Cosimplicial objects and little n-cubes. I. James E. McClure and Jeffrey H. Smith* Department of Mathematics, Purdue University 150 N. University Street West Lafayette, IN 47907-2067 January 19, 2004 Abstract In this paper we show that if a cosimplicial space has a certain kind o* *f combinatorial structure then its total space has an action of an operad weakly equivalen* *t to the little n-cubes operad. Our results are also valid for cosimplicial spectra. 1 Introduction. The little n-cubes operad Cn was introduced by Boardman and Vogt in [6] (except* * that they used the terminology of theories rather than that of operads) as a tool for und* *erstanding n-fold loop spaces. They showed that for any topological space Y the n-fold loo* *p space nY has an action of Cn. In the other direction, May showed in [19] that if Z is a* * space with an action of Cn then there exists a space Y such that the group completion of Z* * is weakly equivalent to nY . In the 30 years since [19] the operad Cn has played an important role in bot* *h unstable and stable homotopy theory. More recently, it has also been of importance (espe* *cially when n = 2) in quantum algebra and other areas related to mathematical physics (see,* * for example, [10], [14], [15], [24], [25]). In the known applications, Cn can be replaced by any operad weakly equivalen* *t to it; such operads are called En operads. There is a highly developed technology that provides sufficient conditions f* *or a space to have an action by an E1 operad (see [1], for example) or an E1 operad ([19], [* *26]). Much less is known about actions of En operads for 1 < n < 1. In this paper we consider the important special situation where the space (o* *r spectrum) Y on which we want an En operad to act is obtained by totalization from a cosi* *mplicial space (resp., spectrum) Xo. We construct an En operad Dn and we show (Theorem * *9.1) that if Xo has a certain kind of combinatorial structure (we call it a n-struc* *ture) then Dn acts on Tot(Xo). ________________________________ *Both authors were partially supported by NSF grants. The first author would* * also like to thank the Isaac Newton Institute for its hospitality during the time this paper was being* * written. 1 The converse of Theorem 9.1 is not true: a Dn-action on Tot(Xo) does not hav* *e to come from a n-structure on Xo. However, in a future paper we will show that Tot in* *duces a Quillen equivalence between the category of cosimplicial spaces with n-structu* *re and the category of spaces with Dn-action; from this it will follow that if Dn acts on * *a space Y then there is a cosimplicial space Xo with a n structure such that Tot(Xo) is weakl* *y equivalent to Y as a Dn-space. The n = 2 case of Theorem 9.1 was proved in [20] (by a more complicated meth* *od than in the present paper), and it has had useful applications, notably o a proof ([20]) that the topological Hochschild cohomology spectrum of an A* *1 ring spectrum R has an action of D2 (this is the topological analog of Deligne'* *s Hochschild cohomology conjecture [8]), and o a proof by Dev Sinha that for k 4 a space closely related to the space o* *f knots in Rk is a 2-fold loop space. The methods we use in this paper are quite general and apply to other catego* *ries of cosimplicial objects besides the categories of cosimplicial spaces and spectra.* * In the sequel to this paper [22] we will apply our methods to the category of cosimplicial ch* *ain complexes. Remark 1.1. Throughout this paper we will use the following conventions for cos* *implicial spaces. (a) We define to be the category of nonempty finite totally ordered sets (* *this is equiv- alent to the category usually called ). We write [m] for the finite totally o* *rdered set {0, . .,.m}. (b) A cosimplicial space Xo is a functor from to spaces. If S is a nonemp* *ty finite totally ordered set we write XS for the value of Xo at S, except that we write * *Xm instead of X[m]. Here is an outline of the paper. As an introduction to the ideas we begin in Sections 2 and 3 with the n = 1 * *(that is, the A1 ) case. In Section 2 we recall the monoidal structure on the category of c* *osimplicial spaces due to Batanin [2]. In Section 3 we give a very simple proof of the fact* * (first shown in [3] and [20]) that if Xo is a monoid with respect to then Tot(Xo) is an A1* * space; the proof is based on an idea due to Beilinson ([12, Section 2]). We also give (in * *Remark 3.3) an explicit description of the combinatorial structure on Xo that constitutes a* * -monoid structure. Our treatment of the E1 case is precisely parallel, with a symmetric monoida* *l structure in place of the monoidal structure (but for technical reasons we need to us* *e augmented cosimplicial spaces instead of cosimplicial spaces; see Remark 6.11). As a prel* *ude, in Sec- tion 4 we give a technically convenient reformulation of the concept of symmetr* *ic monoidal structure; we also define a more general concept (functor-operad) which is used* * in later sec- tions as a way of interpolating between monoidal and symmetric monoidal structu* *res. (The definition of functor-operad was discovered independently, in a different conte* *xt, by Batanin [4]). In Section 5 we digress to offer motivation for the definition of ; the * *definition itself, and the verification that is indeed a symmetric monoidal structure, is given * *in Section 6. The main result in Section 7 (Theorem 7.1) is that if Xo is a commutative mo* *noid with 2 respect to then Tot(Xo) is an E1 space. Section 7 also gives an explicit des* *cription of the combinatorial structure on Xo that constitutes a commutative -monoid struc* *ture. In Section 8 we define for each n a functor-operad n; the special case n = * *1 is the sym- metric monoidal structure , and the special case n = 1 is essentially the same* * (see Remark 8.7) as the monoidal structure . (In the special case n = 2, a construction is* *omorphic to 2 was discovered independently by Tamarkin in unpublished work.) In Section 9* * we use the functor-operad n to construct an ordinary (topological) operad Dn. The mai* *n theorem in Section 9 (Theorem 9.1) says that Dn is weakly equivalent to the little n-cu* *bes operad Cn and that if Xo is a n-algebra then Dn acts on Tot(Xo). Section 9 also gives* * an explicit description of the combinatorial structure on Xo that constitutes an action of * * n on Xo. In Section 10 we restrict to the special case n = 2: we show that an action* * of 2 is essentially the same thing as a more familiar structure, namely an operad with * *multiplication as defined in [9]. It seems likely that there is an analogous result for n > 2,* * perhaps using Batanin's concept of n-operad [4]. In Section 11 we show that the operad Dn acts naturally on nY for all Y . The next two sections contain material which is used in the proofs of Theore* *ms 7.1 and 9.1 and may also be of independent interest. Let Yk0denote the 0-th space of th* *e cosimplicial space ( o) k. In Section 12 we prove that ( o) k is isomorphic as a cosimplicia* *l space to the Cartesian product o x Yk0. We also show that Yk0has a canonical cell struc* *ture and that it is contractible (which completes the proof of Theorem 7.1). In Section * *13 we show that nk( o, . .,. o) is isomorphic as a cosimplicial space to the Cartesian pr* *oduct of its 0-th space with o; this is used in the proof of Theorem 9.1 (the n = 2 case is* * the "fiberwise prismatic subdivisionü sed in [20]). In Section 14 we use a technique of Clemens Berger [5] to show that the oper* *ad Dn defined in Section 9 is weakly equivalent to Cn; this completes the proof of Th* *eorem 9.1. The basic idea of Berger's technique is to compare the two operads by writing t* *hem as homotopy colimits, over the same indexing category, of contractible operads. Finally, in Section 15 we observe that there is a variant of Tot which prese* *rves weak equivalences, and we show that Theorems 7.1 and 9.1 have analogs for this versi* *on of Tot. We would like to thank the referee for a very careful reading of the paper a* *nd for several useful suggestions. 2 A monoidal structure on the category of cosimplicial spaces. We begin with some motivation. We are concerned with the question of when Tot * *of a cosimplicial space has an A1 structure. This question is formally analogous to * *the question of when the normalization of a cosimplicial abelian group has an A1 structure * *(we will explore this analogy further in [22]; also see [21, Sections 3 and 4]). We take* * as our starting point the fact that for any space W , the normalized cochain complex of W has a* * (strictly associative) product, namely the cup product. The normalized cochain complex of* * W is the normalization of the cosimplicial abelian group SoW defined by SoW = Map (SoW, Z), 3 where SoW is the singular complex of W and Map is set maps. We therefore examin* *e the relationship between the cup product and the cosimplicial structure maps of SoW* * . The cup product is defined as usual for x 2 SpW and y 2 SqW by (x ` y)(oe) = x(oe(0, . .,.p)) . y(oe(p, . .,.p + q)) Here oe 2 Sp+qW , . is multiplication in Z, and oe(0, . .,.p) (resp., oe(p, . .* *,.p + q)) is the restriction of oe to the subsimplex of p+qspanned by the vertices 0, . .,.p (r* *esp., p, . .,.p + q). We note that ` is related to the coface and codegeneracy operations by the * *following formulas: æ dix ` y if i p (2.1) di(x ` y) = i-p x ` d y if i > p (2.2) dp+1x ` y = x ` d0y æ six ` y if i p - 1 (2.3) si(x ` y) = i-p x ` s y if i p Now let us return to the category of cosimplicial spaces. Formulas (2.1), (2* *.2)and (2.3) motivate the following definition. Definition 2.1. Let Xo and Y obe cosimplicial spaces. Xo Y ois the cosimplici* *al space whose m-th space is _ ! a Xp x Y q = ~ p+q=m (where ~ is the equivalence relation generated by (x, d0y) ~ (d|x|+1x, y)). The* * cosimplicial operators are given by æ (dix, y) if i |x| di(x, y) = i-|x| (x, d y) if i > |x| and æ (six, y) if i |x| - 1 si(x, y) = i-|x| (x, s y) if i |x| We leave it to the reader to check that the cosimplicial identities are sati* *sfied and that the following holds. Proposition 2.2. is a monoidal structure for the category of cosimplicial spa* *ces, with unit the constant cosimplicial space that has a point in every degree. 4 There is also a monoidal structure for cosimplicial spectra: one simply re* *places the Cartesian products in Definition 2.1 by smash products. We conclude this section with some observations about Kan extensions. This m* *aterial will not be needed logically for the rest of the paper, but it provides useful * *motivation for the constructions in Sections 6 and 8. Recall the conventions in Remark 1.1. * * ` ` If S1, S2, . .,.Sk are finite totally ordered sets there is a unique total o* *rder on S1 . . .Sk for which the inclusion maps into the coproduct are order-preserving and every * *element of Siis less than every element of Sj for i < j. Let : xk ! ` ` be the functor which takes (S1, . .,.Sk) to S1 . . .Sk with this total order. The following fact was first noticed by Cordier and Porter (unpublished). Proposition 2.3. Let Xo1, . .,.Xokbe cosimplicial spaces and let Xo1~x. .~.xXok* *denote the composite Xo1x...xXok x xk -------! Topx . .x.Top -! Top. Then Xo1 . . .Xokis naturally isomorphic to the left Kan extension Lan (Xo1~x.* * .~.xXok). Before giving the proof we mention an important consequence. Remark 2.4. Let * be the functor from cosimplicial spaces to bicosimplicial sp* *aces defined by ` ( *(Xo))S,T= XS T It is a general fact about Kan extensions [17, beginning of Section X.3] that L* *an is the left adjoint of *. This implies that there is a natural 1-1 correspondence between * *maps ff : Xo Y o! Zo and consistent collections of maps ` effS,T: XS x Y T! ZS T where öc nsistent" means that every pair of ordered maps f : S ! S0, g : T ! T * *0induces a commutative diagram effS,T ` XS x Y T _____//ZS T f*xg*|| |(f|`g)* 0 fflffl|0effS0fflffl|,T0` XS x Y T ____//ZS0 T0 5 Proof of Proposition 2.3. For simplicity we assume k = 2. For m 0 let [m] denote the set {0, . .,.m}. Every object in is canonical* *ly isomorphic to one of the form [m] so it suffices to show that the two functors in question* * are naturally isomorphic on the full subcategory of with these objects. Fix m. According to [17, Equation (10) on page 240], the left Kan extension,* * evaluated at [m], can be calculated as follows. Let C be the category of objects -over [* *m]: an object in C is a pair consisting of an object ([n], [n0]) in x and a morphism ([n* *], [n0]) ! [m] in ; a morphism in C is a morphism (f, g) in x making the evident triangle* * commute. Then Lan (Xo1~xXo2) is the colimit of the composite Xo1~xXo2 C ! x -----! Top where the first map is the evident forgetful functor. Now let C0be the full sub* *category of C consisting of pairs (([p], [q]), f : [p + q + 1] ! [m]) where f is a surjecti* *on, p + q is either m - 1 or m, and if p + q = m then f(p) = f(p + 1). Note that an object in C0is * *determined by p and q; to simplify the notation we denote the object by (p, q). If p + q =* * m - 1 there is exactly one morphism from the object (p, q) to (p + 1, q) and exactly one fr* *om (p, q) to (p, q + 1), and there are no other non-identity morphisms in C0. The map from * *(p, q) to (p + 1, q) is the last coface map on [p] and the identity on [q], while the map* * from (p, q) to (p, q + 1) is the identity on [p] and the zeroth coface map on [q]. From this i* *t is clear that the m-th space of Xo1 Xo2is the colimit of the composite Xo1~xXo2 C0 C ! x -----! Top In particular there is a natural map Xo1 Xo2! Lan (Xo1~xXo2). By [17, Section 9.3], this will be an isomorphism if C0is cofinal in C, that is* *, if for each c 2 C the under-category c # C0 is connected. The under-categories can be described * *explicitly: each is a nonempty full subcategory of C0with set of objects of the form {(p, q) | p p0, q q0} for some p0 and q0; clearly all such categories are connected. * * __ * *|__| 3 A sufficient condition for Tot (Xo ) to be an A1 space. In this section we will prove Theorem 3.1. If Xo is a monoid with respect to then Tot(Xo) has an A1 structu* *re. Remark 3.2. (a) Previous proofs of Theorem 3.1 were given by Batanin [3, Theore* *ms 5.1 and 5.2] and by us [20, Theorem 2.4]. (b) The theorem and its proof are also valid for cosimplicial spectra. 6 Remark 3.3. Definition 2.1 implies that Xo is a monoid with respect to if and* * only if there are maps `: Xp x Xq ! Xp+q for all p, q 0 satisfying equations (2.1), (2.2), (2.3), the associativity co* *ndition (3.1) (x ` y) ` z = z ` (y ` z) and the unit condition: there is an element e 2 X0 such that (3.2) x ` e = e ` x = x for all x. The rest of this section is devoted to the proof of Theorem 3.1. Let o denote the cosimplicial space whose m-th space is the simplex m , wi* *th the usual cofaces and codegeneracies. By definition, Tot(Xo) is Hom ( o, Xo) (where Hom d* *enotes the space of cosimplicial maps). Definition 3.4. (a) For each k 0 let A(k) be the space Tot(( o) k). (b) If f 2 A(k) and gi2 A(ji) for 1 i k define fl(f, g1, . .,.gk) 2 A(j1* * + . .+.jk) to be the composite f o k g1 ... gko (j1+...+j ) o -! ( ) ------! ( ) k Theorem 3.1 is an immediate consequence of our next result. Proposition 3.5. (a) Let A be the sequence of spaces A(k), k 0, with the oper* *ations fl : A(k) x A(j1) x . .x.A(jk) ! A(j1 + . .+.jk) defined above. Then A is an operad. (b) If Xo is a monoid with respect to then A acts on Tot(Xo). (c) A is an A1 operad. Proof. Part (a) is clear. For part (b), given f 2 A(k) and x1, . .,.xk 2 Tot(Xo) define f(x1, . .,.xk)* * 2 Tot(Xo) to be the composite f o k x1 ... xk o k ~ o o -! ( ) ------! (X ) -! X where ~ is the monoidal structure map of Xo. This construction gives maps A(k) x (Tot(Xo))k ! Tot(Xo) which fit together to give an action of A on Tot(Xo). For part (c) we need to show that A(0) is a point (which is obvious) and tha* *t each space A(k) is contractible. First consider the case k = 1. If f and g are two cosimpl* *icial maps from o to o, then tf + (1 - t)g will again be a cosimplicial map for each 0 t * *1 (because the cosimplicial structure maps of o are affine) and so we can use the straight-li* *ne homotopy to contract Hom( o, o) to a point. The case k 2 is now immediate from Lemma* * 3.6 * * __ below. * *|__| 7 Lemma 3.6. For each k 1, ( o) k is isomorphic as a cosimplicial space to o. Proof. It suffices to do the case k = 2; the general case follows by induction.* * First we define maps fm : ( o o)m ! m for m 0 by fm ((s0, . .,.sp), (t0, . .,.tq)) = 1_2s0, . .,.1_2(sp + t0), . * *.,.1_2tq These are well defined and fit together to give a cosimplicial map f : o o * *! o. Next define gm : m ! ( o o)m as follows: given (u0, . .,.um ) 2 m , choose the smallest p for which 1 u0 + . .+.up __ 2 and let gm (u0, . .,.um ) = [(2u0, . .,.1 - 2u0 - . .-.2up-1), (2u0 + . .+.2up - 1, 2u* *p+1, . .,.2um )] The gm are continuous, they fit together to give a cosimplicial map g : o ! o* * o, and * * __ f and g are mutually inverse. * * |__| Remark 3.7. Lemma 3.6 is due to Grayson [11, Section 4]. 4 Functor-operads. The purpose of this section is to describe a general setting in which there are* * analogs of Definition 3.4 and Proposition 3.5(a) and (b) (see Definition 4.3 and Propositi* *ons 4.4 and 4.6). Given a category C let Cxk denote the k-fold Cartesian product. For each per* *mutation oe 2 k we define oe# : Cxk ! Cxk to be the functor taking (A1, . .,.Ak) to (Aff(1), . .,.Aff(k)). In order to motivate the definition of functor-operad, let us consider the s* *ituation in which C has a symmetric monoidal structure . For each k 0 define a functor Fk : Cxk ! C by Fk(X1, . .,.Xk) = X1 (X2 (X3 . .).) 8 MacLane's coherence theorem [17] implies that there are canonical natural isomo* *rphisms oe* : Fk ! Fk O oe# and j1,...,jk: Fk(Fj1, . .,.Fjk) ! Fj1+...jk satisfying certain consistency conditions. The following definition is an abstr* *act version of this situation, except that instead of requiring the 's to be natural isomorph* *isms we allow them merely to be natural transformations. Definition 4.1. Let C be a category enriched over Top. A functor-operad F in C* * is a sequence of continuous functors Fk : Cxk ! C together with (i) for each oe 2 k, a continuous natural isomorphism oe* : Fk ! Fk O oe# (ii) for each choice of j1, . .,.jk 0, a continuous natural transformation j1,...,jk: Fk(Fj1, . .,.Fjk) ! Fj1+...jk such that (a) F1 is the identity functor, and the natural transformations 1,...,1: Fk(F1, . .,.F1) ! Fk k : F1(Fk) ! Fk are equal to the identity. (b) All diagrams of the following form commute: Fk(Fj1(Fi11, . .,.Fi1j1), . .,.Fjk(Fik1, ._.,.Fikjk))//_Fj1+...+jk(Fi11,* * . .,.Fikjk) Fk( ,...,|)| || fflffl| fflffl| Fk(Fi11+...+i1j1, . .,.Fik1+...+ikjk)_______//Fi11+...+ikjk (c) (oeø)* = ø*oe* for all oe, ø 2 k (d) Let øi2 jifor 1 i k and let ø be the image of (ø1, . .,.øk) under t* *he map j1x . .x. jk! j1+...+jk Then the following diagram commutes: Fk(Fj1, . .,.Fjk)________//_Fj1+...+jk Fk(fi1*,...,fik*)|| || fflffl| fi fflffl| Fk(Fj1O ø1#, . .,.FjkO øk#)_*//Fj1+...+jkO ø# 9 (e) Let oe 2 k and let ~oebe the permutation in j1+...+jkwhich permutes th* *e blocks {1, . .,.j1}, . .,.{j1+ . .+.jk-1+ 1, . .,.j1+ . .+.jk} in the same way that oe* * permutes the numbers 1, . .,.k. Then the following diagram commutes Fn(Fj1, . .,.Fjk)_______//Fj1+...+jk ff*|| ~ff*|| fflffl| fflffl| Fn(Fjoe(1), . .,.Fjoe(k))_O/~oe#/Fj1+...+jkO ~oe# Remark 4.2. (a) Batanin [4] has independently proposed a similar but more gener* *al defi- nition: if O is an operad in the category of categories Batanin defines an inte* *rnal operad in O to be a collection consisting of an object ak in O(k) for each k 0 and morp* *hisms oe* : ak ! ~oe(ak) for each oe 2 k (where ~oedenotes the action of oe 2 k on O(k)) and j1,...,jk: fl(ak, aj1, . .,.ajk) ! aj1+...+ajk for each j1, . .,.jk 0 (where fl is the structure map of the operad O) satisf* *ying the analogs of properties (a)-(e) in Definition 4.1. A functor-operad in C is then an inter* *nal operad in the endomorphism operad of C. (b) If B is an operad in the category Top we can define a functor-operad F i* *n Top by Fk(X1, . .,.Xk) = Bk x X1 x . .x.Xk with the obvious structure maps. Definition 4.3. Let F be a functor-operad in C and let A be an object of C. (a) Define FA to be the collection of spaces FA(k) = Hom (A, Fk(A, . .,.A)), k 0. (b) Give FA(k) the action induced by the oe*. (c) Define 1 2 FA(1) to be the identity map of A. (d) For each choice of j1, . .,.jk 0 define fl : FA(k) x FA(j1) x . .F.A(jk) ! FA(j1 + . .+.jk) to be the composite Hom (A, Fk(A, . .,.A)) x Hom (A, Fj1(A, . .,.A)) x . .H.om(A, Fjk(A, . .,.A))* * ! Hom(A, ) Hom (A, Fk(Fj1(A, . .,.A), . .,.Fjk(A, . .,.A))) ------! Hom (A, Fj1+...+j* *k(A, . .,.A)) Proposition 4.4. These choices make FA an operad. The proof is an easy verification. 10 Definition 4.5. Let F be a functor-operad in C. An algebra over F is an object* * X of C together with continuous maps k : Fk(X, . .,.X) ! X for k 0 such that (a) 1 is the identity map. (b) The following diagram commutes for each choice of j1, . .,.jk 0 Fk(Fj1(X, . .,.X), . .,.Fjk(X, . .,.X))//_Fj1+...+jk(X, . .,.X) Fk( j1,...,|jk)| |j1+...+jk| fflffl| fflffl| Fk(X, . .,.X)___________k__________//_X (c) k O oe* = k for all oe 2 k. Now let A be an object of C, let X be an algebra over F, and for each k 0 * *define `k : FA(k) x Hom (A, X)k ! Hom (A, X) to be the composite Hom(A, k) Hom (A, Fk(A, . .,.A)) x Hom (A, X)k ! Hom (A, Fk(X, . .,.X)) -------! Hom (* *A, X) Proposition 4.6. The maps `k make Hom (A, X) an algebra over the operad FA. Again, the proof is an easy verification. Definition 4.7. A functor-operad F is strict if the natural transformations j1* *,...,jkare isomorphisms. Proposition 4.8. If F is a strict functor-operad, then F2 is a symmetric monoid* *al structure for C with identity object F0. The commutative monoids with respect to this st* *ructure are the same as the algebras over F. Once more, the proof is an easy verification. 5 A family of operations in So W In Section 6 we will define a symmetric monoidal product on the category of a* *ugmented cosimplicial spaces. In this section we pause to offer motivation for this def* *inition. The results in this section are not needed logically for later sections. The definition of the monoidal product was motivated in Section 2 by the p* *roperties of the cup product in SoW . The cup product is part of a larger family of opera* *tions in SoW whose properties could be used as the basis for a definition of . However, thi* *s larger family is rather inconvenient to work with (because the analog of equation (2.2)for th* *e larger family is complicated) so we will use a related family which has somewhat simpler prop* *erties. We begin with a variant of the cup product. Given x 2 SpW and y 2 SqW we def* *ine x t y 2 Sp+q+1W 11 by (x t y)(oe) = x(oe(0, . .,.p)) . y(oe(p + 1, . .,.p + q)) (note that, in contrast to the cup product, the vertex p is not repeated). This operation is related to the coface and codegeneracy operations in SoW * *by the following equations: æ dix t y if i p + 1 (5.1) di(x t y) = i-p-2 x t d y if i > p + 1 æ six t y if i < p (5.2) si(x t y) = i-p-1 x t s y if i > p Note that there is no analog for t of equation (2.2). The operations ` and t determine each other: x t y = (dp+1x) ` y = x ` d0y x ` y = sp(x t y) Now observe that equations (5.1)and (5.2)can be used as the basis for a char* *acterization of -monoids: Remark 2.4 implies that Xo is a -monoid if and only if there are* * maps t : Xp x Xq ! Xp+q+1 satisfying (5.1), (5.2), the associativity condition (5.3) x t (y t z) = (x t y) t z and the unit condition: there exists e 2 X0 with (5.4) sp(x t e) = s0(e t x) = x (compare this to Remark 3.3). In the remainder of this section we will define a family of operations in So* *W which generalize t; the definition of in Section 6 will be suggested by the propert* *ies of this family. First we need to be a little more explicit in our description of SoW . Recal* *l the conventions in Remark 1.1. Given a nonempty finite set totally ordered set T we let T be t* *he convex hull of T ; in particular, [m]is the usual m . We define STW to be the set of* * all continuous maps T ! W (in particular, S[m]W is what we have been calling Sm W ) and STW t* *o be Map (STW, Z) (so S[m]W is the same as Sm W ). Given a map oe : T ! W and a subset U of T let oe(U) denote the restriction* * of oe to the sub-simplex of T spanned by the vertices in U. Suppose we are given a function f : T ! {1, . .,.k} 12 -1 and elements xi2 Sf (i)W for 1 i k. We can define an element (x1, . .,.xk) 2 STW by (x1, . .,.xk)(oe) = x1(oe(f-1(1)) . x2(oe(f-1(2)) . . ...xk(oe(f-1* *(k)) where . denotes multiplication in Z. This procedure gives a natural transformat* *ion -1(1) f-1(k) T : Sf W . . .S W ! S W Remark 5.1. In the special case where f is the function from {0, . .,.p + q + 1* *} to {1, 2} which takes {0, . .,.p} to 1 and {p + 1, . .,.p + q + 1} to 2, we have (x, y* *) = x t y Next we describe the relation between the operations and the cosimplicia* *l structure maps of SoW . Proposition 5.2. Let _________h________//0 T III tT III tttt fIII$$I zzttgtt {1 . .,.k} be a commutative diagram, where h is a map in (i.e., an order-preserving map)* *. For each i 2 {1 . .,.k} let hi: f-1(i) ! g-1(i) be the restriction of h. Then the diagram -1(1) f-1(k) __//_T Sf W . . .S W S W (h1)* ... (hk)*|| |h*| fflffl| fflffl| -1(1) g-1(k) _//_T0 Sg W . . .S W S W commutes. The proof is an immediate consequence of the definitions. In the special cas* *e of Remark 5.1 we recover equations (5.1)and (5.2). 6 A symmetric monoidal structure on the category of augmented cosimplicial spaces. From now on we will work with augmented cosimplicial spaces (the reason for thi* *s is given in Remark 6.11). 13 Definition 6.1. An augmented cosimplicial space is a functor Xo from + to Top,* * where + is the category of finite totally ordered sets (including the empty set). Our goal in this section is to construct a symmetric monoidal product in t* *he category of augmented cosimplicial spaces. We will do this by constructing a strict func* *tor-operad and letting be 2; see Proposition 4.8. The basic idea in defining k(Xo1, . .,.Xok) is that we build it from formal* * symbols (x1, . .,.xk), where f : T ! {1, . .,.k} (cf. Section 5). In order to get a* * cosimplicial object we have to build in the cosimplicial operators, so we consider symbols o* *f the form h*((x1, . .,.xk)) where h : T ! S is an order-preserving map; such a symbol will represent a poi* *nt in the S-th space k(Xo1, . .,.Xok)S. We want to require these symbols to satisfy * *the relation in Proposition 5.2, and the most efficient way to do this is by means of a Kan * *extension (Definition 6.4); a more elementary description of k is given in equation 6.1. Here are the formal definitions: Definition 6.2. Let k 0. Define ~kto be the set {1, . .,.k} when k 1 and th* *e empty set when k = 0. Definition 6.3. Let Qk be the category whose objects are pairs (f, S), where S * *is an object of + and f is a map of sets from S to ~k, and whose morphisms are commutative * *triangles _____h____// S == T == f==OEOE=g ~k where h is a map in +. There is a forgetful functor : Qk ! + which takes (f, S) to S, and a func* *tor from Qk to the k-fold Cartesian product ( +)xk which takes (f, S) to the k-tuple (f-1(1* *), . .,.f-1(k)). Definition 6.4. For each k 0 define a functor k as follows. Given augmented * *cosimplicial spaces Xo1, . .,.Xok, let Xo1~x. .~.xXokdenote the composite Xo1x...xXok x ( +)xk -------! Topx . .x.Top -! Top. We define the augmented cosimplicial space k(Xo1, . .,.Xok) to be the Kan exte* *nsion Lan ((Xo1~x. .~.xXok) O ) Remark 6.5. (a) 0 is the augmented cosimplicial space which takes every S to a* * point (because a Cartesian product indexed by the empty set is a point). (b) The adjointness property of Lan [17, beginning of Section X.3] implies * *that a map k(Xo1, . .,.Xok) ! Y ois the same thing as a collection of maps -1(1) f-1(k) T : Xf1 x . .x.Xk ! Y , one for each f : T ! ~k, such that the analog of Proposition 5.2 is satisfied. 14 Our next goal is to specify the structure maps oe* and j1,...,jkof the func* *tor-operad . For each of these we will use [17, Equation (10) on page 240] to write the rele* *vant Kan extension as a colimit, and we will then use the following observation, whose p* *roof is left to the reader. Lemma 6.6. Let A and B be categories and let G : A ! Top and H : B ! Top be fun* *ctors. Each pair consisting of a functor K : A ! B and a natural transformation : G * *! H O K induces a map colimA G ! colimBH We begin by constructing the transformation oe*. Let Xo1, . .,.Xokbe augment* *ed cosim- plicial spaces and let S be a totally ordered finite set; we want to construct oe* : k(Xo1, . .,.Xok)S ! k(Xoff(1), . .,.Xoff(k))S Let A1 be the category whose objects are the diagrams ~kofoT___h_//S where T is a totally ordered finite set, f is a map of sets and h is an ordered* * map; we denote such a diagram by (f, h). A morphism from (f, h) to (f0, h0) is a commutative d* *iagram ~kofoT__h__//_S = || g|| = || fflffl|fflffl|fflffl|f00 ~koo_T 0h__//S where g is an ordered map. Let G1 : A1 ! Top Q f-1(i) be the functor which takes (f, h) to Xi . By [17, Equation (10) on page 240] we have (6.1) k(Xo1, . .,.Xok)S = colimA1G1 Q f-1(i) Next let oe 2 k and let H1 be the functor which takes (f, h) to Xff(i); we h* *ave k(Xoff(1), . .,.Xoff(k))S = colimA1H1 Definition 6.7. The map oe* : k(Xo1, . .,.Xok)S ! k(Xoff(1), . .,.Xoff(k))S is induced by the functor K1 : A1 ! A1 which takes (f, h) to (oe-1 O f, h) and * *the natural transformation 1 : G1 ! H1 O K1 which takes (x1, . .,.xk) to (xff(1), . .,.xff(k)). 15 Next let j1, . .,.jk 0, let Xo1, . .,.Xoj1+...+jkbe augmented cosimplicial* * spaces and let S be a finite totally ordered set. We want to construct j1,...,jk: k( j1(Xo1, . .)., . .,. jk(. .,.Xoj1+...+jk))S ! j1+...+jk(Xo* *1, . .,.Xoj1+...+jk)S First observe that k( j1(Xo1, . .)., . .).S= colimA1H, where A1 is the category defined above and H is the functor which takes (f, h) * *to Yk -1(i) ji(Xoj1+...+ji-1+1, . .,.Xoj1+...+ji)f i=1 Thus a point in k( j1(Xo1, . .)., . .).Sis an equivalence class represented by* * a diagram ~kofoT___h_//S -1(i) together with points xi 2 ji(Xoj1+...+ji-1+1, . .).f for 1 i k. Similar* *ly, each xi is represented by a diagram ~jifiTioo_hi//_f-1(i) together with a point j1+...+jiY -1 x0i2 Xfip(p-j1-...-ji-1) p=j1+...+ji-1+1 We can assemble this information into a diagram ___________ ffl ` ` e (6.2) _1 + . .+._koo_~_1 . . .~_koo__U _ || g || fflffl|f fflffl| ~koo_______T _h__//S Q (fflOe)-1(p) and`a point x`2 Xp . Here Ø is the unique ordered bijection, _ takes ~_i* *to i, U is Ti, e is fi, the restriction of g to Tiis hi, and we give U the unique tot* *al order for which g and the inclusions of the Tiare ordered maps. Let A2 be the category of diagrams of the form (6.2)for which g and h are or* *dered; an ob- ject of A2 will be denoted (e, f, g, h). What we have shown so far is that k( * *j1(Xo1, . .)., . .).S Q (fflOe)* *-1(i) is the colimit over A2 of the functor G2 which takes (e, f, g, h) to Xi * *. Next we note that j1+...+jk(Xo1, . .,.Xoj1+...+jk)S is the colimit over the category B of d* *iagrams ___________ f h _1 + . .+._koo_T ____//S Q f-1(i) (denoted (f, h)) of the functor H2 which takes (f, h) to Xi . Now let K2 : A2 ! B take (e, f, g, h) to (Ø O e, h O g); note that H2 O K2 = G2. 16 Definition 6.8. The map j1,...,jk: k( j1(Xo1, . .)., . .).S! j1+...+jk(Xo1, . .,.Xoj1+.* *..+jk)S is induced by the functor K2 : A2 ! B and the identity natural transformation f* *rom G2 to H2 O K2. Finally, let K3 : B ! A2 be the functor which takes (f, h) to (f, _ O f, id,* * h). Then G2O K3 = H2 and we can let 3 : H2 ! G2O K3 be the identity natural transformat* *ion; the result is a natural transformation : j1+...+jk(Xo1, . .,.Xoj1+...+jk)S ! k( j1(Xo1, . .)., . .)* *.S With these definitions it is easy to check that oe*, and are natural in * *S, that conditions (a)-(e) of definition 4.1 are satisfied, and that is inverse to . We have no* *w shown Theorem 6.9. The collection k, k 0, with the structure maps oe* and define* *d above, is a strict functor-operad in the category of augmented cosimplicial spaces. In* * particular_ 2 is a symmetric monoidal product with unit 0. * * |__| Remark 6.10. If A is any symmetric monoidal category with the property that the* * sym- metric monoidal product preserves colimits (which is automatic when A is a clos* *ed sym- metric monoidal category) then Theorem 6.9 and its proof are valid for the cate* *gory of augmented cosimplicial objects in A, with Cartesian products in Top replaced by* * the sym- metric monoidal product in A. Remark 6.11. All of the constructions in this section can be imitated for the c* *ategory of ordinary (nonaugmented) cosimplicial spaces, provided that the maps f in the de* *finition of Qk are required to be surjective. In this setting the product 2 is still a* *ssociative and commutative, but it is not unital: 0 is empty, and neither 0 nor any other co* *simplicial space is an identity object for 2. 7 A sufficient condition for Tot (Xo ) to be an E1 space. If Xo is an augmented cosimplicial space, we define Tot(Xo) to be the usual Tot* * of the cosimplicial space obtained by restricting Xo to . Equivalently, Tot(Xo) is Hom ( o, Xo) where Hom denotes maps of augmented cosimplicial spaces and we extend o to an * *aug- mented cosimplicial space by setting ; = ;. Now apply Proposition 4.4, letting F be the functor operad constructed in * *Section 6 and A the augmented cosimplicial space o. This gives an operad D with k-th * *space D(k) = Tot( k( o, . .,. o)). Recall that we have defined to be 2; this is a symmetric monoidal product* * on the category of augmented cosimplicial spaces. 17 Theorem 7.1. (a) D is an E1 operad. (b) If Xo is a commutative monoid with respect to (equivalently, if Xo is * *an algebra over ) then D acts on Tot(Xo). Proof. Part (b) is immediate from Propositions 4.6 and 4.8. Part (a) will be p* *roved in * * __ Section 12. * * |__| Remark 7.2. The analog of Theorem 7.1(b) is valid, with the same proof, when Xo* * is an augmented cosimplicial spectrum. In the remainder of this section we use the adjointness property of Lan [17* *, beginning of Section X.3] to give an explicit characterization of commutative -monoids. Definition 7.3. Let Xo be an augmented cosimplicial space. A < >-structure on X* *o consists of a map -1(1) f-1(2) T : Xf x X ! X for each totally ordered set T and each f : T ! {1, 2}. Definition 7.4. A < >-structure on Xo is consistent if for every commutative di* *agram _______h______//0 T EE yT EEE yyyy f EE""E __ygyy {1, 2} the diagram -1(1) f-1(2)__//T Xf x X X (h1)*x(h2)*|| |h*| fflffl| |fflffl -1(1) g-1(2)_//_T0 Xg x X X commutes, where hiis the restriction of h to f-1(i). Recall the notation of Definition 6.4. Since Lan is left adjoint to *, a m* *ap Xo Xo ! Xo is the same thing as a natural transformation (Xox~Xo) O ! Xo O and it's easy to check that this is the same thing as a consistent < >-structur* *e on Xo. It remains to translate the commutativity, associativity and unitality conditions * *satisfied by a commutative -monoid into this language. Definition 7.5. A < >-structure on Xo is commutative if the diagram -1(1) f-1(2)_//_T Xf x X X fi|| |=| fflffl| fflffl| -1(2) f-1(1)_//_T Xf x X X commutes, where ø is the switch map and t is the transposition of {1, 2}. 18 For the associativity condition we need some notation. Let T be a totally o* *rdered set and let g : T ! {1, 2, 3} be a function. Define ff : {1, 2, 3} ! {1, 2} by ff(1) = 1, ff(2) = 1, ff(3) = 2 and define fi : {1, 2, 3} ! {1, 2} by fi(1) = 1, fi(2) = 2, fi(3) = 2. Let g1 be the restriction of g to g-1{1, 2}* * and let g2 be the restriction of g to g-1{2, 3}. Definition 7.6. A < >-structure on Xo is associative if, with the notation abov* *e, the diagram -1(1) g-1(2) g-1(3) x1//_g-1{1,2}g-1(3) Xg x X x X X x X 1x|| || fflffl| fflffl| -1(1) g-1{2,3}________//_T Xg x X X commutes for every choice of T and of g : T ! {1, 2, 3}. Definition 7.7. A < >-structure on Xo is unital if there is an element " 2 X; w* *ith the property that if f : T ! {1, 2} takes all of T to 1 then (x, ") = x for all * *x and if f takes all of T to 2 then (", x) = x for all x. Proposition 7.8. A commutative -monoid structure on Xo determines, and is dete* *rmined by, a < >-structure on Xo which is consistent, commutative, associative and uni* *tal. The proof is a routine verification using the definitions in Section 6. Remark 7.9. The operations on SoW defined in Section 5 give a consistent, c* *ommuta- tive, associative and unital < >-structure on SoW . 8 A filtration of by functor-operads. In this section we describe a filtration of by functor-operads n; the operad* * associated to n will turn out to be equivalent to the little n-cubes operad Cn. We begin with some motivation. If T is a totally ordered set and f : T ! {1* *, 2} is a function, the two totally ordered sets f-1(1) and f-1(2) are mixed together to * *form T . The amount of mixing can be measured by the number of times the value of f switches* * from 1 to 2 or from 2 to 1 as one moves through the set T . The idea in the definition* * of n is to control the amount of mixing that is allowed. Definition 8.1. Let T be a finite totally ordered set, let k 2, and let f : T* * ! ~k. We define the complexity of f as follows. If k is 0 or 1 the complexity is 0. If k* * = 2 let ~ be the equivalence relation on T generated by a ~ b if a is adjacent to b and f(a) = f(b) and define the complexity of f to be the number of equivalence classes minus 1.* * If k > 2 define the complexity of f to be the maximum of the complexities of the restric* *tions f|f-1(A) as A ranges over the two-element subsets of ~k. 19 Remark 8.2. If k = 2 the complexity of f is exactly the amount of mixing in f a* *s discussed above. The definition of complexity is suggested by [23]; the reason we use it * *here is that it is well-adapted to the proofs of Theorems 8.5 and 9.1(a). There may be othe* *r ways of defining complexity that would also lead to Theorems 8.5 and 9.1(a), although t* *his seems unlikely. Now fix n 1. Recall the category Qk from Definition 6.3. Definition 8.3. Let Qnkbe the full subcategory of Qk whose objects are pairs (f* *, S) where f has complexity n. Let 'n : Qnk! Qk be the inclusion. Definition 8.4. For each n 1 and each k 0 define a functor nkas follows. G* *iven aug- mented cosimplicial spaces Xo1, . .,.Xok, let Xo1~x. .~.xXokbe the functor defi* *ned in Definition 6.4 and let nk(Xo1, . .,.Xok) be the Kan extension Lan O'n((Xo1~x. .~.xXok) O O 'n) Theorem 8.5. For each n 1, n is a (non-strict) functor-operad in the categor* *y of augmented cosimplicial spaces. Proof. We define oe* as in the proof of Theorem 6.9, using the fact that the co* *mplexity of oe-1 O f is the same as that of f. We define j1,...,jkas in the proof of Theo* *rem 6.9, but we must verify that if the complexities of e|e-1(~_i)and f in diagram (6.2)are * * n then the complexity of Ø`O e`will be also be n. For this we need to show that for each* * two-element subset A of ~_1 . . .~_kthe complexity of Ø O e|e-1(A)will be n; but this is * *true when A is contained in some ~_i(because the complexity of e|e-1(~_i)is n) and it is * *also true if A is * * __ not contained in any ~_i(because the complexity of f is n). * * |__| Remark 8.6. If A is any symmetric monoidal category satisfying the hypothesis o* *f Remark 6.10 then Theorem 8.5 and its proof are valid for the category of augmented cos* *implicial objects in A, with Cartesian products in Top replaced by the symmetric monoidal* * product in A. Remark 8.7. In the special case n = 1, the functor-operad 1 is closely related* * to the monoidal product defined in Section 2. First observe that order-preserving ma* *ps f : T ! ~k have filtration 1 and that every map of filtration 1 can be written uniquely as* * the composite of an order-preserving map and a permuation of ~k. If we use order-preserving * *maps in Definitions 8.3 and 8.4 instead of maps of filtration 1, we get a nonsymmetric * *strict functor- operad which is related to both and 1: (a) The restriction of k to the category of (unaugmented) cosimplicial spac* *es is naturally isomorphic to the iterated -product k. (b) 1kis naturally isomorphic to a k O oe# ff2 k That is, 1 is obtained by extending the nonsymmetric functor-operad in the o* *bvious way to a (symmetric) functor-operad. 20 n 9 An operad which acts on Tot of a -algebra. Applying Proposition 4.4 with F = n and A = o we get an operad Dn with k-th s* *pace Dn(k) = Tot( nk( o, . .,. o)). Theorem 9.1. (a) Dn is weakly equivalent in the category of operads to Cn. (b) If Xo is an algebra over n then Dn acts on Tot(Xo). The statement of part (a) means that there is a chain of operads and weak eq* *uivalences of operads Dn . .!.Cn Part (b) of the Theorem is immediate from Propositions 4.6 and 4.8. Part (a) wi* *ll be proved in Section 14. Remark 9.2. The analog of Theorem 9.1(b) is valid, with the same proof, when Xo* * is an augmented cosimplicial spectrum. In the remainder of this section we give an explicit characterization of n-* *algebras, anal- ogous to that given in Section 7 for commutative -monoids. Definition 9.3. Let Xo be an augmented cosimplicial space. An n-structure on Xo* * consists of a map -1(1) f-1(k) T : Xf x . .x.X ! X for each totally ordered set T , each k 0, and each f : T ! ~kwith complexity* * n. Definition 9.4. An n-structure on Xo is consistent if, for every commutative di* *agram _____h____//0 T == T == """" f==OEOE=g""""" k~ in which f and g have complexity n, the diagram Qk f-1(i)_//_T i=1X X Q(h | || i)*| h*| Q fflffl| fflffl| k g-1(i)//_T0 i=1X X commutes, where hiis the restriction of h to f-1(i). It's easy to check (using the fact that Lan is left adjoint to *) that a c* *onsistent n- structure on Xo is the same thing as a collection of maps nk(Xo, . .,.Xo) ! Xo, one for each k 0. It remains to translate the rest of the definition of n-al* *gebra into this language. 21 Definition 9.5. An n-structure on Xo is commutative if, for each f with complex* *ity n and each oe 2 k, the diagram Q k f-1(i)______//_T i=1X X| s|| |=| Q fflffl| -1 fflffl| k f-1(ff(i))//_T i=1X X commutes (where s is the evident permutation of the factors). For the next definition we need some notation. Suppose we are given a partia* *lly ordered set T , numbers k, j1, . .,.jk 0, and maps f : T ! ~k and gi: f-1(i) ! ___i P for 1 i k. Let j = ji. The maps gidetermine a map g : T ! ~_ in an evident way; the formula for g is X g(a) = gi(a) + ji0 if a 2 f-1(i) i0//_Qk f-1(i) i=1 b=1X i i=1X = || || Q fflffl| fflffl| j g-1(c)_______//T c=1X X In_order_to state the unitality condition we need some more notation. If i * *2 ~klet ~i: k - 1! ~kbe the order-preserving monomorphism whose image does not contain * *i. Definition 9.7. An n-structure on Xo is unital if there is an element " 2 X; wi* *th the following property: <~iO f>(x1, . .,.xi-1, ", xi, . .,.xk-1) = (x1, . .,.xi-1, xi, . .* *,.xk-1) _____ for all f : T ! k - 1with complexity n, all i 2 ~k, and all choices of x1, . * *.,.xk-1. Proposition 9.8. A n-algebra structure on Xo determines, and is determined by,* * an n- structure on Xo which is consistent, commutative, associative and unital. The proof is a routine verification using the definitions in Section 6. 22 10 Example: the cosimplicial space associated to a nonsymmetric operad with multiplication. Let us say that an augmented cosimplicial space Xo is reduced if X; is a point. In this section we specialize to the case n = 2. We show that a 2 structur* *e on a reduced augmented cosimplicial space is the same thing as a ön nsymmetric opera* *d with multiplication." First recall [18, Definition II.1.14] that the definition of nonsymmetric op* *erad is obtained from the usual definition of operad by deleting all references to symmetric gro* *ups. Let Ass be the nonsymmetric operad whose k-th space is a point for all k 0. The next definition is due to Gerstenhaber and Voronov [9]. Definition 10.1. A nonsymmetric operad with multiplication is a nonsymmetric op* *erad O together with a morphism Ass ! O. Remark 10.2. i) It is easy to check that a morphism Ass ! O is the same thing a* *s a pair of elements ~ 2 O(2), e 2 O(0) satisfying fl(~, ~, id) = fl(~, id, ~) and fl(~, e, id) = fl(~, id, e) = id where fl is the composition operation and id is the identity element of the non* *symmetric operad O. ii) An important example of a nonsymmetric operad with multiplication (in th* *e category of abelian groups) is O(k) = Hom Z(A k, A) where A is an associative ring. Here the composition operation is the obvious o* *ne, id2 O(1) is the identity map of the ring A, ~ 2 O(2) is the multiplication of the ring A* *, and e 2 O(0) is the identity element of the ring A. This example is related to the Hochschi* *ld cochain complex. The rest of this section is devoted to the proof of: Proposition 10.3. A 2 structure on a reduced augmented cosimplicial space Xo d* *eter- mines, and is determined by, a structure of nonsymmetric operad with multiplica* *tion on the sequence of spaces Xk, k 0. First suppose that Xo is a reduced augmented 2-algebra; we need to define t* *he compo- sition operation fl, the identity element id2 X1, and elements ~ 2 X2 and e 2 X* *0. For the composition operation, first note`that`if U1, U2, . .,.Um are finite* * totally ordered sets there is a unique total order on U1 . . . Um for which the inclusion map* *s into the 23 coproduct are order-preserving and every element of Ui is less than every eleme* *nt of Uj for i < j. Now let k, j1, . .,.jk 0 be given, and let T be the totally ordered set a a a a a a {0} [j1] {1} [j2] . . . [jk] {k} _____ Let f : T ! k + 1 be the map that takes each set {i} to 1 and each [ji] to i + * *1; thus f-1(1) = [k] and f-1(i) = [ji-1] for i > 1. Let ~ be the equivalence relation o* *n T generated by: x ~ y if x is adjacent to y in the total order and f(x) 6= f(y). Let S be t* *he quotient T= ~. S inherits a total order from T and has j1 + . .+.jk + 1 elements. Let * *h be the composite T ! S ! [j1 + . .+.jk], where the first map is the quotient map and the second is the unique order-pres* *erving bijection. Finally, let fl : Xk x Xj1x . .X.jk! Xj1+...+jk be the composite h* O . Next let " be the unique element of X; and define the elements e, idand ~ by* * e = d0", id= d0e, and ~ = d0id. It is easy to check that these definitions give the sequence of spaces Xk, k* * 0 a structure of nonsymmetric operad with multiplication. In the other direction, suppose that O is a nonsymmetric operad with multipl* *ication. We begin by recalling a standard piece of notation: if x 2 O(k), y 2 O(j) and 1 * *i k, define x Oiy = fl(x, id, . .,.y, . .i.d) where the y is preceded by i - 1 copies of id. This gives a map Oi: O(k) x O(j) ! O(k + j - 1) which inherits associativity and unital properties from fl (see [18, page 46] f* *or details). We can now define the augmented cosimplicial space Oo associated to O. Let * *O; be a single point ". Define the cosimplicial structure maps by letting d0" = e and, * *if x 2 Op with p 0, 8 < ~ O2 x if i = 0 dix = x Oi~ if 0 < i < p + 1 : ~ O 1 x if i = p + 1 six= x Oi+1e. (This definition is motivated by the definition of the structure maps in the Ho* *chschild cochain complex.) It remains to define operations on Oo. For this we need some preliminary definitions. Define t : Op x Oq ! Op+q+1 by x t y = fl(~, x, d0y) 24 Remark 10.4. A function f : T ! ~kis the same thing as a finite sequence with v* *alues in ~k. If f : [p + q + 1] ! ~2is the sequence 1 . .1.2 . .2.with 1 repeated p + 1 time* *s and 2 repeated q + 1 times then will be defined to be t. Next define f~l: O(k) x O(j1) x . .x.O(jk) ! O(2k + j1 + . .+.jk) by ~fl(x, y1, . .,.yk) = fl(x, d0dj1+1y1, . .,.d0djk+1yk) _____ Remark 10.5. If f : [2k + j1 + . .+.jk] ! k + 1is the sequence 12 . .2.13 . .3.1 . .1.k + 1 . .k.+ 1 1, with each i > 1 repeated ji-1times, then will be defined to be ~fl. Next note that if S is a finite totally ordered set there is a canonical hom* *eomorphism OS ~=O||S||, where ||S|| is the number of elements in S minus 1. Now let f : T ! ~k be a map of complexity 2. We will define by induction on ||T ||. First of all, if T is empty, then is defined to be the unique map O; x . .x.O; ! O; If T has a single element then is the canonical homeomorphism O; x . .x.OT x . .~.=OT Next define a segment of f to be a subset S of T such that f has the same va* *lue on the minimal and maximal elements of S, and define a maximal segment to be a segment* * which is not properly contained in any other segment. Let S1, . .,.Sr denote the maximal* * segments of f; then T is the union of the Sj, and the fact that f has complexity 2 imp* *lies that the Sj are disjoint. Also note that each f-1(i) is contained in some Sj. If r > 1 (that is, if f has more than one maximal segment), let g1, . .,.gr * *be the restric- tions of f to S1, . .,.Sr and define to be the composite Yk Yr Y Qr -1(i) g-1(i) j=1S Sr t T Of ~= O j -----! O 1x . .x.O -! O i=1 j=1i2gj(Sj) If r = 1 let j be the value of f at the minimum and maximum elements of T an* *d let t0, . .,.t||f-1(j)||be the elements of f-1(j) in increasing order. For each l f* *rom 1 to ||f-1(j)|| let Ulbe the set {tl-1< t < tl}; then T is the disjoint union of f-1(j) and the* * sets Ul, and each f-1(i) is contained in one of the pieces of this disjoint union. Let glbe * *the restriction of f to Ul. We define to be the composite Yk Y Y Q Y -1(i) f-1(j) g-1(i)1x lf-1(j) U ~flT Of ~=O x O l -----! O x O l-! O i=1 l i2gl(Ul) l Now it is straightforward, although tedious, to check that the operations that we have defined satisfy the hypothesis of Proposition 9.8 with n = 2. 25 n 11 Example: Y One of the basic properties of the little n-cubes operad Cn is that it acts nat* *urally on n-fold loop spaces. In this section we show that the operad Dn has the same property: Proposition 11.1. Dn acts naturally on n-fold loop spaces. First we need to give a cosimplicial model for the n-th loop space of a poin* *ted space Y . Let Snobe the quotient of noby its (n-1)-skeleton; then Snois a pointed simpli* *cial set whose realization is the n-sphere. Let LoYbe the cosimplicial space Map *(Sno, Y ) (w* *hen n = 1, this is just the usual geometric cobar construction on Y ). Lemma 11.2. Tot(LoY) is homeomorphic to nY . Proof. This is easy from the definition of Tot, using the fact that Snohas only* * one non- * * __ degenerate simplex other than the basepoint. * * |__| It only remains to show that LoYis a n-algebra. First we observe that any s* *implicial set Zo has co-operations bfc : ZT ! Zf-1(1)x . .x.Zf-1(k) for all f : T ! ~k, given by bfc(x) = (h*1(x), . .,.h*k(x)) where hi : f-1(i) ! T is the inclusion. It is easy to check that these co-oper* *ations have properties dual to Definitions 9.4, 9.5, 9.6 and 9.7. Lemma 11.3. If f has complexity n then the map bfc : SnT! Snf-1(1)x . .x.Snf-1(k) factors through the wedge Snf-1(1)_ . ._.Snf-1(k) Proof. First observe that f can be thought of as a sequence of length |T | with* * values in ~k, and that (because f has complexity n) this sequence has no subsequence of len* *gth n + 2 of the form ijij . ... Next recall that an element of nTis a function OE : T ! [n], and that this * *element is in the (n - 1)-skeleton if and only if OE is not onto. Let OE : T ! [n] be onto. T* *he entries of bfc(OE) 2 nf-1(1)x . .x. nf-1(k) are the restrictions OE|f-1(i). If two of these restrictions (say when i = a a* *nd i = b) were onto, then the sequence corresponding to f would have a subsequence aba . .o.r * *bab . .o.f length n + 2, which is impossible. Thus all but one of the entries of bfc(OE) m* *ust be in the * * __ (n - 1)-skeleton of no, which proves the lemma. * * |__| Using Lemma 11.3, we see that any f of complexity n induces a map Y f-1(i) ` : LY = Map *( Snf-1(i), Y ) ! Map *(SnT, Y ) = LTY These maps satisfy Definitions 9.4, 9.5, 9.6 and 9.7 because the bfc's satisfy * *the duals of those definitions. This completes the proof of the Proposition. 26 12 The structure of k( o, . . ., o). Throughout this section we write Ykofor the augmented cosimplicial space k( o,* * . .,. o) and YkSfor the value of Ykoat the finite totally ordered set S. We want to inve* *stigate the structure of YkSand Yko. Recall (equation (6.1)) that YkS is colimA1 G1, where A1 is the category def* *ined just before equation (6.1)and G1 is the functor which takes the diagram ~koof_T __h_//S Q -1 to 1 i k f (i). Notation 12.1. A diagram of the form ~kofoT__h__//S, where h is ordered, will be denoted from now on by (f, T, h). * *Q -1 The elements of YkSare equivalence classes of pairs ((f, T, h), u)Pwith u 2 * * i f (i); we think of u as a tuple indexed by T , subject to the condition that a2f-1(i)ua* * = 1 for each i 2 ~k. Note that f must be surjective because ; = ;. Definition 12.2. (a) A diagram (f, T, h) is nondegenerate if T = [m] for some * *m, f is surjective, and for each j < m either f(j)Q6= f(j + 1) or h(j) 6= h(j + 1). -1(i) (b) A pair ((f, T, h), u) with u 2 i f is nondegenerate if (f, T, h) is* * nondegenerate and ua 6= 0 for all a 2 T . Proposition 12.3. Each point in YkSis represented by a unique nondegenerate pai* *r. Corollary 12.4. YkS is a CW complex with one cell of dimension m + 1 - k for ea* *ch nondegenerate (f, [m],Qh); the characteristic map of the cell corresponding to * *(f, [m], h) is -1(i) S a homeomorphism from f to the closure of the cell (and thus Yk is a regu* *lar_CW complex). * *|__| Proof of 12.3. We define a function from pairs to pairs as follows. Given a* * pair ((f, T, h), u), let ~ be the equivalence relation on T generated by a ~ b if a is adjacent tofb,(a) = f(b), and h(a) = h(b) and let T1 be the subset {a 2 T | ua 6= 0}. Let T^ be T1= ~. Then f and h induce maps f^: T^ ! ~kand ^h: T^ ! S. Also, let Q ^-1 P ß : T1 ! ^Tbe the projection and define ^u2 i f (i)by ^uc= a2i-1(c)ua for* * c 2 ^T. Finally, let m = |T^| - 1 and let g : [m] ! ^Tbe the unique ordered bijection. * * We define ((f, T, h), u) = ((f^O g, [m], ^hO g), ^uO g). The proposition is immediate fr* *om the following properties of : 27 (i) ((f, T, h), u) is nondegenerate. (ii) ((f, T, h), u) and ((f, T, h), u) represent the same point in YkS. (iii) If ((f, T, h), u) and ((f0, T 0, h0), u0) represent the same point in * *YkSthen ((f, T, h), u) = ((f0, T 0, h0), u0) * * __ * *|__| Our next goal is to show for each k that Ykois isomorphic as an augmented co* *simplicial space to o x Yk0(this is the analog for of Lemma 3.6). Definition 12.5. For each S, jS is the unique map S ! [0]. This will also be de* *noted by j when S is clear from the context. Definition 12.6. Define !S : YkS! S x Yk0 by letting the projection on the second factor be the map j* induced by j and l* *etting the projection onPthe first factor take the equivalence class of ((f, T, h), u) to * *the element v 2 S with va = 1_k b2h-1(a)ub. The !S fit together to give a cosimplicial map ! : Yko! o x Yk0 Proposition 12.7. ! is an isomorphism of augmented cosimplicial spaces. Proof. The diagram S YkSA_____!____//_ S x Yk0 AAA vvvv ''*A__AA vvi2v zzvv Yk0 commutes, where ß2 is the projection. We begin by showing (1) for each point y 2 Yk0the map j-1*(y) ! ß-12(y) induced by !S is a bije* *ction. For this, it suffices to show S i1 (2) the composite j-1*(y) -!-! S x Yk0-! S is a bijection. So let y be a point of Yk0and let ((f, [m], j[m]), u) be the nondegenerate pair which represents it. We will define an inverse ~ : S ! j-1*(y) 28 of ß1 O !S as follows. Let v 2 S. For each j 2 [m], let aj 2 S be the smallest* * element for which X 1 Xj va __ ui a aj k i=0 Define a totally ordered set T by adjoining to S an immediate successor of aj, * *denoted ~aj, for each j. Define g : T ! [m] by g(b) = j if ~aj-1 b aj. Define f0 : T ! ~k* *to be f O g. Define h : T ! S by h(b) = b if b 2 S and h(~aj) = aj. Define 8 P P >< ji=0ui- k a: a aj i=0 kvb otherwise We define ~(v) to be the point represented by the pair ((f0, T, h), u0). It is * *easy to check that this point is in j-1*(y) (this amounts to showing that ((f0, T, jT), u0) = ((f* *, [m], j[m]), u)) and that ~ is an inverse of ß1 O !S; this completes the proof of (2). Next let e be a cell of Yk0and let ~ebe its closure. Then j-1*(~e) is a fini* *te union of closed cells of YkS, and in particular it is compact. This together with (1) implies t* *hat !S induces a homeomorphism j-1*(~e) ! ß-12(~e) Since the closure of each cell of S x Yk0is contained in a set of the form ß-1* *2(~e), it follows that (!S)-1 is continuous on the closure of each cell of S x Yk0, and from thi* *s it follows * * __ that !S is a homeomorphism. * * |__| We can now complete the proof of Theorem 7.1(a) by showing: Corollary 12.8. Yk0is contractible for each k 0. Proof. First observe that if A is a space then k(Xo1, . .,.Xoi, . .,.Xok) x A ~= k(Xo1, . .,.Xoix A, . .,.Xok) (because xA preserves colimits). Thus we have Yk0x Yj0 = k( o, . .,. o)0 x Yj0 k( o, . .,. o x Yj0, . .,. o)0 k( o, . .,. j( o, . .)., . .).0 by Proposition 5.3 k+j-1( o, . .,. o)0 by Theorem 6.9 = Yk0+j-1 It therefore suffices to prove the corollary when k = 2. Y20has the special pr* *operty that the (n - 1)-skeleton is contained in the closure of either of the two n-cells. * *Corollary 12.4 implies that the closure of a cell is contractible, so the inclusion of the (n * *- 1)-skeleton is nullhomotopic for each n. Thus any map from a sphere into Y20is nullhomotopic, * *so Y20is * * __ contractible. * * |__| 29 13 The structure of nk( o, . . ., o). For use in Section 14, we prove the analogs for nkof the results of Section 12. Fix n and denote the augmented cosimplicial space nk( o, . .,. o) by Zok; t* *hus Dn(k) = Tot(Zok).QRecall Notation 12.1. The elements of ZSkare equivalence classes of p* *airs ((f, T, h), u) -1(i) with u 2 i f , where f has complexity n. We define nondegenerate pairs exactly as in Definition 12.2, and the proof o* *f Proposition 12.3 goes through to show * * __ Proposition 13.1. Each point in ZSkis represented by a unique nondegenerate pai* *r. |__| Corollary 13.2. ZSkis a CW complex with one cell of dimension m + 1 - k for each nondegenerate (f, [m], h) for which f has complexity n; theQcharacteristic ma* *p of the -1(i) cell corresponding to (f, [m], h) is a homeomorphism from f to the closur* *e of_the cell. * * |__| Proposition 13.1 also gives a useful relationship between Zokand the augment* *ed cosim- plicial space Ykodefined in Section 12: Corollary 13.3. The map ZSk! YkSis a monomorphism for all S, and the diagram ZSk____//YkS | | | | fflffl| fflffl| Z0k____//_Yk0 * * __ is a pullback. * * |__| Next we define ! : Zok! o x Z0k as in Section 12:Pthe projection of !S on S takes the equivalence class of ((f* *, T, h), u) to v, where va = 1_k b2h-1(a)ub, and the projection of !S on Z0kis j* (see Notatio* *n 12.5). The diagram Zok_!__// o x Z0k | | | | fflffl| fflffl| Yko_!__// o x Yk0 commutes, and this together with Corollary 13.3 implies Proposition 13.4. ! : Zok! ox Z0kis an isomorphism of augmented cosimplicial s* *paces. 30 14 Proof of Theorem 9.1(a). In this section we prove Theorem 9.1(a). As motivation for the method, recall * *that one way to show that two spaces are weakly equivalent is to show that they have con* *tractible open covers with the same nerve, or more generally to show that they can be dec* *omposed into homotopy colimits of contractible pieces over the same indexing category. * *We will show that the cosimplicial space nk( o, . .,. o) can be decomposed as a homotopy co* *limit of contractible cosimplicial spaces indexed over a certain category Knkconsidered * *by Berger [5]; Berger has shown that Cn(k) is a homotopy colimit of contractible pieces indexe* *d by Knk, and from this we will deduce Theorem 9.1(a). We begin by recalling some definitions from [5] (but our notation differs so* *mewhat from that in [5]). Definition 14.1. For each k 0, let P2~kbe the set of subsets of ~kthat have t* *wo elements. Definition 14.2. (a) Let Kk be the set whose elements are pairs (b, T ), where * *b is a function from P2~kto the nonnegative integers and T is a total ordering of ~k. We give* * Kk the partial order for which (a, S) (b, T ) if a({i, j}) b({i, j}) for each {i, * *j} 2 P2~kand a({i, j}) < b({i, j}) for each {i, j} with i < j in the order S but i > j in th* *e order T . Let Knk be the subset of pairs (b, T ) such that b{i, j} < n for each {i, j} 2 P2~k. Th* *e set Knkinherits an order from Kk. (b) Let K denote the collection of partially ordered sets Kk, k 0, and let* * Kn denote the collection of partially ordered sets Knk, k 0. It is shown in [5] that K is an operad in the category of partially ordered * *sets with the following structure maps. The right action of k on Kk is given by (b, T )æ = (b O æ2, T æ) where æ2: P2~k! P2~kis the function æ2({i, j}) = {æ(i), æ(j)} and where i < j i* *n the total order T æ if æ(i) < æ(j) in the total order T . The operad composition Kk x Ka1x . .x.Kak! K ai takes ((b, T ); (b1, T1), . .,.(bk, Tk)) to the pair (b(b1, . .,.bk), T (T1, . * *.,.Tk)), where b(b1, . .,.bk) is the function which takes {r, s} to æ __ bi({r, s})if {r, s} ai b({i, j})if r 2 __ai, s 2 __ajand i 6= j ` __ and T (T1, . .,.Tk) is the total order of aifor which r < s if either r < s i* *n the order Tior r 2 __ai, s 2 __ajand i < j. Note that, for each n, Kn is a suboperad of K. Let us write N for the functor that takes a partially ordered set to the geo* *metric realization of its nerve. Then N Kn is an operad of spaces and Berger shows ([5, Theorem 1.* *16]) that it is weakly equivalent to the little n-cubes operad Cn. To complete the proof * *of Theorem 9.1(a) it therefore suffices to show that N Kn is weakly equivalent to Dn. We * *will do this by finding a homotopy colimit decomposition of the functor-operad n; first we * *need some definitions. 31 Definition 14.3. Let f :[m] ! ~k. Then (bf, Tf) 2 Kk is the pair where bf({i, j* *}) is one less than the complexity of the restriction of f to a map f-1({i, j}) ! {i, j} and w* *here i < j in the total order Tf if the smallest element of f-1(i) is less than the smalle* *st element of f-1(j). Recall the definition of the category Qk (Definition 6.3). Definition 14.4. For each pair (b, T ) 2 Kk let Q(b,T)be the full subcategory o* *f Qk whose objects are the maps f with (bf, Tf) (b, T ). Let '(b,T): Q(b,T)! Qk be the inclusion functor. The next definition uses the notation of Definition 6.4. Definition 14.5. Let Xo1, . .,.Xokbe augmented cosimplicial spaces. (a) For each (b, T ) 2 Kk, define (b,T)(Xo1, . .,.Xok) to be the Kan extens* *ion Lan ((Xo1~x. .~.xXok) O O '(b,T)) (b) Define n(Xo1, . .,.Xok) to be hocolimKnk (b,T)(Xo1, . .,.Xok) Lemma 14.6. n is a functor-operad. Proof. First note that the natural transformations defining the functor-operad * * restrict to natural transformations (14.1) oe* : (b,T)! (b,T)ffO oe# for oe 2 k and (14.2) : (b,T)( (b1,T1), . .,. (bk,Tk)) ! (b(b1,...,bk),T(T1,...* *,Tk)) Next recall the definition of hocolim given in [13, Section 19.1]: if A is a* * category and F : A ! Top is a functor then hocolimA F = F A U where A denotes the coend and U is the contravariant functor A ! Top which tak* *es an object a 2 A to N(a # A). Now let oe 2 k and observe that oe induces a functor oe# : ((b, T ) # Knk) ! ((b, T )oe # Knk) We define oe* : nk(Xo1, . .,.Xok) ! nk(Xoff(1), . .,.Xoff(k)) 32 to be the map induced by the collection of maps ff*xN(ff#) o o * * n (b,T)(Xo1, . .,.Xok) x N((b, T ) # Knk) ------! (b,T)ff(Xff(1), . .,.Xff(k))* * x N((b, T )oe # Kk) Finally, we define the structural map : nk( nj1, . .,. njk) ! nj1+...+jk to be the map induced by the collection of maps Yk ~ (b,T)( (b1,T1), . .,. (bk,Tk)) x N((b, T ) # Knk) x N((bi, Ti) # Knji) -=! i=1 i Yk * * j xN(fl#) (b,T)( (b1,T1), . .,. (bk,Tk)) x N ((b, T ) # Knk) x ((bi, Ti) # Knji* *) -----! i=1 (b(b1,...,bk),T(T1,...,Tk))x N((b(b1, . .,.bk), T (T1, .* * .,.Tk)) # Knj1+...+jk) where fl# is induced by the composition map Yk fl : Knkx Knji! Knj1+...+jk i=1 * * __ of the Cat-operad Kn. * * |__| Now let Bn be the operad obtained by applying Proposition 4.4 with F = n an* *d A = o. To complete the proof of Theorem 9.1(a) it remains to show: Lemma 14.7. (a) There is a weak equivalence of operads Bn ! Dn (b) There is a weak equivalence of operads Bn ! N Kn For the proof of part (a), we first observe that Qnkis the union of Q(b,T)fo* *r (b, T ) 2 Knk; it follows that nk(Xo1, . .,.Xok) = colimKnk (b,T)(Xo1, . .,.Xok) for all Xo1, . .,.Xok. The projection from hocolim to colim gives a map of func* *tor-operads n ! n and an induced map of the associated operads: OE : Bn ! Dn 33 We need to show that for each k 0 the map OE(k) : Bn(k) ! Dn(k) is a weak equivalence of spaces. Recall from Proposition 13.4 that there is an * *isomorphism of cosimplicial spaces nk( o, . .,. o) ~= o x nk( o, . .,. o)0 The proof of Proposition 13.4 shows that for each (b, T ) we have an isomorphis* *m of cosim- plicial spaces (b,T)( o, . .,. o) ~= o x (b,T)( o, . .,. o)0 It follows that we have homeomorphisms Dn(k) Tot( o) x colimKnk (b,T)( o, . .,. o)0 and (14.3) Bn(k) Tot( o) x hocolimKnk (b,T)( o, . .,. o)0 Thus it suffices to show that the map hocolimKnk (b,T)( o, . .,. o)0 ! colimKnk (b,T)( o, . .,. o)0 is a weak equivalence, and this follows from a standard fact about homotopy col* *imits [13, Theorem 19.9.1]; the "Reedy cofibrancy" condition needed for [13, Theorem 19.9.* *1] is satis- fied in our case because the map [ (b0,T0)( o, . .,. o)0 ! (b,T)( o, . .,. o)0 (b0,T0)<(b,T) is the inclusion of a sub-CW-complex (cf. Corollary 13.2). Next we prove part (b). Consider the map _(k) : Bn(k) = Hom ( o, nk( o, . .,. o)) ! Hom ( o, nk(*, . .,.*)) = N* * Knk where the arrow is induced by the projection o ! * and the second equality fol* *lows from the fact that nk(*, . .,.*) is the constant cosimplicial space with value N Kn* *k. It is easy to check that the collection {_(k)} is an operad map; it remains to show that each* * _(k) is a weak equivalence. Using equation (14.3)and the fact that Tot( o) is contractibl* *e it suffices to show that the map hocolimKnk (b,T)( o, . .,. o)0 ! hocolimKnk (b,T)(*, . .,.*)0 = hocolim* *Knk* (where the arrow is induced by o ! *) is a weak equivalence; and this is a con* *sequence of [13, Remark 18.5.4] and the following lemma. 34 Lemma 14.8. For each choice of b and T the space (b,T)( o, . .,. o)0 is weakly* * equivalent to a point. Proof. The proof is by induction on k. Since the map (14.1)is an isomorphism we may assume that T is the standard t* *otal order on ~k. For each f : [m] ! ~kwe define c(f) : [m + 1] ! ~kto be the function which t* *akes 1 to 1 and p to f(p - 1) if p > 1. This construction gives a functor, also called c,* * from Q(b,T)to itself. Next let C : ! be the functor which takes [m] to [m + 1] and takes a mor* *phism h : [m] ! [n] to the morphism C(h) : [m + 1] ! [n + 1] defined by æ 0 if p = 0 C(h)(p) = h(p - 1) + 1 if p > 0 We can define a map ff : (b,T)( o, . .,. o)0 ! (b,T)( o O C, o, . .,. o)0 -1(i) as follows: if f : [p] ! ~kand ui2 f for 1 i k, let ff take the equival* *ence class of (f, u1, . .,.un) to that of (f, d0u1, u2, . .,.uk). We can also define a map fi : (b,T)( o O C, o, . .,. o)0 ! (b,T)( o, . .,. o)0 by letting fi take the equivalence class of (f, u1, . .,.un) to that of (cf, u1* *, . .,.uk). It is easy to check that ff and fi are well-defined and that fi O ff is the identi* *ty; that is, (b,T)( o, . .,. o)0 is a retract of (b,T)( o O C, o, . .,. o)0. It therefore* * suffices to show that the latter is weakly equivalent to a point. But o O C is isomorphic to the degreewise cone on o, and in particular the* *re is a homotopy equivalence of cosimplicial spaces from o O C to a point, so we have (b,T)( o O C, o, . .,. o)0 ' (b,T)(*, o, . .,. o)0 and an inspection of Definition 14.5 shows that (b,T)(*, o, . .,. o) ~= (b0,T0)( o, . .,. o) where b0is the restriction of b to P2(~k- {1}) and T 0is the restriction of T t* *o ~k- {1}. The inductive hypothesis shows that (b0,T0)( o, . .,. o)0 is weakly equivalent to * *a point, and * * __ this concludes the proof. * * |__| 15 A homotopy-invariant version of Tot. First recall ([13, Theorem 11.6.1]) that there is a model category structure fo* *r cosimplicial spaces in which the weak equivalences are the degreewise weak equivalences and * *the fibrations are the degreewise fibrations. In particular, every object is fibrant. Let ~ ob* *e any cofibrant resolution of o with respect to this model structure. 35 Definition 15.1. Let Xo be a cosimplicial space. gTot(Xo) is defined to be Hom * *( ~o, Xo). Since every cosimplicial space is fibrant, a weak equivalence Xo ! Y oalways* * induces a weak equivalence gTot(Xo) ! gTot(Y o). Definition 15.2. Let ~Dnbe the operad obtained by applying Proposition 4.4 with* * F = n and A = ~ o Our goal in this section is to prove the analog of Theorem 9.1. Theorem 15.3. (a) ~Dnis weakly equivalent in the category of operads to Cn. (b) If Xo is an algebra over n then ~Dnacts on gTot(Xo). Remark 15.4. Note that this theorem includes the analog of Theorem 7.1 as a spe* *cial case. Moreover, the proof we will give can easily be modified to prove the analog of * *Proposition 3.5. Before beginning the proof we give some background information which is of i* *nterest in its own right. We begin with a more explicit description of nk( ~o, . .,.~ o). First obser* *ve that ~ ois a Reedy-cofibrant cosimplicial space by [13, Proposition 15.6.3(2)]. It follows t* *hat ~ ox . .x.~ o is a Reedy-cofibrant multicosimplicial space. Next observe that we can extend t* *he definition of nkto k-fold multicosimplicial spaces Y o,...,oby replacing Xo1~x. .~.xXokin* * Definition 8.4 by Y o,...,o. Now fix a finite totally ordered set S. For each m k - 1 let * *Im be the set of nondegenerate diagrams (f, [m], h) where f has complexity n (see Notation * *12.1 and Definition 12.2(a); note that there cannot be any nondegenerate (f, [m], h) if * *m < k - 1). Recall the definition of latching object ([13, Definition 15.2.5]). Lemma 15.5. Let Y o,...,obe a Reedy-cofibrant k-fold multicosimplicial space. D* *efine a Vk-1 = Y 0,...,0 (f,[m],h)2Ik-1 and define Vm inductively for m k by the pushout diagram ` Lf __________//Vm-1 | | | | ` fflffl| fflffl| -1(1),...,f-1(k)//_ Y f Vm where the coproducts are taken over (f, [m], h) 2 Im and Lf is the latching obj* *ect Lf-1(1),...,f-1(k)(Y o,...,o). Then each Vm-1 ! Vm is a cofibration, and [ nk(Y o,...,o)S = Vm . m k-1 36 * * __ Proof. This follows by the proof of Corollary 13.2. * * |__| Next we give a homotopy-invariance property for nk. Lemma 15.6. Let Y o,...,o! Zo,...,obe a weak equivalence of Reedy-cofibrant k-f* *old multi- cosimplicial spaces. Then the induced map nk(Y o,...,o) ! nk(Zo,...,o) is a weak equivalence. * * __ Proof. This is an easy consequence of Lemma 15.5. * * |__| Remark 15.7. The analog of Lemma 15.6 for the functors (b,T)defined in Section* * 14 is also true, with the same proof. Now we turn to the proof of Theorem 15.3. Part (b) is immediate from Proposi* *tion 4.6. For part (a), recall the functor-operad n and the operad N Kn defined in Secti* *on 14. Let B~nbe the operad obtained by applying Proposition 4.4 with F = n and A = ~ o. * *It suffices to show Lemma 15.8. (i) There is a weak equivalence of operads ~Bn! D~n (ii) There is a weak equivalence of operads ~Bn! N Kn Proof of Lemma 15.8. For part (i), note that the projection from hocolim to co* *lim gives a map of functor-operads n ! n and an induced map of the associated operads: ~OE: ~Bn! ~Dn We need to show that for each k 0 the map ~OE(k) : ~Bn(k) ! ~Dn(k) is a weak equivalence of spaces. Because gTotis homotopy-invariant, it suffices* * to show that the map nk( ~o, . .,.~ o)S ! nk( ~o, . .,.~ o)S 37 is a weak equivalence for each S. Consider the commutative diagram nk( ~o, . .,.~_o)S//_ nk( ~o, . .,.~ o)S | | | | fflffl| fflffl| nk( o, . .,. o)S__// nk( o, . .,. o)S where the vertical maps are induced by the projection ~ o! o. We have shown in* * the proof of Lemma 14.7(a) that the lower horizontal map is a weak equivalence. The secon* *d vertical map is a weak equivalence by Lemma 15.6, and the first vertical map is a weak e* *quivalence by Remark 15.7 and passage to hocolim. This completes the proof of part (i). For part (ii), let D be the 0-th space of ~ o. If Co is any constant cosimpl* *icial space there is a canonical homeomorphism gTot(Co) Map (D, C0) Now nk(*, . .,.*) is the constant cosimplicial space with value N Knk, so we h* *ave gTot( nk(*, . .,.*)) Map (D, NKnk). Thus gTot( nk(*, . .,.*)) is an operad weakly equivalent to N Kn. It therefore * *suffices to show that the map ~Bn(k) = gTot( nk( ~o, . .,.~ o)) ! gTot( nk(*, . .,.*)) induced by the projection ~ o! * is a weak equivalence for each k. 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