Higher homotopy operations and cohomology

HIGHER HOMOTOPY OP ERA TIONS AND COHOMOLOGY D A VID BLANC, MARK W. JOHNSON, AND J AMES M. TURNER Abstract. W e explain how higher homotopy oper ations, defined top olo gically , may be identified under mild assumptions with (the la st of ) the Dwyer-Kan-Smith cohomolog ical obstr uctions to rectifying homotopy-comm utative diagrams. Intr oduction The first secondary homoto py op erations to b e defined w ere T o da brack ets, whic h app eared (in [T1]) in the early 1950’s – at ab o ut the s ame time as the secondary cohomology op erations o f Adem and Masse y (in [Ad ] and [MU]). The definition was later extended t o higher order homoto p y and cohomology op eratio ns (see [Sp, Ma, Kl]), which hav e b een used extensiv ely in algebraic top ology , star t ing with T o da’s o wn calculations of the homotop y groups of spheres in [T2 ]. In [BM], a “ top ological” definition o f higher homotopy o p erations based on the W - construction of Boardman and V ogt, w as giv en in the fo rm of an obstruction theory for rectifying diagrams. The same definition may b e used also for higher cohomology op erations. This w as recen tly mo difie d in [BC] t o take account o f the fact tha t, in practice, higher order op erat io ns, b oth in ho motop y and in cohomology , o c cur in a p ointe d contex t, whic h some what simplifies their definition and t r eat ment. Earlier, in [DKSm2], Dwy er, Kan, a nd Smith ga v e a n obstruction theory for rec- tifying a dia g ram ˜ X : K → ho T in the homotopy category of top olo gical spaces b y making it “infinitely-homotopy comm utative”: the precise statemen t inv olv es the simplicial function complexes map( ˜ X u , ˜ X v ) for all u, v ∈ O = Ob j ( K ), whic h constitute an ( S , O )-category C X (see § 3.11 and Section 4). Their results are th us stated in terms of ( S , O )-categories (simplicially enric hed categories with ob ject set O ). In particular, the obstructions tak e v alues in the corresp o nding ( S , O )- cohomology groups (see [D KSm1, § 2.1]). The purp os e of the presen t note is to explain the relation b et w een these t w o ap- proac hes. Because the W -construction, and thus higher op erations, are defined in terms of cubical sets, it is conv enien t to w ork cubically thro ughout. In this language, ( S , O )-cohomology is replaced b y the (equiv alen t) ( C , Γ)-cohomology (see § 2.25), and our main result (Theorem 4.14 b elow ) may b e stat ed roughly as follows : Assume giv en a directed graph Γ without lo ops (cf. § 3.1 3) of length n + 2, ha ving initial no de v init and terminal no de v fin , and let M b e a cubically enric hed p ointe d mo del category . Date : July 2 9 , 2008 ; revised Apr il 20, 2009. 1991 Mathematics Su bje ct Classific at ion. Primary: 55 Q35; seconda r y: 55N99, 55S2 0 , 1 8G55. Key wor ds and phr ases. Higher homotopy o p e r ations, S O -coho mology , homotopy-commutativ e diagram, rectificatio n, obstruction. 1 2 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER Theorem A . F or e ach p oin te d diagr am ˜ X : Γ → ho M , ther e is a natur al p o inte d c orr esp ondenc e Φ b etwe e n the p ossib l e values of the final D wyer-Kan-Smith ob struc- tion to r e ctifying ˜ X , i n the ( C , Γ) -c ohom olo gy gr oup H n (Γ , π n − 1 C X ) , and the n -th or der homotopy op er ation h h ˜ X i i , a subset of [Σ n − 1 ˜ X ( v init ) , ˜ X ( v fin )] . 0.1. R em a rk. The fact that Φ is p ointe d implies that, not surprisingly , the t w o differen t obstructions to r ectification v anish sim ultaneously . Our ob jectiv e here is to explicitly iden t ify eac h v alue of a higher homotopy op eration (with its usual indeterminacy) with a ( C , Γ)-coho mo lo gy class for Γ. In [BB], a relat io nship b et w een ( S , O )-cohomology and t he cohomolo g y o f a Π- algebra is describ ed. Since the lat t er is a purely algebraic concept, w e hope that to- gether with the presen t result this will provide a systematic w a y to apply homological- algebraic methods to interpret and calculate hig her homotopy a nd cohomolo gy o p er- ations. 0.2 . Notation. The category of compactly generated top olog ical spaces is denoted b y T , and that of p oin ted connected compactly generated spaces b y T ∗ ; their homotopy categories are denoted by ho T and ho T ∗ , resp ectiv ely . The catego ries o f (p oin ted) simplical sets will b e denoted b y S (resp., S ∗ ), those of groups, ab elian groups, and group oids by G p , A b G p , a nd G pd , resp ectiv ely . C at de notes the catego r y o f small categories. If hV , ⊗i is a mono ida l category , w e denote b y V - C at the collection of all (not necessarily small) categories enriche d ov er V (see [Bor2, § 6.2]). A category K is called p ointe d if it ha s a zer o obje ct 0 – that is, 0 is b oth initial and final. In suc h a K , a map factoring through 0 is called a nul l ( or zer o ) map, and since there is a unique suc h map b etw een any t w o ob jec ts, K is enric hed ov er p oin ted sets. 0.3. R em ark. It will b e conv enien t at times to w ork with non-unital categories – that is, categories whic h need not hav e iden tit y maps. These hav e b een studied in the litera t ur e under v a rious names, b eginning with the semi-c ate gories of V.V. V agner (see [V]). The enric hed v ersion appear s, e.g., in [BBM]. 0.4 . Organization. Section 1 prov ides a review of cubical sets and their homoto p y theory . Section 2 discusses cubically enric hed categories, as a replacemen t for t he ( S , O )-categories of D wye r and Ka n, and describ es their mo del category structure (Theorem 2.2 1). In Section 3 w e giv e a “top ological” definition of p oin te d higher homotop y op erations in terms of diagrams indexed b y certain finite categories called lattices. Fina lly , in Section 4 the Dwy er-Kan-Smith obstruction theory is describ ed and the main result (Theorem 4.14 and Coro lla ry 4.15) is prov ed. 0.5. A cknow le dgeme nts. This researc h w as supported b y BSF gran t 2 0 06039; the third author was a lso supp orted by NSF gran t DMS-020664 7 and a Calvin Researc h F ello wship (SDG). 1. Cubical se ts Ev en tho ug h the o bstruction theory of Dwy er, Kan, and Smith w as originally de- fined simplicially , for our purp o ses it app e ars more economical to w ork cubically . This is b ec ause cubical sets are the natural setting for the W -construction of Boardman and V ogt, whic h w as used fo r constructing higher homotopy op erations in [BM] and HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 3 [BC]. Since our goal is to iden tify these op erations with the cohomological obstruc- tions of D wye r-Kan-Smith, w e simplify the exp osition b y framing their theory in cubical terms as w ell. Because cubical ho motop y theory is less familiar than the sim- plicial vers ion, and the relev ant information and definitions a re scattered throughout the literature, we summarize them here. 1.1. D efinition. Let  denote the Box c ate gory , whose ob jects a re the abstract cub es {I n } ∞ n =0 (where I := { 0 , 1 } and I 0 is a single p oint). The mo r phisms of  are generated b y the inclusions d i ε : I n − 1 → I n and pro jections s i : I n → I n − 1 for 1 ≤ i ≤ n and ε ∈ { 0 , 1 } . One can iden tify  with a category of to p ological cub es, where I n corresp onds to [0 , 1] n (an n -fo ld pro duct o f unit in terv a ls), the linear map d i ε : [0 , 1] n − 1 → [0 , 1] n is defined ( t 1 , . . . , t n − 1 ) 7→ ( t 1 , . . . , t i − 1 , ε , t i , . . . , t n − 1 ), and s i : [0 , 1] n → [0 , 1] n − 1 is defined b y omitting t he i -th co ordinate. A contra v ariant functor K :  op → S et is called a cubic al set (or cubical complex), and w e write K n for the set K ( I n ) of n -cub es (or n - cells) of K . The ( i, ε )- fac e map d ε i : K n → K n − 1 and the i -th de gener a cy s i : K n − 1 → K n are induced b y d i ε and s i , resp ectiv ely . A cubical set K is called finite if all but finitely man y n -cub es of K are degenerate (that is, in the image of some s i ). The category of cubical sets is denoted by C . See [KP, I, § 5], [BH1, § 1], or [FRS]. Sev eral obv ious constructions carry ov er from simplicial sets: for example, the n - truncation functor τ n on cubical sets has a left adjoint, and comp osing the tw o yields the cubical n - skeleton functor sk c n : C → C . Th us sk c n K is generated (under the degeneracies) by the k -cub es of K for k ≤ n . 1.2. N ot ation. There is a standard em b edding of  in C , in whic h I n ∈  is take n to the standard n -cub e I n ∈ C (with one non- degenerate cell in dimension n , and all its faces). Applying sk c n to the standard ( n + 1)- cub e I n +1 , we obtain its b oundary ∂ I n +1 := sk c n I n +1 . By omitting the d ε i -face from ∂ I n +1 , we obta in the ( i, ε )- squar e horn ⊓ n,ε i . 1.3. R e m ark. There is also a v ersion of cubical sets without degeneracies, sometimes called semi-cubic al s ets , but these ar e not suitable for homotop y theoretic purp oses (cf. [An1]). On the other hand, Br own a nd Higgins hav e prop osed adding further “adjacen t degeneracies”, called c onne ctions (see [BH1, § 1] and [G M]). These ha v e pro v ed useful in v arious contexts (see, e.g., [An2 , BH2]). 1.4 . The cubical enric hmen t of C . As a functor category , all limits and colimits in C are defined lev elwise . In particular, the k -cubes o f a give n cubical set K ∈ C ( k ≥ 0) form a catego ry C K (under inclusions), and K ∼ = colim I k ∈C K I k . Ho w eve r, it turns out the pro ducts in C do not behav e w ell with resp ect to realiza- tion (see Remark 1.1 0 b elo w), so another monoidal op eration is needed: 1.5. Definition. If K and L are t wo cubical sets, their c ubic al tensor K ⊗ L ∈ C is defined K ⊗ L := colim I j ∈C K , I k ∈C L I j + k . This defines a symmetric monoidal structure ⊗ : C × C → C on cubical sets (see [J3, § 3]). 4 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER More generally , let hV , ⊗ i be a monoidal category with (finite) colimits – for example, hT , ×i , hS , × i , or hC , ⊗i – and assume we hav e “standard cub es” in V , defined by a ( f aithful) mono idal functor T : h  , × i → hV , ⊗i – that is, a compatible choice of “standard cub es” T I n in V . Giv en a (finite) cubical set K , for an y X ∈ V define X ⊗ K := colim C K T X , where the diagra m T X : C K → V is defined by T X I n := X ⊗ T I n . 1.6. Definition. F or hV , ⊗i as ab ov e, t he cubic al mapping c omplex map c V ( X , Y ) ∈ C is defined for any X , Y ∈ V by setting map c V ( X , Y ) n := Hom C ( X ⊗ T I n , Y ) , with the cubical structure inherited f r o m I n ∈  (cf. [K]). W e shall generally abbreviate map c V ( X , Y ) to V c ( X , Y ). In particular, when V is C itself, this mak es hC , ⊗ , I 0 , C c i in to a symmetric monoidal closed category (see [Bor2, § 6.1]). 1.7 . Comparison t o S . Cubical sets are related to simplicial sets by a pair of adjoin t functors (1.8) C T ⇋ S cub S . The triangulation functor T is defined T K := colim I n ∈C K ∆[1] n (compare D efinition 1.5), where ∆[1] n = ∆[1] × . . . × ∆[1] is the standard simplicial n -cub e . The cubic al singular functor S cub : S → C = S et  op is defined a dj oin tly b y ( S cub X )( I n ) := Hom S ( T I n , X ). This is a singular- realization pair in the sense of [DK4]; comp osing (1.8) with the usual adjoin t pair: (1.9) S |−| ⇋ S T yields a similar adjunction to top olog ical spaces. 1.10. R ema rk. Note that T : hC , ⊗i → hS , ×i is strongly monoidal (cf. [Bor2, § 6.1]), in that there is a natural isomorphism (1.11) T ( K ⊗ L ) ∼ = ( T K ) × ( T L ) . On the other hand, S cub : hS , × i → hC , ⊗i is not strongly monoidal, as w e now sho w: as a righ t adjoint, S cub comm utes with (lev elwise) pro ducts up to natural isomorphism, so S cub ( X × Y ) ∼ = S cub ( X ) × S cub ( Y ) . Th us, if S cub w ere strongly monoidal, one w ould ha v e a lev elwise isomorphism S cub ( X ) ⊗ S cub ( Y ) ∼ = S cub ( X ) × S cub ( Y ) . Note this is unlik ely , since a n n -cub e of K ⊗ L corresponds to a pair consis ting of a j -cub e of K (for some 0 ≤ j ≤ n ) and an ( n − j )-cub e o f L , while a n n -cub e of K × L corresp onds to a pair consis ting of an n -cub e of K and an n -cub e of L . In fact, K × L is in general not ev en homotop y equiv alen t to K ⊗ L for K , L ∈ C , – for example, T ( I 1 × I 1 ) ≃ S 1 in S while T ( I 1 ⊗ I 1 ) ∼ = ∆[1] × ∆[1] (see [J1, § 1, Remark 8]). HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 5 Nev ertheless, since I 0 is bo t h terminal in C and the unit for ⊗ , the pro jec tions π K : K ⊗ L → K ⊗ I 0 ∼ = K and π L : K ⊗ L → I 0 ⊗ L ∼ = L induce a natur a l map (1.12) ϑ : K ⊗ L → K × L , whic h is symmetric monoidal in the sense that it comm utes with the obv ious asso- ciativit y and switc h-map isomorphism s. 1.13. F a ct ([J3, § 3]) . F or any L ∈ C , the f unctor − ⊗ L pr eserves monom orphisms in C . 1.14. R emark. Note t ha t − ⊗ I n preserv es colimits, since it has a right adjoin t (defined b y constructing the cubical set of maps b et w een t w o cubical sets as one do es in S – see [J3 , § 4]). Finally , observ e that the cubical mapping complex for S (Definition 1.6) is simply map c S ( − , − ) = S cub map S ( − , − ). 1.15 . The mo d el category. Cubical sets were used quite early on as mo dels for top ological spaces – see [Se ], [EM], [Mu], [Mc1], [P1, P2], and esp ecially [K1 , K2]. How ev er, it w as Grothendiec k,in [G], who suggested that more generally presheaf categories mo deled on certain “test categories” D can serv e a s mo dels for the homotop y catego ry of top o logical spaces. Cisinski, in his thesis [C], carried out this program for D =  (see also t he exp o sition in [J3]). The mo d el catgeory structure is v ery similar to t he analogous one for simplicial sets ( D = ∆): 1.16. Definition. A map f : K → L in C is a) a we ak e quivalenc e if T f : T K → T L is a weak equiv a lence in S (or equiv alently , if | T f | is a w eak eq uiv alence of top ological spaces); b) a c ofibr a tion if it is a monomorphism. c) a fibr ation if it ha s the righ t lifting prop erty (RLP) with respect to all acyclic cofibrations (i.e., those whic h are also we ak equiv alences) – that is, if in all comm uting squares in C : (1.17) A g / / i   K f   B ˜ h > > h / / L where i is an acyclic cofibration, a map ˜ h : B → K exists making the full diagram comm ute. The mo del category define d here is prop er, b y [J3, Theorem 8.2]. 1.18. Definition. The cubic al spher es are S n := S cub (∆[ n ] /∂ ∆[ n ]) for n ≥ 1, with the obv ious basep oint. These corepresen t the homotopy gr oups π n ( − ) := [ S n , − ] ∗ . Similarly , S 0 := I 0 ∐ {∗} corepresen ts π 0 , and a map f : K → L in C ∗ is a w eak equiv alence if and only if it induces a π n -isomorphism fo r a ll n ≥ 0. Note that we may define the fundamen tal group o id ˆ π 1 K of an unp o inted cubical set K ∈ C as for simplicial sets or top olo gical spaces (cf. [Hig, Chapter 2]). 6 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER 1.19. R em a rk . In ana lo gy with the case of simplicial sets (see [GJ, Ch. I]) o ne can sho w that cofibrations whic h are w eak equiv alences ar e the same as the ano dyne maps – t ha t is the closure of the set of inclusions of the form (1.20) i : ⊓ n,ε i ֒ → I n +1 (see § 1.2) under cobase change, retracts, copro duc ts, and coun table comp ositions (see [J3, § 4]). F urthermore, the fibran t ob jects and the fibratio ns in C can a lso b e c haracterized by Kan conditions – ha ving the RLP with resp ect to maps of the form (1.20) (see [K1] and [J3 , Theorem 8.6]). As noted ab ov e ( § 1.6), C is a symmetric monoidal closed category (enric hed o v er itself ), with cubical mapping complexes C c ( − , − ). As shown in [J1, § 3]), it also satisfies the cubical analogue of Quillen’s Axiom SM7 (cf. [Q, I I, § 2]), so C deserv es to b e called a cubic al mo del category . In particular, if L is a fibra nt (Ka n) cubical set, the function complex C c ( K , L ) is fibran t, to o, for any (necessarily cofibrant) K ∈ C . Finally , the fo llo wing result sho ws that C indeed serv es as a mo del for the usual homotop y category of to p ological spaces : 1.21. Prop osition (Cf. [J3 , Theorem 8.8]) . The adjoint functors of (1.8) induc e e quivalenc es of homotopy c ate gorie s ho C ∼ = ho S (so to gether with the p air ( 1.9) , we have ho C ∼ = ho T ). Note that since I 0 is a final o b ject in C , the under category C ∗ := I 0 / C of p oin ted cubical sets constitutes a p ointed vers ion of C , and w e ha v e: 1.22. F act. T her e is a mo del c ate gory structur e on C ∗ , with the same we ak e quiva- lenc es, fibr ations, and c ofi b r ations as C . Pr o of. See [Ho, Prop os ition 1.1.8 ].  1.23 . Spherical mo de l categories. Lik e man y other mo del categories, C ∗ enjo ys a collection of additional useful prop erties tha t w ere axiomatized in [Bl, § 1] under the name o f a spheric al mo del category . This means that: (a) C ∗ has a set A of spheric al obje cts : cofibran t homotopy cogroup ob jects (namely , the cubical spheres A = { S n } ∞ n =1 – Definition 1.18). F urthermore, a map f : K → L in C ∗ is a w eak equiv alence if and o nly if [ A, f ] is an isomorphism for all A ∈ A . (b) Each K ∈ C ∗ has a functorial Postnikov tower o f fibrations: (1.24) . . . → P n K p ( n ) − − → P n − 1 K p ( n − 1) − − − → · · · → P 0 K , as w ell as a w eak equiv alence r : K → P ∞ K := lim n P n K and fibrations r ( n ) : P ∞ K → P n K suc h that r ( n − 1) = p ( n ) ◦ r ( n ) for all n , and r ( n ) # : π k P ∞ K → π k P n K is an isomorphism for k ≤ n and zero for k > n . (c) F or ev ery gro up oid Λ, there is a functorial classifying obje ct B Λ with B Λ ≃ P 1 B Λ and fundamen ta l group oid ˆ π 1 B Λ ∼ = Λ, unique up to homoto p y . (d) Given a group oid Λ and a Λ-mo dule G (t hat is, an ab elian group ob ject o v er Λ), for each n ≥ 2 there is a functoria l extende d G -Eil e nb er g-Mac L ane HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 7 obje ct E = E Λ ( G, n ) in C ∗ /B Λ, unique up to homotop y , equipped with a section s for ( r (1) ◦ r ) : E → P 1 E ≃ B Λ, suc h that π n E ∼ = G as Λ-mo dule s and π k E = 0 for k 6 = 0 , 1 , n . (e) F or ev ery n ≥ 1, there is a f unctor that assigns t o eac h K ∈ C ∗ a homotopy pull-bac k square (1.25) P n +1 K PB p ( n +1) / /   P n K k n   B Λ / / E Λ ( M , n + 2) called an n -th k -inv ariant squar e f o r K , where Λ := ˆ π 1 K , M := π n +1 K , and p ( n +1) : P n +1 K → P n K is the giv en fibration of the P ostnik ov tow er. The map k n : P n K → E Λ ( M , n + 2 ) is called the n -th (functorial) k - inva riant for K . 1.26. Prop osit ion. Th e c ate go ry C ∗ is spheric al. Pr o of. All the prop e rties for C ∗ follo w from F act 1.22, and the ana logous results for S ∗ or T ∗ (see [BJT, Theorem 3.1 5 ]). Note that homotopy groups for cubical sets app ear in [K1, K2], while (minimal, and thus non-functorial) Postnik o v tow ers for cubical sets we re constructed b y P ostnik ov in [P1, P2]. F or functorial cubic al P ostnik ov tow ers, let the n -cosk eleton functor cosk c n : C → C be the righ t adjoin t to sk c n , with r ( n ) : Id → cosk c n the ob vious nat ura l transformation, and similarly for C ∗ . By construction, r ( n ) is an isomorphism in dimensions ≤ n . If K ∈ C ∗ is fibran t, so is cosk c n K , a nd π i cosk c n K = 0 for i > n , since sk c n S i = ∗ for i > n . Thus if K ′ → K is a functor ia l fibrant replacemen t, and w e c hange (1.27) K ′ . . . → cosk c n +1 K ′ → cosk c n K ′ → cosk c n − 1 K ′ . . . functorially in to a tow er o f fibrations, we obtain (1.2 4 ). F or (strictly) functorial Eilen berg- Mac Lane o b jects , use [BDG, Prop. 2 .2], and apply S cub . F or f unctorial k -in v ariants in C ∗ , use the construction in [BDG, § 5-6] (whic h w orks in C ∗ , to o).  1.28. R emark. In general, the maps cosk c n K → cosk c n − 1 K in (1.27) (adjoin t to the inclusion of sk eleta) are not fibrations (though the original construction of Kan, when applied to a fibrant cubical set K , yields a tow er of fibrations with no further mo dification – see, e.g., [GJ, VI, § 2]). Ho w eve r, if w e a re only in terested in a sp ecific P ostnik ov section P n , as lo ng as K is fibran t w e can use cosk c n +1 K as a fibrant mo d el for P n K , a nd need only mo dify the next section if w e w a n t p ( n +1) : P n +1 K → P n K to b e a fibrat io n. 2. Cubicall y e nriched ca tegories In [DK2], D wy er a nd Kan sho w ed ho w any mo del catego r y (mor e generally , a n y small category M equipped with a class of we ak equiv alences) can b e enric hed b y simplicial function complexe s, so that the resulting simplic ially enric hed category en- co des the homotop y theory of M (see R emark 3.10 b elo w). Th us the category s C at 8 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER of simplicial small categories can b e t ho ugh t of as a “univ ersal mo del category”, pro- viding a setting fo r a “homotopy theory o f homotop y theories”. Other suc h univ ersal mo dels w ere later pro vided in [DKSm2, § 7], [R], and [Be]. An imp ortan t sub cat ego ry of s C at consists of t ho se simplicial categories with a fixed set of ob jects. This is a sp ecial case o f the followin g: 2.1. Definition. F or an y set O , denote by O - C at the category of all small categories D with O := Ob j D . More g enerally , assume Γ ∈ O - C at is a small category , p ossibly non-unital, and let hV , ⊗ i b e a mono ida l category . A ( V , Γ)- c ate gory is a category D ∈ O - C at enric hed ov er V , with mapping ob jec ts map v D ( − , − ) ∈ V , suc h that (2.2) Hom Γ ( u, v ) = ∅ ⇒ map v D ( u, v ) is the initial ob ject in V . Th us when V is p o in ted, w e require map v D ( u, v ) = ∗ whene v er Hom Γ ( u, v ) = ∅ . The category of all ( V , Γ)-categories will b e denoted b y ( V , Γ)- C at . The mor- phisms in ( V , Γ)- C at a re enric hed functors whic h are the iden tity on O . When Hom Γ ( u, v ) is nev er empt y (so that w e ma y disregard condition (2.2)) w e write ( V , O )- C at instead of ( V , Γ)- C at . Dwy er a nd Kan call these O - diagr ams in V . 2.3. R emark. If Γ is non-unital, Hom Γ ( u, u ) ma y b e empt y , in whic h case map v D ( u, u ) will b e empt y , if V = S et or S . This is allow ed in the enric hed v ersion of semi- categories (see R emark 0.3). How ev er, the discussion b e lo w can b e readily carried out in t he con text o f ordinary (enric hed) catego r ies, at the cost of pay ing atten tion to units. Thus if V is p ointed, Hom( u, u ) has (at least) tw o maps: the iden tit y and the zero map; t hese will coincide o f u is the zero ob ject. W e shall in fact concen trate on the case where Γ has no self-maps u → u – e.g., a non-unital par tially ordered set. The main examples of hV , ⊗i to k eep in mind are h S et, ×i , h G p, ×i , h G pd, ×i , hS , ×i , a nd hC , ⊗i . 2.4 . ( S , O ) -categories. Although w e shall b e mainly concerned with ( C , Γ)- categories, w e first recall the more familiar simplicial v ersion: Note that when V = S , an ( S , Γ)-category can b e thought of a s a simplicial ob ject o v er O - C at (or ( S et, Γ) - C at ). Th us eac h M • ∈ ( S , O )- C at is a simplicial catego r y with fixed ob ject set O in each dimension, and all face and degeneracy f unctors are the iden tit y on ob jects (cf. [DK1 , § 1.4]). 2.5. F act. The for getful functor U : C at → D i G to the c ate g o ry of dir e cte d gr ap hs has a le ft adjoint F : D i G → C at , the free categor y functor (cf. [Ha] ). 2.6. Definition. A simplicial category E • ∈ ( S , O )- C at is fr e e if eac h category E n , and eac h degeneracy functor s j : E n → E n +1 , is in t he essen tial image of the functor F . The pair of adjoin t functors of F act 2.5 defines a comona d F U : C at → C at , and th us f or each small category D , an augmen ted simplicial category E • → D with E n := ( F U ) n +1 D . If D ∈ ( S et, Γ)- C at , then E • ∈ ( S , Γ)- C at . W e denote this canonical fr e e s implicial r esolution o f D by F s D . 2.7. R em ark. In [D K1, § 1], Dwye r and Kan define a mo del category structure on ( S , O )- C at (also v alid for ( S , Γ)- C at ), whic h turns out t o b e a r esolution mo del HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 9 c ate gory in the sense of [Bo u] ( see also [J2], [DKSt, § 5] and [BJT , § 2]). The spherical ob jects for ( S , O )- C at (cf. § 1.23(a)) are o b jects o f the form M • := S n ( u,v ) for n ≥ 1 a nd Hom Γ ( u, v ) 6 = ∅ , defined by: (2.8) M ( u ′ , v ′ ) = ( S n for u ′ = u and v ′ = v ∗ otherwise, One can also show that ( S , O )- C at and ( S , Γ)- C at are spherical – t ha t is, endo w ed with the additional structure described in § 1.23 (of which only the existence of mo dels is guaran teed in a resolution mo del category). 2.9 . The mo del categor y ( C , Γ) - C at . In the case of ( C , Γ)-catego r ies, the situatio n is somewhat complicated by the fact that they cannot simply b e view ed as cubical ob jects in C at , b ecause ⊗ , and thus the comp osition maps, are not defined dimension wise (see Remark 1 .1 0). Berger and Mo e rdijk hav e defined a mo de l categor y structure for algebras o v er coloured op erads in a suitable symmetric monoidal mo del cat ego ry , whic h applies in part icular to ( C , Γ)- C at (see [BM2], and compare [BM1]). Ho w ev er, in this pap e r w e only need to conside r ( C , Γ)-categories for a sp ecial t ype of category Γ, f or whic h it is easy to describe an explicit mo del cat ego ry structure in whic h W Γ is cofibran t: 2.10. D efinition. A small non-unital category Γ will b e called a quasi-lattic e if it has no self-maps; in this case there is a par t ia l ordering on O = Ob j ( Γ ) , with u ≺ v if and only if Hom Γ ( u, v ) 6 = ∅ , and w e require in additio n that Γ b e lo c al ly fin i te in the sens e that for any u ≺ v in O , the interv al Seg[ u, v ] := { w ∈ O | u  w  v } is finite. 2.11. Example. The simplest example is a line ar lattic e of length n + 1, whic h w e denote b y Γ n +1 : this consists of a single comp osable ( n + 1)-chain: v init = ( n + 1 ) φ n +1 − − − → n φ n − → ( n − 1 ) → · · · → 2 φ 2 − → 1 φ 1 − → 0 = v fin . Another example is a comm uting square: v init φ ′ / / φ ′′   v ′ ψ ′   v ′′ ψ ′′ / / v fin Observ e that fo r categories o f diagrams indexed on a directed Reedy category (i.e., one for which the “inv erse sub cat ego ry” is trivial), the R eedy mo del structure (cf. [Hir, § 15.2.2]) a grees with t he pro jectiv e mo del structure. In this situation, cofibrations of diagrams are t ho se morphisms whose “lat ching maps” are all cofibrations in the target category , while fibrations and w eak equiv alences of diagrams are defined ob jectwis e. Our curren t con text is sufficien tly similar to allo w an analogous inductiv e argument, dep ending on the follo wing analog of Reedy’s latching ob jects and maps: 2.12. Definition. Given a quasi-lattice Γ, a map F : A → B in ( C , Γ)- C at , and u ≺ v in O , the c o mp osition c ate gory ( J A , B ( u,v ) , < ) is a partia lly ordered set, whose ob jects a re pairs h ω , X i , where ω is a chain h u = w 0 ≺ w 1 ≺ . . . ≺ w k − 1 ≺ w k = v i 10 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER in h O , ≺i , and the index X is either A or B . W e omit the copy of the trivial chain h u ≺ v i indexed by B . The partial order is defined by setting h ω , X i ≤ h ω ′ , X ′ i whenev er ω ′ is a (not necessarily prop er) subchain of ω , and either X = X ′ or X = A , X ′ = B . The corresp onding c omp osition diagr am D = D A , B ( u,v ) : J A , B ( u,v ) → C is defined b y sending h ω , X i to N k j =1 X c ( w j − 1 , w j ). The morphisms are generated b y the follo wing t w o t yp es of maps: (i) If ω ′ is obtained from ω b y o mitting in ternal no de w j (1 < j < k ), the map D h ω , X i → D h ω ′ , X i is Id ⊗ · · · ⊗ cmp X ( w j − 1 ,w j ,w j +1 ) · · · ⊗ Id, where cmp X ( w j − 1 ,w j ,w j +1 ) : X c ( w j − 1 , w j ) ⊗ X c ( w j , w j +1 ) → X c ( w j − 1 , w j +1 ) is the cubical comp osition map in X ∈ { A , B } ; (ii) The map D h ω , A i → D h ω , B i is N k i =1 F ( w i − 1 ,w i ) . Note tha t F ( u,v ) : A c ( u, v ) → B c ( u, v ), tog ether with the composition maps of B ending in B c ( u, v ), induce a map ϕ ( u,v ) : colim D A , B ( u,v ) → B c ( u, v ). In par ticular, when Seg [ u, v ] = { u, v } is minimal, colim D A , B ( u,v ) is simply A c ( u, v ) and ϕ ( u,v ) is F ( u,v ) : A c ( u, v ) → B c ( u, v ). W e now provide the details of the mo del cat ego ry structure on ( C , Γ) - C at – in ter alia, in order to a llow the r eader to v erify t hat the construction works in the non-unital setting: 2.13. Lemma. If Γ is a quasi-lattic e , the c a te gory ( C , Γ) - C at has al l lim its and c olimits. Pr o of. F or an y small category Γ, the limits in ( C , Γ)- C at are constructed b y taking the limit at each ( u, v ) ∈ O 2 , with comp ositions defined for the pro duc t Q i ∈ I A i b y the ob vious maps: ( Y i ∈ I A i [ u, w ]) ⊗ ( Y i ∈ I A i [ w , v ]) → Y i ∈ I ( A i [ u, w ] ⊗ A i [ w , v ]) cmp ( u,w,v ) − − − − − − → Y i ∈ I A i [ u, v ] , and similarly for the other limits. F or t he colimits, note that ⊗ is defined as a colimit (cf. Definition 1.5), so it comm utes with colimits in C . F or ( C , Γ)-categories { A i } i ∈ I , the copro duct D := ` i ∈ I A i is defined by induction on the cardinality of Seg[ u, v ] in h O , ≺i . When Seg[ u , v ] = { u, v } is minimal, w e let D ( u, v ) := ` i ∈ I A i ( u, v ). In general, set D ( u, v ) := a i ∈ I A i ( u, v ) ∐ a u ≺ w ≺ v D ( u, w ) ⊗ D ( w , v ) , with the obv ious (ta utological) comp osition on the righ t-hand summands. No w giv en maps F : A → B and G : A → E in ( C , Γ)- C at , the pushout P O is once more defined b y induction on the cardinality of Seg [ u, v ], as follows : In the initial case, when Seg[ u , v ] is minimal, P O ( u , v ) is simply the pushout of E ( u, v ) ← A ( u, v ) → B ( u , v ) in C . In the induction step, w e let J = J P O ( u,v ) denote the union of the comp o sition categories J A , B ( u,v ) , J A , E ( u,v ) , and J B , P O ( u,v ) (see Definition 2.12). Th us the o b jects of HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 11 J are pairs h ω , X i , where ω is a c hain h u = w 0 ≺ w 1 ≺ . . . w k = v i and X ∈ { A , B , E , P O } , a gain omitting h u ≺ v , P O i . Again J is a partia lly ordered set, with the order r elatio n defined to b e the union of t ho se for ( A , B ), ( A , E ), and ( B , P O ). The comp osition diagrams D A , B ( u,v ) , D A , E ( u,v ) , D B , P O ( u,v ) , and D E , P O ( u,v ) fit together to f orm a comp os ition diagram D P O ( u,v ) : J P O ( u,v ) → C . The last t wo diagra ms a re w ell-defined, b ecause w e omit the trivial chain h u ≺ v , P O i , a nd all other v alues of D B , P O ( u,v ) and D E , P O ( u,v ) ha v e already b e en defined b y our induction assumption. W e no w let P O ( u , v ) b e the colimit in C of the diagram D P O ( u,v ) : J P O ( u,v ) → C . The constructions of the copro ducts and pushouts implies that all colimits exist in ( C , Γ)- C at , b y the dual o f [Bor1, Thm. 2 .8 .1 & Prop. 2.8.2].  2.14. Definition. Let Γ b e a quasi-lattice, and let A and B b e ( C , Γ)-catego ries. A map F : A → B in ( C , Γ)- C at is (a) a we ak e quivalen c e if F ( u,v ) : A c ( u, v ) → B c ( u, v ) is a w eak equiv alence in C (see § 1.15) for any u ≺ v in O . (b) a fibr ation if F ( u,v ) : A c ( u, v ) → B c ( u, v ) is a (Kan) fibration in C for all u ≺ v in O . (c) a (acycli c ) c o fibr ation if for all u ≺ v in O the maps F ( u,v ) : A c ( u, v ) → B c ( u, v ) a nd ϕ ( u,v ) : colim D A , B ( u,v ) → B c ( u, v ) a r e (acyclic) cofibrations in C . 2.15. R emark . A straightforw ard induction sho ws that the acyclic cofibrations so defined are precisely tho se cofibrations whic h are weak equiv alences. The follo wing lemmas sho w that these c hoices yield a mo del category structure on ( C , Γ)- C at : 2.16. Lemma. I f Γ is a quasi-lattic e, F : A → B is a c ofibr ation and P : D → E is an fibr ation in ( C , Γ) - C at , an d either F or P is a we ak e quivale nc e, then ther e is a lifting ˆ H in any c om mutative sq uar e (2.17) A G / / F   D P   B H / / ˜ H > > E . Pr o of. W e c ho ose ˜ H ( u,v ) : B c ( u, v ) → D c ( u, v ) b y induction on the car dina lity of the in terv a l Seg[ u, v ] in h O , ≺i : When Seg [ u, v ] = { u, v } is minimal, w e simply c hoo se a lift ˜ H ( u,v ) in: A c ( u,v ) G ( u,v ) / / F ( u,v )   D c ( u,v ) P ( u,v )   B c ( u,v ) H ( u,v ) / / ˜ H ( u,v ) 7 7 E c ( u,v ) using the fact that F ( u,v ) is a cofibratio n a nd P ( u,v ) an acyclic fibration in C (see (1.17) ab o v e). 12 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER In the induction step, a ssume w e hav e c hosen compatible lifts ˜ H ( u ′ ,v ′ ) for a ll prop e r subin terv a ls Seg [ u ′ , v ′ ] ⊂ Seg [ u, v ]. These yield a ma p ˆ G making the follo wing solid square comm ute in C : colim D A , B ( u,v ) ϕ ( u,v )   ˆ G / / D c ( u,v ) P ( u,v )   B c ( u, v ) H ( u,v ) / / ˜ H ( u,v ) 6 6 E c ( u,v ) and since ϕ ( u,v ) is a cofibration by Definition 2.14 , and P ( u,v ) is a n acyclic fibration b y assumption, the lifting ˜ H ( u,v ) exists. The same argumen t sho ws that there exists a lift ing in (2.17) when F : A → B is an acyclic cofibration and P : D → E is a fibration.  2.18. Lemma. If Γ is a quasi-lattic e, any map F : A → B in ( C , Γ) - C at factors as: (2.19) A I   @ @ @ @ @ @ @ F / / B D P ? ? ~ ~ ~ ~ ~ ~ ~ wher e I is a c ofibr ation and P is a fib r ation; and we c an r e quir e either I or P to b e a we ak e quivalenc e. Pr o of. 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, c ho ose an y factorization: A c ( u,v ) I ( u,v ) # # F F F F F F F F F F ( u,v ) / / B c ( u,v ) D c ( u,v ) P ( u,v ) ; ; x x x x x x x x x where I ( u,v ) is an acyclic cofibration and P ( u,v ) is a fibratio n in C . No w assume b y induction tha t we ha v e c hosen compatible fa cto r izat io ns (2.20) A c ( u,w ) ⊗ A c ( w, v ) ζ A ( u,v ) / / I ( u,w ) ⊗   I ( w,v )   Col ( A ) ( u, v ) ω A / / φ ( u,v )   A c ( u,v ) η ( u,v )   D c ( u,w ) ⊗ D c ( w, v ) ζ D ( u,v ) / / P ( u,w ) ⊗   P ( w,v )   Col ( D ) ( u, v ) θ ( u,v ) / / ψ ( u,v )   P O ( u,v ) ξ ( u,v )   B c ( u,w ) ⊗ D c ( w, v ) ζ B ( u,v ) / / Col ( B ) ( u, v ) ω B / / B c ( u,v ) where each cubical set Col ( E ) ( u, v ) (f or E = A , B , D ) is the colimit o v er all prop er subin terv a ls Seg[ u ′ , v ′ ] ⊂ Seg [ u, v ] and u ′ ≺ w ≺ v ′ of the diagr a m of comp osition maps cmp E ( u ′ ,w ,v ′ ) : E c ( u ′ , w ) ⊗ E c ( w , v ′ ) → E c ( u ′ , v ′ ) . HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 13 in eac h row, ζ E : E c ( u,w ) ⊗ E c ( w, v ) → Col ( E ) ( u, v ) is the structure map for the colimit, while ω E : Col ( E ) ( u, v ) → E c ( u,v ) is induced by the comp ositions. The cubical set P O ( u,v ) is the pushout of the upp er right-hand square, with structure maps η ( u,v ) and θ ( u,v ) , and ξ ( u,v ) is induced on the pushout by F ( u,v ) and the maps ψ ( u,v ) (from the naturality of the colimit) and ω B . Note that the map I ( u,w ) ⊗ I ( w, v ) is an acyclic cofibration in C (see F act 1.13), so the induced map φ ( u,v ) is, to o, as is η ( u,v ) , by cobase c hange. The map P ( u,w ) ⊗ P ( w, v ) , as well as the induced map ψ ( u,v ) , comes from the compatible factorizat io ns (2.20). Finally , c ho ose a factorizatio n P O ( u, v ) ζ ( u,v ) & & L L L L L L L L L L ξ ( u,v ) / / B c ( u, v ) D c ( u, v ) P ( u,v ) 9 9 s s s s s s s s s s where ζ ( u,v ) is an acyclic cofibration and P ( u,v ) is a fibratio n in C . This defines the cubical set D c ( u, v ), whic h is equipped with compo sition maps cmp ( u,w ,v ) := ζ ( u,v ) ◦ θ ( u,v ) ◦ ζ D ( u,v ) . Setting I ( u,v ) := ζ ( u,v ) ◦ η ( u,v ) yields the required acyclic cofibration, and since ξ ( u,v ) is induced by F , w e hav e P ( u,v ) ◦ I ( u,v ) = F ( u,v ) , as required. The same construction, mutatis mutandis , yields a factorization (2.19) where I is a cofibration a nd P an acyclic fibration.  2.21. Theorem. If Γ is a q uasi - l a ttic e, Definition 2.14 pr ovides a mo del c ate go ry structur e on ( C , Γ) - C at . Pr o of. The catego r y ( C , Γ)- C at is complete and co complete by Lemma 2.13. The classes of w eak equiv alences and fibrations a re clearly closed under comp ositions, and include all isomorphisms. The same ho lds for cofibrations b y an induction argumen t. Also, if t w o out of the three maps F , G , and G ◦ F are we ak equiv alences, so is the third. The lifting prop erties for (co)fibrations are in Lemma 2.16, and the factorizations are give n by Lemma 2.18.  As exp ected, the t w o ke y types o f ( V , Γ)- cat ego ries ar e relat ed by suitable functors (compare [Bor2, Prop. 6.4.3]): 2.22. Prop osition. F or any quasi-la ttic e Γ , the functors T : C → S and S cub : S → C of (1.8) extend to func tors ( C , Γ) - C at ⇋ ( S , Γ) - C at . F uthermor e, this is a s tr ong Quil l e n p air (cf. [Hir, § 8.5.1] ) , and desc ends to an adjunction at the le v e l of homotopy c ate gories. Pr o of. The functor T extends to ( C , Γ)- C at b y (1.11). F or S cub , given A s ∈ ( S , Γ)- C at , with comp osition ξ : A s ( u, w )) × A s ( w , v ) → A s ( u, v ) w e define the comp osition map cmp ( u,w ,v ) : S cub ( A s ( u, w )) ⊗ S cub ( A s ( w , v )) → S cub ( A s ( u, v )) for the ( C , Γ)-category S cub A s to b e the comp osite S cub ξ ◦ ϑ (see (1 .12)). As S cub is a strong right Quillen functor, it follo ws from the definitions that the exte nsion is also strong right Quillen.  14 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER 2.23 . Semi-spherical st ructure on ( C , Γ) - C at . The discussion ab o ve , including the mo del category structures, is v alid when we replace C or S b y their p oin ted v ersions (see [Ho, Prop osition 1.1.8]). Moreov er, ev en though we cannot construct en try-wise spheres for ( C , Γ)-categories as in (2.8), the category ( C , Γ )- C at may b e called semi-spheric al , in the sense of having the rest of the spherical structure described in § 1.23, as follows : 2.24. Definition. Giv en a quasi-latt ice Γ and a ( C ∗ , Γ) - category A , its fundamental gr oup oid is the ( G pd, Γ)-category obtained b y applying t he fundamen tal gro up oid functor ˆ π 1 to A . Not e t hat b ecause ˆ π 1 : C → G pd factors through T : C → S , using (1.1 1) w e see that ˆ π 1 A is indeed a ( G pd , Γ ) - category (cf. [Bo r2, Prop. 6.4.3]). Similarly , fo r eac h n ≥ 2 the functor π n , applied en trywis e to A , yields a ( G p, Γ)- category , whic h is actually a ( ˆ π 1 A - M od, Γ)- catego ry (see Definition 2.1). Note that, as for top ological spaces, π n A is a mo dule o v er ˆ π 1 A . a) Eac h ( C ∗ , Γ) - category A has a functoria l P ostnik ov to w er, obtained b y apply- ing the functors P n of § 1.24 to eac h A c ( u, v ), and using P n ( A ( u, v )) ⊗ P n ( A ( v , w )) → P n ( P n ( A ( u, v )) ⊗ P n ( A ( v , w ))) ∼ = P n ( A ( u, v ) ⊗ A ( v , w )) → P n ( A ( u, w )) . b) F or ev ery ( G pd, Γ)-category Λ, there is a functorial classifying obje c t B Λ ∈ ( C ∗ , Γ) - C at . c) Give n a ( G pd, Γ)-catego ry Λ, and a Λ-mo dule G (i.e., an ab elian g r o up ob ject in ( S et, Γ)- C at/ Λ), fo r eac h n ≥ 2 there is a functorial extende d G -Eile n b er g- Mac L ane obje ct E Λ ( G, n ) in ( C ∗ , Γ)- C at/B Λ. d) F or n ≥ 1, there is is a functorial k -inv ariant square for A as in (1.25 ). All t hese pro p erties are straigh tforw ard fo r ( S ∗ , Γ) - C at (b y a pplying the a na logous functors fo r S ∗ comp onen t wise ), and they may b e transfered to ( C ∗ , Γ)- C at using Prop osition 2.2 2. 2.25. D efinition. Give n a ( G pd, Γ)- category Λ, a Λ-mo dule G , a ( C ∗ , Γ) - category A , and a twisting map p : A → B Λ, w e define the n -th ( C , Γ)- c ohomolo gy g r oup of A with co efficien ts in G to b e H n Λ ( A , G ) := [ A , E Λ ( G, n )] ( C , Γ)- C at/B Λ . 2.26. R emark. T ypically , w e ha v e Λ = ˆ π 1 A , with the ob vious map p . More generally , in [D K Sm1] Dwy er, Kan, a nd Smith give a definition of the ( S , O )- cohomology of any ( S , O )-category with co effi cien t s in a Λ- mo dule G ; and there is also a relativ e v ersion, for a pair ( A , B ) (cf. [DKSm1, § 2.1]). It is straig h tforw ard to v erify that the tw o definitions of cohomology coincide (when they are both defined) under the corresp ondence of Prop osition 2.2 2. 3. La ttices and higher homotopy ope ra tions W e can no w define higher homotop y op erations as obstructions to rectifying a homotop y comm utative diagram X : K → ho T , using the approac h of [BM ], with the mo dification in the p oin ted case given in [BC ]. F or this purp ose, it is conv enien t to w ork with a sp ecific cofibrant cubical resolution of the indexing category K . W e need mak e no sp ecial assum ptions ab out K at this stage. HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 15 Boardman and V o gt originally defined t heir “ ba r construction” W K t o p ologically (see [BV, I II, § 1]). The ( C , O )-v ersion ma y be desc rib ed as follo ws: 3.1. Definition. The W-c onstruction on a small category K with O = Ob j K is the ( C , O )-category W K , with the cubical mapping complex W K ( a, b ) f or ev ery a, b ∈ Ob j ( K ), constructed a s follo ws: F or ev ery comp osable sequence (3.2) f • = ( a = a n +1 f n +1 − − → a n f n − → a n − 1 . . . a 1 f 1 − → a 0 = b ) of length n + 1 in K , there is a n n - cub e I n f • in W K ( a, b ), sub jec t to t w o conditions: (a) The i -th 0-face of I n f • is iden tified with I n − 1 f 1 ◦ ... ◦ ( f i · f i +1 ) ◦ ...f n +1 , that is, we carry out the i -th comp osition in the sequence f • (in the category K ). (b) The cubical composition W K ( a 0 , a i ) ⊗ W K ( a i , a n +1 ) → W K ( a 0 , a n +1 ) = W K ( a, b ) iden t ifies the “pro duc t” ( n − 1 )-cub e I i f 0 ◦ ... ◦ f i ⊗ I n − i − 1 f i +1 ◦ ... ◦ f n +1 with the i -th 1-face of I n f • . 3.3. Notation. Note the three differen t kinds of comp osition t ha t o ccur in W K : (a) The internal comp osition of K is denoted b y f · g , or simply f g . (b) The cubic al comp osition of W K , denoted b y f ⊗ g , whic h corresp onds to the ⊗ -pro duct of the associated cubes. (c) The p otential comp osition o f W K , denoted by f ◦ g , is the heart of the W -construction: it pro vides another dimension in the cub e f o r the homotopies b et w een f ⊗ g and f · g . Th us a comp osable sequence f • as in (3.2) (indexing a cub e in W K ) will b e denoted in f ull by f 1 ◦ . . . ◦ f n +1 ; the comp os ed map f 1 f 2 · · · f n +1 : a → b in K is denoted b y comp( f • ); and the cubical comp osite f 1 ⊗ f 2 ⊗ · · · ⊗ f n +1 will b e denoted by ⊗ f • (as an index fo r a suitable cub e in W K ). 3.4. Definition. The minimal v ertex of I n f • is I 0 comp( f • ) , whic h is in the image of all 0-fa ce maps. The opp osite maximal v ertex, in the image of all 1- face maps, is indexed b y ⊗ f • according to the conv en tion a b ov e, with I 0 f 1 ⊗ I 0 f 2 ⊗ · · · ⊗ I 0 f n +1 iden t ified with I 0 ⊗ f • under the iterated cubical comp ositions. If w e think of a small category K as a constan t cubical category in ( C , O )- C at for O = Ob j K , there is an ob vious map of ( C , O )-categories γ c : W K → K , and follo wing w ork of [Le] and [Co] w e sho w: 3.5. Lemma. The map T γ c : T W K → T K = K may b e identifie d with γ s : F s K → K (se e § 1.7 ff.). Pr o of. Consider an individual cub e I n φ • of W K : this is isomorphic to W Γ n +1 , where Γ n +1 (Example 2.11) consists of a comp osable sequence of n + 1 maps: ( n + 1 ) φ n +1 − − − → n φ n − → ( n − 1 ) → · · · → 2 φ 2 − → 1 φ 1 − → 0 . 16 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER ( φ 1 )( φ 2 )( φ 3 ) t ✛ (( φ 1 ))(( φ 2 )( φ 3 )) t ( φ 1 )( φ 2 φ 3 ) ✻ (( φ 1 )( φ 2 ))(( φ 3 )) t ( φ 1 φ 2 )( φ 3 ) ✛ (( φ 1 φ 2 )( φ 3 )) t ( φ 1 φ 2 φ 3 ) ✻ (( φ 1 )( φ 2 φ 3 )) ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❦ (( φ 1 )( φ 2 )( φ 3 )) ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ((( φ 1 ))(( φ 2 )( φ 3 ))) ((( φ 1 )( φ 2 ))(( φ 3 ))) Figure 1. The triangulated 2- cub e F s Γ 3 The free simplicial resolution of F s Γ n +1 is the triangulation of the n -cub e I n φ • b y n ! n -simplices, corresp onding to the p oss ible f ull paren thesizations of φ • (see Figure 1). This ma y b e iden tified canonically with the standard triangulatio n ∆[1] n ∈ S o f I n ∈ C (see [BB, § 3]), thu s indeed identifyin g F s K with T W K .  3.6. Prop osition. If Γ is a quasi - l a ttic e, the map of ( C , Γ) -c ate gories γ c : W Γ → Γ is a c ofibr ant r esolution. Pr o of. The map o f ( S , O )-categories γ s : F s K → K is a w eak equiv alence, since F s is defined b y a comonad (see [CP, § 1]). Th us F s K is indeed a free simplicial resolution of K (see [DK1, § 2.4], [CP, § 2], and [BM, § 2.21]). Hav ing iden tified γ s : F s K → K with T γ c : T W K → T K = K , it follows from Prop osition 2.22 tha t γ c is a w eak equiv alence. By construction, each comp osition 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 eve ry “pro duct” cub e I n f • ⊗ I k g • ∼ = I n + k f • ⊗ g • ⊆ I n + k +1 f • ◦ g • it is the inclusion of a 1-face. Th us the map ϕ ( u,v ) : colim D ∗ , B ( u,v ) → B c ( u, v ) of Definition 2.12 is just the inclusion o f the sub-cubical complex consisting of all the 1-faces, whic h is a cofibration (in fact, an ano dyne map). This sho ws that W K is cofibrant.  3.7 . Rect ifying homotopy comm utativ e diagrams. W e can use the cofibran t resolution W K → K to study the rectification of a homotop y-comm utativ e diagram ˜ X : K → ho M in some mo del category M (suc h as T o r T ∗ ). Since the 0-sk eleton of W K is isomorphic to F K , choo sing an arbitrar y repre- sen ta t ive X 0 ( f ) for each homotop y class ˜ X ( f ) for eac h morphism f of K , yields a lifting of ˜ X to X 0 : sk c 0 W K → M . Note that a c hoice of a 0-realizatio n X 0 : F K → M is equiv alen t to c ho osing basep oin ts in eac h relev ant comp onent of eac h M c ( u, v ), although of course this cannot b e done coheren tly unless ˜ X is rectifiable. HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 17 3.8. R emark. Our goal is to extend X 0 o v er the sk eleta of W K . How ev er, the “naiv e” cubical sk eleton functor sk c k : C → C ( § 1.1) is not monoidal with resp ect to ⊗ (unlik e the simplicial analogue), so it do es not comm ute with comp osition maps. Nev ertheless, one can define a k -sk eleton functor for ( C , O )-categories in general; when Γ is a quasi-lattice ( § 2.24) and A is a cofibran t ( C , Γ) - category (suc h as W Γ), sk c k A can b e defined b y simply including all ⊗ -pro duct cubes of i -cub es in A with i ≤ k . Of course, if A is n -dimensional ( t ha t is, has no no n- degenerate i -cub es f o r i > n ), then sk c n A = A a g rees with the na iv e n -sk eleton. If M is cubically enriched ( § 1.6), extending X 0 to a cubical functor X 1 : sk c 1 W K → M is equiv alen t to c ho osing homotopies b e t w een eac h ˜ X ( f 1 ◦ f 2 ) and ˜ X ( f 1 ) ◦ ˜ X ( f 2 ), since the 1-cub es of W K corresp ond to a ll p ossible (tw o term) factorizations of maps in K . Extending X 1 further to X 2 : sk c 2 W K → M means c ho osing homotopies b et w een the homotopies for three-fold comp ositions, and so on. This is the idea underlying a fundamen tal result o f Boardman and V o g t: 3.9. Theorem ([BV, Cor. 4.21 & Thm. 4.49]) . A dia gr am ˜ X : K → ho T lifts to T if and only if it extends to a simplicial functor X ∞ : W K → T . 3.10. R emark. In fact, for our purp oses we do not hav e to assume tha t the category M is cubically enrich ed, or ev en has a mo del catego ry structure: all w e need is for M to ha v e a suitable class of weak equiv a lences W , f r om whic h w e can construct a n ( S , O )- category L ( M , W ) as in [DK2, § 4], and then the corresp onding ( C , O )-category S cub L ( M , W ) by Prop osition 2.2 2. Note that when M and W a r e p ointed, the construction of Dwye r and Kan is naturally p oin ted, to o. Ho w eve r, to av oid excessiv e v erbiage w e shall assume for simplicit y tha t M is a cubically enric hed mo de l categor y . W e do not actually need the full (usually large) category M (o r S cub L ( M , W )), since w e can mak e use of the follo wing: 3.11. Definition. Give n a diag ram ˜ X : K → ho M for a mo del categor y M ∈ C - C at , let C X b e the smallest ( C , K )-category inside M thro ugh whic h an y lift of ˜ X to X : K → M factors. This means that C X is the ( C , K )-category having cubical mapping space s C X ( X u , X v ) := ( M c ( X u , X v ) if u ≺ v in O := Ob j K ∅ otherwise. This is a sub-cubical category of M . F or simplicit y , we further reduce the mapping spaces of C X so that t hey consist only of those comp o nen ts of M c ( X u , X v ) whic h are actually hit b y ˜ X , so that π 0 C X = K . In particular, if K is the partially ordered set h O , ≺i , w e ma y assume the mapping spaces o f C X are connected (when they are not empt y). 3.12 . Poin ted diagrams. W e w ant to understand the relationship b etw een tw o p ossible wa ys to describ e the (final) obstruction to the existence o f an extension X ∞ : top ologically and cohomologically . Unfortunately , ev en though these obstructions can b e defined for quite general K , they do not alw a ys coincide; this can b e seen b y comparing the sets in which they take v a lue. Ho w eve r, we are in fact only in terested in the cases where the obstruction can naturally b e thought of as the h igher homotopy op er ation asso ciated to the da t a 18 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER ˜ X : K → ho T . The usual man tra sa ys that suc h an op eration is defined when “a lo w er order op eration v anishes for tw o (or more) reasons”. Indeed, the example of the usual T o da brac ket sho ws that the pro blem cannot b e stated simp ly in t erms of rectifying a homotop y-commutativ e diagram, since an y diagram index ed b y a linear indexing category Γ n as ab o ve can alw a ys b e rectified: what w e w ant is to realize certain n ull-homotopic maps b y zer o maps (see [BM, § 3.12]). This suggests that w e restrict atten tion to p ointe d dia g rams, and to the follo wing sp ecial type o f index ing category: 3.13. Definition. A lattic e is a finite quasi-lattice Γ ( § 2.10) equipp ed with a (we ak l y) initial ob ject v init and a (we akly) final ob ject v fin , satisfying: (a) There is a unique φ max : v init → v fin . (b) F or eac h v ∈ Ob j Γ, there is at least one map v init → v and at least one map v → v fin . A comp osable sequence of n arrow s in Γ will b e called an n - ch ain . The maximal o ccuring n (necessarily fo r a c hain from v init to v fin , factorizing φ max ) is called the length of Γ. 3.14. R em ark. Note that if the length o f Γ is n + 1, then W Γ is n -dimensional, in the sense that the cubical f unction complex W Γ( v init , v fin ) has dimension n , and dim( W Γ( u , v )) < n for any other pair u, v in Γ. 3.15. Definition. W e shall mainly b e in terested in the case when Γ is p o in ted (in whic h case necessarily φ max = 0). A nul l se quenc e in Γ is then a comp osable sequence f • :=  a n +1 f n +1 − − → a n f n − → a n − 1 . . . a 1 f 1 − → a 0  with comp( f • ) = 0, but no constituen t f i is zero. It is called r e d uc e d if all adjacen t comp ositions f i +1 · f i ( i = 1 , . . . , n ) are zero. An n -cub e I n f • in W Γ indexed b y a (reduced) null sequenc e is called a (reduced) n ul l cub e . As noted ab ov e, we wan t to concen t r a te on t he pro blem of replacing null-homotopic maps with zero maps, giv en a p ointed diagram ˜ X : Γ → ho M whic h comm utes up to p o in ted homotopy . W e shall therefore assume from now on that all other (non- zero) triangles in the diagra m commute strictly . How ev er, since the non- zero ma ps in Γ do not f orm a sub-category , we shall need the following: 3.16. Definition. The unp ointe d version U p ( K ) of a p o inted category K is defined as follow s: if K ∼ = F ( K ) /I for some set of relations I in the fr ee categor y F ( K ), then the ob jects of U p ( K ) are those of K , except f o r the zero o b jects , and U p ( K ) := F ( K ′ ) / ( I ∩ F ( K ′ )), where K ′ is obtained from the underlying gra ph K of K by omitting all zero ob jects and maps. The inclusion K ′ ֒ → K induces a functor ι : U p ( K ) → K . Essen tially , U p ( K ) is t he full sub category of K o mitting 0 and all maps into or out of the zero ob ject 0. Ho wev er, if the comp osite f · g : a → b is zero in K with f 6 = 0 6 = g , t hen w e add a new (non-zero) map ϕ : a → b in U p ( K ) (with ι ( ϕ ) = 0), to serv e a s the comp osite in U p ( K ) of f and g . 3.17 . Defining higher op erations. HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 19 From no w on we assume give n a p oin ted lattice Γ and a diagram up-to-homoto p y ˜ X : Γ → ho M in to a p oin ted cubically enric hed mo del category M . Setting Γ ′ := U p (Γ), we also assume that the comp osite ˜ X ◦ ι lifts to a strict diagram X ′ : Γ ′ → M . F o r simplicit y we also denote t he factorization of X ′ though C X ( § 3.11) b y X ′ : Γ ′ → C X . Our goal is to extend X ′ to a p ointed diagram X : Γ → C X . (Note tha t X ′ itself cannot b e p ointed in o ur sense, but it still takes v a lues in the p ointed catego ry M ). Ob viously , if X ′ do es extend to suc h an X , ev ery map ϕ ∈ Γ ′ whic h facto r s through 0 in Γ m ust b e (weakly ) n ull-homotopic in M . Thus , w e additio na lly include this restriction o n the original data as part of our assumptions. Our approac h is to extend X ′ b y induction o v er the sk eleta of W Γ, where w e actually need: 3.18. Definition. G iven Γ and X ′ as ab ov e, for each k ≥ 0 the r el a tive k -skeleton for (Γ , Γ ′ ), denoted b y sk c k (Γ , Γ ′ ), is the pushout: sk c k W Γ ′ sk c k ι / / sk c k γ c   sk c k W Γ   Γ ′ / / sk c k (Γ , Γ ′ ) in ( C , Γ)- C at (cf. Lemma 2.13), where γ c : W Γ → Γ is the augmen tation o f Prop osition 3.6 . Note that the natural inclusions sk c k − 1 ֒ → sk c k induce maps sk c k − 1 (Γ , Γ ′ ) → sk c k (Γ , Γ ′ ). A map of ( C , Γ ) -categories X ′ k : sk c k (Γ , Γ ′ ) → C X extending X ′ : Γ ′ → C X is called k - al lowable . In particular, if Γ is a lattice of length n + 1, b y Remark 3.14 W (Γ , Γ ′ ) := sk c n (Γ , Γ ′ ) is the pushout W Γ ′ ι / / γ c   W Γ   Γ ′ / / W (Γ , Γ ′ ) . 3.19. R emark. X ′ extends canonically to a p o inted map X 0 : sk c 0 W Γ → C X , b ecause sk c 0 W Γ is a free catego ry , and the only new o b ject is 0. T o gether with γ c this determines a canonical 0-allo w able extension X ′ 0 : sk c 0 (Γ , Γ ′ ) → C X . If Γ is a lattice of length n + 1, in order to rectify X ′ w e w an t to extend X ′ 0 inductiv ely ov er the relative sk eleta sk c k (Γ , Γ ′ ) to an n -allow able map X ′ ∞ : W (Γ , Γ ′ ) → C X – equiv alen tly , a map X ∞ : W Γ → C X whic h agrees with the initial X ′ : Γ ′ → C X . Recall that b ecause dim W Γ = dim W (Γ , Γ ′ ) = n , X ′ n is actually X ′ ∞ in the sense of Theorem 3.9, so this yields a rectification of X ′ for suitable M (suc h a s T ∗ ). W e assumed in § 3.1 7 that X ′ : Γ ′ → C X tak es ev ery map ϕ ∈ Γ ′ whic h factors through 0 in Γ t o one whic h is n ull- homotopic in M . Therefore, b y choosing n ull-homotopies for all suc h maps w e see tha t X ′ 0 alw a ys extends non-c an o n ic al ly to a 1- allo w able X ′ 1 : sk c 1 (Γ , Γ ′ ) → C X . 20 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER Ho w eve r, in general there are obstructions to obtaining k -allow able extensions for k ≥ 2. These are complic ated to define “top ologically” (see [BM] and [BC]). F ortu- nately , in or der to define the higher homotop y op eration a sso ciated to X ′ , w e only need to consider the last obstruction. That is, we assume w e hav e already pro duced an ( n − 1)-allow able extension X ′ n − 1 : sk c n − 1 (Γ , Γ ′ ) → C X , and wan t t o extend it to X ′ n . It may b e p ossible to do so in differen t w ays. In order to define the set h h X ′ i i o f “la st obstructions”, w e need the followin g: 3.20. Lemma. Assume that Γ = Γ n +1 is a c omp osa ble ( n + 1) - c hain f • ( § 2.11) and that the i -th adjac ent c om p osition f i · f i +1 6 = 0 in Γ , and let f ′ • := ( f 1 , . . . , f i − 1 , f i · f i +1 , f i +2 , . . . f n ) . L et ι : I n f ′ • ֒ → I n +1 f • b e the inclusion of the i -th z e r o fac e. L et ˜ Γ b e the line ar lattic e c orr esp onding to f ′ • . Then for any X : Γ ′ → C X , the in clusion ι : ˜ Γ ֒ → Γ ind uc es a one-to-o ne c orr es p ondenc e b etwe en the set of extensions of X ′ 0 : sk c 0 (Γ , Γ ′ ) → C X to W Γ and the extensions of ˜ X ′ 0 : sk c 0 ( ˜ Γ , Γ ′ ) → C X to W ˜ Γ . Pr o of. The i -th dimension of I n f • corresp onds to the i -th a djacen t comp osition f i · f i +1 in the ( n + 1)-c hain f • , and if this comp osite is not zero, then X ′ n , b eing allo w a ble, is constan t along this dimension. Th us the pro jection ρ : I n f • → I n − 1 f ′ • induces the inv erse t o ι ∗ .  3.21. Pr op osition. L et Γ b e a lattic e of leng th n + 1 and X ′ : Γ ′ → C X a diagr am. L e t J Γ b e the set of len gth n + 1 r e duc e d nul l se quenc e s of Γ (Definition 3.15). Ther e i s a natur al c orr esp ondenc e b etwe en ( n − 1) - a l lowable extensions X ′ n − 1 : sk c n − 1 (Γ , Γ ′ ) → C X of X ′ and ma p s F X ′ n − 1 : W f • ∈ J Γ Σ n − 1 X ′ ( v init ) → X ′ ( v fin ) , such that F X ′ n − 1 is nul l-hom otopic if and only if X ′ n − 1 extends to sk c n (Γ , Γ ′ ) . Pr o of. In order to extend X ′ n − 1 : sk c n − 1 (Γ , Γ ′ ) → C X to sk c n (Γ , Γ ′ ), w e m ust c ho ose extensions to the n - cub es of W Γ. These o ccur only in the full mapping complex W Γ( v init , v fin ), a nd are in one-to-one corresp ondence with those decomp ositions f • =  v init = a n +1 f n +1 − − → a n f n − → a n − 1 . . . a 1 f 1 − → a 0 = v fin  of φ max : v init → v fin whic h are of maximal length n + 1. Note that the minimal v ertex of I n f • is indexed by φ max = 0; t he maximal ve rtex is I 0 ⊗ f • (Definition 3.4 ). By Lemma 3.20 we need only consider those maximal decomp ositions f • for whic h eve ry adjacent comp o sition f i · f i +1 = 0 . In this case, w e may assume that an y facet I n − 1 f ′ • of I n f • whic h touc hes t he v ertex lab eled b y φ max = 0 has a t least one factor of f ′ • equal to 0 (in Γ), so X ′ n − 1 | I n − 1 f ′ • = 0. Th us X ′ n − 1 | I n f • is giv en b y a map in M F ′ ( X ′ n − 1 ,I n f • ) : X ′ ( v init ) ⊗ ∂ I n → X ′ ( v fin ) whic h sends X ′ ( v init ) ⊗ I 0 φ max and ∗ X ( v init ) ⊗ I n to ∗ X ( v fin ) , so it induces ˜ F ( X ′ n − 1 ,I n f • ) : X ′ ( v init ) ∧ S n − 1 → X ′ ( v fin ). Note further that any tw o suc h n -cub es I n f • and I n g • ha v e distinct maximal v ertices I 0 f • and I 0 g • , so they can only meet in facets adjacent to the minimal v ertex, where ˜ H v anishes. Th us altogether X ′ n − 1 is describ ed by a map (3.22) F X ′ n − 1 : _ f • ∈ J Γ Σ n − 1 X ′ ( v init ) → X ′ ( v fin ) , HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 21 where J Γ is the set of length n + 1 reduced null seque nces of Γ. Clearly , F X ′ n − 1 is n ull-homo t o pic if and only if X ′ n − 1 extends to all of W Γ, since W Γ( v init , v fin ) is map( C ( W f • ∈ J Γ Σ n − 1 X ′ ( v init )) , X ′ ( v fin )), up to homotop y , where C K is t he cone on K .  3.23. Definition. The n -th order p ointe d higher homo topy op er ation h h X ′ i i asso ci- ated to X ′ : Γ ′ → C X as ab ov e is defined to b e the subset: (3.24) h h X ′ i i ⊆ " _ f • ∈ J Γ Σ n − 1 X ′ ( v init ) , X ′ ( v fin ) # ho M consisting of all maps F X ′ n − 1 as a b o v e, f o r all p o ssible choices of ( n − 1)-allow able extensions X ′ n − 1 , of X ′ . W e sa y the op eration va n ishes if this set contains the zero class. 4. Cohomology and re ctifica tion The appro ac h o f Dwy er, Kan, and Smith to realizing a homotop y-comm utativ e diagram ˜ X : Γ → ho M is also based on Theorem 3.9, whic h sa ys that ˜ X can b e rectified if and only if it extends to W Γ. W e do not a ctually need the full force of their theory , which is wh y w e can w ork in an arbitrary p ointed mo del categor y M , rather than just T ∗ (see also Remark 3.10 ). Essen tially , they define the (p ossibly empt y) mo duli space hc ˜ X to be the nerve of the category of all p ossible rectifications of ˜ X (cf. [DKSm2, § 2.2]), and hc ∞ ˜ X is the space of all ∞ -homo t o p y comm utativ e lifts of ˜ X in (the simplicial v ersion of ) map C - C at ( W Γ , M ) = map ( C , Γ)- C at ( W Γ , C X ) ( § 3.11). They then show t hat hc ˜ X is (w eakly) homotopy equiv alen t t o hc ∞ ˜ X (see [D KSm2, Theorem 2.4]). Th us the realization problem is equiv alen t to finding suitable elemen ts in ma p ( C , Γ)- C at ( W Γ , C X ). Dwy er, Kan, and Smith also consider a relativ e v ersion, where ˜ X has a lr eady b een rectified to Y : Θ → M for some sub-category Θ ⊆ Γ ( see [DKSm2, § 4]). W e shall in fact need only the case Θ = Γ ′ and Y = X ′ , so w e w an t an elemen t in map ( C , Γ)- C at ( W (Γ , Γ ′ ) , C X ) (see § 3.18). 4.1 . The to wer. If Γ is a quasi-latt ice, ( C , Γ)- C at has a semi-spherical mo del cate- gory structure (see § 2.9 and § 2.2 3 ). Therefore, the P ostnik ov to w er { P m C X } ∞ m =0 of the ( C , Γ)-category C X allo ws us to define hc m ˜ X := map ( C , Γ)- C at ( W (Γ , Γ ′ ) , P m − 1 C X ) for m ≥ 1. Note that P 0 C X is homotopically trivial – that is, eac h comp onent of eac h mapping space ( P 0 C X )( u, v ) is contractible – so hc 1 ˜ X is, to o. More- o v er, ˜ X : Γ → ho M (o r X ′ : Γ ′ → C X ) determines a canonical “ tautological” comp onen t o f hc 1 ˜ X – namely , the comp o nen t of the map f X 1 : W ( Γ , Γ ′ ) → P 0 C X , corresp onding to the canonical 0-allow able extension X ′ 0 : sk c 0 (Γ , Γ ′ ) → C X of § 3.19. Because C X is w eakly equiv alent to the limit o f its P ostnik ov tow er ( § 1.23(b)), the space hc ∞ ˜ X is the homotop y limit of the tow er: (4.2) hc ∞ ˜ X → . . . → hc n ˜ X → hc n − 1 ˜ X . . . → hc 1 ˜ X . In general, t here are lim 1 problems in determining the comp onen ts of hc ∞ ˜ X (see [DKSm1, § 4.8]), but these will not b e relev ant to us here, b ecause of the follo wing: 22 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER 4.3. Lemma. If Γ has length n + 1 , the tower (4.2) is c onstant fr om hc n − 1 ˜ X up. Pr o of. W e ma y assume that C X is fibran t (e.g., if ˜ X v is a cubical Kan complex for each v ∈ O ). Then sk c n (Γ , Γ ′ ) = W Γ b y Remark 3.14, where in this case w e are using the naiv e n -sk eleton (see R emark 3.8) whic h is left adj o in t to the n -cosk eleton functor. By Remark 1.28, we ma y use the latter for P n − 1 C X . Th us the choice s o f n -allo w able extensions X ′ n : sk c n (Γ , Γ ′ ) = W Γ → C X of ˜ X are in natural one-to- one corresp ondence with lifts f X n : W ( Γ , Γ ′ ) → P n − 1 C X of f X 1 .  4.4 . The obstruct ion theory. In view of the a b ov e discussion, the realization problem for ˜ X : Γ → ho M – and in part icular, the p ointed vers ion fo r X ′ : Γ ′ → M (see § 3.17 ) – can b e solv ed if one can successiv ely lift the elemen t f X 1 ∈ hc 1 ˜ X t hro ugh the to w er (4.2). In fact, w e do not really need t he (simplicial or cubical) mapping spaces hc m ˜ X := map ( C , Γ)- C at ( W (Γ , Γ ′ ) , P m − 1 C X ) at all – we simply need to lift the maps g X m : W ( Γ , Γ ′ ) → P m − 1 C X in the Pos tnik o v tow er for C X . Let k m − 1 : C X → E G ( π m C X , m + 1) b e the ( m − 1 ) -st k -in v arian t for C X , where G := ˆ π 1 C X (see § 2.2 3 ff. ). Giv en a lifting g X m , compo sing it with k m − 1 yields a map h ( X m ) : W (Γ , Γ ′ ) → E G ( π m C X , m + 1): W (Γ , Γ ′ ) g X m ! ! ^ X m +1 % % p * * P m C X / /   P m − 1 C X k m − 1   B G s / / E G ( π m C X , m + 1) pro j i i T o iden tify h ( X m ) as an elemen t in the appropria te cohomology group (D efi- nition 2.25), note that in this case the t wisting map p : W (Γ , Γ ′ ) → B G fa cto r s through ˆ π 1 X n : ˆ π 1 W (Γ , Γ ′ ) → ˆ π 1 P m − 1 C X = ˆ π 1 C X = G , and b y Propo sition 3.6, the fundamen tal group oid ˆ π 1 W (Γ , Γ ′ ) = Γ is discrete. Th us [ h ( X m )] takes v alue in H m +1 Γ ( W (Γ , Γ ′ ); π m C X ), which w e abbreviate to H m +1 (Γ; π m C X ). The lifting prop ert y for a fibration se quence (ov er B G ) then yields: 4.5. P rop osition ([DKSm2, Prop. 3.6]) . The map g X m lifts to ^ X m +1 in hc m +1 ˜ X if and o nly if [ h ( X m )] vanis h es in H m +1 (Γ; π m C X ) . 4.6 . Relating t he tw o obstructions. In order to see how the tw o obstructions w e hav e describ ed are related, we need some more notation: F or a p oin ted lattice Γ of length n + 1, let \ W (Γ , Γ ′ ) denote the sub-( C , Γ)-category of W (Γ , Γ ′ ) obtained from sk c n − 1 (Γ , Γ ′ ) b y adding all unreduced null n -cubes (Def- inition 3.1 5). By Lemma 3.20, any ( n − 1)- allo w able extension X ′ n − 1 : sk c n − 1 (Γ , Γ ′ ) → C X extends canonically to b X : \ W (Γ , Γ ′ ) → C X . If i n : sk n \ W (Γ , Γ ′ ) → \ W (Γ , Γ ′ ) HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 23 and i : \ W (Γ , Γ ′ ) → W (Γ , Γ ′ ) are the inclusions, w e th us ha v e a comm utativ e diagram in ( C , Γ)- C at : sk c n − 1 \ W (Γ , Γ ′ ) sk c n − 1 i =Id   i n − 1 / / X ′ n − 1 ) ) S S S S S S S S S S S S S S S S S \ W (Γ , Γ ′ ) b X   sk c n − 1 (Γ , Γ ′ ) X ′ n − 1 / / C X Because Γ is a lattice of length n + 1, W Γ is n -dimensional. F urthermore, if w e break up an y chain in Γ into disjoin t sub-chains o f length k and ℓ ( k + ℓ = n + 1), the resulting comp osite cub e ha s dimension ( k − 1) + ( ℓ − 1) = n − 1. Th us the only non- degenerate n -cubes in W Γ are indecomposable in W Γ( v init , v fin ), whic h implies that sk c n − 1 (Γ , Γ ′ ) is in fact defined using the naive ( n − 1)-sk eleton (see Remark 3.8). Th us b y adjoin tness (using R emark 1.28) w e ha ve: (4.7) \ W (Γ , Γ ′ ) i   b X / / C X r   W (Γ , Γ ′ ) f X n / / cosk c n − 1 C X = P n − 2 C X in whic h r is the fibratio n r ( n − 1) = p ( n − 1) of § 1.23(b). No w let R Γ b e the ( C , Γ)-category of all r e duc e d n ull ( n − 1)- spheres (that is, b oundaries of the reduced n ull n -cub es) in W Γ. Th us: (4.8) R c Γ ( u, v ) = ( S f • ∈ J Γ ∂ I n f • if ( u, v ) = ( v init , v fin ) ∅ otherwise (in the notation of (3.22)) . 4.9. F act. Ther e is a homotopy c ofibr ation se quenc e of ( C , Γ) -c ate gories (4.10) R Γ j − → \ W (Γ , Γ ′ ) i − → W (Γ , Γ ′ ) . Pr o of. By definition of a po in ted lattice, all the n -cub es of W Γ (and th us of W (Γ , Γ ′ )) are null cub es. Thu s the map i : \ W (Γ , Γ ′ ) → W (Γ , Γ ′ ) is actually an isomorphism in all mapping slots except ( u, v ) = ( v init , v fin ), where the n -cells attac hed via j provide the missing (necessarily reduced) n ull n -cub es.  24 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER 4.11. D efinition. Let Γ b e a p oin ted lattice of length n + 1, C X a ( C , Γ)-category , and define J Γ as in Prop osition 3.21. T o eac h comm uting square: (4.12) \ W (Γ , Γ ′ ) i   ˆ h / / C X r   W (Γ , Γ ′ ) h / / P n − 2 C X in ( C , Γ)- C at , w e assign the comp osite k n − 2 · h in H n (Γ , π n − 1 C X ). Denote b y K n ( C X ) the subset of H n (Γ , π n − 1 C X ) consisting of all suc h elemen ts k n − 2 · h . Finally , define Φ n : K n ( C X ) → Q f • ∈ J Γ π n − 1 C X ( v init , v fin ) b y assigning to (4.12 ) the homotop y class of the comp osite σ := ( ˆ h · j )( v init , v fin ) : R Γ ( v init , v fin ) → C X ( v init , v fin ). 4.13. Lemma. The map Φ n is wel l- d efine d. Pr o of. F reuden thal susp ension g iv es an isomorphism [ R Γ ( v init , v fin ) , C X ( v init , v fin )] ho C ∼ = − → [Σ R Γ , E ˆ π 1 W Γ ( π n − 1 C X , n )] ho ( C , Γ)- C at , so Φ n ma y b e equiv a len tly defined b y a ssigning to the comp osite k n − 2 · h the extension e = Σ σ in the follo wing diagram: \ W (Γ , Γ ′ ) i   ˆ h / / C X p   W Γ ∂   h / / k n − 2 · h ) ) T T T T T T T T T T T T T T T T T T P n − 2 C X k n − 2   Σ R Γ e / / E G ( π n − 1 C X , n ) where W Γ ∂ − → Σ R Γ Σ j − → Σ \ W (Γ , Γ ′ ) is the contin uation of t he cofibra tion sequence of (4.10). Here w e used the fact that R Γ is concen trated in the ( v init , v fin ) slot, b y (4.8 ). Note that the extension e (and t h us σ = Φ n ( k n · h ), the adjo in t of e with re- sp ect to the (Σ , Ω) adjunction) is uniquely determined up to homotopy , since [Σ \ W (Γ , Γ ′ ) , E ˆ π 1 W Γ ( π n C X , n + 1)] = 0 fo r dimension reasons.  Our main result, Theorem A of the In tro duction, is no w a consequence of the follo wing Theorem and Corolla r y: 4.14. Theorem. Give n X ′ : Γ ′ → M as in § 3.17, the map Φ n is a p oin ted c orr esp ondenc e b e twe en the set of elements of K n ( C X ) obtaine d fr om c ommuting squar es of the form (4.7) and h h X ′ i i of (3.24) – that is, Φ n ( α ) = 0 if and only if α = 0 . HIGHER HOMOTOPY OPERA TIO NS AND COHOMOLOGY 25 Pr o of. By Prop osition 4 .5 , the comp osite h ( X n − 1 ) := k n − 2 · [ X n − 1 is the obstruction to extending b X to X n : W (Γ , Γ ′ ) → C X , and since R Γ j   σ / / E G ( π n − 1 C X , n − 1) ℓ   \ W (Γ , Γ ′ ) i   b X / / C X p   W Γ ∂   \ X n − 1 / / X n 5 5 P n − 2 C X k n − 2   Σ R Γ e / / E G ( π n − 1 C X , n ) comm utes, with the left v ertical column a cofibration and the righ t v ertical column a fibration sequence, the fact that e = 0 ⇔ σ = 0 implies that the comp osite 0 = e · ∂ = k n − 2 · [ X n − 1 = h ( X n − 1 ). Conv ersely , if X n exists, then b X · j = X n · i · j = 0, so ℓ · σ = 0 , and sinc e π n − 1 ℓ is an isomorphism, σ = 0.  4.15. C or ollary . Th e Dwyer-Ka n-Smith ob s truction class [ h ( X n − 1 )] of Pr op osition 4.5 is zer o in H n Γ ( W Γ; π n − 1 C X ) if and only i f the c orr es p onding hom otopy class F X ′ n − 1 is nul l. Ther efor e, h h X ′ i i vanishes if and only if K n ( C X ) c ontains 0 . 4.16. R e mark. 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T o da, Comp osition metho ds in the homotopy gr oups of spher es , Adv. in Math. Study 49 , Princeto n U. Press, P rinceton, 1962 . [V] V.V. V a gner, “The theory of relations and the algebra of partial mappings” , in The ory of Semigr oups and Applic ations, I Izdat. Sa ratov. Univ., Sa ratov”, 196 5 , pp. 3-17 8. 28 D. BLAN C, M.W. JOHNS ON, AND J.M. TURNER Dep ar tment of Ma thema tics, University of Haif a, 31905 Haif a, Israel E-mail addr ess : bl anc@m ath.ha ifa.ac.il Dep ar tment of Ma thema tics, Penn St a te Al toona, Al toona, P A 1 6601-3760, USA E-mail addr ess : mw j3@ps u.edu Dep ar tment of Ma thema tics, Cal vin Col lege, Grand Rapids, MI, USA E-mail addr ess : jt urner @calvi n.edu