HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY DAVID BLANC, MARK W. JOHNSON, AND JAMES M. TURNER Abstract. We explain how higher homotopy operations, defined topologicall* *y, may be identified under mild assumptions with (the last of) the Dwyer-Kan* *-Smith cohomological obstructions to rectifying homotopy-commutative diagrams. Introduction The first secondary homotopy operations to be defined were Toda brackets, whi* *ch appeared (in [T1 ]) in the early 1950's - at about the same time as the secon* *dary cohomology operations of Adem and Massey (in [Ad ] and [MU ]). The definition * *was later extended to higher order homotopy and cohomology operations (see [Sp , Ma* * , Kl]), which have been used extensively in algebraic topology, starting with Tod* *a's own calculations of the homotopy groups of spheres in [T2 ]. In [BM ], a "topological" definition of higher homotopy operations based on * *the W - construction of Boardman and Vogt, was given in the form of an obstruction theo* *ry for rectifying diagrams. The same definition may be used also for higher cohomo* *logy operations. This was recently modified in [BC ] to take account of the fact th* *at, in practice, higher order operations, both in homotopy and in cohomology, occur in* * a pointed context, which somewhat simplifies their definition and treatment. Earlier, in [DKSm2 ], Dwyer, Kan, and Smith gave an obstruction theory for * *rec- tifying a diagram "X : K ! ho T in the homotopy category of topological spac* *es by making it "infinitely-homotopy commutative": the precise statement involves * *the simplicial function complexes map (X"u, "Xv) for all u, v 2 O = Obj (K), w* *hich constitute an (S, O)-category CX (see x3.11 and Section 4). Their results* * are thus stated in terms of (S, O)-categories (simplicially enriched categories wit* *h object set O). In particular, the obstructions take values in the corresponding (S, * *O)- cohomology groups (see [DKSm1 , x2.1]). The purpose of the present note is to explain the relation between these two * *ap- proaches. Because the W -construction, and thus higher operations, are defined* * in terms of cubical sets, it is convenient to work cubically throughout. In this l* *anguage, (S, O)-cohomology is replaced by the (equivalent) (C, )-cohomology (see x2.25)* *, and our main result (Theorem 4.14 below) may be stated roughly as follows: Assume given a directed graph without loops (cf. x3.13) of length n + 2, * *having initial node vinitand terminal node vfin, and let M be a cubically enriched poi* *nted model category. ___________ Date: July 29, 2008; revised April 20, 2009. 1991 Mathematics Subject Classification. Primary: 55Q35; secondary: 55N99, 55* *S20, 18G55. Key words and phrases. Higher homotopy operations, SO-cohomology, homotopy-c* *ommutative diagram, rectification, obstruction. 1 2 D. BLANC, M.W. JOHNSON, AND J.M. TURNER Theorem A. For each pointed diagram "X: ! ho M, there is a natural pointed correspondence between the possible values of the final Dwyer-Kan-Smith obstr* *uc- tion to rectifying X", in the (C, )-cohomology group Hn( , ssn-1CX ), and t* *he n-th order homotopy operation <>, a subset of [ n-1X"(vinit), X"(vfin)]. 0.1. Remark. The fact that is pointed implies that, not surprisingly, the two* * different obstructions to rectification vanish simultaneously. Our objective here is to e* *xplicitly identify each value of a higher homotopy operation (with its usual indeterminac* *y) with a (C, )-cohomology class for . In [BB ], a relationship between (S, O)-cohomology and the cohomology of a - algebra is described. Since the latter is a purely algebraic concept, we hope t* *hat to- gether with the present result this will provide a systematic way to apply homo* *logical- algebraic methods to interpret and calculate higher homotopy and cohomology ope* *r- ations. 0.2. Notation. The category of compactly generated topological spaces is denote* *d by T , and that of pointed connected compactly generated spaces by T*; their homot* *opy categories are denoted by ho T and ho T*, respectively. The categories of (po* *inted) simplical sets will be denoted by S (resp., S*), those of groups, abelian gro* *ups, and groupoids by Gp, AbGp , and Gpd, respectively. Cat denotes the categor* *y of small categories. If is a monoidal category, we denote by V-Cat the collection of al* *l (not necessarily small) categories enriched over V (see [Bor2 , x6.2]). A category K* * is called pointed if it has a zero object 0 - that is, 0 is both initial and final. In * *such a K, a map factoring through 0 is called a null (or zero) map, and since there is a * *unique such map between any two objects, K is enriched over pointed sets. 0.3. Remark. It will be convenient at times to work with non-unital categories * * - that is, categories which need not have identity maps. These have been studied* * in the literature under various names, beginning with the semi-categories of V.V. * *Vagner (see [V ]). The enriched version appears, e.g., in [BBM ]. 0.4. Organization. Section 1 provides a review of cubical sets and their homoto* *py theory. Section 2 discusses cubically enriched categories, as a replacement fo* *r the (S, O)-categories of Dwyer and Kan, and describes their model category structure (Theorem 2.21). In Section 3 we give a "topological" definition of pointed hig* *her homotopy operations in terms of diagrams indexed by certain finite categories c* *alled lattices. Finally, in Section 4 the Dwyer-Kan-Smith obstruction theory is descr* *ibed and the main result (Theorem 4.14 and Corollary 4.15) is proved. 0.5. Acknowledgements. This research was supported by BSF grant 2006039; the third author was also supported by NSF grant DMS-0206647 and a Calvin Research Fellowship (SDG). 1.Cubical sets Even though the obstruction theory of Dwyer, Kan, and Smith was originally de- fined simplicially, for our purposes it appears more economical to work cubical* *ly. This is because cubical sets are the natural setting for the W -construction of Boar* *dman and Vogt, which was used for constructing higher homotopy operations in [BM ] * *and HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 3 [BC ]. Since our goal is to identify these operations with the cohomological ob* *struc- tions of Dwyer-Kan-Smith, we simplify the exposition by framing their theory in cubical terms as well. Because cubical homotopy theory is less familiar than th* *e sim- plicial version, and the relevant information and definitions are scattered thr* *oughout the literature, we summarize them here. 1.1. Definition. Let denote the Box category, whose objects are the abstract * *cubes {In}1n=0 (where I := {0, 1} and I0 is a single point). The morphisms of a* *re generated by the inclusions di": In-1 ! In and projections si : In ! In-1 * * for 1 i n and " 2 {0, 1}. One can identify with a category of topological cubes, where In correspo* *nds to [0, 1]n (an n-fold product of unit intervals), the linear map di": [0, 1]n-* *1 ! [0, 1]n is defined (t1, . .,.tn-1) 7! (t1, . .,.ti-1, ", ti, . .,.tn-1), and si: [0,* * 1]n ! [0, 1]n-1 is defined by omitting the i-th coordinate. A contravariant functor K : op ! Set is called a cubical set (or cubical com* *plex), and we write Kn for the set K(In) of n-cubes (or n-cells) of K. The (i, ")* *-face map d"i: Kn ! Kn-1 and the i-th degeneracy si: Kn-1 ! Kn are induced by di" and si, respectively. A cubical set K is called finite if all but finitely ma* *ny n-cubes of K are degenerate (that is, in the image of some si). The category of cubic* *al sets is denoted by C. See [KP , I,x5], [BH1 , x1], or [FRS ]. Several obvious constructions carry over from simplicial sets: for example, t* *he n- truncation functor on on cubical sets has a left adjoint, and composing the * *two yields the cubical n-skeleton functor skcn: C ! C. Thus skcnK is generated * *(under the degeneracies) by the k-cubes of K for k n. 1.2. Notation. There is a standard embedding of in C, in which In 2 is ta* *ken to the standard n-cube In 2 C (with one non-degenerate cell in dimension n, * *and all its faces). Applying skcn to the standard (n + 1)-cube In+1, we obtai* *n its boundary @In+1 := skcnIn+1. By omitting the d"i-face from @In+1, we obtain* * the (i, ")-square horn un,"i. 1.3. Remark. There is also a version of cubical sets without degeneracies, some* *times called semi-cubical sets, but these are not suitable for homotopy theoretic pur* *poses (cf. [An1 ]). On the other hand, Brown and Higgins have proposed adding further "adjacent degeneracies", called connections (see [BH1 , x1] and [GM ]). These* * have proved useful in various contexts (see, e.g., [An2 , BH2 ]). 1.4. The cubical enrichment of C. As a functor category, all limits and colimi* *ts in C are defined levelwise. In particular, the k-cubes of a given cubical set * * K 2 C (k 0) form a category CK (under inclusions), and K ~=colimIk2CK Ik. However, it turns out the products in C do not behave well with respect to re* *aliza- tion (see Remark 1.10 below), so another monoidal operation is needed: 1.5. Definition. If K and L are two cubical sets, their cubical tensor K L 2* * C is defined K L := colimIj2CK , Ik2CLIj+k . This defines a symmetric monoidal structure : C x C ! C on cubical sets (s* *ee [J3, x3]). 4 D. BLANC, M.W. JOHNSON, AND J.M. TURNER More generally, let be a monoidal category with (finite) colimits * *- for example, , , or - and assume we have "standard cub* *es" in V, defined by a (faithful) monoidal functor T : < , x > ! - that * *is, a compatible choice of "standard cubes" T In in V. Given a (finite) cubical se* *t K, for any X 2 V define X K := colimCK TX , where the diagram TX : CK ! V is defined by TX In := X T In. 1.6. Definition. For as above, the cubical mapping complex map cV(X, Y* * ) 2 C is defined for any X, Y 2 V by setting map cV(X, Y )n := Hom C(X T In, Y ) , with the cubical structure inherited from In 2 (cf. [K ]). We shall gener* *ally abbreviate map cV(X, Y ) to Vc(X, Y ). In particular, when V is C itself, this makes into a symmet* *ric monoidal closed category (see [Bor2 , x6.1]). 1.7. Comparison to S. Cubical sets are related to simplicial sets by a pair of * *adjoint functors T (1.8) CS S . cub The triangulation functor T is defined T K := colimIn2CK [1]n (compare Defini* *tion 1.5), where [1]n = [1] x . .x. [1] is the standard simplicial n-cube. The cu* *bical op n singular functor Scub : S ! C = Set is defined adjointly by (ScubX)(I ) := Hom S(T In, X). This is a singular-realization pair in the sense of [DK4 ]; c* *omposing (1.8) with the usual adjoint pair: |-| (1.9) S T S yields a similar adjunction to topological spaces. 1.10. Remark. Note that T : ! is strongly monoidal (cf. [Bor2 , * *x6.1]), in that there is a natural isomorphism (1.11) T (K L) ~= (T K) x (T L) . On the other hand, Scub : ! is not strongly monoidal, as we now show: as a right adjoint, Scub commutes with (levelwise) products up to natu* *ral isomorphism, so Scub(X x Y ) ~=Scub(X) x Scub(Y ) . Thus, if Scub were strongly monoidal, one would have a levelwise isomorphism Scub(X) Scub(Y ) ~=Scub(X) x Scub(Y ) . Note this is unlikely, since an n-cube of K L corresponds to a pair consi* *sting of a j-cube of K (for some 0 j n) and an (n - j)-cube of L, while an n-c* *ube of K x L corresponds to a pair consisting of an n-cube of K and an n-cube of L. * *In fact, K x L is in general not even homotopy equivalent to K L for K, L 2* * C, - for example, T (I1 x I1) ' S1 in S while T (I1 I1) ~= [1] x [1] (see * *[J1, x1, Remark 8]). HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 5 Nevertheless, since I0 is both terminal in C and the unit for , the proje* *ctions ssK : K L ! K I0 ~=K and ssL : K L ! I0 L ~=L induce a natural map (1.12) # : K L ! K x L , which is symmetric monoidal in the sense that it commutes with the obvious asso- ciativity and switch-map isomorphisms. 1.13. Fact ([J3, x3]). For any L 2 C, the functor - L preserves monomorphis* *ms in C. 1.14. Remark. Note that - In preserves colimits, since it has a right adjo* *int (defined by constructing the cubical set of maps between two cubical sets as one does in S - see [J3, x4]). Finally, observe that the cubical mapping complex * *for S (Definition 1.6) is simply map cS(-, -) = Scubmap S (-, -). 1.15. The model category. Cubical sets were used quite early on as models for topological spaces - se* *e [Se], [EM ], [Mu ], [Mc1 ], [P1 , P2], and especially [K1 , K2 ]. However, it was Gr* *othendieck,in [G ], who suggested that more generally presheaf categories modeled on certain * *"test categories" D can serve as models for the homotopy category of topological spac* *es. Cisinski, in his thesis [C ], carried out this program for D = (see also t* *he exposition in [J3]). The model catgeory structure is very similar to the analog* *ous one for simplicial sets (D = ): 1.16. Definition. A map f : K ! L in C is a) a weak equivalence if T f : T K ! T L is a weak equivalence in S (or equivalently, if |T f| is a weak equivalence of topological spaces); b) a cofibration if it is a monomorphism. c) a fibration if it has the right lifting property (RLP) with respect to al* *l acyclic cofibrations (i.e., those which are also weak equivalences) - that is, * *if in all commuting squares in C: g A _____//K>>_ "h_______ (1.17) i||_____|f|__ fflffl|fflffl|____ B __h__//L where i is an acyclic cofibration, a map "h: B ! K exists making the f* *ull diagram commute. The model category defined here is proper, by [J3, Theorem 8.2]. 1.18. Definition. The cubical spheres are Sn := Scub( [n]=@ [n]) for n 1, wi* *th the obvious basepoint. These corepresent the homotopy groups ssn(-) := [Sn, -* *]*. Similarly, S0 := I0 q {*} corepresents ss0, and a map f : K ! L in C* * * is a weak equivalence if and only if it induces a ssn-isomorphism for all n 0. Note that we may define the fundamental groupoid ^ss1K of an unpointed cubi* *cal set K 2 C as for simplicial sets or topological spaces (cf. [Hig , Chapter 2* *]). 6 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 1.19. Remark. In analogy with the case of simplicial sets (see [GJ , Ch. I]) o* *ne can show that cofibrations which are weak equivalences are the same as the anodyne * *maps - that is the closure of the set of inclusions of the form (1.20) i : un,"i,! In+1 (see x1.2) under cobase change, retracts, coproducts, and countable compositions (see [J3, x4]). Furthermore, the fibrant objects and the fibrations in C can a* *lso be characterized by Kan conditions - having the RLP with respect to maps of the fo* *rm (1.20) (see [K1 ] and [J3, Theorem 8.6]). As noted above (x1.6), C is a symmetric monoidal closed category (enriched ov* *er itself), with cubical mapping complexes Cc(-, -). As shown in [J1, x3]), it * *also satisfies the cubical analogue of Quillen's Axiom SM7 (cf. [Q , II, x2]), so C * *deserves to be called a cubical model category. In particular, if L is a fibrant (Kan) * *cubical set, the function complex Cc(K, L) is fibrant, too, for any (necessarily cof* *ibrant) K 2 C. Finally, the following result shows that C indeed serves as a model for the u* *sual homotopy category of topological spaces: 1.21. Proposition (Cf. [J3, Theorem 8.8]). The adjoint functors of (1.8) indu* *ce equivalences of homotopy categories hoC ~=ho S (so together with the pair (* *1.9), we have hoC ~=ho T ). Note that since I0 is a final object in C, the under category C* := I0=C * * of pointed cubical sets constitutes a pointed version of C, and we have: 1.22. Fact. There is a model category structure on C*, with the same weak equ* *iva- lences, fibrations, and cofibrations as C. Proof.See [Ho , Proposition 1.1.8]. 1.23. Spherical model categories. Like many other model categories, C* enjoys a collection of additional use* *ful properties that were axiomatized in [Bl, x1] under the name of a spherical model category. This means that: (a) C* has a set A of spherical objects: cofibrant homotopy cogroup objec* *ts (namely, the cubical spheres A = {Sn}1n=1 - Definition 1.18). Furthermor* *e, a map f : K ! L in C* is a weak equivalence if and only if [A, f] i* *s an isomorphism for all A 2 A. (b) Each K 2 C* has a functorial Postnikov tower of fibrations: p(n) p(n-1) (1.24) . .!.PnK --! Pn-1K ---! . .!.P0K , as well as a weak equivalence r : K ! P1 K := limnPnK and fibrations r(n): P1 K ! PnK such that r(n-1)= p(n)O r(n) for all n, and r(n)#: sskP1 K ! sskPnK is an isomorphism for k n and zero for k > n. (c) For every groupoid , there is a functorial classifying object B with B ' P1B and fundamental groupoid ^ss1B ~= , unique up to homotopy. (d) Given a groupoid and a -module G (that is, an abelian group object over ), for each n 2 there is a functorial extended G-Eilenberg-Mac Lane HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 7 object E = E (G, n) in C*=B , unique up to homotopy, equipped with a section s for (r(1)O r) : E ! P1E ' B , such that ssnE ~=G as -modul* *es and sskE = 0 for k 6= 0, 1, n. (e) For every n 1, there is a functor that assigns to each K 2 C* a homoto* *py pull-back square p(n+1) Pn+1K__________//_PnK (1.25) |||_PB_| |kn| fflffl| fflffl| B ______//E (M, n + 2) called an n-th k-invariant square for K, where := ^ss1K, M := ssn+1K, and p(n+1): Pn+1K ! PnK is the given fibration of the Postnikov tower. The map kn : PnK ! E (M, n + 2) is called the n-th (functorial) k-invaria* *nt for K. 1.26. Proposition. The category C* is spherical. Proof.All the properties for C* follow from Fact 1.22, and the analogous res* *ults for S* or T* (see [BJT , Theorem 3.15]). Note that homotopy groups for cubi* *cal sets appear in [K1 , K2 ], while (minimal, and thus non-functorial) Postnikov t* *owers for cubical sets were constructed by Postnikov in [P1 , P2]. For functorial cubical Postnikov towers, let the n-coskeleton functor coskcn* *: C ! C be the right adjoint to skcn, with r(n) : Id ! coskcn the obvious natur* *al transformation, and similarly for C*. By construction, r(n) is an isomorphi* *sm in dimensions n. If K 2 C* is fibrant, so is coskcnK, and ssicoskcnK = 0 * * for i > n, since skcnSi = * for i > n. Thus if K0 ! K is a functorial fib* *rant replacement, and we change (1.27) K0. .!.coskcn+1K0 ! coskcnK0 ! coskcn-1K0. . . functorially into a tower of fibrations, we obtain (1.24). For (strictly) functorial Eilenberg-Mac Lane objects, use [BDG , Prop. 2.2],* * and apply Scub. For functorial k-invariants in C*, use the construction in [BDG* * , x5-6] (which works in C*, too). 1.28. Remark. In general, the maps coskcnK ! coskcn-1K in (1.27) (adjoint to the inclusion of skeleta) are not fibrations (though the original constructi* *on of Kan, when applied to a fibrant cubical set K, yields a tower of fibrations with* * no further modification - see, e.g., [GJ , VI, x2]). However, if we are only in* *terested in a specific Postnikov section Pn, as long as K is fibrant we can use coskc* *n+1K as a fibrant model for PnK, and need only modify the next section if we want p(n+1): Pn+1K ! PnK to be a fibration. 2. Cubically enriched categories In [DK2 ], Dwyer and Kan showed how any model category (more generally, any small category M equipped with a class of weak equivalences) can be enriched by simplicial function complexes, so that the resulting simplicially enriched cate* *gory en- codes the homotopy theory of M (see Remark 3.10 below). Thus the category sCat 8 D. BLANC, M.W. JOHNSON, AND J.M. TURNER of simplicial small categories can be thought of as a "universal model category* *", pro- viding a setting for a "homotopy theory of homotopy theories". Other such unive* *rsal models were later provided in [DKSm2 , x7], [R ], and [Be ]. An important subcategory of sCat consists of those simplicial categories w* *ith a fixed set of objects. This is a special case of the following: 2.1. Definition. For any set O, denote by O-Cat the category of all small cate* *gories D with O := Obj D. More generally, assume 2 O-Cat is a small category, poss* *ibly non-unital, and let be a monoidal category. A (V, )-category is a ca* *tegory D 2 O-Cat enriched over V, with mapping objects map vD(-, -) 2 V, such that (2.2) Hom (u, v) = ; ) map vD(u, v) is the initial objectVin. Thus when V is pointed, we require map vD(u, v) = * whenever Hom (u, v) = ;. The category of all (V, )-categories will be denoted by (V, )-Cat. The * *mor- phisms in (V, )-Cat are enriched functors which are the identity on O. When Hom (u, v) is never empty (so that we may disregard condition (2.2)) * *we write (V, O)-Cat instead of (V, )-Cat. Dwyer and Kan call these O-diagrams* * in V. 2.3. Remark. If is non-unital, Hom (u, u) may be empty, in which case map * *vD(u, u) will be empty, if V = Set or S. This is allowed in the enriched version of * *semi- categories (see Remark 0.3). However, the discussion below can be readily carr* *ied out in the context of ordinary (enriched) categories, at the cost of paying att* *ention to units. Thus if V is pointed, Hom (u, u) has (at least) two maps: the ident* *ity and the zero map; these will coincide of u is the zero object. We shall in fact concentrate on the case where has no self-maps u ! u - * * e.g., a non-unital partially ordered set. The main examples of to keep in m* *ind are , , , , and . 2.4. (S, O)-categories. Although we shall be mainly concerned with (C, )-categories, we first recal* *l the more familiar simplicial version: Note that when V = S, an (S, )-category can be thought of as a simplicial ob* *ject over O-Cat (or (Set, )-Cat). Thus each Mo 2 (S, O)-Cat is a simplicial categ* *ory with fixed object set O in each dimension, and all face and degeneracy functors* * are the identity on objects (cf. [DK1 , x1.4]). 2.5. Fact. The forgetful functor U : Cat ! DiG to the category of directed g* *raphs has a left adjoint F : DiG ! Cat, the free category functor (cf. [Ha ]). 2.6. Definition. A simplicial category Eo 2 (S, O)-Cat is free if each categor* *y En, and each degeneracy functor sj : En ! En+1, is in the essential image of the f* *unctor F . The pair of adjoint functors of Fact 2.5 defines a comonad F U : Cat ! Cat, * * and thus for each small category D, an augmented simplicial category Eo ! D with En := (F U)n+1D. If D 2 (Set, )-Cat, then Eo 2 (S, )-Cat. We denote th* *is canonical free simplicial resolution of D by FsD. 2.7. Remark. In [DK1 , x1], Dwyer and Kan define a model category structure on (S, O)-Cat (also valid for (S, )-Cat), which turns out to be a resolution m* *odel HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 9 category in the sense of [Bou ] (see also [J2], [DKSt , x5] and [BJT , x2]). T* *he spherical objects for (S, O)-Cat (cf. x1.23(a)) are objects of the form Mo := Sn(u,v)f* *or n 1 and Hom (u, v) 6= ;, defined by: ( Sn for u0= u and v0= v (2.8) M(u0, v0) = * otherwise, One can also show that (S, O)-Cat and (S, )-Cat are spherical - that is, endowed with the additional structure described in x1.23 (of which only the exi* *stence of models is guaranteed in a resolution model category). 2.9. The model category (C, )-Cat. In the case of (C, )-categories, the situation is somewhat complicated by t* *he fact that they cannot simply be viewed as cubical objects in Cat, because , and t* *hus the composition maps, are not defined dimensionwise (see Remark 1.10). Berger and Moerdijk have defined a model category structure for algebras over coloured operads in a suitable symmetric monoidal model category, which applies in parti* *cular to (C, )-Cat (see [BM2 ], and compare [BM1 ]). However, in this paper we onl* *y need to consider (C, )-categories for a special type of category , for which it i* *s easy to describe an explicit model category structure in which W is cofibrant: 2.10. Definition. A small non-unital category will be called a quasi-lattice * *if it has no self-maps; in this case there is a partial ordering on O = Obj ( ), with * *u v if and only if Hom (u, v) 6= ;, and we require in addition that be locally* * finite in the sense that for any u v in O, the interval Seg[u, v] := {w 2 O | u w* * v} is finite. 2.11. Example. The simplest example is a linear lattice of length n + 1, whic* *h we denote by n+1: this consists of a single composable (n + 1)-chain: OEn+1 OEn OE2 OE1 vinit= (n + 1) ---! n -! (n - 1) ! . .!. 2 -! 1 -! 0 = vfin. Another example is a commuting square: 0 vinitOE__//v0 OE00|| |_0| fflffl|_fflffl|00 v00____//_vfin Observe that for categories of diagrams indexed on a directed Reedy category * *(i.e., one for which the "inverse subcategory" is trivial), the Reedy model structure * *(cf. [Hir, x15.2.2]) agrees with the projective model structure. In this situation, cofibr* *ations of diagrams are those morphisms whose "latching maps" are all cofibrations in the * *target category, while fibrations and weak equivalences of diagrams are defined object* *wise. Our current context is sufficiently similar to allow an analogous inductive a* *rgument, depending on the following analog of Reedy's latching objects and maps: 2.12. Definition. Given a quasi-lattice , a map F : A ! B in (C, )-Cat, * *and u v in O, the composition category (JA,B(u,v), <) is a partially ordered * *set, whose objects are pairs , where ! is a chain 10 D. BLANC, M.W. JOHNSON, AND J.M. TURNER in , and the index X is either A or B. We omit the copy of the trivial * *chain indexed by B. The partial order is defined by setting whenever !0 is* * a (not necessarily proper) subchain of !, and either X = X 0 or X = A, X 0= B. The corresponding composition diagram D = DA,B(u,v): JA,B(u,v)! C is defin* *ed by N k sending to j=1 X c(wj-1, wj). The morphisms are generated by the following two types of maps: (i)If !0 is obtained from ! by omitting internal node wj (1 < j < k), * *the map D ! D is Id . . .cmp X(wj-1,wj,wj+1). .I.d, where cmp X(wj-1,wj,wj+1): X c(wj-1, wj) X c(wj, wj+1) ! X c(wj-1, wj+1) is the cubical composition map inN X 2 {A, B}; (ii)The map D ! D is ki=1F(wi-1,wi). Note that F(u,v): Ac(u, v) ! Bc(u, v), together with the composition maps o* *f B ending in Bc(u, v), induce a map '(u,v): colimDA,B(u,v)! Bc(u, v). In parti* *cular, when Seg[u, v] = {u, v} is minimal, colimDA,B(u,v)is simply Ac(u, v) and* * '(u,v) is F(u,v): Ac(u, v) ! Bc(u, v). We now provide the details of the model category structure on (C, )-Cat - inter alia, in order to allow the reader to verify that the construction works * *in the non-unital setting: 2.13. Lemma. If is a quasi-lattice, the category (C, )-Cat has all limits * *and colimits. Proof.For any small category , the limits in (C, )-Cat are constructed byQta* *king the limit at each (u, v) 2 O2, with compositions defined for the product i* *2IAi by the obvious maps: Y Y Y cmp(u,w,v)Y ( Ai[u, w]) ( Ai[w, v]) ! (Ai[u, w] Ai[w, v]) ------! Ai[* *u, v] , i2I i2I i2I i2I and similarly for the other limits. For the colimits, note that is defined as a colimit (cf. Definition 1.5), s* *o it commutes`with colimits in C. For (C, )-categories {Ai}i2I, the coproduct D* * := i2IAi is defined by induction on the cardinality`of Seg[u, v] in . * * When Seg[u, v] = {u, v} is minimal, we let D(u, v) := i2IAi(u, v). In general, * *set a a D(u, v) := Ai(u, v) q D(u, w) D(w, v) , i2I u w v with the obvious (tautological) composition on the right-hand summands. Now given maps F : A ! B and G : A ! E in (C, )-Cat, the pushout PO is once more defined by induction on the cardinality of Seg[u, v], as follows: In the initial case, when Seg[u, v] is minimal, PO (u, v) is simply the p* *ushout of E(u, v) A(u, v) ! B(u, v) in C. In the induction step, we let J = JPO(u,v)denote the union of the composition categories JA,B(u,v), JA,E(u,v), and JB,PO(u,v)(see Definition 2.12). Thu* *s the objects of HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 11 J are pairs , where ! is a chain and X 2 {A, B, E, PO }, again omitting . Again J is a partially ordered* * set, with the order relation defined to be the union of those for (A, B), (A, E),* * and (B, PO ). The composition diagrams DA,B(u,v), DA,E(u,v), DB,PO(u,v), and DE,PO(u* *,v)fit together to form a composition diagram DPO(u,v): JPO(u,v)! C. The last two diagrams a* *re well-defined, because we omit the trivial chain , and all other v* *alues of DB,PO(u,v)and DE,PO(u,v)have already been defined by our induction assumption.* * We now let PO (u, v) be the colimit in C of the diagram DPO(u,v): JPO(u,v)! C. The constructions of the coproducts and pushouts implies that all colimits ex* *ist in (C, )-Cat, by the dual of [Bor1 , Thm. 2.8.1 & Prop. 2.8.2]. 2.14. Definition. Let be a quasi-lattice, and let A and B be (C, )-categori* *es. A map F : A ! B in (C, )-Cat is (a) a weak equivalence if F(u,v): Ac(u, v) ! Bc(u, v) is a weak equivalence* * in C (see x1.15) for any u v in O. (b) a fibration if F(u,v): Ac(u, v) ! Bc(u, v) is a (Kan) fibration in C f* *or all u v in O. (c) a (acyclic) cofibration if for all u v in O the maps F(u,v): Ac(u, * *v) ! Bc(u, v) and '(u,v): colimDA,B(u,v)! Bc(u, v) are (acyclic) cofibratio* *ns in C. 2.15. Remark. A straightforward induction shows that the acyclic cofibrations so defined are precisely those cofibrations which are weak equivalences. The following lemmas show that these choices yield a model category structure* * on (C, )-Cat: 2.16. Lemma. If is a quasi-lattice, F : A ! B is a cofibration and P : D * *! E is an fibration in (C, )-Cat, and either F or P is a weak equivalence, then * *there is a lifting H^ in any commutative square __G__// A >D>__ "H_______ (2.17) F ||______P||_ fflffl|_fflffl|___ B __H__//E . Proof.We choose "H(u,v): Bc(u, v) ! Dc(u, v) by induction on the cardinality* * of the interval Seg[u, v] in : When Seg[u, v] = {u, v} is minimal, we simply choose a lift "H(u,v)in: G(u,v) c Ac(u,v)_________//D(u,v) H"(u,v)__77__________ F(u,v)|| __________|P(u,v)|_ fflffl|_________fflffl| Bc(u,v)_________//_Ec(u,v) H(u,v) using the fact that F(u,v) is a cofibration and P(u,v) an acyclic fibration i* *n C (see (1.17) above). 12 D. BLANC, M.W. JOHNSON, AND J.M. TURNER In the induction step, assume we have chosen compatible lifts H"(u0,v0)for al* *l proper subintervals Seg[u0, v0] Seg[u, v]. These yield a map ^Gmaking the followin* *g solid square commute in C: ^G c colim DA,B(u,v)_______//D(u,v)66_ "H(u,v)_____________ '(u,v)|| ____________P(u,v)||__ _____ fflffl|_____ fflffl| Bc(u, v)___H________//Ec(u,v) (u,v) and since '(u,v)is a cofibration by Definition 2.14, and P(u,v)is an acyclic fi* *bration by assumption, the lifting "H(u,v)exists. The same argument shows that there exists a lifting in (2.17) when F : A ! B is an acyclic cofibration and P : D ! E is a fibration. 2.18. Lemma. If is a quasi-lattice, any map F : A ! B in (C, )-Cat facto* *rs as: _____F______// A B @@ "??" (2.19) @@I@@ """ OO@"" P D where I is a cofibration and P is a fibration; and we can require either I or P* * to be a weak equivalence. Proof.Again construct D, I, and P in (2.19) by induction on the cardinality of Seg[u, v]. When Seg[u, v] = {u, v} is minimal, choose any factorization: F(u,v) c Ac(u,v)_____________//_B(u,v) FFFI(u,v)FF xx;;x FF xxxx F##F xx P(u,v) Dc(u,v) where I(u,v) is an acyclic cofibration and P(u,v) is a fibration in C. Now assume by induction that we have chosen compatible factorizations iA(u,v) !A Ac(u,w) Ac(w,v)________//Col(A)(u, v)________//_Ac(u,v) I(u,w)||I(w,v)|| OE(u,v)|| j(u,v)|| fflffl|fflffl|iD fflffl| ` fflffl| (2.20) Dc(u,w) Dc(w,v)_____(u,v)//_Col(D)(u,_v)(u,v)//_PO(u,v) P(u,w)||P(w,v)|| _(u,v)|| ,(u,v)|| fflffl|fflffl|iB(u,v) fflffl| B fflffl| Bc(u,w) Dc(w,v)________//_Col(B)(u,_v)!______//_Bc(u,v) where each cubical set Col(E)(u, v) (for E = A, B, D) is the colimit over a* *ll proper subintervals Seg [u0, v0] Seg[u, v] and u0 w v0 of the diagram of compo* *sition maps cmp E(u0,w,v0): Ec(u0, w) Ec(w, v0) ! Ec(u0, v0) . HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 13 in each row, iE : Ec(u,w) Ec(w,v)! Col(E)(u, v) is the structure map for the* * colimit, while !E : Col(E)(u, v) ! Ec(u,v)is induced by the compositions. The cubical * *set PO (u,v) is the pushout of the upper right-hand square, with structure maps j(* *u,v) and `(u,v), and ,(u,v) is induced on the pushout by F(u,v) and the maps _(* *u,v) (from the naturality of the colimit) and !B . Note that the map I(u,w) I(w,v)is an acyclic cofibration in C (see Fact 1.13)* *, so the induced map OE(u,v) is, too, as is j(u,v), by cobase change. The map P(u,w)* * P(w,v), as well as the induced map _(u,v), comes from the compatible factorizations * *(2.20). Finally, choose a factorization ,(u,v) PO (u, v)_________________//_Bc(u, v) LLLi(u,v)LL sss99s LLL ssss L&&L ss P(u,v) Dc(u, v) where i(u,v) is an acyclic cofibration and P(u,v) is a fibration in C. This d* *efines the cubical set Dc(u, v), which is equipped with composition maps cmp (u,w,v):= i(u,v)O `(u,v)O iD(u,v). Setting I(u,v):= i(u,v)O j(u,v) yields the required acyclic cofibration, and s* *ince ,(u,v) is induced by F , we have P(u,v)O I(u,v)= F(u,v), as required. The same construction, mutatis mutandis, yields a factorization (2.19) wher* *e I is a cofibration and P an acyclic fibration. 2.21. Theorem. If is a quasi-lattice, Definition 2.14 provides a model catego* *ry structure on (C, )-Cat. Proof.The category (C, )-Cat is complete and cocomplete by Lemma 2.13. The classes of weak equivalences and fibrations are clearly closed under compositio* *ns, and include all isomorphisms. The same holds for cofibrations by an induction argum* *ent. Also, if two out of the three maps F , G, and G O F are weak equivalences, so is the third. The lifting properties for (co)fibrations are in Lemma 2.16, and* * the factorizations are given by Lemma 2.18. As expected, the two key types of (V, )-categories are related by suitable f* *unctors (compare [Bor2 , Prop. 6.4.3]): 2.22. Proposition. For any quasi-lattice , the functors T : C ! S and Scub* * : S ! C of (1.8) extend to functors (C, )-Cat (S, )-Cat. Futhermore, this* * is a strong Quillen pair (cf. [Hir, x8.5.1]), and descends to an adjunction at the* * level of homotopy categories. Proof.The functor T extends to (C, )-Cat by (1.11). For Scub, given As 2 (S, )-Cat, with composition , : As(u, w)) x As(w, v) ! As(u, v) we define * *the composition map cmp(u,w,v): Scub(As(u, w)) Scub(As(w, v)) ! Scub(As(u, v)) * *for the (C, )-category ScubAs to be the composite Scub, O # (see (1.12)). A* *s Scub is a strong right Quillen functor, it follows from the definitions that the ext* *ension is also strong right Quillen. 14 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 2.23. Semi-spherical structure on (C, )-Cat. The discussion above, including the model category structures, is valid when * *we replace C or S by their pointed versions (see [Ho , Proposition 1.1.8]). Moreov* *er, even though we cannot construct entry-wise spheres for (C, )-categories as in (2.* *8), the category (C, )-Cat may be called semi-spherical, in the sense of having the r* *est of the spherical structure described in x1.23, as follows: 2.24. Definition. Given a quasi-lattice and a (C*, )-category A, its fundam* *ental groupoid is the (Gpd , )-category obtained by applying the fundamental groupo* *id functor ^ss1to A. Note that because ^ss1: C ! Gpd factors through T : C ! * *S, using (1.11) we see that ^ss1A is indeed a (Gpd , )-category (cf. [Bor2 , Pro* *p. 6.4.3]). Similarly, for each n 2 the functor ssn, applied entrywise to A, yields a * *(Gp , )- category, which is actually a (^ss1A-Mod, )-category (see Definition 2.1). No* *te that, as for topological spaces, ssnA is a module over ^ss1A. a) Each (C*, )-category A has a functorial Postnikov tower, obtained by ap* *ply- ing the functors Pn of x1.24 to each Ac(u, v), and using Pn(A(u, v)) Pn(A(v, w))! Pn(Pn(A(u, v)) Pn(A(v, w))) ~=Pn(A(u, v) A(v, w)) ! Pn(A(u, w)) . b) For every (Gpd , )-category , there is a functorial classifying object* * B 2 (C*, )-Cat. c) Given a (Gpd , )-category , and a -module G (i.e., an abelian group ob* *ject in (Set, )-Cat= ), for each n 2 there is a functorial extended G-Eile* *nberg- Mac Lane object E (G, n) in (C*, )-Cat=B . d) For n 1, there is is a functorial k-invariant square for A as in (1.* *25). All these properties are straightforward for (S*, )-Cat (by applying the ana* *logous functors for S* componentwise), and they may be transfered to (C*, )-Catusi* *ng Proposition 2.22. 2.25. Definition. Given a (Gpd , )-category , a -module G, a (C*, )-categ* *ory A, and a twisting map p : A ! B , we define the n-th (C, )-cohomology group* * of A with coefficients in G to be Hn(A, G) := [A, E (G, n)](C, )-Cat=B. 2.26. Remark. Typically, we have = ^ss1A, with the obvious map p. More generally, in [DKSm1 ] Dwyer, Kan, and Smith give a definition of the * *(S, O)- cohomology of any (S, O)-category with coefficients in a -module G; and there* * is also a relative version, for a pair (A, B) (cf. [DKSm1 , x2.1]). It is strai* *ghtforward to verify that the two definitions of cohomology coincide (when they are both defi* *ned) under the correspondence of Proposition 2.22. 3. Lattices and higher homotopy operations We can now define higher homotopy operations as obstructions to rectifying a homotopy commutative diagram X : K ! ho T , using the approach of [BM ], with the modification in the pointed case given in [BC ]. For this purpose, it is co* *nvenient to work with a specific cofibrant cubical resolution of the indexing category K* *. We need make no special assumptions about K at this stage. HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 15 Boardman and Vogt originally defined their "bar construction" W K topologica* *lly (see [BV , III, x1]). The (C, O)-version may be described as follows: 3.1. Definition. The W-construction on a small category K with O = Obj K is the (C, O)-category W K, with the cubical mapping complex W K(a, b) for ev* *ery a, b 2 Obj (K), constructed as follows: For every composable sequence fn+1 fn f1 (3.2) fo = (a = an+1 --! an -! an-1 . .a.1-! a0 = b) of length n + 1 in K, there is an n-cube Info in W K(a, b), subject to t* *wo conditions: (a) The i-th 0-face of Info is identified with In-1f1O...O(fi.fi+1)O...fn+1, * *that is, we carry out the i-th composition in the sequence fo (in the category K). (b) The cubical composition W K(a0, ai) W K(ai, an+1) ! W K(a0, an+1) = W K(a, b) identifies the "product" (n - 1)-cube Iif0O...Ofi In-i-1fi+1O...Ofn+1w* *ith the i-th 1-face of Info. 3.3. Notation. Note the three different kinds of composition that occur in W K: (a) The internal composition of K is denoted by f . g, or simply fg. (b) The cubical composition of W K, denoted by f g, which corresponds to the -product of the associated cubes. (c) The potential composition of W K, denoted by f O g, is the heart of t* *he W -construction: it provides another dimension in the cube for the homoto* *pies between f g and f . g. Thus a composable sequence fo as in (3.2) (indexing a cube in W K) will * *be denoted in full by f1 O . .O.fn+1; the composed map f1f2. .f.n+1: a ! b in* * K is denoted by comp (fo); and the cubical composite f1 f2 . . .fn+1 wil* *l be denoted by fo (as an index for a suitable cube in W K). 3.4. Definition. The minimal vertex of Info is I0comp(fo), which is in the * *image of all 0-face maps. The opposite maximal vertex, in the image of all 1-face map* *s, is indexed by fo according to the convention above, with I0f1 I0f2 . . .I0f* *n+1 identified with I0 fo under the iterated cubical compositions. If we think of a small category K as a constant cubical category in (C, O)-C* *at for O = Obj K, there is an obvious map of (C, O)-categories flc : W K ! K, and following work of [Le ] and [Co ] we show: 3.5. Lemma. The map T flc : T W K ! T K = K may be identified with fls : FsK* * ! K (see x1.7 ff.). Proof.Consider an individual cube InOEo of W K: this is isomorphic to W n+* *1, where n+1 (Example 2.11) consists of a composable sequence of n + 1 maps: OEn+1 OEn OE2 OE1 (n + 1) ---! n -! (n - 1) ! . .!. 2 -! 1 -! 0 . 16 D. BLANC, M.W. JOHNSON, AND J.M. TURNER ((OE1))((OE2)(OE3)) (OE1)(OE2)(OE3)t_______________________________oet(OE1)(OE2OE3)|| |6QQk __________________ |6 | Q | | Q Q | (((OE1))((OE2)(OE3)|))| | Q |_________________ | | | Q | | Q | | Q | | | ((OE1)(OE2))((O|E3)) ((OE1)(OE2)(OE3)) |((OE1)(OE2OE3)) | | | Q | | Q Q | | __________________ Q | | Q | | | (((OE1)(OE2))((OE3)))| Q | | |_________________ | Q | |t Q | (OE1OE2)(OE3)________________________________oet(OE1OE2OE3) ((OE1OE2)(OE3)) Figure 1. The triangulated 2-cube Fs 3 The free simplicial resolution of Fs n+1 is the triangulation of the n-cube * *InOEoby n! n-simplices, corresponding to the possible full parenthesizations of OEo * * (see Figure 1). This may be identified canonically with the standard triangulation [1]n 2 S* * of In 2 C (see [BB , x3]), thus indeed identifying FsK with T W K. 3.6. Proposition. If is a quasi-lattice, the map of (C, )-categories flc :* * W ! is a cofibrant resolution. Proof.The map of (S, O)-categories fls : FsK ! K is a weak equivalence, sin* *ce Fs is defined by a comonad (see [CP , x1]). Thus FsK is indeed a free simpl* *icial resolution of K (see [DK1 , x2.4], [CP , x2], and [BM , x2.21]). Having ident* *ified fls : FsK ! K with T flc : T W K ! T K = K, it follows from Proposition 2.22 that * * flc is a weak equivalence. By construction, each composition map W K(a, b) W K(b, c) ! W K(a, c) of W K is an inclusion of a sub-cubical complex, since on every "product" cube Info * *Ikgo~= In+kfo goIn+k+1foOgoit is the inclusion of a 1-face. Thus the map '(u,v): coli* *mD*,B(u,v)! Bc(u, v) of Definition 2.12 is just the inclusion of the sub-cubical complex c* *onsisting of all the 1-faces, which is a cofibration (in fact, an anodyne map). This show* *s that W K is cofibrant. 3.7. Rectifying homotopy commutative diagrams. We can use the cofibrant resolution W K ! K to study the rectification of a homotopy-commutative diagram "X: K ! hoM in some model category M (such as T or T*). Since the 0-skeleton of W K is isomorphic to F K, choosing an arbitrary r* *epre- sentative X0(f) for each homotopy class "X(f) for each morphism f of K, yie* *lds a lifting of X" to X0 : skc0W K ! M. Note that a choice of a 0-realization X0 : F K ! M is equivalent to choosi* *ng basepoints in each relevant component of each Mc(u, v), although of course th* *is cannot be done coherently unless X" is rectifiable. HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 17 3.8. Remark. Our goal is to extend X0 over the skeleta of W K. However, t* *he "naive" cubical skeleton functor skck: C ! C (x1.1) is not monoidal with resp* *ect to (unlike the simplicial analogue), so it does not commute with composition map* *s. Nevertheless, one can define a k-skeleton functor for (C, O)-categories in gen* *eral; when is a quasi-lattice (x2.24) and A is a cofibrant (C, )-category (such a* *s W ), skckA can be defined by simply including all -product cubes of i-cubes in A w* *ith i k. Of course, if A is n-dimensional (that is, has no non-degenerate i-cube* *s for i > n), then skcnA = A agrees with the naive n-skeleton. If M is cubically enriched (x1.6), extending X0 to a cubical functor X1 : skc1W K ! M is equivalent to choosing homotopies between each X"(f1 O f2) and "X(f1) O X"(f2), since the 1-cubes of W K correspond to all possible (two te* *rm) factorizations of maps in K. Extending X1 further to X2 : skc2W K ! M means choosing homotopies between the homotopies for three-fold compositions, and so * *on. This is the idea underlying a fundamental result of Boardman and Vogt: 3.9. Theorem ([BV , Cor. 4.21 & Thm. 4.49]). A diagram X" : K ! ho T lifts to T if and only if it extends to a simplicial functor X1 : W K ! T . 3.10. Remark. In fact, for our purposes we do not have to assume that the categ* *ory M is cubically enriched, or even has a model category structure: all we need is f* *or M to have a suitable class of weak equivalences W, from which we can construct an (S* *, O)- category L(M, W) as in [DK2 , x4], and then the corresponding (C, O)-categ* *ory ScubL(M, W) by Proposition 2.22. Note that when M and W are pointed, the construction of Dwyer and Kan is naturally pointed, too. However, to avoid exce* *ssive verbiage we shall assume for simplicity that M is a cubically enriched model ca* *tegory. We do not actually need the full (usually large) category M (or ScubL(M, W)), since we can make use of the following: 3.11. Definition. Given a diagram X" : K ! ho M for a model category M 2 C-Cat, let CX be the smallest (C, K)-category inside M through which any l* *ift of "Xto X : K ! M factors. This means that CX is the (C, K)-category having cubical mapping spaces ( Mc(Xu, Xv) ifu v in O := Obj K CX (Xu, Xv) := ; otherwise. This is a sub-cubical category of M. For simplicity, we further reduce the mapping spaces of CX so that they co* *nsist only of those components of Mc(Xu, Xv) which are actually hit by X", so that ss0CX = K. In particular, if K is the partially ordered set , we may a* *ssume the mapping spaces of CX are connected (when they are not empty). 3.12. Pointed diagrams. We want to understand the relationship between two possible ways to describe the (final) obstruction to the existence of an extens* *ion X1 : topologically and cohomologically. Unfortunately, even though these obstructio* *ns can be defined for quite general K, they do not always coincide; this can be se* *en by comparing the sets in which they take value. However, we are in fact only interested in the cases where the obstruction can naturally be thought of as the higher homotopy operation associated to the data 18 D. BLANC, M.W. JOHNSON, AND J.M. TURNER "X: K ! ho T . The usual mantra says that such an operation is defined when "a lower order operation vanishes for two (or more) reasons". Indeed, the example* * of the usual Toda bracket shows that the problem cannot be stated simply in terms * *of rectifying a homotopy-commutative diagram, since any diagram indexed by a linear indexing category n as above can always be rectified: what we want is to re* *alize certain null-homotopic maps by zero maps (see [BM , x3.12]). This suggests that we restrict attention to pointed diagrams, and to the foll* *owing special type of indexing category: 3.13. Definition. A lattice is a finite quasi-lattice (x2.10) equipped with a* * (weakly) initial object vinit and a (weakly) final object vfin, satisfying: (a) There is a unique OEmax : vinit! vfin. (b) For each v 2 Obj , there is at least one map vinit! v and at least * *one map v ! vfin. A composable sequence of n arrows in will be called an n-chain. The maximal occuring n (necessarily for a chain from vinit to vfin, factorizing OEmax) * * is called the length of . 3.14. Remark. Note that if the length of is n + 1, then W is n-dimensi* *onal, in the sense that the cubical function complex W (vinit, vfin) has dimension* * n, and dim(W (u, v)) < n for any other pair u, v in . 3.15. Definition. We shall mainly be interested in the case when is pointed (* *in which case necessarily OEmax = 0). A null sequence in is then a composable s* *equence i f f f j fo := an+1 -n+1-!an -n!an-1 . .a.1-1!a0 with comp (fo) = 0, but no constituent fi is zero. It is called reduced if all* * adjacent compositions fi+1. fi (i = 1, . .,.n) are zero. An n-cube Info in W inde* *xed by a (reduced) null sequence is called a (reduced) null cube. As noted above, we want to concentrate on the problem of replacing null-homot* *opic maps with zero maps, given a pointed diagram X" : ! ho M which commutes up to pointed homotopy. We shall therefore assume from now on that all other (n* *on- zero) triangles in the diagram commute strictly. However, since the non-zero m* *aps in do not form a sub-category, we shall need the following: 3.16. Definition. The unpointed version Up(K) of a pointed category K is defi* *ned as follows: if K ~=F (K)=I for some set of relations I in the free category * * F (K), then the objects of Up(K) are those of K, except for the zero objects, and Up(* *K) := F(K0)=(I \ F(K0)), where K0 is obtained from the underlying graph K of K by omitting all zero objects and maps. The inclusion K0 ,! K induces a functor ' : Up(K) ! K. Essentially, Up(K) is the full subcategory of K omitting 0 and all maps in* *to or out of the zero object 0. However, if the composite f . g : a ! b is zero i* *n K with f 6= 0 6= g, then we add a new (non-zero) map ' : a ! b in Up(K) (wi* *th '(') = 0), to serve as the composite in Up(K) of f and g. 3.17. Defining higher operations. HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 19 From now on we assume given a pointed lattice and a diagram up-to-homotopy "X : ! ho M into a pointed cubically enriched model category M. Setting 0 := Up( ), we also assume that the composite X" O ' lifts to a strict dia* *gram X0 : 0 ! M. For simplicity we also denote the factorization of X0 though CX (x3.11) by X0 : 0! CX . Our goal is to extend X0 to a pointed diagram X : ! CX . (Note that X0 itself cannot be pointed in our sense, but it still takes values in the pointed* * category M). Obviously, if X0 does extend to such an X, every map ' 2 0 which facto* *rs through 0 in must be (weakly) null-homotopic in M. Thus, we additionally incl* *ude this restriction on the original data as part of our assumptions. Our approach is to extend X0 by induction over the skeleta of W , where * *we actually need: 3.18. Definition. Given and X0 as above, for each k 0 the relative k-skel* *eton for ( , 0), denoted by skck( , 0), is the pushout: skck' c skckW 0_____//_skkW skckflc|| || fflffl| c fflffl| 0_______//_skk( , 0) in (C, )-Cat (cf. Lemma 2.13), where flc : W ! is the augmentation of Proposition 3.6. Note that the natural inclusions skck-1,! skck induce maps skck-1( , 0) ! skck( , 0). A map of (C, )-categories X0k: skck( , 0) ! CX extending X0 : * * 0! CX is called k-allowable. In particular, if is a lattice of length n+1, by Remark 3.14 W ( , 0) := s* *kcn( , 0) is the pushout ___'___//_ W 0 W |flc| || fflffl| fflffl| 0 _____//_W ( , 0) . 3.19. Remark. X0 extends canonically to a pointed map X0 : skc0W ! CX , because skc0W is a free category, and the only new object is 0. Together wi* *th flc this determines a canonical 0-allowable extension X00: skc0( , 0) ! CX . If is a lattice of length n + 1, in order to rectify X0 we want to ext* *end X00 inductively over the relative skeleta skck( , 0) to an n-allowable map* * X01 : W ( , 0) ! CX - equivalently, a map X1 : W ! CX which agrees with the initial X0 : 0! CX . Recall that because dim W = dim W ( , 0) = n, X0n* * is actually X01 in the sense of Theorem 3.9, so this yields a rectification of * *X0 for suitable M (such as T*). We assumed in x3.17 that X0 : 0 ! CX takes every map ' 2 0 which factors through 0 in to one which is null-homotopic in M. Therefore, by choos* *ing null-homotopies for all such maps we see that X00 always extends non-canonical* *ly to a 1-allowable X01: skc1( , 0) ! CX . 20 D. BLANC, M.W. JOHNSON, AND J.M. TURNER However, in general there are obstructions to obtaining k-allowable extension* *s for k 2. These are complicated to define "topologically" (see [BM ] and [BC ]).* * Fortu- nately, in order to define the higher homotopy operation associated to X0, we* * only need to consider the last obstruction. That is, we assume we have already produced an (n - 1)-allowable extension X0n-1: skcn-1( , 0) ! CX , and want to extend it to X0n. It may be possible* * to do so in different ways. In order to define the set <> of "last obstructions"* *, we need the following: 3.20. Lemma. Assume that = n+1 is a composable (n + 1)-chain fo (x2.11) and that the i-th adjacent composition fi. fi+16= 0 in , and let f0o:= (f1, . .,.fi-1, fi. fi+1, fi+2, . .f.n) . Let ' : Inf0o,! In+1fo be the inclusion of the i-th zero face. Let " be the li* *near lattice corresponding to f0o. Then for any X : 0! CX , the inclusion ' : " ,! * *induces a one-to-one correspondence between the set of extensions of X00: skc0( , 0) * *! CX to W and the extensions of "X00: skc0(" , 0) ! CX to W ". Proof.The i-th dimension of Info corresponds to the i-th adjacent composition fi. fi+1 in the (n + 1)-chain fo, and if this composite is not zero, then * *X0n, being allowable, is constant along this dimension. Thus the projection ae : Info! I* *n-1f0o induces the inverse to '*. 3.21. Proposition. Let be a lattice of length n + 1 and X0 : 0 ! CX a diagram. Let J be the set of length n + 1 reduced null sequences of (Def* *inition 3.15). There is a natural correspondence betweenW(n-1)-allowable extensions X0n* *-1: skcn-1( , 0) ! CX of X0 and maps FX0n-1: fo2J n-1X0(vinit) ! X0(vfin* *), such that FX0n-1 is null-homotopic if and only if X0n-1 extends to skcn( ,* * 0). Proof.In order to extend X0n-1: skcn-1( , 0) ! CX to skcn( , 0), we must* * choose extensions to the n-cubes of W . These occur only in the full mapping complex W (vinit, vfin), and are in one-to-one correspondence with those decompositio* *ns i f f f j fo = vinit= an+1 -n+1-!an -n!an-1 . .a.1-1!a0 = vfin of OEmax : vinit! vfin which are of maximal length n + 1. Note that the mini* *mal vertex of Info is indexed by OEmax = 0; the maximal vertex is I0 fo (Definitio* *n 3.4). By Lemma 3.20 we need only consider those maximal decompositions fo for which every adjacent composition fi. fi+1 = 0. In this case, we may assume th* *at any facet In-1f0oof Info which touches the vertex labeled by OEmax = 0 has * *at least one factor of f0o equal to 0 (in ), so X0n-1|In-1= 0. Thus X0n-1|In is g* *iven by f0o fo a map in M F(0X0n-1,Inf): X0(vinit) @In ! X0(vfin) which sends X0(vinit) * * I0OE o * * max and *X(vinit) In to *X(vfin), so it induces "F(X0n-1,Info): X0(vinit) ^ S* *n-1 ! X0(vfin). Note further that any two such n-cubes Info and Ingo have distinct maximal vertices I0fo and I0go, so they can only meet in facets adjacent to the min* *imal vertex, where "H vanishes. Thus altogether X0n-1 is described by a map ` (3.22) FX0n-1: n-1X0(vinit) ! X0(vfin) , fo2J HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 21 where J is the set of length n + 1 reduced null sequences of . Clearly, * *FX0n-1 is null-homotopicWif and only if X0n-1 extends to all of W , since W (vi* *nit, vfin) is map (C( fo2J n-1X0(vinit)), X0(vfin)), up to homotopy, where CK is the co* *ne on K. 3.23. Definition. The n-th order pointed higher homotopy operation <> as* *soci- ated to X0 : 0! CX as above is defined to be the subset: " # ` (3.24) <> n-1X0(vinit), X0(vfin) fo2J hoM consisting of all maps FX0n-1 as above, for all possible choices of (n - 1)-a* *llowable extensions X0n-1, of X0. We say the operation vanishes if this set contains* * the zero class. 4. Cohomology and rectification The approach of Dwyer, Kan, and Smith to realizing a homotopy-commutative diagram "X: ! ho M is also based on Theorem 3.9, which says that X" can be rectified if and only if it extends to W . We do not actually need the full * *force of their theory, which is why we can work in an arbitrary pointed model category M, rather than just T* (see also Remark 3.10). Essentially, they define the (possibly empty) moduli space hcX" to be the n* *erve of the category of all possible rectifications of X" (cf. [DKSm2 , x2.2]), an* *d hc1 X" is the space of all 1-homotopy commutative lifts of X" in (the simplicial versi* *on of) map C-Cat(W , M) = map (C,-)Cat(W , CX ) (x3.11). They then show that hcX" * * is (weakly) homotopy equivalent to hc1 X" (see [DKSm2 , Theorem 2.4]). Thus t* *he realization problem is equivalent to finding suitable elements in map (C, )-Cat* *(W , CX ). Dwyer, Kan, and Smith also consider a relative version, where X" has already be* *en rectified to Y : ! M for some sub-category (see [DKSm2 , x4]). * *We shall in fact need only the case = 0 and Y = X0, so we want an element in map (C,-)Cat(W ( , 0), CX ) (see x3.18). 4.1. The tower. If is a quasi-lattice, (C, )-Cat has a semi-spherical model* * cate- gory structure (see x2.9 and x2.23). Therefore, the Postnikov tower {P mCX }1m* *=0 of the (C, )-category CX allows us to define hcm X" := map (C,-)Cat(W ( , 0), P* * m-1CX ) for m 1. Note that P 0CX is homotopically trivial - that is, each comp* *onent of each mapping space (P 0CX )(u, v) is contractible - so hc1X" is, too* *. More- over, X" : ! ho M (or X0 : 0 ! CX ) determines a canonical "tautologica* *l" component of hc 1"X - namely, the component of the map Xf1 : W ( , 0) ! P 0CX* * , corresponding to the canonical 0-allowable extension X00: skc0( , 0) ! CX o* *f x3.19. Because CX is weakly equivalent to the limit of its Postnikov tower (x1.23* *(b)), the space hc1 "X is the homotopy limit of the tower: (4.2) hc1 X" ! . .!.hcnX" ! hcn-1X" . .!.hc1X" . In general, there are lim1 problems in determining the components of hc1 X" (see [DKSm1 , x4.8]), but these will not be relevant to us here, because of t* *he following: 22 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 4.3. Lemma. If has length n + 1, the tower (4.2) is constant from hcn-1X" up. Proof.We may assume that CX is fibrant (e.g., if "Xv is a cubical Kan com* *plex for each v 2 O). Then skcn( , 0) = W by Remark 3.14, where in this case we* * are using the naive n-skeleton (see Remark 3.8) which is left adjoint to the n-cosk* *eleton functor. By Remark 1.28, we may use the latter for P n-1CX . Thus the choices* * of n-allowable extensions X0n: skcn( , 0) = W ! CX of X" are in natural one-* *to-one correspondence with lifts fXn: W ( , 0) ! P n-1CX of fX1. 4.4. The obstruction theory. In view of the above discussion, the realization problem for "X : ! ho M * * - and in particular, the pointed version for X0 : 0 ! M (see x3.17) - can be solved if one can successively lift the element Xf1 2 hc 1"X through the tow* *er (4.2). In fact, we do not really need the (simplicial or cubical) mapping spa* *ces hcm "X:= map (C,-)Cat(W ( , 0), P m-1CX ) at all - we simply need to lift t* *he maps gXm: W ( , 0) ! P m-1CX in the Postnikov tower for CX . Let km-1 : CX ! EG (ssm CX , m + 1) be the (m - 1)-st k-invariant for CX * *, where G := ^ss1CX (see x2.23 ff.). Given a lifting gXm, composing it with km-1 y* *ields a map h(Xm ) : W ( , 0) ! EG (ssm CX , m + 1): __________gXm_______________________________________* *_____________________________________________________________________________* *__________________________________________ W ( , 0) __________________________________________* *___________________________________________________________________ __________________ ____________________________________* *______ ______^Xm+1______________________________________________* *_________________ _________________________________________________________* *_____________ _________%%________________!!___________________________* *_________________________________ __________________________________//_ mm-1 _____________________________________PP CX CX __________________________| p______________________________________________________* *____________________|k| ____________________________________________________* *___|fflffl|| m-1 ___**_____________________________________________* *_________________________fflffl|s//_G BG ii____E__(ssm_CX_,_m_+_1)______________ _________________________________________* *_____________________________________________________________________________* *___ proj To identify h(Xm ) as an element in the appropriate cohomology group (Defi- nition 2.25), note that in this case the twisting map p : W ( , 0) ! BG fac* *tors through ^ss1Xn: ^ss1W ( , 0)! ^ss1Pm-1 CX= ^ss1CX= G, and by Proposition 3.6* *, the fundamental groupoid ^ss1W ( , 0)= is discrete. Thus [h(Xm )] takes va* *lue in Hm+1 (W ( , 0); ssm CX ), which we abbreviate to Hm+1 ( ; ssm CX ). The lifting property for a fibration sequence (over BG) then yields: 4.5. Proposition ([DKSm2 , Prop. 3.6]). The map gXm lifts to ^Xm+1 in hc* *m+1 "X if and only if [h(Xm )] vanishes in Hm+1 ( ; ssm CX ). 4.6. Relating the two obstructions. In order to see how the two obstructions we have described are related, we ne* *ed some more notation: For a pointed lattice of length n+1, let W "( , 0)denote the sub-(C, )-ca* *tegory of W ( , 0) obtained from skcn-1( , 0) by adding all unreduced null n-cubes* * (Def- inition 3.15). By Lemma 3.20, any (n-1)-allowable extension X0n-1: skcn-1( , 0* *) ! CX extends canonically to bX : W "( , 0)! CX . If in : sknW "( , 0)! W "* *( , 0) HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 23 and i : W "( , 0)! W ( , 0) are the inclusions, we thus have a commutative diagram in (C, )-Cat: in-1 skcn-1W"( , 0)_________//"W ( , 0) SSSSX0n-1S skcn-1i=Id|| SSSSSS |bX| fflffl| SSSSS))Sfflffl| skcn-1( , 0)____________//_CX X0n-1 Because is a lattice of length n + 1, W is n-dimensional. Furthermore,* * if we break up any chain in into disjoint sub-chains of length k and ` (k + ` = n * *+ 1), the resulting composite cube has dimension (k - 1) + (` - 1) = n - 1. Thus the only non-degenerate n-cubes in W are indecomposable in W (vinit, vfin), * *which implies that skcn-1( , 0) is in fact defined using the naive (n - 1)-skele* *ton (see Remark 3.8). Thus by adjointness (using Remark 1.28) we have: Xb W "( , 0)__________________//CX (4.7) i|| r|| fflffl| fflffl| W ( , 0)_____f____//coskcn-1CX = Pn-2CX Xn in which r is the fibration r(n-1)= p(n-1) of x1.23(b). Now let R be the (C, )-category of all reduced null (n - 1)-spheres (t* *hat is, boundaries of the reduced null n-cubes) in W . Thus: ( S @Inf if(u, v) = (vinit, vfin) (4.8) Rc(u, v) = fo2J o ; otherwise (in the notation of (3.22)) . 4.9. Fact. There is a homotopy cofibration sequence of (C, )-categories j i 0 (4.10) R -! W "( , 0)-! W ( , ) . Proof.By definition of a pointed lattice, all the n-cubes of W (and thus of W ( , 0)) are null cubes. Thus the map i : W "( , 0)! W ( , 0) is actua* *lly an isomorphism in all mapping slots except (u, v) = (vinit, vfin), where the * *n-cells attached via j provide the missing (necessarily reduced) null n-cubes. 24 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 4.11. Definition. Let be a pointed lattice of length n + 1, CX a (C, )-ca* *tegory, and define J as in Proposition 3.21. To each commuting square: ^h W"( , 0)______//CX | (4.12) i || r|| fflffl| fflffl| W ( , 0)_h__//Pn-2CX in (C, )-Cat, we assign the composite kn-2 . h in Hn( , ssn-1CX ). Den* *ote by Kn(CX ) the subset of Hn(Q, ssn-1CX ) consisting of all such elements k* *n-2 . h. Finally, define n : Kn(CX ) ! fo2J ssn-1CX (vinit, vfin) by assigning to (4* *.12) the homotopy class of the composite oe := (^h.j)(vinit, vfin) : R (vinit, vfin) ! C* *X (vinit, vfin). 4.13. Lemma. The map n is well-defined. Proof.Freudenthal suspension gives an isomorphism ~= ^ssW [R (vinit, vfin), CX (vinit, vfin)]hoC -! [ R , E 1 (ssn-1CX , n)]ho(C* *, )-Cat, so n may be equivalently defined by assigning to the composite kn-2 . h * *the extension e = oe in the following diagram: ^h W "( , 0)_______________//CX | i|| p|| fflffl| h fflffl| W _______________//_TTTPn-2CX | TTTTTT @|| TTTkn-2.hTTTTkn-2|| fflffl| TT)) fflffl| R _____e______//_______________________EG (ssn-1CX ,* * n) j where W @-! R -! W "( , 0) is the continuation of the cofibration sequen* *ce of (4.10). Here we used the fact that R is concentrated in the (vinit, vf* *in) slot, by (4.8). Note that the extension e (and thus oe = n(kn . h), the adjoint of e with * *re- spect to the ( , ) adjunction) is uniquely determined up to homotopy, since [ W "( , 0), E^ss1W(ssnCX , n + 1)] = 0 for dimension reasons. Our main result, Theorem A of the Introduction, is now a consequence of the following Theorem and Corollary: 4.14. Theorem. Given X0 : 0 ! M as in x3.17, the map n is a pointed correspondence between the set of elements of Kn(CX ) obtained from commuting squares of the form (4.7) and <> of (3.24) - that is, n(ff) = 0 * * if and only if ff = 0. HIGHER HOMOTOPY OPERATIONS AND COHOMOLOGY 25 Proof.By Proposition 4.5, the composite h(Xn-1) := kn-2 . [Xn-1 is the obstruc* *tion to extending bX to Xn : W ( , 0) ! CX , and since R _________oe_//_EG (ssn-1CX , n - 1) | j || `| fflffl| b fflffl|| W"( , 0)______X__________//CX55_______ ___________|_______ i || ______Xn_________p||____ fflffl|_______________ fflffl| W _________________//_Pn-2CX | "Xn-1 @ || kn-2|| fflffl| fflffl| R ______e_______//EG (ssn-1CX , n) commutes, with the left vertical column a cofibration and the right vertical co* *lumn a fibration sequence, the fact that e = 0 , oe = 0 implies that the composi* *te 0 = e.@ = kn-2.X[n-1 = h(Xn-1). Conversely, if Xn exists, then Xb.j = Xn.i.j * *= 0, so ` . oe = 0, and since ssn-1` is an isomorphism, oe = 0. 4.15. Corollary. The Dwyer-Kan-Smith obstruction class [h(Xn-1)] of Propositi* *on 4.5 is zero in Hn(W ; ssn-1CX ) if and only if the corresponding homotopy c* *lass FX0n-1 is null. Therefore, <> vanishes if and only if Kn(CX ) conta* *ins 0. 4.16. Remark. 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TURNER Department of Mathematics, University of Haifa, 31905 Haifa, Israel E-mail address: blanc@math.haifa.ac.il Department of Mathematics, Penn State Altoona, Altoona, PA 16601-3760, USA E-mail address: mwj3@psu.edu Department of Mathematics, Calvin College, Grand Rapids, MI, USA E-mail address: jturner@calvin.edu