation developed in x1 for the enrichment, tenso* *ring and cotensoring of C over T. Before we start we should stress that the constructions in this paper only ap* *ply to reduced operads and cooperads. That is, those for which the unit (or counit) map is an * *isomorphism. This is reflected in several places, most notably in the fact that our trees ar* *e not allowed to have vertices with only one incoming edge (see Remark 3.2). It is a necessary c* *ondition for our construction of the cooperad structure on B(P ). 4.1. Definition of the bar construction. We give two definitions of the bar con* *struction for an operad. The first is somewhat informal and relies on C being the categor* *y of based spaces, but captures how we really think about these objects. The second is a p* *recise formal definition as a coend in the category C. Definition 4.1. Let P be a reduced operad in T. The bar construction on P is th* *e symmetric sequence B(P ) defined as follows. A general point p in B(P )(A) consists of: o an isomorphism class of A-labelled trees: T 2 T(A); o a weighting of T ; o for each (internal) vertex v of T , an point pv in the based space P (i(* *v)) (recall that i(v) is the set of incoming edges of the vertex v); subject to the following identifications: o if pv is the basepoint in P (i(v)) for any v then p is identified with t* *he basepoint *; o if the internal edge e has length zero, we identify p with the point q g* *iven by - the tree T=e; 18 MICHAEL CHING - the weighting on T=e in which an edge has the same length as the cor* *responding edge of T in the weighting that makes up p;4 - quOvgiven by the image under the composition map P (i(u)) ^ P (i(v)) ! P (i(u O v)) of (pu, pv) (notice that i(u O v) = i(u) Ov i(v)); - qt= pt for the other vertices t of T=e; o if a root or leaf edge has length zero, p is identified with *. A bijection oe : A ! A0gives us an isomorphism oe* : B(P )(A) ! B(P )(A0) by re* *labelling of the leaves of the tree. In this way, B(P ) becomes a symmetric sequence in T. Example 4.2. Consider B(P )(1). There is only one tree with a single leaf and * *only one weighting on it. It has no vertices so B(P )(1) does not depend at all on P . * * With the basepoint (which is disjoint in this case because nothing is identified to it) * *we get B(P )(1) = S0. Next consider B(P )(2). Again there is only one tree, but this time it has a * *vertex (with two incoming edges) and the space of ways to weight the tree is 1 = [0, 1]. Ma* *king all the identifications we see that B(P )(2) = P (2), the (reduced) suspension of P (2). Definition 4.3 (The functors PA). A key ingredient of the general definition of* * the bar construction is that an operad P in C determines a functor PA(-) : T(A)op! C. where T(A), as always, is the poset of isomorphism classes of A-labelled trees * *ordered by edge collapse. For a tree T we define ^~ PA(T ) := P (i(v)) vertices v in T where we recall that i(v) is the set of incoming edges to the vertex v. If e is* * an internal edge in T with endpoints u and v then there is a partial composition map P (i(u)) Z P (i(v)) ! P (i(u O v)). This then induces a map PA(T ) ! PA(T=e) that makes PA(-) into a functor as claimed. Recall from Definition 3.10 that we have a functor ~w(-) : T(A) ! T given by taking the space of weightings on a tree, modulo those for which a roo* *t or leaf edge has length zero. ___________ 4This is the inverse image under the injective map w(T=e) ! w(T) of the weighting corresponding to p. The condition that e has length zero says * *precisely that this weighting is in the image of this map. OPERADIC BAR CONSTRUCTIONS 19 Definition 4.4 (Formal definition of the bar construction). The bar constructio* *n of the reduced operad P is the symmetric sequence B(P ) defined by Z T2T(A) B(P )(A) := ~w(-) T(A)PA(-) = w~(T ) PA(T ). This is the coend in C of the bifunctor w~(-) PA(-) : T(A) x T(A)op! C. (See [14] for a full treatment of coends.) The definition of the coend is a co* *limit over a category whose objects are morphisms in T(A) and we will write the coend above * *as colim ~w(T ) PA(T 0) T T02T(A) when we need to manipulate it. A bijection A ! A0induces an isomorphism of categories T(A) ! T(A0). If T 7! * *T 0under this isomorphism then PA(T ) = PA(T 0) and ~w(T ) = ~w(T 0). Therefore we get * *an induced isomorphism B(P )(A) ! B(P )(A0). This makes B(P ) into a symmetric sequence in* * C. Remark 4.5. To see that our two definitions of the bar construction are equival* *ent when C = T, recall that the coend is a quotient of the coproduct ` ~w(T ) PA(T ). T2T(A) That is, a point consists of a weighted tree together with elements of the P (i* *(v)) for vertices v subject to some identifications. The maps PA(T ) ! PA(T=e) and ~w(T=e) ! ~w(T* * ) encode the identifications made in Definition 4.1. Example 4.6 (The associative operad). An important source of operads in based s* *paces is to take operads in unbased spaces (with symmetric monoidal structure given by c* *artesian product) and add a disjoint basepoint to each term. Here we do a calculation w* *ith an example of this type. Let Ass be the operad for associative monoids in unbased spaces. This is give* *n by Ass(n) := n (with the discrete topology and regular n-action). The composition maps are th* *e inclusions given by identifying rx n1x . .x. nr with a subgroup of n1+...+nr. Let us calculate B(Ass+) (where (-)+ denotes ad* *ding a disjoint basepoint). The points pv 2 Ass+(i(v)) in Definition 4.1 basically determine an order on * *the incoming edges to vertices of a tree. This allows us to identify a point in B(Ass+)(n) w* *ith a planar weighted tree with leaves labelled 1, . .,.n. This breaks B(Ass+)(n) up into a * *wedge of n! terms, each corresponding to an ordering of the leaves of the trees involved. With a little bit of work, it can be seen that each of these terms is an (n-1* *)-sphere. Think of constructing a planar weighted tree with leaves labelled in a fixed order (s* *ay, 1, . .,.n) by the following method. Connect the first leaf to the root with an edge of len* *gth 1. Then attach the second leaf at some point along the edge already drawn. Attach the t* *hird leaf at some point along the path from the second leaf to the root, and so on. The spac* *e of choices made in doing all this is [0, 1]n-1 and we obtain precisely the planar weighted* * trees we want 20 MICHAEL CHING in this manner (see Figure 5). The root edge or a leaf edge will have length ze* *ro if and only if at least one of our choices was either 0 or 1. Hence the space we want is o* *btained by identifying the boundary of [0, 1]n-1 to a basepoint. This gives Sn-1. 1 2 3 1 2 3 Figure 5. Constructing planar weighted trees Therefore we have B(Ass+)(n) = Sn-1 ^ ( n)+ where n acts trivially on the Sn-1 term and by translation on the non-basepoin* *ts of ( n)+. We can also picture what happens for n = 3 in terms of sticking together the * *spaces w~(T ) ^ AssA(T )+ for T 2 T(3). The ~w(T ) are the quotients of the spaces pic* *ture in Figure 4 by the subspaces outlined in bold. To make up B(Ass+)(3) we need six copies o* *f the 1- simplex (corresponding to the points in Ass(3)) and twelve copies of the 2-simp* *lex. (There are four points in Ass(2) x Ass(2) and three trees of this type.) These fit tog* *ether to form six disjoint copies of the following picture, one for each permutation of 1, 2,* * 3. The type of tree used to form each part is shown. 1 2 3 1 2 3 1 2 3 When we collapse the bold subspaces to the basepoint we get a wedge of six copi* *es of S2 as expected. Remark 4.7. Finally we should note that the W -construction of Boardman and Vog* *t (also called the bar construction) is defined in a very similar manner to B(P ). It * *uses slightly different spaces of trees and produces an operad instead of a cooperad. See [16* *] for details. 4.2. Relation to the simplicial bar construction. In this section we show that * *B(P ) is isomorphic to the geometric realization of the standard simplicial bar const* *ruction on the reduced operad P . This simplicial bar construction can be defined for any * *augmented monoid in a monoidal category.5 In our case, the monoidal category is symmetric* * sequences in C under the composition product. We have already remarked, in Proposition 2.* *6, that an operad is precisely a monoid for this product. ___________ 5See [15, II.2.3] for a discussion of different forms of the simplicial bar c* *onstruction. OPERADIC BAR CONSTRUCTIONS 21 Definition 4.8 (Simplicial bar construction). Let P be a reduced operad in C. T* *he simplicial bar construction Bo(P ) is the simplicial object in the category of symmetric s* *equences on C with Bk(P ) = P_O_._.O.P-z___". k For i = 1, . .,.k - 1 face maps di: P_O_._.O.P-z___"! P_O_._.O.P-z___" k k-1 are given by composing the ithand (i + 1)thterms using the operad composition P* * O P ! P . The maps d0 and dk are given by applying the augmentation map P ! I to the firs* *t and last copies of P respectively. Degeneracy maps sj : P_O_._.O.P-z___"! P_O_._.O.P-z___" k k+1 are given for j = 0, . .,.k by using the unit map I ! P to insert a copy of P b* *etween the jthand (j + 1)thterms. Remark 4.9. All that is really needed for this definition is that P be augmente* *d. However, we need P to be reduced to make the following identification of the simplicial * *bar construction with B(P ) as defined previously. Proposition 4.10. Let P be a reduced operad in C. Then the geometric realizatio* *n6 of Bo(P ) is isomorphic to the bar construction B(P ). Proof.We give the proof for C = T (which is the only case we require in this pa* *per) based on the informal description of B(P ) in Definition 4.1. The same idea could be * *used to write a proof that works for any C using the formal definition of B(P ) as a coend. The idea is that the iterated composition products that make up the simplicia* *l bar con- struction can be thought of in terms of sequences of partitions which in turn a* *re related to trees of the type we are using to define B(P ). We first give an explicit description of the n-simplices in Bo(P )(A). These * *are given by the object P_O_._.O.P-z___"(A). n Delving into the definition of O we can write this as a coproduct over all sequ* *ences of partitions of the set A: ^0= ~0 ~1 . . .~n-1 ~n = ^1 where ~ ~ if ~ is finer than ~ (if two elements of A are in the same block in* * ~, they are also in the same block in ~) and ^0, ^1are the minimal and maximal partitions w* *ith respect to this order. The terms in the coproduct are appropriate smash products of the* * P (r). We get a factor of P (r) every time one of the blocks of one of the partitions bre* *aks up into r blocks in the next partition along. A point in the geometric realization |Bo(P )| can be represented by a point i* *n the topological n-simplex n together with a choice of sequence of partitions as described abov* *e and a point in the appropriate smash product of the spaces P (r). ___________ 6The geometric realization of a simplicial symmetric sequence is defined poin* *twise: |X|(A) = |X(A)|. Note that a simplicial symmetric sequence is the same thing as a symmetric sequ* *ence of simplicial objects. 22 MICHAEL CHING A sequence of partitions determines an A-labelled tree T as follows. Take a * *vertex for each block of each ~ifor i = 0, . .,.n - 1. Add a root and a leaf for each elem* *ent of A. Two vertices are joined by an edge if they come from consecutive partitions of the * *sequence and the block for one is contained in the block for the other. Finally we add a roo* *t edge from the ~0 vertex to the root and a leaf edge from each leaf to the corresponding ~* *n-1 vertex. (Notice that vertices in this tree might have only one input edge - let's allow* * this for the moment.) A point in n determines a weighting on the tree we have just constructed. T* *hinking of n as the subspace of Rn+1 with x0 + . .+.xn = 1 and xi 0, we get a weight* *ing by giving the root edge length x0, the edges connecting the vertices for ~i-1to th* *e vertices for ~i length xi and the leaf edges length xn. We can now remove the vertices with * *only one input edge, connecting their input and output edges. This gives us a point in w* *(T ) for some tree T in the sense of Definition 3.1. Finally notice that because P (1) = S0 (as P is reduced), the smash product o* *f spaces P (r) determined by the sequence of partitions is precisely PA(T ). Therefore we actu* *ally obtain a point in B(P )(A). It remains to show that this process sets up a homeomorphism between |Bo(P )(* *A)| and B(P )(A). There are a couple of key steps. Firstly the degeneracy maps in the s* *implicial bar construction are isomorphisms on terms in the coproduct. These correspond to in* *serting lots of vertices with one input edge in our trees, which are then removed by the def* *inition. So we only have to worry about the identifications made by the face maps. But the fac* *e maps are given by removing partitions from the sequences which corresponds to edge colla* *pse. Hence the identifications made in defining B(P ) are the same as those in defining th* *e realization of Bo(P ). This completes the proof. 4.3. Cooperad structure on the bar construction. Up to this point, all we have * *done is identify the simplicial bar construction on a reduced operad in terms of tre* *es. The main point of this paper is that this identification allows us to see that there is * *a cooperad structure on the bar construction. In this section we describe this structure. The key to* * getting the cooperad cocomposition maps is the process of grafting of trees. Definition 4.11 (Tree grafting). Let T be an A-labelled tree, U a B-labelled tr* *ee and a an element of A. We define the grafting of U onto T at a to be the tree T [a U * *obtained by identifying the root edge of U to the leaf edge of T corresponding to a. Fig* *ure 6 below illustrates this process. 2 3 1 a 2 3 1 7! T U T [a U Figure 6. Tree grafting OPERADIC BAR CONSTRUCTIONS 23 We denote the newly identified edge by ea. Every other edge of T [aU comes ei* *ther from T or from U. The vertices of T [aU are the vertices of T together with the vert* *ices of U (and they have the same number of incoming edges). Finally there is a natural A [a B* *-labelling of T [a U got by combining the labellings of T and of U. We say that an A [a B-labelled tree is of type (A, B) if it is of the form T * *[a U for an A-labelled tree T and a B-labelled tree U. The next lemma says that an A [a B-l* *abelled tree is a grafting in at most one way. This is trivial but crucial to the const* *ruction of the cooperad structure maps below. Lemma 4.12. For any A [a B-labelled tree V there is at most one pair (T, U) su* *ch that V = T [a U. Proof.In the grafted tree T [aU the `upper' endpoint of the edge ea is a vertex* * whose `parent leaves' are labelled precisely by the elements of B. There can be at most one s* *uch vertex in V and cutting along that produces the trees T, U that make up V . Definition 4.13 (Cocomposition maps for B(P )). We have to define maps (4.14) B(P )(A [a B) ! B(P )(A) Z B(P )(B) for finite sets A, B and a 2 A. A point p in B(P )(A [a B) consists of a weigh* *ted tree V labelled by A [a B together with elements of pv 2 P (i(v)) for vertices v of V * *. We treat two cases: (1) If V is not of the form T [a U for an A-lablled tree T and a B-labelled * *tree U, then we will map p to the basepoint on the right-hand side of (4.14). (2) If V is of this form (that is, it is of type (A, B)) then things are mor* *e interesting. Below we describe how the map (4.14) is defined in this case. Since V is of type (A, B), Lemma 4.12 tells us that there is a unique A-labelle* *d tree T and a unique B-labelled tree U such that V = T [aU. We use these trees as the basis f* *or elements q 2 B(P )(A) and r 2 B(P )(B) respectively. What remains to be seen is how the * *weighting and vertex labels of V determine weightings and vertex labels for T and U. The vertex labels are easy because the vertices of T [aU consist of the verti* *ces of each of T and U with the same numbers of input edges. Therefore we take qv := pv 2 P (i(v)) for vertices v of T and ru := pu 2 P (i(v)) for vertices u of U. The way in which a weighting on T [a U determines weightings on T and U is th* *e key part of our construction. This comes about via a map (4.15) w~(T [a U) ! ~w(T ) ^ ~w(U) (recall that ~w(-) is the space of weightings on a tree with those that have ze* *ro length root or leaf edges identified to a basepoint). So take a weighting of T [a U. Define a weighting on T by giving the edges t* *he same lengths they had in T [a U and giving the leaf edge for a the necessary length * *to make the root-leaf distances equal to 1. Next define a weighting on U by taking the len* *gths from T [a U and scaling up by a constant factor to make the root-leaf distances equa* *l to 1 (the length of the root edge of U comes from the length of the edge ea in T [a U). T* *he scaling 24 MICHAEL CHING factor is the inverse of the total length of the U part of T [a U. The only tim* *e this doesn't work is if all the U-edges in T [a U (including ea) are of length zero. However* * in that case the weighting we just defined on T has a leaf edge of length zero and so is the* * basepoint in w~(T ). This is enough to define a map of the form (4.15). The only thing lef* *t to check is that if a leaf or root edge of T [a U is of length zero then the same is true o* *f either of the chosen weightings on T and U. This is clear. Figure 7 illustrates a particula* *r case of the map (4.15). 1 2 3 a 3 1 2 x _* *x_ x + y x + y x* *+y y 7! _y_ z z x+y T [a U T U Figure 7. The map ~w(T [a U) ! ~w(T ) ^ ~w(U) This completes the definition of the cooperad structure maps (4.14): B(P )(A [a B) ! B(P )(A) Z B(P )(B) given, in summary, by: ( q = (T, {pv}v2T), r = (U, {pv}v2U)if V = T [a;U p = (V, {pv}) 7! * otherwise. with the weightings on T, U given by the map (4.15) just constructed. We still have to check that these maps are well-defined. To see this we have * *to look at the identifications made in the definition of B(P )(A [a B): o if pv equals the basepoint in P (i(v)) for any vertex v 2 V then the sam* *e will be true of the corresponding vertex in either T or U. Hence such a p maps to the* * basepoint; o if an interior edge e of the tree V underlying the point p is of length * *zero, p is identified with another point p0as described in Defintion 4.1. We have various poss* *ibilities: (1)V is not of the form T [aU in which case neither is V=e and both p a* *nd p0map to the basepoint; (2)V = T [a U and e corresponds to an internal edge of T . In this case* *, the points q and q0will be identified via the collapse of that edge, and the po* *ints r and r0 will be equal. So p and p0map to the same element of B(P )(A) Z B(P * *)(B); (3)V = T [a U and e corresponds to an internal edge of U. This is simil* *ar to case (2); (4)V = T [a U and e is the edge ea obtained from identifying the root e* *dge of U with the a-leaf edge of T . In this case V=e is no longer of the for* *m T [a U and so p0 maps to the basepoint. But in the weighting on U determined b* *y that on T [a U the root edge has length scaled up from the length of ea w* *hich is OPERADIC BAR CONSTRUCTIONS 25 therefore zero. So the point r is the basepoint in B(P )(B) and so p* * also maps to the basepoint. o we have already checked in the definition of the map (4.15) that if a ro* *ot or leaf edge in p is of length zero, then the same is true of at least one of q and r* *. Therefore such a p maps to the basepoint in B(P )(A) Z B(P )(B). This completes the check that our maps (4.14) are well-defined. The final piec* *e of the cooperad structure for B(P ) is a counit map B(P )(1) ! S0. But we already saw* * that B(P )(1) ~=S0 (in the based space case) so our counit is this isomorphism. Note* * that this means B(P ) turns out to be a reduced cooperad. Example 4.16. The map B(P )({1, 2, 3}) ! B(P )({a, 3}) ^ B(P )({1, 2}) is pictured in Figure 7. The left-hand side (with vertices labelled by element* *s of P (2)) represents a point p of B(P )({1, 2, 3}). The two trees on the right-hand side * *(with vertices labelled by those same elements in the obvious way) represent the image of p in* * B(P )({a, 3})Z B(P )({1, 2}). In this example, all points that are based on trees of shapes ot* *her than that shown are mapped to the basepoint. We will save for later the task of checking that these maps do indeed give us* * a cooperad structure. First we translate Definition 4.13 into the category-theoretic langu* *age needed to define the cocomposition maps for a general C. To do this, we notice that the `* *ungrafting' process more-or-less makes our categories T(A) into a cooperad of categories. T* *o make this precise, we describe an `add a disjoint basepoint' functor for categories. Definition 4.17 (Categories with initial objects). Write Cat+ for the category * *in which an object is a (small) category C+ together with an initial object * such that Hom* * C+(X, *) is empty for all X 6= * and Hom C+(*, *) consists only of the identity morphism. T* *he morphisms in Cat+ are functors that preserve the initial objects. There is a functor from the category Catof all (small) categories to Cat+ giv* *en by adding an initial object with the correct morphisms to a category C to obtain C+. Note* * that every object in Cat+ can be obtained in this way, but not every morphism in Cat+ is g* *iven by adding an initial object to a morphism in Cat. Define a symmetric monoidal product ^ on Cat+ by: C+ ^ D+ := C+ x D+=C+ _ D+. where the wedge product is the disjoint union with the initial objects identifi* *ed and the quotient identifies this wedge product to the initial object of the smash produ* *ct. Notice that if C, D 2 Catthen C+ ^ D+ = (C x D)+ The unit for this product is the category 1+ with two objects and a single morp* *hism between them. In particular we write T(A)+ for the category formed by adding an initial obj* *ect to our poset of A-labelled trees T(A). The reason for making all these new definitions* * is then the following result. Proposition 4.18. The categories T(A)+ form a reduced cooperad in the Cat+. 26 MICHAEL CHING Proof.The cocomposition maps have the form T(A [a B)+ ! T(A)+ ^ T(B)+ = (T(A) x T(B))+ and are given by `ungrafting' trees. Take V 2 T(A [a B). If V is a tree of type* * (A, B) we map it to the pair (T, U) where T, U are the unique trees that graft together t* *o give V (see Lemma 4.12). If V is not of type (A, B) (or is the initial object) we map it to* * the initial object of the right-hand side. First we must check that we have indeed given a functor here. Suppose that V* * V 0 in T(A [A B). The only interesting case is when V is of type (A, B), so maps * *to a pair (T, U) on the right-hand side. We have to show two things: that V 0is also of t* *ype (A, B) with decomposition (T 0, U0) and then that T T 0and U U0. Well, let ea be * *the edge in V at which the grafting took place. Since V is obtained from V 0by a sequenc* *e of edge collapses, ea must come from an edge ea0in V 0that is not collapsed in this seq* *uence. This edge breaks V 0into two parts and we can write V 0= T 0[a0U0 for some trees T 0* *, U0 with some labellings (a priori, not necessarily by A and B). But it is now clear tha* *t U0 must yield U after undergoing some edge collapses. So U0 2 T(B) and U U0. Similarly, T 0* *2 T(A) and T T 0(after relabelling a0by a). Notice that T(1)+ is isomorphic to the unit 1+ for the symmetric monoidal str* *ucture on Cat+. We take as unit map the (unique) isomorphism 1+ ! T(1)+. It still remains to check that the cooperad axioms do indeed hold for our coc* *omposition maps. This is simple and we leave it to the reader. Remark 4.19. The original categories T(A) in fact already form an operad in Cat* *with com- position maps given by grafting rather than ungrafting. This operad structure i* *s effectively what is used by Boardman and Vogt to define their W -construction. The next step is to show that the bar construction can be defined as a coend * *in T(A)+ instead of T(A). Lemma 4.20. Let P be a reduced operad in C. The functors w~(-) and PA(-) on T(* *A) naturally extend to functors ~w(-) : T(A)+ ! T and PA(-) : (T(A)+)op! C and we have Z T2T(A)+ B(P )(A) = ~w(T ) PA(T ). Proof.We set ~w(*) = *T and PA(*) = *C with the obvious definition on morphisms* * (using the fact that *T is an initial object in T and *C is a terminal object in C). I* *t is then clear that * 2 T(A)+ does not contribute anything to the coend which therefore reduce* *s to the previous definition of B(P )(A). The maps (4.15) of Definition 4.13 are still the key ingredients in construct* *ing the cooperad maps for B(P ). Lemma 4.21. The maps w~(T [a U) ! ~w(T ) ^ ~w(U) OPERADIC BAR CONSTRUCTIONS 27 previously defined form part of a natural transformation T(A [a B)+ O | OOOOO~w(-) | OOO | O'' | + o7T7 | ooo | oooo . fflffl|oo~w(-)^w~(-) T(A)+ ^ T(B)+ Proof.The bottom functor here is defined in the obvious way on T(A) x T(B) and * *sends * to *. For V 2 T(A [a B) not of type (A, B), the corresponding part of the n* *atural transformation is ~w(V ) ! *. The only really interesting naturality square comes from V V 0with V 0of ty* *pe (A, B) and V not. The square that must commute in this case is w~(V )_______/~w(V/0) | | | | | | | fflffl| fflffl| * _____/~w(T/0) ^ ~w(U0) This is the content of part (4) of the checking we did towards the end of Defin* *ition 4.13: from any weighting on V , the weighting we get on V 0will have length zero for * *the edge connecting the T 0-part to the U0-part. Hence the root edge of the correspondin* *g weighting on U0 will have length zero. So we map into the basepoint of ~w(T 0) ^ ~w(U0). We have a corresponding result for the functors PA(-) of Definition 4.4. Lemma 4.22. Let P be a reduced operad in C. Then there is a natural transformat* *ion T(A [a B)op+ TTTPA[aB(-) | TTTTT**T || + jjj4C4j fflffl|jjjjjPA(-)ZPB(-) (T(A)+ ^ T(B)+)op Proof.In other words, given V 2 T(A [a B) we have maps PA[aB(V ) ! PA(T ) Z PB(U) when V = T [a U. There are obvious isomorphisms that we take for these maps. * * The naturality squares are easily seen to commute. Again the only one that seems li* *ke it might be interesting is for V V 0with V 0of type (A, B) and V not. But in fact this* * square just 28 MICHAEL CHING turns out to be PA[aB(V 0)_____/PA[aB(V/) | || | | | | | | fflffl| fflffl| PA(T 0) Z PB(U0)_____/*/_ which is not so interesting after all. Finally, we can define the cocomposition maps for the cooperad B(P ). Definition 4.23 (Formal construction of cocomposition maps for B(P )). Let P be* * a reduced operad in C and let B(P ) be the symmetric sequence of Definition 4.4. The coco* *mposition map B(P )(A [a B) ! B(P )(A) Z B(P )(B) is given by the following sequence of maps: B(P )(A [a B) = colim w~(V ) PA[aB(V 0) V V 02T(A[aB)+ ! colim (w~(T ) ^ ~w(U)) (PA(T 0) Z PB(U0* *)) (T,U) (T0,U0)2T(A)+^T(B)+ ! colim (w~(T ) PA(T 0)) Z (w~(U) PB(U0* *)) (4.24) (T,U) (T0,U0)2T(A)+^T(B)+ ` ' ` ' ! colim ~w(T ) PA(T 0)Z colim ~w(U) PB(U0) T T02T(A)+ U U02T(B)+ = B(P )(A) Z B(P )(B) The first map here comes from combining the natural transformations of Lemmas 4* *.21 and 4.22. The second is given by the transformation d of Definition 1.10. It is for* * precisely this reason that the axiom giving us d is necessary. The third map is given by unive* *rsal properties of colimits. This completes the construction of the cooperad structure maps for* * B(P ). The next task is to check that the maps we have described actually do make B(* *P ) into a cooperad. That is, we must check the duals of axioms (1)-(4) from Definition 2.* *2. The key step is to see that the maps (4.15) satisfy corresponding conditions. Lemma 4.25. Let T, U, V be A-, B- and C-labelled trees respectively and let a, * *a02 A, b 2 B. Let I denote the unique *-labelled tree. Recall that ~w(I) = S0. Then the follo* *wing diagrams commute: (1) ~w(T [a U [bV )_______/~w(T/[a U) ^ ~w(V ) | | | | | | | | fflffl| fflffl| ~w(T ) ^ ~w(U [bV_)___/~w(T/) ^ ~w(U) ^ ~w(V ) OPERADIC BAR CONSTRUCTIONS 29 (2) w~(T [a U [a0V )_______/~w(T/[a_U) ^ ~w(V ) | | | | | | | | fflffl| fflffl| w~(T [a0V ) ^ ~w(U)____/~w(T/)_^ ~w(U) ^ ~w(V ) (3) ~w(T [a I)_____/~w(T/) ^ ~w(I) JJ | JJJ | JJJ | ~=JJJ | JJJ | J$$Jfflffl| ~w(T ) (4) ~w(I [* T )____/~w(I)/^ ~w(T ) JJ | JJJ | JJJ | ~=JJJ | JJJ | J$$Jfflffl| ~w(T ) Proof.The argument for diagram (1) is contained in Figure 8. A point in ~w(T [a* * U [bV ) comes from a weighting of the grafted tree T [a U [b V . The top-left corner o* *f Figure 8 shows such a tree with some lengths labelled: o u is the length of the root edge; o v is the distance from the root vertex to the lower vertex of the edge t* *hat joins U to T (there may be intermediate vertices along this route, we let v deno* *te the total distance); o w is the length of the edge that joins U to T ; o x is the distance from the upper vertex of that edge to the lower vertex* * of the edge that joins V to U; o y is the length of the edge that joins V to U; o z is the remaining distance to any of the leaves of V . Figure 8 shows that whichever way we map our weighted tree around diagram (1) w* *e get the same result. (Note that if y + z or w + x + y + z equals to zero, then z =* * 0 and we are the basepoint in every corner of diagram (1).) We therefore conclude that d* *iagram (1) commutes. Diagram (2) is similar to (1) but easier. For diagram (3), notice that the im* *age in ~w(T ) of a weighting of T [aI will be effectively the same weighting. The image in ~w* *(I) = S0 will be the non-basepoint unless the leaf edge for a has length zero. But if this is* * the case our starting point was the basepoint in ~w(T [a I). This shows that the diagram com* *mutes. For diagram (4), the image in ~w(T ) of a weighting of I [* T will again be t* *he very same weighting (no scaling up is necessary). The image in w~(I) = S0 will always be* * the non- basepoint. Therefore this diagram also commutes. 30 MICHAEL CHING V z y + z y _z_y+z U x U x 7- ! w w V _y_y+z T v T v u u 7- ! 7- ! _z_y+z ___z__ w+x+y+z V w + x + y + z w + x + y + z _y_y+z V ___y__w+x+y+z y+z _____* *_w+x+y+z 7- ! T v U ___x__w+x+y+z T v U _* *__x__w+x+y+z ___w__ u w+x+y+z u ___w_* *_w+x+y+z Figure 8. Commutativity of diagram (1) of Lemma 4.25 We are now in a position to state the main result of this paper. Theorem 4.26. Let P be a reduced operad in the symmetric monoidal T-category C.* * The maps of Definition 4.13 make B(P ) into a reduced cooperad in C. Proof.We give the formal argument for the maps of Definition 4.23. To fit the * *relevant diagrams onto a page we need some new notation. Let's write ~w(T, U) := ~w(T ) ^ ~w(U) and PA,B(T 0, U0) := PA(T 0) Z PB(U0). Figure 9 then shows the diagram that has to commute for the dual of axiom (1) o* *f Definition 2.2 to hold for B(P ). The key to showing that this commutes is putting colim ~w(T, U, V ) PA,B,C(T 0, U0, V 0) T T02T(A)+ U U02T(B)+ V V 02T(C)+ into the center of the square. We've connected this to the top and left sides o* *f the square using maps similar to the first map in Definition 4.23. We've connected it to t* *he right and OPERADIC BAR CONSTRUCTIONS 31 ~w(Q, V * *) (w~(Q) PA[aB(Q0))Z colim ~w(S) PA[aB[bC(S0)_____// colim 0* * 0______// colim 0 S S02T(A[aB[bC)+ Q Q02T(A[aB)+PA[aB,C(Q * *, V ) Q Q02T(A[aB)+ (w~(V ) PC(V )) OO V V 02T(C)+ * * V V 02T(C)+ O OO * * OO OO OO * * OO OO OO * * OO OO OO * * OO OO OO * * OO OO O * * O OO fflfflO * * fflfflO fflffl * * 0, U0))Z w~(T, R) colim w~(T, U, V * *) colim (w~(T, U) PA,B(T colim 0 0___________/T/T02T(A)PA,B,C(T 0, U* *0, V_0)___/T/T02T(A)_ (w~(V ) PC(V 0)) T T02T(A)+ PA,B[bC(T , R ) U U02T(B)+ * * U U02T(B)+ R R02T(B[bC)+ V V 02T(+C)+ * * V V 02T(+C)+ O O * * O OO OO * * O OO OO * * OO OO O * * OO OO OO * * OO OO OO * * OO OO fflfflO * * fflffl fflffl * * 0))Z (w~(T ) PA(T 0))Z (w~(T ) PA(T 0))Z colim (w~(T ) PA(* *T 0 colim 0 _____/T/T02T(A) (w~(U, V ) PB,* *C(U0, V_0))__//_ colim (w~(U) PB(U ))Z T T02T(A)+ (w~(R) PB[bC(R )) U U02T(B)+ * * T T02T(A)+(w~(V ) P (V 0)) R R02T(B[bC)+ V V 02T(+C)+ * * U U02T(B)+ C * * V V 02T(C)+ Figure 9. The commutative diagram that verifies the first as* *sociativity axiom for the cooperad structure on B(P ) in the proof of Theorem 4.26 32 MICHAEL CHING bottom sides using maps of the form d from Definition 1.10. It's then enough to* * show that the four smaller squares commute. The top-left square commutes because of diagram (1) in Lemma 4.25. The bottom* *-left and top-right squares commute because of the naturality of the transformations d. T* *he bottom- right square commutes because it is an example of the associativity axiom we re* *quired of our d transformations in 1.10. This completes the verification of the dual of axiom (1) of Definition 2.2. F* *or axiom (2) the argument is similar, but using diagram (2) of Lemma 4.25. For the duals of * *axioms (3) and (4) we use the unit axiom for the transformations d together with diagrams * *(3) and (4) of Lemma 4.25. We leave the reader to fill in the details of these proofs. 5. Cobar constructions for reduced cooperads The cobar construction for a cooperad is strictly dual to the bar constructio* *n for an operad. More precisely, recall that a cooperad Q in a category C is the same thing as a* *n operad Qop in the opposite category Cop. The cobar construction on Q is then defined to b* *e the bar construction on Qop. This bar construction is a cooperad in Cop and hence an op* *erad in C. In symbols: (Q) := B(Qop)op. It will be useful for us to have a more explicit description of the cobar const* *ruction. Definition 5.1 (Cobar construction on a cooperad). Being dual to the bar constr* *uction, the cobar construction is defined as an end rather than a coend. Let Q be a coo* *perad in C. Then for each finite set A, Q determines a functor QA(-) : T(A) ! C by QA(T ) = Q(i(v1)) Z . .Z.Q(i(vn)) where v1, . .,.vn are the vertices of T . This is a functor because the cocompo* *sition maps for Q give us maps QA(T=e) ! QA(T ). (Recall that the corresponding functor for an operad was defined on T(A)op.) T* *he cobar construction (Q) is then the symmetric sequence with Z (Q)(A) := Map T(A)(w~(-), QA(-)) = Map C(w~(T ), QA(T )). T(A) This is the end of the bifunctor T(A)opx T(A) ! C given by (T, U) 7! Map C(w~(T ), QA(U)) where Map Cdenotes the cotensoring structure for C over T (and hence the tensor* *ing structure for Cop). Remark 5.2. The cobar construction (Q) on a reduced cooperad Q in based spaces* * is isomorphic to the totalization of a cosimplicial cobar construction that comes * *from looking at Q as a comonoid (see Remark 2.14). This is dual to the result that B(P ) is * *isomorphic to the simplicial bar construction (Proposition 4.10). OPERADIC BAR CONSTRUCTIONS 33 The operad structure maps for (Q) are strictly dual to the cooperad maps for* * B(P ). They are again given by the process of breaking an (A [a B)-labelled weighted t* *ree into its A-labelled and B-labelled parts (when this is possible, and by * when it is not* *). The proof of the following result is dual to that of Proposition 4.26. Corollary 5.3. Let Q be a reduced cooperad in the symmetric monoidal T-category* * C. Then the cobar construction (Q) is a reduced operad in C. 6. Duality for operads and cooperads In this section we examine how the bar and cobar constructions relate to the * *`duality' functor D : T op! C; X 7! Map C(X, S) where S is the unit of the symmetric monoidal structure on C. The case to keep * *in mind is C = Sp in which case S is the sphere spectrum and this duality functor is Spani* *er-Whitehead duality. Lemma 6.1. Let Q be a cooperad of based spaces. Then DQ is an operad in C. Proof.The composition maps for DQ are given by Map C(Q(A), S) Z Map C(Q(B), S) ! Map C(Q(A) ^ Q(B), S) ! Map C(Q(A [a B), S* *). The first map is the natural transformation constructed in Proposition 1.12 (it* *'s the dis- tributive map d for Cop). The second comes from the corresponding cocomposition* * map for Q. Remark 6.2. The dual of an operad need not in general be a cooperad because the* * map d need not in general have an inverse. However when it does we have a nice dual* *ity result connecting the bar and cobar constructions. For this to work we need to put the* * following condition on the spaces that make up our operad. Definition 6.3. Two based spaces X, Y are compatibly dualizable in C if the map d : Map C(X, S) Z Map C(Y, S) ! Map C(X ^ Y, S) is an isomorphism. Proposition 6.4. Let P be an operad in based spaces whose terms (that is, the P* * (A) for finite sets A) are pairwise compatibly dualizable. Then DP has a natural cooper* *ad structure. Moreover, we have an isomorphism of operads in C: DB(P ) ~= (DP ). Proof.The cooperad structure maps for DP are constructed in the same way as the* * operad structure maps for DQ in 6.1 but using the inverse of the relevant map d provid* *ed by the `compatibly dualizable' hypothesis. The second part relies on the descriptions of the bar and cobar constructions* * as coends and ends respectively. The coend B(P ) is a colimit: B(P )(A) = colim~w(T ) ^ PA(T 0) T T0 34 MICHAEL CHING where the colimit is taken over all inequalities of trees in T(A). Therefore DB(P )(A) = Map C(colimw~(T ) ^ PA(T 0), S) ~=lim Map C(w~(T ) ^ PA(T 0), S) ~=lim Map C(w~(T ), Map C(PA(T 0), S)) ~=lim Map C(w~(T ), (DP )A(T 0)) The last identity again uses the `compatibly dualizable' hypothesis in the form: Map C(P (i(v1)) ^ . .^.P (i(vn)), S) ~=Map C(P (i(v1)), S) Z . .Z.Map C(P (* *i(vn)), S). The final line of this calculation is precisely the limit that defines (DP ). * * We leave the reader to check that this is an isomorphism of operads. Remark 6.5. The only case of this result we will use in this paper is when all * *the terms of the operad P are S0. These are pairwise compatibly dualizable in any C because Map C(S0, C) ~=C for any C 2 C. Remark 6.6. Replacing C with Cop we obtain dual results. These concern the fun* *ctor S : X 7! X S, the `suspension spectrum' functor. We find that if Q is a coope* *rad in based spaces then SQ is a cooperad in C. If P is an operad whose terms are pairwise c* *ompatibly dualizable then SP is an operad in C and SB(P ) ~=B(SP ). We have now reached the stage where we can apply our constructions to Goodwil* *lie's calculus of functors (see x8). Before doing so, we extend our bar and cobar con* *structions to modules and comodules. This will then allow us to construct modules over the de* *rivatives of the identity. 7. Bar constructions for modules and comodules In this section we extend the bar and cobar constructions to modules and como* *dules. We show that there is a bar construction on left (respectively right) modules over* * a reduced operad P that yields left (respectively right) comodules over the cooperad B(P * *). Dually, there is a cobar construction on left (respectively right) comodules over a red* *uced cooperad Q that yields left (respectively right) modules over the operad (Q). These are* * special cases of two-sided bar and cobar constructions. Given a reduced operad P with right m* *odule R and left module L, we will define a two-sided bar construction B(R, P, L). Taki* *ng either R or L to be the unit symmetric sequence I will yield the promised one-sided cons* *tructions for individual modules. The two-sided construction is isomorphic to the standar* *d simplicial two-sided bar construction (see Definition 7.9) but, in order to get the comodu* *le structure, we have reinterpreted this in terms of trees. Most of the material in this section is a straightforward generalization of t* *hat of xx3-5. First, in x7.1 we describe the more general species of tree necessary for the d* *efinitions of the two-sided constructions. In x7.2 we give these definitions and show that the ba* *r construction of x4.1 is a special case. In x7.3 we construct the maps that make the bar cons* *truction on a module into a comodule, and dually, the cobar construction on a comodule into a* * module. As previously, C denotes a symmetric monoidal T-category with null object *. OPERADIC BAR CONSTRUCTIONS 35 7.1. Generalized trees. To accommodate the presence of the P -modules R and L i* *n the two-sided bar construction, we need to make two changes to our notion of tree, * *one at the root level and one at the leaf level: (1) we allow the root element of a tree to have more than one incoming edge; (2) we allow the leaves of a tree to have repeated labels, that is, an A-lab* *elling is a surjection from A to the set of leaves, rather than a bijection. We will refer to this notion as a `generalized tree', or sometimes just a `tree* *' if the context makes it clear that we mean the generalized version. The following definition m* *akes things precise. Definition 7.1 (Generalized trees). Let A be a finite set. A generalized A-lab* *elled tree consistst of: o a poset T with a unique minial element r (the root) satisfying condition* *s (2) and (3) of Definition 3.1; o a surjection ' from the finite set A to the set of maximal elements (the* * leaves) of T . We use letters T, U, . .t.o denote generalized trees, usually taking the labell* *ing map ' for granted. We write Tree(A) for the set of isomorphism classes of generalized A-l* *abelled trees. All the terminology of Definition 3.1 applies equally well to generalized trees. Edge collapse for generalized trees is defined in exactly the same way as for* * the trees of x3 except that now we allow ourselves to collapse root edges as well as internal e* *dges. To get the right category structure on Tree(A) we need a way to collapse leaf edges as* * well. The following definition provides this. Definition 7.2 (Bud collapse). A bud in a generalized tree T is a vertex all o* *f whose incoming edges are leaf edges. Equivalently, a bud is a maximal vertex. If b is* * a bud in T , a b-leaf is a leaf of T that is attached to b. Given a generalized A-labelled tree T and a bud b 2 T , we define a generaliz* *ed A-labelled tree Tb which is obtained from T by bud collapse. The underlying poset of Tb i* *s obtained from T by removing the b-leaves. This makes b into a leaf in Tb. The A-labellin* *g on Tbis that of T for the leaves that still remain with b inheriting the labels of its old l* *eaves. Formally, we are composing the A-labelling on T with the surjection from the leaves of T * *to the leaves of Tb that sends the b-leaves in T to b. Visually, we can think of this process* * as collapsing all the leaf edges attached to b (see Figure 10). {1, 3} 2 4 4 {1, 2, 3} b _ T Tb Figure 10. An example of bud collapse for generalized {1, 2, 3, 4}-labelle* *d trees 36 MICHAEL CHING Definition 7.3 (The categories Tree(A)). If T and T 0are generalized A-labelled* * trees, we say that T T 0is T can be obtained from T 0by a sequence of edge collapses (o* *f either internal or root edges) or bud collapses. This makes the set Tree(A) of isomorp* *hism classes of generalized A-labelled trees into a poset and hence a category. Standard A-l* *abelled trees (as defined in x3) are also generalized A-labelled trees and T(A) is a full sub* *category of Tree(A). See Figure 11 for pictures of Tree(1) and Tree(2). 1 {1, 2} 1 2 1 2 Tree(1) Tree(2) Figure 11. Tree(1) and Tree(2) (the arrows represent the direction of the morphisms in Tree(2)) Definition 7.4 (Weightings on generalized A-labelled trees). We don't need to c* *hange the definition of a weighting for generalized trees: it is an assignment of lengths* * to the edges of a tree such that the root-leaf distances all equal 1. As before, we write w(T )* * for the space of weightings on the generalized tree T . The following result generalizes Lemm* *a 3.9. Lemma 7.5. Let T be a generalized A-labelled tree with n vertices. Then w(T ) i* *s homeo- morphic to Dn and the boundary @w(T ) ~=Sn-1 consists of those points in which * *some edge of T has length zero. Proof.The labelling plays no role in the space of weightings so we can ignore i* *t. Picture T as a collection of (non-generalized)Ptrees T1, . .,.Tk attached at their roots.* * Suppose Tj has nj vertices so that n = nj. Then we have w(T ) ~=w(T1) x . .x.w(Tk) ~=Dn1x Dnk ~=Dn. Under this decomposition, a point is in the boundary of w(T ) if and only if an* *y of it is in the boundary of any of the w(Tj). That is, if and only if any of the edges of T* * has length zero. Definition 7.6 (The functor w(-) on Tree(A)). The functor w(-) : T(A) ! U of De* *finition 3.10 can be extended to all of Tree(A). To do this, we have to say what happens* * when we apply w(-) to a morphism Tb ! T coming from a bud collapse (for b a bud in a tr* *ee T ). Given a weighting of Tb we get a weighting of T by giving length zero to all th* *e leaf edges attached to b. This defines a map w(Tb) ! w(T ) and it is not hard to see that this does indeed give us a functor w(-) : Tree(A) ! U as claimed. Adding a disjoint basepoint we get a functor w(-)+ : Tree(A) ! T. OPERADIC BAR CONSTRUCTIONS 37 7.2. The two-sided bar construction. Along with the spaces of weightings the ke* *y parts of the bar construction on an operad were functors PA(-) : T(A)op! C. The appropriate generalizations of these to functors on Tree(A)opare as follows. Definition 7.7. Let P be a reduced operad in C with right module R and left mod* *ule L. We define functors (R, P, L)A : Tree(A)op! C by7 ~^ ^~ -1 (R, P, L)A(T ) := R(i(r)) Z P (i(v)) Z L(' l). verticesv2T leavesl2T Recall that i(v) denotes the set of incoming edges to a vertex v 2 T . Here ' * *denotes the labelling surjection from A to the set of leaves of T , so that '-1l is the set* * of labels attached to the leaf l. To complete the definition, we have to give the effect of (R, P, L)A(-) on mo* *rphisms in Tree(A). Notice that Tree(A) is generated by the morphisms corresponding to: (1) collapse of root edges; (2) collapse of internal edges; (3) bud collapse. We will describe the effect of (R, P, L)A(-) on each of these types of generati* *ng morphism and then check that they are compatible. (1) Suppose first that e is a root edge of the generalized A-labelled tree T* * . Then we have a morphism T=e ! T corresponding to collapsing e. Applying (R, P, * *L)A(-) we should get a morphism (R, P, L)A(T ) ! (R, P, L)A(T=e). This is given by the map R(i(r)) Z P (i(v)) ! R(i(r O v)) that comes from the right P -module structure on R. Here v is the upper * *endpoint of the edge e in T . Notice that r O v is the root element in T=e. (2) Now suppose that e is an internal edge of T . The morphism (R, P, L)A(T ) ! (R, P, L)A(T=e) is then given (as in Definition 4.4) by the partial composition map P (i(u)) Z P (i(v)) ! P (i(u O v)) for the operad P where u, v are the endpoints of e. (3) Finally, suppose that b is a bud in the generalized A-labelled tree T . * *We have to give a map (R, P, L)A(T ) ! (R, P, L)(Tb). This comes from the map P (i(b)) Z L('-1l1) Z . .Z.L('-1lr) ! L('-1b) ___________ 7It is a serendipitous fact of our terminology for trees that the right modul* *e R relates to the roots of our trees and the left module L relates to the leaves. 38 MICHAEL CHING that is part of the left P -module structure on L. Here l1, . .,.lr are * *the b-leaves in T and we have r a '-1b = '-1li i=1 from the definition of bud collapse. The associativity conditions for R and L to be P -modules ensure that these cho* *ices do indeed determine a functor Tree(A)op! C. Definition 7.8 (Two-sided bar construction). Let P be a reduced operad in C wit* *h right module R and left module L as above. The bar construction on P with coefficient* *s in R and L is the symmetric sequence B(R, P, L) defined by the coends Z T2Tree(A) B(R, P, L)(A) := w(T )+ (R, P, L)A(T ) for finite sets A. A bijection A ! A0determines an isomorphism of categories Tr* *ee(A) ! Tree(A0) under which our functors wA(-), wA0(-) and (R, P, L)A, (R, P, L)A0 cor* *respond. It therefore induces an isomorphism B(R, P, L)(A) ! B(R, P, L)(A0). So we do indee* *d have a symmetric sequence B(R, P, L). There is a more informal description of this bar construction that generalize* *s that of B(P ) from Definition 4.1. For a finite set A, a point in B(R, P, L)(A) consists of o a weighted generalized A-labelled tree T ; o a point in R(i(r)) where r is the root of T ; o for each vertex v 2 T , a point in P (i(v)); o for each leaf l 2 T , a point in L('-1l) where '-1l is the set of labels* * attached to l. These are subject to identifications that tell us what happens when the lengths* * of some of the edges tend to zero. When a root edge tends to zero we use the right P -modu* *le structure map for R. When an internal edge tends to zero we use the composition map for P* * . When a collection of leaf edges attached to a bud tend to zero (note that the leaf e* *dges attached to a particular bud must all have the same length in a weighting) we use the le* *ft P -module structure for L. Finally, of course, we identify to the basepoint in B(R, P, L)* *(A) if any of the chosen points in R(i(r)), P (i(v)), L('-1l) are the basepoint there. We now recall the simplicial version of the two-sided bar construction for an* * operads and modules over them. Definition 7.9 (Simplicial two-sided bar construction). Let P be an operad in C* * with right module R and left module L. The simplicial bar construction on P with coeffici* *ents in L and R is the simplicial object Bo(R, P, L) in the category of symmetric sequenc* *es in C with Bn(R, P, L) := R O P_O_._.O.P-z___"O L. n The face maps di: Bn(R, P, L) ! Bn-1(R, P, L) for i = 1, . .,.n - 1 are given by the operad composition map P O P ! P . The f* *ace map d0 is given by the right module structure R O P ! R and dn is given by the left modul* *e structure P O L ! L. The degeneracy map sj : Bn(R, P, L) ! Bn+1(R, P, L) OPERADIC BAR CONSTRUCTIONS 39 is given by using the unit map I ! P to insert an extra copy of P between the j* *th and j + 1thterms. Proposition 7.10. Let P be a reduced operad in C with right module R and left m* *odule L. The bar construction of Definition 7.8 is isomorphic to the geometric realiz* *ation of the simplicial bar construction: B(R, P, L) ~=|Bo(R, P, L)|. Proof.This is an straightforward extension of the argument of Proposition 4.10. Our first example of the two-sided bar construction is that the reduced bar c* *onstruction a lone operad is a special case. Example 7.11. Let P be a reduced operad in C and take R = L = I the unit symmet* *ric sequence. Recall that I is a left and right module over any augmented operad. I* *t is easy to see from the definitions that for the simplicial bar constructions we have. Bo(I, P, I) ~=Bo(P ). This tells us that B(I, P, I) ~=B(P ) but we can see this directly as well. First notice that ( PA(T ) if T 2 T(A); (I, P, I)A(T ) ~= * otherwise. 8 This means that only the objects T 2 T(A) contribute to the calculation of th* *e coend in Definition 7.8. However, we still have to take into account morphisms U ! T wit* *h U =2T(A). This amounts to collapsing to the basepoint those weighted trees in which eithe* *r the root edge or a leaf edge has length zero (since these are the images of the maps w(U* *) ! w(T )). All together this tells us that B(I, P, I)(A) is equal to the coend Z T2T(A) ~w(T ) PA(T ) where ~w(T ) is the quotient of w(T ) by the weightings which have either root * *or leaf edge of length zero. This is precisely B(P )(A). Therefore we have B(I, P, I) ~=B(P ) a* *s claimed. Example 7.12. It is easy to see that B(R, P, L)(1) = R(1) Z L(1). We have alrea* *dy seen (Figure 11) that there are three objects in Tree(2). From this we see that B(R,* * P, L)(2) is the homotopy pushout of the following diagram R(1) Z P (2) Z L(1) Z L(1)___/R(1)/Z L(2) | | | fflffl| R(2) Z L(1) Z L(1) ___________ 8Because I(n) = * for n > 1, we have (I, P, I)A(T) = * whenever T has more th* *an one root edge, or when any leaf has more than one label. These are precisely the generalized A-la* *belled trees not in T(A). For T 2 T(A) we have (I, P, I)A(T) = I(1) Z PA(T) Z I(1) Z . .Z.I(1) ~=PA(1). 40 MICHAEL CHING If R = L = I, the bottom-left and top-right objects are * and the top-left obje* *ct is P (2). So we recover B(P )(2) = B(I, P, I)(2) = P (2). Definition 7.13 (Bar constructions for modules). Let P be a reduced operad in C* * and let R be a right P -module. We define the bar construction on R by B(R) := B(R, P, I) where I as previously is the unit for the composition product of symmetric sequ* *ences. If L is a left P -module, its bar construction is B(L) := B(I, P, L) We trust that it will not be confusing to use the same notation for the bar con* *struction of right and left modules. Example 7.14. Applying Example 7.12 to the one-sided case we see that B(R)(1) ~=R(1); B(R)(2) ~=hocofib(R(1) Z P (2) ! R(2)) and B(L)(1) ~=L(1); B(L)(2) ~=hocofib(P (2) Z L(1) Z L(1) ! L(2)). Definition 7.15 (Cobar constructions for comodules). All the constructions of t* *his section can be applied to operads and modules in Cop, that is, to cooperads and comodul* *es in C. We summarize the results. If Q is a reduced cooperad in C with left comodule L and right comodule R, th* *e formula of Definition 7.7 defines functors (R, Q, L)A(-) : Tree(A) ! C for each finite set A and we define the cobar construction on Q with coefficien* *ts in R and L to be the symmetric sequence (R, Q, L) with Z (R, Q, L)(A) := Map C(w(T )+, (R, Q, L)A(T )). T2Tree(A) This is isomorphic to the totalization of the two-sided cosimplicial cobar cons* *truction on Q with coefficients in R and L. The cobar construction on R is (R) := (R, Q, I) and the cobar construction on L is (L) := (I, Q, L). Example 7.16. Taking R = L = I we recover the cobar construction of x5: (I, Q, I) ~= (Q). Example 7.17. Taking the duals of the results of Example 7.12 we see that (R, Q, L)(1) ~=R(1) Z L(1) OPERADIC BAR CONSTRUCTIONS 41 and that (R, Q, L)(2) is the homotopy pullback of R(1) Z L(2) | | | | fflffl| R(2) Z L(1) Z L(1)_____/R(1)/Z Q(2) Z L(1) Z L(1) In particular: (R)(1) ~=R(1); (R)(2) ~=hofib(R(2) ! R(1) Z Q(2)) and (L)(1) ~=L(1); (L)(2) ~=hofib(L(2) ! Q(2) Z L(1) Z L(1)). 7.3. Structure maps for bar constructions on modules. In this section we use si* *milar methods to x4.3 to show that the bar construction on a P -module (that is, a si* *ngle left or right module) is a comodule over the cooperad B(P ). The key to this is the con* *struction of a general map of the form (7.18) B(R, P, L) ! B(R, P, I) "OB(I, P, L) where "Ois the composition of symmetric sequences defined using the product in * *C rather than the coproduct (see Remark 2.14). Taking R = I and recalling that B(I, P, I* *) = B(P ) we obtain a left B(P )-comodule structure on B(L) = B(I, P, L). Similarly, taki* *ng L = I we get a right B(P )-comodule structure on B(R) = B(R, P, I). Notice that taking R* * = L = I we recover the cooperad structure on B(P ). The definition of the map (7.18) is a relatively straightforward generalizati* *on of the co- operad structure on B(P ). We start by describing the grafting and ungrafting p* *rocesses for generalized trees. Definition 7.19 (Basic grafting for generalized trees). Let T be a generalized * *A-labelled tree and U a generalized B-labelled tree and let a be an element of A. We will * *define the grafting of U onto T only if T and U satisfy the following conditions: o the root of U has only one incoming edge; o the leaf of T labelled by a is labelled only by a and no other elements * *of A. In this case, the grafted tree T [a U is defined exactly as in Definition 4.11 * *by identifying the root edge of U to the a-leaf edge of T . Figure 12 gives an example. {1, 2} 3 4 x 4 {1, 2} 3 _ T [x U T U Figure 12. Grafting generalized labelled trees 42 MICHAEL CHING To define the maps (7.18) we will need to graft trees onto all of the leaf ed* *ges of the base tree T . To do this, we must assume that all the leaves of T only have one labe* *l, so that T satisfies the stronger condition for a labelling we required in Definition 3.3.* * Notice also that the trees U we are to graft onto T satisfy the stronger root condition of Defin* *ition 3.1. The following definitions will help us talk about trees of these types. Definition 7.20 (More categories of trees). For a finite set A, we define the f* *ollowing subcategories of Tree(A): Troot(A) := {T 2 Tree(A)| the root of T has only one incoming edge} Tleaf(A) := {T 2 Tree(A)| the leaves of T are labelled bijectively}by.A Notice that T(A) = Troot(A) \ Tleaf(A). ` Definition 7.21 (Grafting and ungrafting generalized trees). Let A = j2JAjbe * *a partition of A into nonempty subsets. Given trees Uj 2 Troot(Aj) and T 2 Tleaf(J), we den* *ote the tree obtained by grafting all the Uj onto T at the appropriate places by T [J Uj. We say that a generalized A-labelled tree is of type {Aj} if it is of the form * *T [J Uj for some such T and Uj. The correct generalization of the functor of Proposition 4.* *18 is then a functor Tree(A)+ ! Tleaf(J)+ ^ Troot(Aj1)+ ^ . .^.Tleaf(Ajr)+ that breaks the tree (T [J Uj) into its components T and the Uj and sends a tre* *e not of type {Aj} to the initial object on the right-hand side. This `ungrafting' funct* *or is the basis of the map (7.18). Our new categories of trees can be used as the base categories for defining t* *he one-sided bar constructions. For this we need the appropriate spaces of weightings. Definition 7.22 (More spaces of weightings). For each finite set A we define a * *functor wleaf(-) : Tleaf(A) ! T where wleaf(T ) is the quotient of w(T ) by the space of weightings in which so* *me leaf edge has length zero, and a functor wroot(-) : Troot(A) ! T where wroot(T ) is the quotient of w(T ) by the space of weightings in which th* *e root edge has length zero. Lemma 7.23. Let P be a reduced operad in C with right module R and left module * *L. Then the one-sided bar constructions are given by Z T2Tleaf(A) B(R)(A) = B(R, P, I)(A) ~= wleaf(T ) (R, P, I)A(T ) and Z T2Troot(A) B(L)(A) = B(I, P, L)(A) ~= wroot(T ) (I, P, L)A(T ). OPERADIC BAR CONSTRUCTIONS 43 Proof.These calculations are similar to that in Example 7.11 where we showed th* *at B(P ) = B(I, P, I). They use the facts that (R, P, I)A(T ) = * for T =2Tleaf(A) and (I, P, L)A(T ) = * for T =2Troot(A). The final piece of the puzzle is the construction of a map analogous to (4.15* *) that tells us how to weight the trees obtained from ungrafting. ` Definition 7.24. Let A = j2JAj be a partition of the finite set A into nonemp* *ty subsets. Given trees T 2 Tleaf(J) and Uj 2 Troot(Aj) we define a map w(T [J Uj)+ ! wleaf(T ) ^ wroot(Uj1) ^ . .^.wroot(Ujr) by the obvious generalization of the construction of the maps ~w(T [a U) ! ~w(T* * ) ^ ~w(U) in Definition 4.13. Definition 7.25. Putting together all these ingredients we construct maps B(R, P, L)(A) ! B(R, P, I)(J) Z B(I, P, L)(Aj1) Z . .Z.B(I, P, L)(Ajr). In an analogous way to Definition 4.23, these come from the maps of Definition * *7.24 together with the isomorphisms (R, P, L)A(T [J Uj) ! (R, P, I)J(T ) Z (I, P, L)Aj1(Uj1) Z . .Z.(I, P, L)* *Ajr(Ujr). Together these maps make up the map of symmetric sequences B(R, P, L) ! B(R, P, I) "OB(I, P, L) as promised. Proposition 7.26. Let P be a reduced operad in C with right module R and left m* *odule L. The maps of Definition 7.25 determine a right B(P )-comodule structure on B(R) * *and a left B(P )-comodule structure on B(L). Proof.Taking L = I in 7.25 we get the right comodule structure on B(R). Taking * *R = I we get the left comodule structure on B(L). We have to check the appropriate as* *sociativity and unit axioms. This is a generalization of the work of x4.3. We leave the rea* *der to write out all the details, including the diagram corresponding to Figure 9. Corollary 7.27. Dually, suppose that Q is a reduced cooperad in C with right co* *module R and left comodule L. Then there is a map (R, Q, I) O (I, Q, L) ! (R, Q, L) that makes (R) into a right (Q)-module (by taking L = I) and (L) into a left* * (Q)- module (by taking R = I). Proof.Apply Proposition 7.26 to Q considered as an operad in Cop. This completes our descriptions of the bar and cobar constructions for operad* *s, cooperads, modules and comodules. We turn now to our main application of this theory - the* * Goodwillie derivatives of the identity functor. 44 MICHAEL CHING 8. Application to the calculus of functors In this section we describe the application of bar and cobar constructions to* * Goodwillie's calculus of homotopy functors. The main result is that the derivatives of the i* *dentity form an operad in spectra. We now assume that C is a suitable category Sp of spectra, f* *or example, the S-modules of [5] (see Example 1.14(2)). Let I : T ! T be the identity functor on based spaces. The Goodwillie derivat* *ives of I can be described in terms of the partition poset complexes [1]. We recall one o* *f the ways to define these. Definition 8.1 (Categories of partitions). A partition of a finite set A is an * *equivalence relation on A. Let K(A) be the poset formed by the partitions of A with ~ ~ i* *f ~ is finer than ~, that is, if the set of relations for ~ is contained in the set of relat* *ions for ~. The category K(A) has an initial object ^0and a terminal object ^1. Let K0(A) = K(A* *) - ^0, the category of proper partitions, and K1(A) = K(A) - ^1, the category of non-trivi* *al partitions. Note that the group A of permutations of A acts on all of these categories in * *a obvious way. Definition 8.2 (The partition poset complexes (A)). For a finite set A, the pa* *rtition poset complex (A) is the geometric realization of the following simplicial set T (A)* *o formed from the nerves of these categories of partitions: NoK(A) T (A)o = ___________________. NoK0(A) [ NoK1(A) So the n-simplices in T (A)o are sequences of n + 1 partitions ~0 ~1 . . .~n with a sequence identified to the basepoint if it does not have both ~0 = ^0and* * ~n = ^1. The face and degneracy maps are given by respectively removing partitions from the * *sequence and repeating terms in the usual way for the nerve of a category. The simplicia* *l set T (A)o is pointed and so its geometric realization (A) is a based space. The action of * *A on K(A) induces an action on (A). Remark 8.3. What we are calling the partition poset complex is the suspension o* *f the complex Kn of [1]. The simplicial set T (n)o is isomorphic to that called Tn in* * Definition 1.1 of [1]. Proposition 8.4 (Arone-Mahowald, [1]). The derivatives of the identity are mode* *lled by the dual spectra of the finite complexes (n) = ({1, . .,.n}): @nI ' Map Sp( (n), S). The action of the symmetric group n on (n) induces an action on the dual spec* *trum and this agrees with the action that comes with the spectrum @nI. The key observation (due to Tom Goodwillie) is that the partition poset compl* *exes can be described as spaces of trees. We can interpret these as the spaces of a bar * *construction. Definition 8.5. Let ComT be the operad in based spaces with ComT(A) := S0 for all finite sets A and with all composition maps equal to the identity on S0. Lemma 8.6. The partition poset complex (A) is homeomorphic to B(ComT)(A). OPERADIC BAR CONSTRUCTIONS 45 Proof.We have already seen that B(ComT) is homeomorphic to the realization of t* *he simpli- cial bar construction on ComT. It is therefore enough to show that the simplici* *al set T (A)o used to define (A) is also given by this simplicial bar construction. An n-simplex in T (A) is an increasing sequence of partitions of A of length * *n-1 (together with a disjoint basepoint). On the other hand the based set of n-simplices in t* *he simplicial bar construction is ComT O . .O.ComT (A). ________-z_______" n But this is equal to the wedge over increasing sequences of partitions of lengt* *h n - 1 of S0. Hence we see that the two sets of n-simplices are the same. The face and degene* *racy maps in each case correspond to removing a partition and repeating a partition respe* *ctively. We therefore have isomorphic simplicial sets. Corollary 8.7. Let @nI denote the model of the nthderivative of the identity gi* *ven by @nI = Map Sp( (n), S). Then we have @nI = (DComT)(n). In particular, the derivatives of the identity form an operad in spectra. We de* *note this operad by @*I. Proof.We have @nI = Map ( (n), S) = DB(ComT)(n) = (DComT)(n) by Lemma 8.6 and Proposition 6.4 (which applies since all the spaces in ComT ar* *e S0). Remark 8.8. The derivatives of the identity are the cobar construction on the c* *ooperad ComSp in spectra with ComSp(A) = DComT(A) = S where S is the sphere spectrum, for all finite sets A and with all cocompositio* *n maps the canonical isomorphisms. This cooperad can be thought of as the cocommutative co* *algebra cooperad for spectra. Remark 8.9. We can use the constructions of x7 to get modules over the operad @* **I. If C is a comodule over ComSp then its cobar construction (C) is a @*I-module. We g* *ive two examples: (1) Let X be a based space. Then the suspension spectrum 1 X is a ComSp-coa* *lgebra (that is, just a commutative coalgebra) with comultiplication given by t* *he (reduced) diagonal map on X: 1 X ! 1 (X ^ X) ~= 1 X ^ 1 X. As remarked in Definition 2.15, a coalgebra over a cooperad Q determines* * a left Q- comodule. Thus we obtain a left ComSp-comodule _1_X_. We now take the * *cobar construction to get a left @*I-module MX := (_1_X_). From the calculations of 7.17 we find that MX (1) = 1 X 46 MICHAEL CHING and MX (2) ~=hofib( inftyX ! 1 X ^ 1 X) ' -1 hocofib( 1 X ! 1 X ^ 1 X) ' -1 1 hocofib(X ! X ^ X) So MX (2) is (up to homotopy and a desuspension) the mapping cone of the* * reduced diagonal on X. Further work is needed to analyze the spectra MX (n) for * *larger n. In x9.7 we will look at ways to calculate the homology of these spectra. (2) A moment's thought will reveal that a right ComSp-comodule is precisely * *the same thing as a functor (FinSets, i) -! Sp where the left-hand side is the category of finite sets with morphisms g* *iven by the surjections. Work in progress by Greg Arone has demonstrated a relations* *hip between such functors and the Goodwillie calculus of homotopy functors F from ba* *sed spaces to spectra. In particular, we hope to show, by combining the cobar const* *ruction with this work, that the derivatives of such an F form a right @*I-module. 9. Homology of the bar and cobar constructions and Koszul duality In this section we look at spectral sequences for calculating the homology of* * the bar and cobar constructions on operads and cooperads in based spaces or spectra. I* *t turns out that we can relate the E1-term of these spectral sequences to the algebraic* * bar and cobar constructions described in, for example, [7] and [6]. This leads to a lin* *k with Koszul duality which says, briefly, that if the homology of the reduced operad P is Ko* *szul, then the homology of B(P ) is its Koszul dual cooperad, and dually, if the homology of t* *he cooperad Q is Koszul then the homology of (Q) is its Koszul dual operad. This supports * *the point- of-view that the bar construction for an operad in based spaces or spectra is t* *he analogue of the Koszul dual for an algebraic operad. Here is a summary of this section. We start in x9.1 by recalling how the homo* *logy (with coefficients in the commutative ring k) of an operad in based spaces or spectra* * has the structure of an operad in graded k-modules. Then in x9.2, the main work of the* * chapter begins and we describe the filtration of the bar construction that gives rise t* *o our spectral sequence and identify the `filtration quotients'. This filtration is based on * *the number of vertices in the trees that underlie the bar construction. We deal immediately w* *ith the two- sided construction of x7.2, recalling that the construction for a lone operad i* *s a special cases of this. As usual, for the cobar construction, we just dualize everything. Th* *at is, we get a cofiltration, or tower, whose inverse limit is the cobar construction and we * *identify the fibres of the stages in this tower. In x9.3 we give conditions under which the * *inclusion maps of the filtrations are cofibrations, thus ensuring that our `filtration quotien* *ts' are actually the homotopy cofibres of filtration. This will allow us later to use our identi* *fication of these quotients to calculate the E1 term in the spectral sequence. This E1 term turns* * out to be given by the algebraic bar construction which we describe in x9.4. We give a de* *finition of this that emphasizes its similarity to the topological version and show that th* *is definition is equivalent to that given by Getzler and Jones [7] and Fresse [6]. Then in x9* *.5 we finally set up the spectral sequence and identify its E1 term with the algebraic bar co* *nstruction as claimed. In x9.6 we look at Koszul operads and prove the result identifying the* * homology of the bar construction on P with the Koszul dual of the homology of P . Finally, * *in x9.7 we OPERADIC BAR CONSTRUCTIONS 47 use our spectral sequences to investigate the homology of the @*I-modules MX co* *nstructed in Remark 8.9(1). 9.1. Homology of topological operads. Throughout this chapter we fix a commutat* *ive ring k and consider the categories Mod k of graded k-modules and Chk of chain c* *omplexes over k. First we describe the symmetric monoidal structure on these categories. Definition 9.1 (Symmetric monoidal structures on Mod k and Chk). The tensor pro* *duct determines a symmetric monoidal structure on graded k-modules with M (M N)r := Mp Nq p+q=r where the graded symmetry isomorphism M N ! N M is given by m n 7! (-1)|m||n|n m and the unit object is the graded module k concentrated in degree 0. If M and N* * are chain complexes with differentials dM and dN respectively, we define a differential * *on M N by dM N (m n) := dM (m) n + (-1)|m|m dN (n). This makes into a symmetric monoidal structure on Chk with the same unit k en* *dowed with the trivial differential. Throughout this section we will use H*(-) to denote the homology with coeffic* *ients in the commutative ring k of an object in C when C is either T or Sp. If C is the * *category T of based spaces, this is the reduced homology.9 If C is a category Sp of spect* *ra, it is the spectrum homology H*(E) = ss*(Hk ^ E). We recall the K"unneth maps for these ho* *mology theories. Proposition 9.2. Let C = T or Sp and take C, D 2 C. Then there is a natural map H*(C) H*(D) ! H*(C Z D) that is an isomorphism if either H*(C) or H*(D) consists of flat k-modules. The* *se maps are symmetric monoidal in the sense that they commute with the associativity and co* *mmutativity isomorphisms in the categories C and Mod k. Definition 9.3. Let M be a symmetric sequence in T or Sp. Then we denote by H*M* * the symmetric sequence of graded k-modules given by H*M(A) := H*(M(A)). The main result of this section is that the homology of a topological operad * *or cooperad is, under suitable conditions, an operad or cooperad in Mod k. Lemma 9.4. Let P be an operad in T or Sp. Then H*P is an operad of graded k-mod* *ules. If P is reduced then so is H*P . If M is a left (respectively, right) P -module* *, then H*M is a left (respectively, right) H*P -module. Let Q be a cooperad in T or Sp such that the homology groups H*(Q(A)) are fla* *t k- modules. Then H*(Q) is a cooperad of graded k-modules that is reduced if Q is. * *If C is a ___________ 9We stress that any homology group of a based space in this paper is meant to* * be the reduced homology. 48 MICHAEL CHING left Q-comodule then H*(C) is a left H*(Q)-comodule. If C is a right Q-comodule* * such that the H*(C(A)) are flat k-modules then H*(C) is a right H*(Q)-comodule. Proof.The operad structure maps are given by the composites H*(P (A)) H*(P (B)) ! H*(P (A) ^ P (B)) ! H*(P (A [a B)) and the unit by the map k ~=H*(S) ! H*(P )(1) where S denotes either S0, the unit of T, or the unit of Sp. To check the oper* *ad axioms we use the associativity and commutativity of the K"unneth formula as stated in* * Proposition 9.2. Clearly, if P is reduced (so that the unit map S ! P (1) is an isomorphism* *) then so is H*P . The structure maps for H*M are defined similarly. In the cooperad case we need the flatness condition. It allows us to define c* *ocomposition maps by H*(Q(A [a B)) ! H*(Q(A) ^ Q(B)) ~=H*(Q(A)) H*(Q(B)) using the inverse of the K"unneth map. The counit map is the composite H*Q(1) ! H*(S) ~=k and again, if Q is reduced, so is H*Q. In the case of a left comodule C we sim* *ilarly get comodule structure maps H*(C(A [a B)) ! H*(C(A) ^ Q(B)) ~=H*(C(A)) H*(Q(B)) where the K"unneth map is an isomorphism without any condition on H*(C(A)) (we * *are still assuming that the H*(Q(B)) are flat). In the right comodule case, we do s* *till need the flatness assumption. Remark 9.5. We can consider cohomology instead of homology in which case the K"* *unneth isomorphism also requires a finite-generation hypothesis. We get the following * *results. If Q is a cooperad in based spaces or spectra then H*(Q) is an operad of graded k-mo* *dules. If P is an operad with the cohomology groups H*(P ) finitely-generated flat k-modu* *les then H*(P ) is a cooperad of graded K-modules. Similar results hold for comodules an* *d modules. 9.2. Filtering the bar construction. The spectral sequence we want to construct* * comes from a filtration on the bar construction by the number of vertices in the unde* *rlying trees. In this section we construct this filtration and calculate the filtration quoti* *ents. Definition 9.6 (Filtration on the category of trees). Write Trees(A) for the su* *bcategory of Tree(A) whose objects are the (isomorphism classes of) trees with less than or * *equal to s vertices. We then have Tree0(A) Tree1(A) . . .Tree|A|-1(A) = Tree(A). Each Trees(A) is an initial subcategory of Tree(A). That is, if U T and T 2 T* *rees(A) then U 2 Trees(A). The filtration `quotients' are the discrete categories Qs(A) := Trees(A) - Trees-1(A) whose objects are the trees with precisely s vertices. For each tree T 2 Tree(A* *) we write |T | for the number of vertices of T . OPERADIC BAR CONSTRUCTIONS 49 Definition 9.7 (Filtration on the two-sided bar construction). Let P be a reduc* *ed operad in C with right module R and left module L. Define Z T2Trees(A) B(R, P, L)s(A) := w(T )+ (R, P, L)A(T ). For varying finite sets A these form a symmetric sequence in C. From the inclu* *sion of categories Trees-1(A) Trees(A) we get natural maps B(R, P, L)s-1(A) ! B(R, P, L)s(A). In the case C = T, it is easy to see that the resulting sequence of maps is a f* *iltration of B(R, P, L)(A) by subspaces. The subspace B(R, P, L)s(A) consists of those poin* *ts repre- sented by trees with less than or equal to s vertices. Example 9.8. The generalized A-labelled trees with no vertices (i.e. only a roo* *t and some leaves) correspond one-to-one with (unordered) partitions of A. We therefore se* *e that B(R, P, L)0 = R O L where O is the composition product of symmetric sequences. Example 9.9. Take R = L = I so that B(R, P, L) = B(P ). We then have B(P )0 = I* * by the previous example. If |A| > 1 there is precisely one (non-generalized) A-lab* *elled tree with only one vertex and we therefore get ( S1 P (A) if |A| >;1 B(P )1(A) = B(P )(1) ~=S if |A| =.1 Recall that S is the unit of the symmetric monoidal category C. We can think of the sequence B(R, P, L)0(A) ! B(R, P, L)1(A) ! . .!.B(R, P, L)(A) as a kind of `cellular' filtration. That is, we obtain B(R, P, L)s(A) by attac* *hing `cells' to B(R, P, L)s-1(A), one for each generalized A-labelled tree T with exactly s ver* *tices. The following proposition makes this precise. Proposition 9.10. There is a pushout square in C of the form ` @w(T )+ (R, P, L)A(T_)___/B(R,/P, L) s-1(A) T2Qs(A) OO O O OO OO OO OOO OO OO OO OO ` fflffl fflfflO w(T )+ (R, P, L)A(T_)_____/B(R,/P, L) s(A) T2Qs(A) where @w(T ) denotes the boundary of the space w(T ). To identify the top horizontal map in this diagram we use the following simpl* *e but impor- tant lemma. 50 MICHAEL CHING Lemma 9.11. Let T be a generalized A-labelled tree. Then @w(T )+ ~=colimw(U)+. U 1, it is concentrated in 1 s |A| - 1. We say that P is Kosz* *ul if Hs,*(B(P )(A), @) = 0 fors 6= |A| - 1 where @ denotes the tree differential on B(P ). Definition 9.42 (Koszul duals of Koszul operads). Let P be a Koszul operad in g* *raded k-modules. The Koszul dual of P is the symmetric sequence K(P ) given by the ho* *mology 66 MICHAEL CHING of the reduced bar construction on P . We grade K(P ) according to the total de* *gree (that is, internal degree plus weight degree) of B(P ): K(P )r(A) = H|A|-1,r+1-|A|(B(P )(A), @). Notice that K(P )(A) is the kernel of the differential B(P )|A|-1,*(A) ! B(P )|* *A|-2,*(A), so there is a natural inclusion K(P ) ! B(P ). Proposition 9.43 (Cooperad structure on the Koszul dual of an operad). Let P be* * a Koszul operad in graded k-modules such that each K(P )(A) is a flat k-module. Then th* *e Koszul dual K(P ) has a natural cooperad structure. Proof.We already know from Definition 9.33 that the bar construction B(P ) has * *a cooperad structure. We get the structure for K(P ) by taking homology. So cocomposition * *maps for K(P ) are given by H(B(P )(A [a B)) ! H(B(P )(A) B(P )(B)) ~=H(B(P )(A)) H(B(P )(B)) where we use the flatness assumption to get the middle isomorphism. We dually define the Koszul property and Koszul dual for cooperads of graded * *k-modules. Definition 9.44 (Koszul cooperads and Koszul duals). Let Q be a reduced coopera* *d of graded k-modules. Then Q is Koszul if the homology of the reduced cobar constru* *ction is concentrated in the lowest18 tree degree. In this case, the Koszul dual of Q is* * the symmetric sequence K(Q) of graded k-modules with K(Q)r(A) := H1-|A|,r+|A|-1( (Q)(A), @*). where @* is the tree differential on (Q). Since K(Q) is the bottom homology gr* *oup of (Q) there is a natural surjection (Q) ! K(Q). Proposition 9.45 (Operad structure on the Koszul dual of a cooperad). Let Q be * *a Koszul cooperad of graded k-modules. Then the Koszul dual K(Q) has a natural operad st* *ructure. Proof.The composition maps for K(Q) are given by H( (Q)(A)) H( (Q)(B)) ! H( (Q)(A) (Q)(B)) ! H( (Q)(A [a B). Notice that we don't need a flatness assumption here. Fresse [6] gives various fundamental results for Koszul duality of operads an* *d cooperads, in particular, the following. Lemma 9.46 (Fresse,[6], Lemma 5.2.10). Let P be a Koszul operad of graded k-mod* *ules such that the k-modules P (A) and K(P )(A) are flat. Then K(P ) is a Koszul coo* *perad and K(K(P )) ~=P as operads. Dually, let Q be a Koszul cooperad of graded k-modules such that th* *e modules Q(A) and K(Q)(A) are flat. If Q is Koszul then its Koszul dual operad K(Q) is a* *lso Koszul and K(K(Q)) ~=Q ___________ 18Recall that the tree grading for the cobar construction is concentrated in * *negative degrees. `Lowest' here means most negative. OPERADIC BAR CONSTRUCTIONS 67 as cooperads. We now give the main result of this section. Proposition 9.47. Let P be a reduced operad in T or Sp such that each object P * *(A) is cofibrant and all homology groups H*P (A) and H*B(P )(A) are flat k-modules. If* * H*P is a Koszul operad then H*B(P ) ~=K(H*P ) as cooperads. Dually, let Q be a reduced cooperad in Sp such that each object Q(A) is fibra* *nt and the homology groups H*Q(A) are flat k-modules. If H*Q is a Koszul cooperad then H* (Q) ~=K(H*Q) as operads. Proof.The cofibrancy and flatness conditions ensure that the spectral sequence * *of Propo- sition 9.38 exists for each finite set A and that H*B(P ) is a cooperad in Mod * *k. We have already seen that the spectral sequence has the form (E1*,*, d1) = (B(H*P )*,*(A), @) =) H*B(P ). Because H*P is Koszul, the homology of the bar construction is concentrated in * *the s = |A| - 1 column. Therefore, the E2-term is concentrated in this column and so th* *e spectral sequence collapses. We then see that HrB(P )(A) ~=E2|A|-1,r-|A|+1~=H|A|-1,r-|A|+1(B(H*P )(A), @) ~=K(H*P )r(A) and so H*B(P ) ~=K(H*P ) as claimed. It follows that the modules K(H*P )(A) are flat so, by Proposition * *9.43, K(H*P ) has a cooperad structure. It remains to show that this cooperad structure agree* *s with that on H*B(P ). The first thing to notice is that the above identification of H*B(P )(A) with* * the submodule K(H*P )(A) on B(H*P )(A) is realized by an edge homomorphism of our spectral se* *quence. This edge homomorphism comes from applying homology to the quotient map ` B(P )(A) ! w_(T ) ^ PA(T ) T2Qs(A) where s = |A|-1. The key property of these maps is that they fit into commutati* *ve diagrams ` B(P )(A [a B) ______________//_ w_(V ) ^ PA[aB(V ) | V 2Qs+s0(A[aB) | | | | | | | | | | fflffl|| fflffl|| ` ` B(P )(A) ^ B(P )(B)_____//_ w_(T ) ^ w_(U) ^ PA(T ) ^ PB(U) T2Qs(A)U2Qs0(B) where the map on the right-hand side is built from the familiar maps w_(T [a U) ! w_(T ) ^ w_(U) 68 MICHAEL CHING and the isomorphisms PA[aB(T [a U) ! PA(T ) PB(U) with terms for trees V not of type19 (A, B) mapping to the basepoint. Taking homology of this diagram, the right-hand side map gives the cooperad s* *tructure on B(H*P ) as described in Lemma 9.35. This shows that the edge homomorphisms o* *f the spectral sequence identify the cooperad structure on H*B(P ) with the restricti* *on of that on B(H*P ). Since the cooperad structure on K(H*P ) is also the restriction of tha* *t on B(H*P ), it follows that H*B(P ) ~=K(H*P ) is an isomorphism of cooperads. The dual result is proved similarly. Example 9.48. We return to the Goodwillie derivatives of the identity functor. * *Recall that @*I ~= (ComSp) where ComSp is the cooperad of spectra with ComSp(A) = S for all A. The homolo* *gy of this cooperad is given by ( k if * =;0 H*(ComSp)(A) = 0 otherwise; for all finite sets A. This is the cooperad of commutative coalgebras in the ca* *tegory of graded k-modules. Fresse shows in [6, x6] (by updating a result of Ginzburg and Kapran* *ov [8]) that this cooperad is Koszul (for k = Q, Fp, Z) with Koszul dual given by a suspensi* *on of the Lie operad. Proposition 9.47 therefore applies and we recover the homology of the d* *erivatives of the identity: ( Lie(n) sgnn if * = 1 - n; H*(@nI) = 0 otherwise. Moreover, we now know that the `Lie' operad structure on this homology is equal* * to that induced by the operad structure on the @nI themselves, completing the main goal* * set out in the introduction to this paper. 9.7. Homology of the @*I-modules MX . In this final section, we use our spectra* *l sequence to investigate the homology of the @*I-module MX associated to a based space X * *as described in Remark 8.9(1). Recall that this module is given by a cobar construction: MX := (I, ComSp, _1_X_). We can describe explicitly the spectral sequence for calculating H*MX (2). The * *cobar con- struction is one-sided and we only have to consider trees for which the root ha* *s a single incoming edge. There are two 2-labelled trees of this type with zero and one v* *ertices re- spectively and a morphism between them. The E1 term in the spectral sequence th* *erefore only has nonzero entries in the columns s = 0 and s = -1. These entries are res* *pectively H*X and H*(X ^ X) ~= H*X H*X with the differential given by the reduced diago* *nal X ! X ^ X. The spectral sequence therefore takes the following form. ___________ 19That is, not obtained by grafting a B-labelled tree onto an A-labelled tree. OPERADIC BAR CONSTRUCTIONS 69 .. . 0 H*X H*X H*X -2 -1 0 This reduces to the long exact sequence of homology determined by the cofibre s* *equence X ! X ^ X ! hocofib(X ! X ^ X). This is consistent with the calculation of MX (2) made in Remark 8.9. Things become more interesting (and much more complicated) for MX (n) when n * *> 2. For n = 3 there are eight trees of interest: 1 2 3 1 2 3 1 2, 3 1, 2, 3 (3 labellings) (3 labellings) and the E1 term of the spectral sequence takes the form 3H*X 3 H*X 3 3H*X 2 H*X -2 -1 0 The differential d1 is built from the reduced diagonal (between pairs of terms * *corresponding to bud collapse) and isomorphisms (between pairs of terms corresponding to coll* *apse of an internal edge). We will close the paper by looking at X = Sr, the r-sphere (for r 2). In th* *is situation the reduced diagonal is zero and there can be no higher differentials or extens* *ions in the spectral sequence. This will allow us to calculate H*MSr in its entirety. Before stating the general result, we look at what happens for MSr(3) (with c* *oefficients in Z). The E1-term of the spectral sequence now looks like 70 MICHAEL CHING Z3 Z 0 3r . . .. .. .. . 0 Z3 0 2r . .. . .. . .. 0 0 Z r -2 -1 0 Consider, for a moment, the spectral sequence for calculating the homology of @* *3I. The only nonzero part of the E1 term is the map Z3 - Z based on the four trees labelled bijectively with the set {1, 2, 3}. We know th* *at the homol- ogy is concentrated in top degree. (This is part of the statement that the coco* *mmutative cooperad is Koszul.) So the E2 term is Z2 0. From this we deduce that the only nonzero homology group of @3I is: H-1(@3I) = Z2 and when we take into account the 3-action, we can write this Lie(3) sgn3. Now, return to the case at hand. The 3r-row of the spectral sequence for H*MS* *r(3) looks exactly the same as the 0-row for H*@3I and we can see that they are based on t* *he same trees. Therefore the homology is the same and we deduce that the E2-term takes * *the form Z2 0 0 3r . . .. .. .. . 0 Z3 0 2r . .. . .. . .. 0 0 Z r -2 -1 0 OPERADIC BAR CONSTRUCTIONS 71 We now see that there can be no higher differentials and that, if r > 1, no ext* *ensions. We therefore conclude that 8 >>Lie(3) sgn3* = 3r - 2; >< Z3 * = 2r - 1; H*(MSr(3)) ~= >>Z * = r; >: 0 otherwise. From our previous calculations we see that ( Z * = 2r - 1 orr; H*(MSr(2)) ~= 0 otherwise. We conclude with the general result. Proposition 9.49. Let Sr be the r-sphere for r 2. The homology groups H*MSr h* *ave the following description. There is a `generators', written (A), in HrMSr(A) for ea* *ch non-empty finite set A. The entire homology H*MSr(A) then has a basis given by all possib* *le iterated brackets of the form [. .[.[(A1), (A2)], (A3)] . .,.(Ak)] where A1, . .,.Ak is a partition of A into non-empty finite subsets and [-, -] * *is a symmetric binary operation of degree -1 satisfying the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 that also captures the H*(@*I)-module structure20 on H*(MSr). Sketch proof.The methods used to calculate the case A = {1, 2, 3} above extend * *to general A. The element [. .[.[(A1), (A2)], (A3)] . .,.(Ak)] comes from the part of the spectral sequence given by the binary A-labelled tre* *e that reflects the structure of this iterated bracket, as in the following picture. A1 A2 A3 Ak . . . . The H*(@*I)-module structure comes from relating the module structures on the* * algebraic and topological cobar constructions. 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MR 2004e:18014 Department of Mathematics, Room 2-089, Massachusetts Institute of Technology,* * Cam- bridge MA 02139 E-mail address: mcching@math.mit.edu