A-infinity monads and completion

Journal of Homotop y and Related St ructures, vol. ??(? ?), ????, pp.1– 23 A ∞ -MONADS AND COMPLETION T I L M A N B A U E R A N D A S S A F L I B M A N Abstract Given an operad A of topological spaces, we consider A - monads in a topological cate gory C . When A is an A ∞ -operad, any A -monad K : C → C can be thought of as a monad up to coherent homotopies. W e define the completion functor with respect to a n A ∞ -monad and prove that it is an A ∞ -monad itself. 1. Introduction The starting point for this pa per is the claim made in [ BK72 , I.5.6] that the R - completion functor ˆ R , where R is a commutative ring, is a monad (triple). This claim was not proved and later retracted by the authors. It is a monad up to ho- motopy , a s was proved in [ L ib03, Bou03 ]. In this pa per we prove that these homo- topies are part of a system of higher coherent homotopies, turning ˆ R into what we call a n A ∞ -monad . In f act we work in a wider context. Let E be a ca tegory which is complete and cocomplete, tensored and cotensor ed over a “ convenient” category S p c of topological spaces in the sense of Steenrod [ Ste67 ] – see Section 2. Assume f urther that E has an enriched mono idal structure ⋄ with unit I . For a (non-symmetric) operad A of topological spa ces, a n A -algebra in ( E , ⋄ , I ) is an object K ∈ E which is equipped with appropriate A -algebra structure ma ps A ( n ) → map E ( K ⋄ . . . ⋄ K , K ) . If A is the a ssociative operad assoc with assoc ( n ) = ∗ for all n , a n A -algebra is just a monoid in ( E , ⋄ , I ) and therefore has an a ssociated cobar constructio n R K : ∆ → E , R K ( n ) = K ⋄ ( n + 1 ) . Its totalization ˆ K = T ot R K , where ∆ is the ca tegory of ordered sets [ n ] = { 0, . . . , n } a nd monotonic maps, is called the completion of K . This constru ction can be generalized to ar b itrary A -a lgebras, as we will show in Se c tion 3 below . The ma in result of this paper is the following completion theorem: Theorem 1.1 . For any o p erad A there exists an operad ˆ A mapping to A with the following property: if K is an A -algebra in ( E , ⋄ , I ) then ˆ K is an ˆ A-algebra and ˆ K → K is a morphism of ˆ A-algebras. If A is an A ∞ -operad th en so is ˆ A. Our principal application of this theorem is when E is the (la rge) category of continuous endofunctors of a complete a nd cocomplete category C tensored and cotensored over S p c , with the monoidal structure given by composition of func- tors. See Section 3.3 below for more details, particularly about the set- theoretic . 2000 Mathematics Subject Classific ation: 55P60, 18D50, 18C15 Key words and p h rases: monad, A ∞ -monad, completion, cobar resolution c  ????, T ilman Bauer and Assaf L ibman. Permission to copy for private use granted. Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 2 difficulties involved. A monoid K in this category is simply a monad, and an A - algebra is what we call an A-m o nad on C . If the spa ces of A are weakly contractible we call K an A ∞ -monad. The resolution R K and the completion ˆ K generalize the classical definitions given by Bousfield and Kan in [ BK72 ] for monads a ssociated to commutative rings. Thus we obtain as an immediate cor ollary of the theorem: Corollary 1.2 . Let K be an A-monad in C . Then ˆ K is an ˆ A-monad and ˆ K → K is a morphism of ˆ A-monad s. If K is an A ∞ -monad, then so is ˆ K . For example, the theorem works in the following setting: • The f unctor K ( X ) = Ω ∞ ( E ∧ X ) for an A ∞ -ring spectrum E . If E is ordi- nary homolog y with coefficients in a ring R then ˆ K is a topological version of the Bousfield-Kan R - completion. For mor e genera l E , it is the Be nde rsky- Thompson completion [ BT00 ]. • The functor ˆ K for an A ∞ -monad K . The output of the theorem, namely ˆ K , can be fed ba ck into it to obtain ˆ ˆ K . In pa rticular , if we start with a completion at a homology theory E a s above, we get a secondary multiplicative Bousfield spectral sequence whose E 1 -term consists of the homotopy groups of iterated completions of a space, and which converges to the homoto py groups of the completion at ˆ K , na mely ˆ ˆ K ( − ) Following Dror and Dwyer in [ DD77 ], the completion procedure can be it- erated transfinitely to obtain a “long tower ” · · · → ˆ ˆ K → ˆ K → K . When E is the ordinary homology with coefficients in F p or a subring of Q , they have shown that if we iterate this often enough, the spectral sequence will eve ntu- ally converge to the homoto py groups of L p X , the localiz ation. It would be very interesting to know if this works for any homology theory E . • The same constr uction works in the category of spectra, where the monad is given by K ( X ) = E ∧ X for an A ∞ -spectrum E . The theorem allows us to con- struct a secondary Adams spectral sequence with E 1 -term constructed f rom the homotopy groups of the iterated E -nilpotent completion a nd converging to the homotopy groups of a second-order completion. In this way , just as for spaces, a long tower of spectra under the E -localization can be constructed, with a higher Adams spectral sequence relating every tower stage to the f ol- lowing one. The crux of the theorem is the assertion that ˆ A is an A ∞ -operad if A is one. Thus, the completion of a monad is not a monad, but the completion of a monad up to coherent homoto pies turns out to be a monad of the same type. W e remark that if E is enriched over s S e t s , the category of simplicial sets, the construction of the operad ˆ A and its action on ˆ K in Se ctions 3–6 work without any change. However , it is not true that ˆ A is an A ∞ -operad of simplicial sets if A has the same property (see Theorem 6. 5 .) For this reason we will only be interested in enrichment over S p c . In the companion paper [ BL09 ] we show that a related result, stronger in som e sense and weaker in another , holds in the simplicial wo rld. W e show there that if the monad under consideration is the free R -module functor for a ring R then there Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 3 is a n explicit, combinatorially defined A ∞ -operad a ction on the cobar construction, not just its totalization. It is worth noting that when ( E , ⋄ , I ) is ( S p c , × , ∗ ) or any other monoidal cate- gory where I is a terminal object then for any A ∞ -algebra K , R K has a c osimplicial retraction to I and thus ˆ K = I . The theorem is thus only making a trivial statement. 2. Enriched homotopy limits The purpose of this section is to set up some notation and recall various ea sy facts about enriched categories. Most of the facts can be f ound in the literature, e .g. [ Bor94, Kel05 ], and a re collected here for the reader ’s convenience. Let S p c denote a complete and cocomplete “convenient” ca te gory of topo logi- cal spaces in the sense of Steenrod [ S te67 ]. Thus, S p c has mapping space s with the right adjunction properties where map ( X , Y ) is the set of all the contin uous func- tions X → Y topologized appropriately . In a ddition, Sp c is large enough to c on- tain the subcategory of CW -complexes and it is equipped with a model structure where fibrations are S erre fibrations and weak equivalences a re weak homotopy equivalences. The prime examples for S p c are the category of compactly gener- ated Hausdorff spa ces [ S te67 ] or the cate gory of k -spaces, [ Lew78 , A ppendix] or [ Hov99 , Section 2. 4]. Let C be a small c ategory enriched over S p c (in short: a S p c -ca tegory). Follow- ing [ HV92 ], we make the following definition. Definition 2.1. A CW-category C is a S p c -c a tegory all of whose morphism spaces map C ( x , y ) a re CW -complexes, and where the ide ntity morphisms are 0-cells. The composition maps a re required to b e cellular maps. Any S p c - c a tegory C can be turned into a CW -category by applying the singular and the realiza tion functor (cf. [ BK72 , VII. § 2]) to its morphism spa ces. W e write C ∗ for the S p c - category of continuous (i.e. Sp c -enriched) functors C → S p c , cf. [ Kel 05 ]. This category has all limits and colimits because they are formed in S p c . C onsequently , the coend F ⊗ C G of F ∈ C op ∗ and G ∈ C ∗ and the end [ F , G ] C of F , G ∈ C ∗ are defined [ Ke l05 , § 2.1 eq. (2.2 ) and § 3.10 eqs. (3.68, 3.70) ]. Every functor J : C → D of small CW -ca tegories induces a continuous restriction functor J ∗ : D ∗ → C ∗ . The left Kan extension of F ∈ C ∗ along J is given by LKan J F = map D ( J ( − ) , − ) ⊗ C F = c ∈ C Z map D ( J ( c ) , − ) × F ( c ) [ Kel0 5 , (4.25) ]. It is left a d joint to J ∗ , i.e. for every G ∈ D ∗ there are natural homeomorphisms map D ∗ ( LKan J F , G ) ∼ = map C ∗ ( F , J ∗ G ) [ Kel05 , Theorem 4.38 ]. (2.2) It follows that the assignment F 7→ LKan J F gives rise to a continuous functor LKan J : C ∗ → D ∗ . (2.3) Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 4 By [ K el05 , § 4 eq. (4 .19)], f or any G ∈ ( D op ) ∗ there is an isomorphism G ⊗ D LKan J F = ( J ∗ G ) ⊗ C F . (2.4) Given an object c ∈ C , we write y c for the functor map C ( c , − ) ∈ C ∗ . It f ollows from (2.2) and Y oneda’s L emma [ K el05 , 2 .4] that LKan J y c = y J c = map D ( J c , − ) . (2.5) Let C 1 , . . . , C n be S p c -c ategories. The ex ternal product of F i ∈ C ∗ i is f ∏ i F i ∈  ∏ i C i  ∗ , defined by ( c 1 , . . . , c n ) 7→ ∏ i F i ( c i ) (2.6) where the product on the right is taken in S p c . This gives a continuous functor e ∏ : ∏ i C ∗ i → ( ∏ i C i ) ∗ . Given f unctors J i : C i → D i between S p c -categories ( i = 1, . . . , n ), we claim that LKan ∏ i J i  f ∏ i F i  = f ∏ i ( LKan J i F i ) . (2.7) Indeed, for any ( d 1 , . . . , d n ) ∈ ∏ i D i , it f ollows f rom Fubini’s theorem and Y oneda’s Lemma [ Kel05 , § 3.3 and § 3.10 eqs. ( 3.63) , (3.67) , § 2.4] tha t LKan ∏ i J i ( f ∏ i F i ) ( d 1 , . . . , d n ) = ( c 1 ,. . ., c n ) ∈ ∏ i C i Z  ∏ i map D i ( J ( c i ) , d i )  × ∏ i F i ( c i ) = ∏ i  c i ∈ C i Z map D i ( J i ( c i ) , d i ) × F i ( c i )  =  f ∏ i LKan J i F i  ( d 1 , . . . , d n ) . For a ny c i ∈ C i , the external product e ∏ i y c i is the functor y ( c 1 ,. . ., c n ) ∈ ( ∏ i C i ) ∗ . Hence (2.5) implies that for any functor Q : ∏ i C i → D , LKan Q ( f ∏ y c i ) = y Q ( c 1 ,. . ., c n ) = map D ( Q ( c 1 , . . . , c n ) , − ) . (2.8) Following Hollender and V ogt [ HV92 ], we d efine the two-sided bar construc- tion B ( G , C , F ) ∈ s S p c of functors F ∈ C ∗ and G ∈ ( C op ) ∗ as the simplicial space whose space of n -simplices is B n ( G , C , F ) = ∐ c n ,. . ., c 0 ∈ C  G ( c 0 ) × n − 1 ∏ i = 0 map C ( c i + 1 , c i ) × F ( c n )  . The nerve of C is the simplicial space B C = B ( ∗ , C , ∗ ) . If G ∈ ( C op × D ) ∗ is a bi- functor , then B ( G , C , F ) inherits the structure of a functor D → s S p c , and similarly for F . W e would like to take the geometric realiza tion of these bar constructio ns. In order for this to be homotopy meaningful, we need the f ollowing result: Lemma 2.9. If th e values of F and G are CW - c omplexes then B ( G , C , F ) is cofibrant in the Reedy model structure on simp licial spaces (c f. [ GJ9 9 , VII. § 2]). Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 5 Proof. W e need to see that the inclusion of the latc hing obj ects φ n : L n B ( G , C , F ) → B n ( G , C , F ) are cofibrations of topological spaces. W e will in fa ct show that φ n is the inclusion of a CW - subcomplex. Its image is given by im ( φ n ) = { ( g , f n , . . . , f 1 , f ) ∈ B n ( G , C , F ) | at lea st one f i is an identity } , and the inclusion im ( φ n ) ⊆ B n ( G , C , F ) is an inclusion of a CW -subcomplex be- cause we assume that all the spaces map C ( − , − ) and the spaces F ( − ) and G ( − ) are CW -complexes and because { id c } is a 0-ce ll in map C ( c , c ) . Thus, im ( φ n ) ⊆ B n ( G , C , F ) is a cofibration. In order to prove that φ n is a cofibration, it only remains to prove that it is a homeomorphism onto its image. First, φ n is injective because the simplicial set un- derlying B ( G , C , F ) , as any simplicial set, is cofibra nt. Furthermore, φ n is a closed map: every d egeneracy map s i is a closed inclusion since { id c } ⊆ ma p C ( c , c ) are closed inclusions, a nd φ n is a quotient of the closed map n − 1 ∐ i = 0 B n − 1 ( G , C , F ) ∑ s i − − → B n ( G , C , F ) by the definition of L n B ( G , C , F ) . Following [ HV92 ] we define the homotopy coend to be the geometric r ealization B ( G , C , F ) = | B ( G , C , F ) | . That is, B ( G , C , F ) = ∆ • ⊗ ∆ B ( G , C , F ) = Z n ∈ ∆ ∆ n × B n ( G , C , F ) , where ∆ • : ∆ → S p c is the standard n -simplex functor . It follows from [ GJ99 , Proposition VII.3. 6] that if F → F ′ and G → G ′ are (object- wise) weak equivalences of f unctors then the resulting B ( G , C , F ) → B ( G ′ , C , F ′ ) is a wea k equivalence of c ofibra nt spaces. Define E C = B ( map C ( − , − ) , C , ∗ ) ∈ s ( C ∗ ) and EC = |E C | ∈ C ∗ . The Reedy cofibrant simplicial space E C ( c ) is augmented by E − 1 C ( c ) = ∗ , and the inclusion of id e ntity morphisms into map C ( c , c ) gives rise to maps s − 1 : E n C ( c ) → E n + 1 C ( c ) for all n > − 1 by map C ( c 0 , c ) × n − 1 ∏ i = 0 map C ( c i + 1 , c i ) → map C ( c , c ) × map C ( c 0 , c ) × n − 1 ∏ i = 0 map C ( c i + 1 , c i ) . The maps s − 1 form a lef t contraction f or E C ( c ) → ∗ , see [ CDI02 , § 3.2 and Propo- sitions 3 .3 a nd 3.5 ], which implies that E C ( c ) is contractible because the geometric realization of the Reedy cofibrant simplicial space E C ( c ) is equivalent to its homo- topy c olimit as a functor ∆ op → S p c . W e rem ark that E C op ⊗ C F = B ( ∗ , C , F ) [ Kel0 5 , § 3.10 eq. (3.72) ]. Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 6 Definition 2.10. The enriched homotopy (co)limit of F ∈ C ∗ [ DF87 ] is defined by holim C F = [ E C , F ] C = Z c ∈ C hom ( EC ( c ) , F ( c ) ) , hocolim C F = ( E C op ) ⊗ C F = Z c ∈ C E C op ( c ) × F ( c ) . Here hom ( − , − ) is the mapping space of two objects in S p c . Therefore hocolim C F = EC op ⊗ C F = | B ( ∗ , C , F ) | This construction is homotopy invariant as detailed in the next result. Proposition 2.11. Let C , D be CW-categories. 1. Let X , Y ∈ C ∗ be functors who se values are CW -co mplexes. A natural transforma- tion T : X → Y induces a natural map hocolim C X → hocolim C Y which is a weak equivalence if T ( c ) is a weak equivalence X ( c ) → Y ( c ) for all c ∈ C. 2. Fix a functor X ∈ C ∗ whose va lues are CW-complexes and consider a continuous functor F : D → C between two categories C and D with the same objects. Assume that F is the id entity on objects and induces weak equivalences o n mapping spaces. Then the natural map hocolim D F ∗ X → hocolim C X is a weak equivalence. Proof. The natural transformation T and the functor F give rise to morphisms of Reedy cofibrant simplicial spaces B ( ∗ , C , X ) T − → B ( ∗ , D , Y ) and B ( ∗ , D , F ∗ ( X )) F − → B ( ∗ , C , X ) which are homotopy equivalences in every simplicial d e gree by the hypotheses on T a nd F . W e obtain weak equivalences by taking geometric realizations by [ GJ99 , Propositio n VII.3.6 ]. Similarly to Lemma 2.9, for any F ∈ C ∗ the cosimplicial spac e [ E C , F ] C is Reedy fibrant. Here there is no restriction on the spaces F ( c ) because every X ∈ S p c is fibrant. It f ollows that holim C F = T ot ( [ E C , F ] C ) is homotopy invariant. Proposition 2.12. Th e inclusion E 0 C → | E C | = E C induces a fibration in S p c map C ∗ ( E C , X ) → ma p C ∗ ( E 0 C , X ) . For c ∈ C , let y c = map C ( c , − ) ∈ C ∗ be the c ontinuous functor corepresented by c . Given a spa ce A we write y c ⊗ A for the functor c ′ 7→ y c ( c ′ ) × A . Lemma 2.13. Let A → B be a cofibration and let X ∈ C ∗ be a c ontinuous functor . Then map C ∗ ( B ⊗ y c , X ) → map C ∗ ( A ⊗ y c , X ) is a fibration for ev ery c ∈ C . Proof. By adjunction and Y oneda’s Lemma [ Kel05 , § 3.10 eq. (3. 72)], the map under consideration is isomorphic to map ( B , X ( c ) ) → map ( A , X ( c ) ) , which is a fibration by hypothesis on A ⊆ B . One can check this by induction on the skeleta of the rela- tive CW -complexes ( B , A ) and applying [ Spa 66 , § 2.8 , Theorem 2] to the inclusions ∂ ∆ n ⊆ ∆ n . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 7 Proof of Proposition 2.1 2 . Let sk n E C denote the (objectwise) n - skeleton of the sim- plicial C -space E C . By [ GJ99 , VII.3.8 ] and Lemma 2.9 there are pushout squares ∐ c ( A c × ∆ n ∪ A c × ∂ ∆ n B c × ∂ ∆ n ) ⊗ y c 0 / /   | sk n − 1 E C |   ∐ c B c × ∆ n ⊗ y c 0 / / | sk n E C | , (2.14) where c = ( c n , . . . , c 0 ) , B c = ∏ n − 1 i = 0 map C ( c i + 1 , c i ) , and A c ⊆ B c is the subset of degenerate n -simplices, i.e. those where at lea st one map is the identity . W e have seen in L emma 2.9 that B C is Reed y cofibrant and, in fact, that the map L n B C → B n C is an inclusion of CW -complexes. M oreover , L n B C = ∐ A c and B n C = ∐ c B c . This implies that the left hand side of (2. 14) satisfies the conditions of Lemma 2.1 3 . Thus, by applying map C ( − , X ) to (2.14), it becomes a pullback square in which one of the sides is a fibration and therefore the side opposite to it, namely map C ∗ ( | sk n E C | , X ) → map C ∗ ( | sk n − 1 E C | , X ) is a lso a fibration. Since the inverse lim it of a tower of fibrations is a fibration and since E C = colim n | sk n E C | , we obtain the required fibra tion map C ∗ ( E C , X ) → map C ∗ ( sk 0 E C , X ) = map C ∗ ( E 0 C , X ) . 3. A ∞ -algebras, A ∞ -monads, and their completions 3.1. Operads and associat ed categorie s Definition 3.1. By an opera d A of topological spa ces we alwa ys mea n a non- symmetric operad as in [ May72 , Definition 3.1 2]. Tha t is, A consists of a sequence of spaces A ( n ) for n > 0 with associative composition oper a tions A ( n ) × A ( k 1 ) × · · · × A ( k n ) → A  ∑ k i  and a b a se point ι ∈ A ( 1 ) which serves a s a unit. W e c all A a n A ∞ -operad if all the spaces A ( n ) are contrac tible CW -complexes. Notation 3.2 . The following coordinate-free description of an operad A will be useful: If S is a finite ordered set, we write A ( S ) for A ( # S ) . If ϕ : S → T is a mono- tonic map of finite ordered sets, we write A ( ϕ ) = ∏ t ∈ T A ( ϕ − 1 t ) , wher e ϕ − 1 t = ϕ − 1 ( { t } ) ⊆ S . Thus, the operad structure is given by maps µ ( T , ϕ ) : A ( T ) × A ( ϕ ) → A ( S ) , or equivalently , by a collection of ma ps µ ( ψ , ϕ ) : A ( ψ ) × A ( ϕ ) → A ( ψ ◦ ϕ ) for a ll S ϕ − → T ψ − → W Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 8 which are given by ∏ w ∈ W A ( ψ − 1 w ) × ∏ t ∈ T A ( ϕ − 1 t ) = ∏ w ∈ W  A ( ψ − 1 w ) × A ( ϕ | ϕ − 1 ψ − 1 w )  ∏ µ ( ψ − 1 w , ϕ | ϕ − 1 ψ − 1 w ) − − − − − − − − − − − − → ∏ w ∈ W A ( ϕ − 1 ψ − 1 w ) . and which are associative and unital. The base points in A ( { s } ) assemble to a base point ι S ∈ A ( id S ) which acts as a unit for µ . Let ∆ + be the essentially small ca tegory of finite ordered sets and monotonic maps, where we allow the empty set as an object, and ∆ the f ull subcategory of nonempty finite ordered sets. W e might as well take ∆ and ∆ + to be the small skeleton of ordered sets [ n ] = { 0, . . . , n } , but the c oordinate-free setting is more natural. Concatenation of ordered sets makes ∆ + into a (non-symmetric) monoidal category; we will denote this monoidal structure by ⊔ , keeping in mind that it is not a categorical coproduct. W e will now “thicken up” the ca te gories ∆ and ∆ + by allowing morphisms to be parameterize d by the spaces of an operad. S uch cate gories were called “categories of operators in standa rd form” in [ BV68 ]. Definition 3 .3. For an operad A , let ∆ ( A ) be its associated category of opera - tors [ BV68, MT78 ] and let ∆ + ( A ) be the obvious enlarged ca tegory containing the empty set. Explicitly , map ∆ + ( A ) ( S , T ) = ∐ ϕ ∈ Hom ∆ + ( S , T ) A ( ϕ ) (cf. Notation 3 .2) In [ MT78 , § 4] it was shown that this is indeed a mon oidal topological category over the category ∆ + of finite ordered set with c oncate nation ⊔ of ordered sets. Namely , the maps A ( S ) → ∗ give rise to a functor π : ∆ + ( A ) → ∆ + . If A is A ∞ then ∆ + ( A ) is a CW -ca te gory (2.1 ). The following pro position is im- mediate from the definitions: Proposition 3.4. 1. map ∆ + ( A ) ( ∅ , S ) = ∏ S A ( ∅ ) for any S ∈ ∆ + ( A ) . If S 6 = ∅ then map ∆ + ( A ) ( S , ∅ ) = ∅ . 2. ∆ + ( assoc ) = ( ∆ + , ⊔ ) where assoc denotes the “ associative operad”, nam ely the operad all of whose spaces are singletons. 3. If A is an A ∞ -operad then π : ∆ + ( A ) → ∆ + induces hom otopy equivalences on the mapping spaces. 3.2. Algebras over operads in monoidal ca tegories Let ( E , ⋄ , I ) be a monoidal ca tegory enriched over S p c such that ⋄ is an enriched functor . For a n object K we write K ⋄ n for K ⋄ · · · ⋄ K ( n times) where K ⋄ 0 is the monoidal identity object I . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 9 Definition 3.5. The object K gives rise to an operad O End ( K ) d e fined as follows. For a totally ordered set S , the S th spa c e O End ( K ) ( S ) is map E ( K ⋄ S , K ) . For a mono- tonic map ϕ : S → T , the composition law is given by (c f . [ May72 , Definition 1. 2]) map E ( K ⋄ T , K ) × ∏ t ∈ T map E ( K ⋄ ϕ − 1 t , K ) monoidal − − − − − → map E ( K ⋄ T , K ) × map E ( K ⋄ S , K ⋄ T ) compose − − − − → map E ( K ⋄ S , K ) . If A is an operad, an A -algebra in ( E , ⋄ , I ) is an object K ∈ E with a morphism of operads A → O End ( K ) . Remark 3.6. In the classical case of A -a lgebras in ( Sp c , × , ∗ ) , there is a n associated monad A such that A -a lgebras are the same as A -algebras in the above sense. This does not hold in general m onoidal categories. In fact, there is no such thing as a f ree A - algebra on an object K . The r eason for this is the failure of the monoidal structure to be “linear ” over space s: Both of the following conditions are sa tisfied in S p c but not in a genera l monoidal c ategory (the second property assumes that the cate gory is tensored over S p c ) : • K ⋄ ( L 1 ⊔ L 2 ) ∼ = ( K ⋄ L 1 ) ⊔ ( K ⋄ L 2 ) • K ⋄ ( X ⊗ L ) ∼ = X ⊗ ( K ⋄ L ) for all spa c e s X . Definition 3. 7 . Let K be an A -algebra in ( E , ⋄ , I ) with structure map σ : A → O End ( K ) . The A - cobar construction of K is the monoidal f unctor R K : ( ∆ + ( A ) , ⊔ , ∅ ) → ( E , ⋄ , I ) defined by setting R K ( S ) = K ⋄ S , a nd on morphism spac es (cf . Notation 3.2) by map ∆ + ( A ) ( S , T ) = ∐ ϕ ∈ ∆ + ( S , T ) ∏ t ∈ T A ( ϕ − 1 t ) ∐ ϕ ∏ t σ ( ϕ − 1 t ) − − − − − − − − → ∐ ϕ ∈ ∆ + ( S , T ) ∏ t ∈ T map ( K ⋄ ϕ − 1 t , K ) ⋄ − → map ( K ⋄ S , K ⋄ T ) . The cont inuity of the maps between the mapping spaces follows f rom the con- tinuity of the opera d map σ and the continuity of the monoidal structur e maps in E . It clearly carrie s identities in ∆ + ( A ) to identities in E because σ carries the base point of A ( { s } ) to the base point of map ( K , K ) . T o see tha t R K respects composi - tion, let us consider morphi sms S a − → T and T b − → W in ∆ + ( A ) . Thus, a = ( a t ) t ∈ T and b = ( b w ) w ∈ W where a t ∈ A ( ϕ − 1 t ) and b w ∈ A ( ψ − 1 w ) for some monotonic functions ϕ : S → T a nd ψ : T → W . Denote f t = σ ( a t ) ∈ map ( K ⋄ ϕ − 1 t , K ) and g w = σ ( b w ) ∈ map ( K ⋄ ψ − 1 w , K ) . L e t • denote the composition operation in A . Since σ is a morphism of operad s it follows from Definition 3 . 5 of O End, Definitio n 3 .7, and by writing T = F w ψ − 1 w that R K ( b ◦ a ) = R K  ( b w • ( a t ) t ∈ ψ − 1 w ) w ∈ W  = ✸ w ∈ W  σ ( b w • ( a t ) t ∈ ψ − 1 w )  = ✸ w ∈ W  g w ◦  ✸ t ∈ ψ − 1 w f t   =  ✸ w ∈ W g w  ◦  ✸ t ∈ T f t  = R K ( b ) ◦ R K ( a ) . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 10 The fact that the mon oidal operation ⋄ in E is a ssociative is the reason that R K is a monoidal functor . The details are straightforward a nd a re left to the read er . Example 3.8. An ordinary monoid K in E is an assoc-algebra. Indeed the images of assoc 0 and assoc 2 in O End ( K ) give the unit η : I → K and the monoid operation µ : K ◦ K → K . Inspection reveals that R K : ∆ + → E is the standard cobar construc- tion with ( R K ) n = K ⋄ ( n + 1 ) and with coface maps d i = K i η K n − i and s i = K i µ K n − i . Definition 3.9. Given an opera d A , let J : ∆ ( A ) → ∆ + ( A ) denote the inclusion of the full subca tegory ∆ ( A ) spa nned by the non-empty sets. T he completion ˆ K ∈ E of an A -algebra K is ˆ K = holim ∆ ( A ) J ∗ R K = [ E ∆ ( A ) , J ∗ R K ] ∆ ( A ) ( cf. Definition 2.10 . ) 3.3. Endofunctor categories and A -monads Our main examples for E a re categories of endofunctors. L et C be a category which is complete a nd cocomplete, and which is tensored by ⊗ C : C × Sp c → C and cotensored by [ − , − ] C : S p c op × C → C . The a d joint map C : C op × C → Sp c of ⊗ C endows C with an enrichment over Sp c . Let E = E ND ( C ) d enote the Ca tegory of a ll the continuous endofunctors of C together with natural transformations between them. The word Category is ca p- italized beca use the natura l tra nsformations between two endofunctors need not form a set. T o overcome this difficulty we pass to a larger universe [ ML98 , I.6]. The morphism Spaces in E ND ( C ) are denoted Map ( K , L ) . Remark 3.1 0 . Thro ughout this paper we will not make an essential use of the larger universe. The introduction of the Ca te gory END ( C ) is only done for the sake of book-keeping and we will never refer to morphism S ets in END ( C ) . One may prefer to consider END ( C ) as a convenient notation f or the classes of con- tinuous endofunctors K on C a nd the classes of natural transformations between them. In f act, we are only going to construct explicit functors D → END ( C ) from small topological ca tegories D . These are the same a s continuous (honest) functors F : D × C → C . Alternatively , these are assignments of e ndofunctors F ( d ) for every object of D and of natural transformations F ( d ) → F ( d ′ ) for every morphism of D such tha t the compositions map D ( d , d ′ ) → Nat ( F ( d ) , F ( d ′ ) ) ev X − − → map C ( F ( d ) ( X ) , F ( d ′ ) ( X )) are continuous functions for all objects X ∈ C . It now becomes obvious that it is just a matter of convenience to assume that END ( C ) is a Category (capitalized or not). The Category END ( C ) is tensored and cotensored over S pc , where for every space A ∈ S p c and every F ∈ END ( C ) we d efine F ⊗ A and [ A , F ] in END ( C ) as the endofunctors defined by x 7 → F ( x ) ⊗ C A and x 7 → [ A , F ( x )] C . The adjoint of the tensor is M a p ( F , G ) = R X ∈ C map C ( F ( X ) , G ( X ) ) . Composition of f unctors equips END ( C ) with an enriched mono idal structur e ( ⋄ , I ) : K ⋄ L = K ◦ L and I = id C . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 11 W e will reserve the symbols ◦ and id for composition of natural tr a nsformations (i.e. Morphisms) and the identity natural transformation in this Category . Explic- itly , Map ( K , L ) × Ma p ( K ′ , L ′ ) ⋄ − → Map ( K ⋄ K ′ , L ⋄ L ′ ) has the effec t ( f , f ′ ) 7→ f ⋄ f ′ = ( L ⋄ f ′ ) ◦ ( f ⋄ K ′ ) = ( f ⋄ L ′ ) ◦ ( K ⋄ f ′ ) . The conti- nuity of K and L guarantees the continuity of the map a bove between the map- ping Spaces. T o see that this gives rise to a mon oidal structur e one calcula tes that id K ⋄ id L = ( L ⋄ id K ) ◦ ( id L ⋄ K ) = id K ⋄ L ◦ id K ⋄ L = id K ⋄ L . Also, given f i : K i → L i and g i : J i → K i where i = 1, 2 we have ( f 1 ⋄ f 2 ) ◦ ( g 1 ⋄ g 2 ) = ( f 1 ⋄ L 2 ) ◦ ( K 1 ⋄ f 2 ) ◦ ( K 1 ⋄ g 2 ) ◦ ( g 1 ⋄ J 2 ) = ( f 1 ⋄ L 2 ) ◦ ( K 1 ⋄ ( f 2 ◦ g 2 ) ) ◦ ( g 1 ⋄ J 2 ) = ( f 1 ⋄ L 2 ) ◦ ( g 1 ⋄ L 2 ) ◦ ( J 1 ⋄ ( f 2 ◦ g 2 ) ) = ( f 1 ◦ g 1 ) ⋄ ( f 2 ◦ g 2 ) . Any K ∈ END ( C ) gives rise to an Operad O End ( K ) as in Definition 3.5, whose n th Space is Map ( K ⋄ n , K ) . Definition 3.11. An A-monad in C is an A -algebra in ( END ( C ) , ⋄ , I ) . Thus, an A - monad in C is an endofunctor of C , namely an object K ∈ END ( C ) together with a Morphism of Operads σ : A → O End ( K ) , see Definition 3.7, where O End ( K ) is an Opera d of S paces and A is an operad of (small) spaces. If one wishes to avoid the set-theoretic difficulties a round END ( C ) mentioned be fore, one can replace the map of Operads σ : A → O End ( K ) with a collection of natural trans- formations of e ndofunctors of C A ( n ) ⊗ C K ⋄ n ( − ) α ( n ) − − → K ( − ) which ma ke K an algebr a over the operad A in a sense similar to [ May72 ]. T o do this, one needs to use the continuity of K , which gives rise to natural ma ps U ⊗ C K ( − ) → K ( U ⊗ C − ) for any topological space U . Note that ordinary monads [ E M6 5 ] (called triples there) are assoc-monads in our terminolog y , a nd the cobar construction R K ( X ) of Definit ion 3.7 and comple- tion ˆ K ( X ) of Definition 3.9 coincide with the usual de finitions [ BK7 2 ]. 4. The extended A ∞ -operad The goal of this section is to constr uct an “e xtended” operad ˜ A associated with any operad A . In Section 5 , this operad will be seen to act natura lly on the comple- tion of a ny A - algebra ( Definition 3 .9.) Througho ut this section we fix an opera d A and let D + denote its category of operators ∆ + ( A ) (Definition 3.3) . W e de note by J : D → D + the inclusion of the full subcategory spanned by the objects ∅ 6 = S ∈ D + . The S -fold product of a category C with itself, where S is a finite ordered set, is denoted C S . Definition 4.1. For a finite ordered set S , the ordinal sum functor is defined by u ( S ) : D + S → D + , ( V s ) s ∈ S 7→ G s ∈ S V s . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 12 The monoidal a xioms for ⊔ readily imply Lemma 4.2. 1. u ( ∅ ) : ∗ → D + is the inclusion of the ⊔ - unit ∅ into D + . 2. u ( ∗ ) = id D + . 3. u ( S ) = u ( T ) ◦  ∏ t ∈ T u ( ϕ − 1 t )  for any m onotonic S ϕ − → T (see Nota tion 3 .2). The functor E D ∈ D ∗ (see Section 2) ca n be extended to a continuous functor g E D ∈ D ∗ + by setting g E D ( ∅ ) = ∅ . In f act, by Pr oposition 3.4 and [ Kel05 , Propo si- tion 4.23] g E D = LKan J E D (4.3) because J is fully faithful. Definition 4.4. For a finite ordered set S define ǫ ( S ) ∈ D ∗ + by ǫ ( S ) = LKan u ( S )  f ∏ s ∈ S g E D  (see Definition 2. 6). Let ϕ : S → T be a monotonic map and write S = F t ∈ T ϕ − 1 t . It follows from (2.7) and f rom [ Kel0 5 , Theorem 4 . 47] that ǫ ( S ) = LKan u ( S ) ( f ∏ S g E D ) = Lemma 4.2 LKan u ( T ) ◦ LKan ∏ t u ( ϕ − 1 t )  f ∏ t ∈ T f ∏ ϕ − 1 t g E D  = (2.7) LKan u ( T )  f ∏ t LKan u ( ϕ − 1 t ) ( f ∏ ϕ − 1 t g E D )  = LKan u ( T )  f ∏ t ∈ T ǫ ( ϕ − 1 t )  (4.5) Definition 4.6. Given a finite ordered set S , define a space ˜ A ( S ) by ˜ A ( S ) = map D ∗ + ( g E D , ǫ ( S ) ) . Analogously to Notation 3. 2, for a monotonic map ϕ : S → T we d efine ˜ A ( ϕ ) by ˜ A ( ϕ ) = ∏ t ∈ T ˜ A ( ϕ − 1 t ) = ∏ t ∈ T map D ∗ + ( g E D , ǫ ( ϕ − 1 t ) ) If | S | = 1 then Le mma 4. 2 (2) implies that ǫ ( S ) = g E D . The identity on g E D equips ˜ A ( S ) with a natural base point. For any ϕ : S → T , d e fine a continuous function ˜ µ ( T , ϕ ) : ˜ A ( T ) × ˜ A ( ϕ ) → ˜ A ( S ) (4.7) as follows. For any element f ∈ ˜ A ( T ) and any element ( g t ) t ∈ T ∈ ˜ A ( ϕ ) recall tha t f and g t are morphisms f : g E D → ǫ ( T ) and g t : g E D → ǫ ( ϕ − 1 t ) . Apply (4.5 ) to define ˜ µ ( T , ϕ ) : ( f , ( g t ) ) 7 → LKan u ( T ) ( f ∏ t ∈ T g t ) ◦ f . The continuity of ˜ µ ( T , ϕ ) follow s f rom the fact tha t L Kan u ( T ) and e ∏ T are continu- ous f unctors ( 2.3, 2.6). Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 13 Following Notation 3 .2, this gives rise to maps ˜ A ( ψ ) × ˜ A ( ϕ ) ˜ µ ( ψ , ϕ ) − − − → ˜ A ( ψ ◦ ϕ ) for a ll S ϕ − → T ψ − → W . (4 .8) That is, ˜ µ ( ψ , ϕ ) = ( ˜ µ ( ψ − 1 w , ϕ | ϕ − 1 ψ − 1 w ) ) w ∈ W The goal of this section is to prove the following two results. Theorem 4 .9. The spaces ˜ A ( S ) t ogether with the maps ˜ µ ( ϕ , ψ ) give rise to an operad ˜ A which we call t h e “ex tended operad” associated t o A. Theorem 4.10. I f A is an A ∞ -operad th en the spa ces ˜ A ( S ) are weakly contract ible. Proof of Theorem 4 .9. Given ϕ : S → T we denote a n element of ˜ A ( ϕ ) by f = ( f t ) t ∈ T for f t ∈ map D ∗ + ( g E D , ǫ ( ϕ − 1 t ) ) . For monotonic maps S ϕ − → T ψ − → W and elements f = ( f t ) ∈ ˜ A ( ϕ ) and g = ( g w ) ∈ ˜ A ( ψ ) , (4 .7) and (4. 8) imply that g ◦ f : = ˜ µ ( g , f ) = ( h w ) where h w = LKan u ( ψ − 1 w )  f ∏ t ∈ ψ − 1 w f t  ◦ g w : g E D → ǫ ( ϕ − 1 ψ − 1 w ) (4.11) Associativity of com position. Let S ϕ − → T ψ − → W θ − → R be maps in ∆ + and consider f = ( f t ) ∈ ˜ A ( ϕ ) , g = ( g w ) ∈ ˜ A ( ψ ) , and h = ( h r ) ∈ ˜ A ( θ ) . Then ( ( h ◦ g ) ◦ f ) r = (4.11) LKan u ( ψ − 1 θ − 1 r )  f ∏ t ∈ ψ − 1 θ − 1 r f t  ◦  LKan u ( θ − 1 r )  f ∏ w ∈ θ − 1 r g w   ◦ h r = Lemma 4.2 LKan u ( θ − 1 r ) ◦ LKan ∏ w ∈ θ − 1 r u ( ψ − 1 w )  f ∏ w ∈ θ − 1 r ( f ∏ t ∈ ψ − 1 w f t )  ◦  LKan u ( θ − 1 r )  f ∏ w ∈ θ − 1 r g w   ◦ h r = (2.7) LKan u ( θ − 1 r )  f ∏ w ∈ θ − 1 r  LKan u ( ψ − 1 w ) ( f ∏ t ∈ ψ − 1 w f t )  ◦ g w  ◦ h r = (4.11) LKan u ( θ − 1 r )  f ∏ w ∈ θ − 1 r ( g ◦ f ) w  ◦ h r = ( h ◦ ( g ◦ f ) ) r . Unitality of composition. There are two cases to check. First consider S id S − → S ϕ − → T and fix f = ( f t ) ∈ ˜ A ( ϕ ) . Since L Kan and e ∏ are functors, it f ollows from Defini- tion 4.4 that ( f ◦ ι S ) t = LKan u ( ϕ − 1 t )  f ∏ s ∈ ϕ − 1 t id g E D  ◦ f t = id ǫ ( ϕ − 1 t ) ◦ f t = f t . Thus, f ◦ ι S = f . Next, consider S ϕ − → T id T − → T and and element f = ( f t ) ∈ ˜ A ( ϕ ) . By Lemma 4 .2, u ( ∗ ) : D + → D + is the id entity , hence ( ι T ◦ f ) t = ( LKa n u ( ∗ ) f t ) ◦ id g E D = f t . Therefore ι T ◦ f = f . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 14 Lemma 4.12. Fix n > 1 and let ∆ × n be t h e n-fold product. T h en for any T ∈ ∆ we ha v e hocolim ( P 1 ,. . ., P n ) ∈ ( ∆ × n ) op ∆ ( P 1 ⊔ · · · ⊔ P n , T ) ≃ ∗ . Proof. When n = 1 this is a triviality because hocolim ∆ op ∆ ( − , T ) = ∆ ( − , T ) ⊗ ∆ op E ∆ = E ∆ ( T ) ≃ ∗ . Given θ ∈ ∆ ( P , T ) we denote by max ( θ ) ∈ T the maximal element in the image of θ . For any t ∈ T we denote by T − t the subset of T consisting of all the elements t ′ ∈ T such that t ′ > t . W e observe that ∆ ( P ⊔ Q , T ) = ∐ θ ∈ ∆ ( P , T ) ∆ ( Q , T − max ( θ )) . The lemma now follows by induction on n be cause hocolim ( P 1 ,. . ., P n ) ∈ ∆ × n ∆  n G i = 1 P i , T  = hocolim P 1 ∈ ∆ hocolim ( P 2 ,. . ., P n ) ∈ ∆ × ( n − 1 ) ∆  P 1 ⊔ n G i = 2 P i , T  = hocolim P 1 ∈ ∆ ∐ θ ∈ ∆ ( P 1 , T ) hocolim ( P 2 ,. . ., P n ) ∈ ∆ × ( n − 1 ) ∆  n G i = 2 P i , T − max ( θ )  | {z } ≃∗ by induct ion ≃ hocolim P 1 ∈ ∆ ∆ ( P 1 , T ) ≃ ∗ . Proof of Theorem 4 .10. Since g E D = LKan J E D , equation (2.2) implies ˜ A ( S ) = map D ∗ + ( g E D , ǫ ( S ) ) = map D ∗ ( E D , J ∗ ( ǫ ( S ) ) ) = holi m D J ∗ ( ǫ ( S ) ) . It is therefore sufficient to prove that the values of J ∗ ( ǫ ( S ) ) are contractible spac e s, that is, we have to show that ǫ ( S ) ( T ) ≃ ∗ for eve ry T 6 = ∅ . First, we note that e ∏ S E D = E D S because E ( D S ) = ( E D ) S and because geo- metric realiza tion commutes with finite products by [ May72 , Theorem 11.5 ] . From Definition 4 .4, (2.7) a nd [ Kel0 5 , Theorem 4.47] it follows that ǫ ( S ) = LKan u ( S )  f ∏ S LKan J E D  = LKan u ( S ) ◦ LKan ∏ S J  f ∏ S E D  = LKan u ( S ) ◦ ( ∏ S J ) ( E D S ) . Set F = u ( S ) ◦ ( ∏ S J ) , thus ǫ ( S ) = ( LKan F E D S ) . By the de finition of LKan and by Fubini’s theorem in enriched categories [ Kel05 , § 2.1 and 3 .3], ǫ ( S ) ( T ) = map D + ( F ( − ) , T ) ⊗ D S E D S = hocolim ( D S ) op map D + ( F ( − ) , T ) . There is an obviou s ordinal sum functor u ′ ( S ) : ∆ S + → ∆ + and inclusion J ′ : ∆ → Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 15 ∆ + which render the following diagrams commutative: D + S ∏ S π   u ( S ) / / D + π   ∆ + S u ′ ( S ) / / ∆ + D J / / π   D + π   ∆ J ′ / / ∆ + . The vertical arrows induce the identity on object sets and are wea k equivalences of categories by Proposition 3 .4(3). Set F ′ : = u ′ ( S ) ◦ ( ∏ S J ′ ) . T his is the functor ∆ × n → ∆ + where ( P 1 , . . . , P n ) 7→ F i P i . There results a weak equivalence of functors D + → S p c map D + ( F ( − ) , T ) ≃ − → ( ∏ S π ) ∗  ∆ + ( F ′ ( − ) , T )  . It follows from Proposition 2.1 1 that there is a weak equivalence ǫ ( S ) ( T ) ≃ hocol im ( ∆ S ) op ∆ + ( F ′ ( − ) , T ) which is weakly equivalent to a point by Lemma 4. 1 2 and the fact that T 6 = ∅ and that ∆ is a full subcategory of ∆ + . 5. The action of ˜ A on the completion As in Subsection 3.2, let ( E , ⋄ , I ) be a complete and cocomplete monoidal cate- gory c ompatibly te nsored and cotensored over Sp c . Througho ut this section we will write f : X → Y for the ele ments of map E ( X , Y ) , cf. [ Ke l05 , 1.3]. Furthermore, for F , G ∈ E and A ∈ S p c we will write F A for [ A , F ] and { F , G } for map E ( F , G ) . In this section we will prove: Theorem 5. 1. Let A be an A ∞ -operad and K be an A- algebra in ( E , ⋄ , I ) (cf. D efini- tion 3.11) . Then ˆ K is an ˜ A-algebra where ˜ A is defined in Definition 4.6. Given A , B ∈ S p c a nd F , G ∈ E we c onsider the map λ F , G A , B defined by A × B ev # A × ev # B − − − − − → { F A , F } × { G B , G } ✸ − → { F A ⋄ G B , F ⋄ G } . (5.2) By adjoining F A ⋄ G B to the le f t a nd then A × B to the right we obtain maps ( F A ⋄ G B ) ⊗ ( A × B ) ρ F , G A , B − − → F ⋄ G and F A ⋄ G B Θ F , G A , B − − → ( F ⋄ G ) A × B . (5.3 ) They are natura l in A , B , F a nd G and we will frequently omit these “d ecorations” from λ F , G A , B . By a djunction, (5.2) is equal to A × B ev # A × B − − − →  ( F ⋄ G ) A × B , F ⋄ G  ( Θ F , G A , B ) ∗ − − − − →  F A ⋄ G B , F ⋄ G  . (5.4) Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 16 W e claim that the natural transformations Θ are a ssociative in the sense that [ A , F ] ⋄ [ B , G ] ⋄ [ C , H ] Θ F , G A , B ⋄ [ C , H ] / / [ A , F ] ⋄ Θ G , H B , C   [ A × B , F ⋄ G ] ⋄ [ C , H ] Θ F ⋄ G , H A × B , C   [ A , F ] ⋄ [ B × C , G ⋄ H ] Θ F , G ⋄ H A , B × C / / [ A × B × C , F ⋄ G ⋄ H ] (5.5) commutes. Indeed, consider first the composition of the arrows a t the top and right. After a djoining A × B × C to the left and F A ⋄ G B ⋄ H C to the right, the adjunc- tions of (5.2) and (5.3) show that this map becomes the composition of the arrows on the lef t and bottom of the following diagram. A × B × C ev # A × ev # B × ev # C / / ev # A × B × ev # C   ( ρ F ⋄ G , H A × B , C ) # * * { F A , F } × { G B , G } × { H C , H } ✸ × Id   ✸ t t { ( F ⋄ G ) A × B , F ⋄ G } × { H C , H } Θ ∗ × Id / / ✸   { F A ⋄ G B , F ⋄ G } × { H C , H } ✸   { ( F ⋄ G ) A × B ⋄ H C , F ⋄ G ⋄ H } ( Θ ⋄ H C ) ∗ / / { F A ⋄ G B ⋄ H C , F ⋄ G ⋄ H } The first square of this diagram commutes by (5 .4) and the second since ✸ is func- torial. Using a simi lar argument on e shows that the adjoint of the composition of the arrows at the left and bottom of (5.5) is also equal to the composition of the arrows at the top and right of the diagram above. It follows that (5 . 5) commutes. Now consider a small CW -ca tegory C ( 2.1) and a functor F : C → E . For any A : C → Sp c we d efine [ A , F ] C = Z c ∈ C [ A ( c ) , F ( c ) ] ∈ E ( cf. Definition 2 .10). The assignment F 7→ [ A , F ] C is a continuous functor because the inverse limit functor is. Given f unctors F i : C i → E ( i = 1, 2), there is a f unctor F 1 ˜ ⋄ F 2 : C 1 × C 2 → E , ( c 1 , c 2 ) 7 → F 1 ( c 1 ) ⋄ F 2 ( c 2 ) . The transformation ˜ ⋄ is natural and associative because the monoidal operation ⋄ is associative. T his construction genera lizes (2 .6). By taking ends, the natural transformations Θ of (5.3) now give rise to natural transformations Θ : [ A 1 , F 1 ] C 1 ⋄ [ A 2 , F 2 ] C 2 → [ A 1 ˜ × A 2 , F 1 ˜ ⋄ F 2 ] C 1 × C 2 . (5.6) where A i ∈ C ∗ i and F i : C i → E f or i = 1 , 2. The associativity ( 5.5) and the naturality Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 17 of limits imply a similar associativity: [ A 1 , F 1 ] C 1 ⋄ [ A 2 , F 2 ] C 2 ⋄ [ A 3 , F 3 ] C 3 Θ ⋄ Id / / Id ⋄ Θ   [ A 1 ˜ × A 2 , F 1 ˜ ⋄ F 2 ] C 1 × C 2 ⋄ [ A 3 , F 3 ] C 3 Θ   [ A 1 , F 1 ] C 1 ⋄ [ A 2 ˜ × A 3 , F 2 ˜ ⋄ F 3 ] C 2 × C 3 Θ / / [ A 1 ˜ × A 2 ˜ × A 3 , F 1 ˜ ⋄ F 2 ˜ ⋄ F 3 ] C 1 × C 2 × C 3 (5.7) W e will now fix an A ∞ -operad A with category of opera tors D + = ∆ + ( A ) (Definition 3.3). The inclusion of the full subca tegory of the non-empty sets is de- noted J : D → D + . W e recall fro m Definition 3 .7 that an A -a lgebra K gives rise to a monoidal f unctor R = R K : D + → E such that R K ( ∗ ) = K . By Definition 3.9 a nd (4.3), ˆ K = [ E D , J ∗ R K ] D = [ g E D , R K ] D + . Since R is monoidal, for ever y finite ordered set S ( see Definition 4. 1 ), R ˜ ⋄ · · · ˜ ⋄ R | {z } | S | t imes = u ( S ) ∗ ( R ) . (5.8) Definition 5.9. For a ny finite ordered set S , define R ( S ) = h f ∏ s ∈ S g E D , e ✸ s ∈ S R i D S + ∈ E In particular , R ( 1 ) = ˆ K . The natural tra nsformations Θ of (5 .6) give rise to natu- ral tr ansformations R ( S ) ⋄ R ( T ) → R ( S ⊔ T ) . It follows from ( 5.7) that for e very monotonic ϕ : S → T there results a well-d efined natural transformation β ϕ : ✸ t ∈ T R ( ϕ − 1 t ) → R ( S ) , (see Notation 3.2). (5.10) It also follows f rom (5. 7) that the transformations β are associative in the sense that for any S ϕ − → T ψ − → W the following square commutes: ✸ w ∈ W ✸ t ∈ ψ − 1 w R ( ϕ − 1 t ) ✸ w ∈ W β ϕ | ϕ − 1 ψ − 1 w / / ✸ w ∈ W R ( ϕ − 1 ψ − 1 w ) β ψ ◦ ϕ   ✸ t ∈ T R ( ϕ − 1 t ) β ϕ / / R ( S ) (5.11) W e see f rom (5.8), from the ad junction of u ( S ) ∗ and L Kan u ( S ) , and f rom Defini- tion 4.4 that R ( S ) = [ ǫ ( S ) , R ] D + . (5.12) Thus, there is a continuous map map D ∗ + ( ǫ ( S ) , ǫ ( T ) ) f 7 → f ∗ − − − − → map ( R ( S ) , R ( T ) ) . Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 18 Lemma 5.13. Let ϕ : S → T be a monotonic map and consider f t : g E D → ǫ ( ϕ − 1 t ) fo r all t ∈ T , cf. Notation 3.2 . Then the following sq uare comm utes: ✸ t ∈ T R ( ϕ − 1 t ) ✸ t ∈ T ( f ∗ t ) / / β ϕ   ✸ t ∈ T R ( 1 ) β id T   R ( S ) ( LK a n u ( T ) ( e ∏ t f t ) ) ∗ / / R ( T ) Proof. Immediate from the naturality of Θ together with (5 .12) and (4 .5). Proof of Theorem 5 .1. For eve ry finite ordered set S define σ S : ˜ A ( S ) → Map ( ✸ S R ( 1 ) , R ( 1 ) ) by the composition map D ∗ + ( g E D , ǫ ( S ) ) f 7 → f ∗ − − − → Map ( R ( S ) , R ( 1 ) ) β ∗ id S − − → Map ( ✸ S R ( 1 ) , R ( 1 ) ) . (5 . 14) W e cla im that these maps form a morphism of operads ˜ A → O End ( ˆ K ) . First, ob- serve that when S = ∗ then β id S = id R ( 1 ) and therefore σ ( id g E D ) = id R ( 1 ) . Thus, σ respects the identity elements of the operads. It remains to pro ve that σ respects the compositio n operation in both operads, which we de note b y • . Consider some ϕ : S → T and elements f ∈ map ( g E D , ǫ ( T ) ) = ˜ A ( T ) , and g t ∈ map ( g E D , ǫ ( ϕ − 1 t ) ) = ˜ A ( ϕ − 1 t ) ( t ∈ T ) . By construction ( Definition 4.6), f • ( g t ) = ( LKa n u ( T ) e ∏ t g t ) ◦ f . Now , by the defi- nition of σ , the monoidality of ⋄ , Le mma 5.13, and ( 5.11), σ ( f ) •  σ ( g t )  t = ( f ∗ ◦ β id T ) ◦  ✸ t ∈ T ( g ∗ t ◦ β id ϕ − 1 t )  = ( f ∗ ◦ β Id T ) ◦  ✸ t ∈ T g ∗ t  ◦  ✸ t ∈ T β id ϕ − 1 t  = f ∗ ◦  LKan u ( T ) f ∏ t g t  ∗ ◦ β ϕ ◦  ✸ t ∈ T β id ϕ − 1 t  =  f • ( g t ) t  ∗ ◦ β id S = σ  f • ( g t ) t  . That is, σ respects the composition operations of the operads. 6. The restricted operad action In Section 5 we showed how ˜ A acts on the completion ˆ K of any A - algebra K . The goal of this sec tion is to construct an operad ˆ A which is A ∞ if A is, together with morphisms of operads ˜ A ← ˆ A → A which make the natural map ˆ K → K defined in (6.7) into a morphism of ˆ A -algebr a s. Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 19 Recall that for any S p c -ca tegory C we write y c for the c orepresentable functor map C ( c , − ) . An element g ∈ map C ( c , c ′ ) gives rise to a natural transformation g ∗ : y c ′ → y c which we a lso denote by y g . Fix an operad A and set D + = ∆ + ( A ) , see Definition 3.3 . Following Nota- tion 3.2 , given a monotonic map ϕ : S → T , we obtain f unctors y ϕ − 1 t . By (4.5, 2 .8), LKan u ( T ) ( e ∏ t y ϕ − 1 t ) = y S . M ore generally , for ψ : S ′ → S we have LKan u ( T )  f ∏ t g ∗ t  =  G t g t  ∗ : y S → y S ′ where g t ∈ map D + ( ψ − 1 ϕ − 1 t , ϕ − 1 t ) . (6.1) Since E 0 D + = ∐ T ∈ D + y T , we obtain from E 0 D + ⊆ E D + a na tural transformation κ { ∗} : y { ∗} → g E D . More generally , from (6 .1) a nd Definition 4.4 we obtain a na tural tr a nsformation κ S : y S → ǫ ( S ) defined by κ S = LKan u ( S )  f ∏ S κ { ∗}  ( S ∈ D + ). (6.2) For every finite ordered set S , we define maps ρ : ˜ A ( S ) = map D ∗ + ( g E D , ǫ ( S ) ) ( κ {∗ } ) ∗ − − − − → map D +  y { ∗} , ǫ ( S )  , ζ : A ( S ) = map D + ( S , ∗ ) = map D ∗ + ( y { ∗} , y S ) ( κ S ) ∗ − − − → map D ∗ +  y { ∗} , ǫ ( S )  . Definition 6.3. For a ny finite ordered set S , define the space ˆ A ( S ) as the pullback ˆ A ( S ) ¯ ζ / / ¯ ρ   ˜ A ( S ) ρ   A ( S ) ζ / / map D ∗ +  y { ∗} , ǫ ( S )  Thus, ˆ A ( S ) is the subspace of ˜ A ( S ) × A ( S ) consisting of the pairs ( f , g ) with f : g E D → ǫ ( S ) and g ∈ A ( S ) = map D + ( S , ∗ ) satisfying f ◦ κ { ∗} = κ S ◦ y g . (6.4) W e equip ˜ A × A with the product operad structure. Theorem 6.5 . The spaces ˆ A ( S ) form a sub-operad of ˜ A × A . In pa rticular the projections ˆ A → ˜ A and ˆ A → A are morph isms of operads. More over , if A is an A ∞ -operad then ˆ A ( S ) is weakly contractible for all S a nd if A ( ∅ ) = ∗ then ˆ A ( ∅ ) = ∗ . Proof. W e first note that if A ( ∅ ) = ∗ then ǫ ( ∅ ) = ∗ by Lemma 4.2(1 ) and Defini- tion 4.4. Therefore all three spaces defining the pullback square of Definition 6.3 are points, whence ˆ A ( ∅ ) = ∗ . Now a ssume that A is a n A ∞ -operad. Consider the inclusion y { ∗} ⊆ E 0 D + = ∐ T ∈ D + y T . By L emma 2 . 12 there is a composite fibration map D ∗ + ( g E D , ǫ ( S ) ) → map D ∗ + ( E 0 D + , ǫ ( S ) ) → map D + ( y { ∗} , ǫ ( S ) ) = ǫ ( S ) ( ∗ ) Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 20 because the second map is isomorphic to the fibra tion ∏ T ∈ D + ǫ ( S ) ( T ) → ǫ ( S ) ( ∗ ) by Y oneda’s Lemma. In the proof of Theorem 4 .10 we have show n that ǫ ( S ) ( ∗ ) is weakly contractible and d educed that the spaces ˜ A ( S ) are weakly contractible. It now follows that the map ρ in the pullback square in Definition 6.3 is an equiv- alence as well as a fibration. Since a pullback of a trivial fibration of spa ces is a trivial fibration, we de duce that ˆ A ( S ) → A ( S ) ≃ ∗ is an equivalence. It remains to show that the spa c es ˆ A ( S ) form a sub-operad of ˜ A × A . W e will let • denote the operad composition in both operads ˜ A and A . Fix a monotonic map ϕ : S → T and consider ( f , g ) ∈ ˆ A ( T ) and ( f t , g t ) ∈ ˆ A ( ϕ − 1 t ) . Set f ′ = f • ( f t ) and g ′ = g • ( g t ) . W e now claim that ( f ′ , g ′ ) ∈ ˆ A ( S ) . First, we note that by (2. 7), Lemma 4 .2(3), the de finition of κ ϕ − 1 t , a nd [ Kel0 5 , Theorem 4.47] , LKan u ( T )  f ∏ t ∈ T κ ϕ − 1 t  = LKan u ( T )  f ∏ t ∈ T LKan u ( ϕ − 1 t )  f ∏ ϕ − 1 t κ { ∗}   = LKan u ( S )  f ∏ S κ { ∗}  = κ S . W e also observe that by Definition 3.3 of the ca tegory D + and (6.1 ), LKan u ( T )  f ∏ t ∈ T g ∗ t  ◦ g ∗ =  G t g t  ∗ ◦ g ∗ =  g • ( g t )  ∗ . (6.6) It follows that ( f ′ , g ′ ) ∈ ˆ A ( S ) be cause (6. 4) applies to ( f , g ) and ( f t , g t ) , hence f ′ ◦ κ { ∗} = LKan u ( T )  f ∏ T f t  ◦ f ◦ κ { ∗} = LKan u ( T )  f ∏ T f t  ◦ κ T ◦ g ∗ = (6.2) LKan u ( T )  f ∏ T ( f t ◦ κ { ∗} )  ◦ g ∗ = LKan u ( T )  f ∏ T ( κ ϕ − 1 t ◦ g ∗ t )  ◦ g ∗ = LKan u ( T )  f ∏ T κ ϕ − 1 t  ◦ LKan u ( T )  f ∏ T g ∗ t  ◦ g ∗ = (6.6) κ S ◦ ( g • ( g t ) ) ∗ = κ S ◦ ( g ′ ) ∗ . Thus, ˆ A is stable under the composition law of ˜ A × A . It remains to show that ˆ A contains the identity element ( id g E D , ι ) of ˜ A × A . Indeed, ι is the identity morphism in map D + ( ∗ , ∗ ) and therefore id g E D ◦ κ { ∗} = κ { ∗} ◦ ι ∗ . Thus, ( id g E D , ι ) ∈ ˆ A ( ∗ ) by (6.4). If K is a n A -algebra, κ { ∗} gives rise to a natural transformation ˆ K = [ g E D , R K ] D + τ = ( κ {∗ } ) ∗ − − − − − − − − − → [ y ∗ , R K ] D + = R K ( ∗ ) = K . (6.7) Proof of Theorem 1 .1. W e consider the morphisms of operad s ˆ A → ˜ A and ˆ A → A obtained from Theorem 6. 5. T ogether with Theorem 5 .1 we obtain a morphism of operads ˆ A → ˜ A → O End ( ˆ K ) , that is, ˆ K is an ˆ A - algebra. T o prove tha t τ : ˆ K → K is a morphism of ˆ A -algebr a s we Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 21 need to prove that that the following diagram is commutative: ˆ A ( S ) σ ˆ K / / ¯ ρ   Map  ✸ S R ( 1 ) , R ( 1 )  τ ∗ / / Map  ✸ R ( 1 ) , K  A ( S ) σ K / / Map  ✸ K , K  ( ✸ τ ) ∗ O O (6.8) By Definition 3.7, K gives rise to a monoidal f unctor R = R K : ∆ + ( A ) → E such that R K ( ∗ ) = K and R ( 1 ) = ˆ K (5.9). The natura lity of the transformation Θ (5.6 ) and the d e finition of β (5.10) imply that the following diagram commutes: ✸ S [ g E D , R ] D + ✸ S τ / / β id S   ✸ S [ y ∗ , R ] D + Θ   σ K ( g ) / / [ y { ∗} , R ] D + [ ǫ ( S ) , R ] D + ( κ S ) ∗ / / [ y S , R ] D + ( y g ) ∗ 7 7 p p p p p p p p p p p (6.9) By Y oneda and because ( y g ) ∗ is isomorphic to σ K ( g ) = R ( g ) : R ( S ) → R ( ∗ ) , ✸ S  y { ∗} , R  D + = ✸ S R ( ∗ ) = R ( S ) = [ y S , R ] D + . The image of ( f , g ) ∈ ˆ A ( S ) ⊆ ˜ A ( S ) × A ( S ) under the top map of Diagram ( 6 .8) is the na tural tr a nsformation ✸ S R ( 1 ) β id S − − → R ( S ) f ∗ − → R ( 1 ) τ = ( κ {∗ } ) ∗ − − − − − → [ y { ∗} , R ] = K . Since f ◦ κ { ∗} = κ S ◦ g ∗ by (6.4), the commutativity of ( 6 .9) implies τ ◦ f ∗ ◦ β id S = ( κ S ◦ g ∗ ) ∗ ◦ β id S = ( y g ) ∗ ◦ κ ∗ S ◦ β id S =  ✸ S τ  ◦ σ K ( g ) . This shows that ( 6.8) c ommutes, hence ˆ K → K is a morphism of ˆ A - algebras. T o resolve the difficulty that ˆ A need not consist of CW -complexes, we a pply the map | Sing ( ˆ A ) | → ˆ A . This is a morphism of opera ds by naturality , since both S ing and | − | commute with products, a nd | S ing ( ∗ ) | = ∗ . References [BL09] T ilman Baue r and Assaf L ibman. A simplicial A ∞ –monad acting on R - resolution s. Homotop y, Homology a nd App lications , 11 ( 2):55 –73, 2009. [BT00] Martin Bendersky a nd Robert D. Thompson. The B ousfield-Kan spectral sequence for periodic homology theories. Amer . J . Math. , 12 2:599 –635 , 2000. [BV68] J. M . Boardman a nd R. M. V ogt. Homotopy-everything H - spaces. Bull. Amer . Math. Soc. , 74:11 17–11 22, 196 8. Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 22 [Bor94] Francis Borceux. Handbook of categ orical algebra. 2 , volume 51 of Ency- clopedia of Mathemat ics and its Applications . Cambridge University Press, Cambridge, 19 94. C ategories a nd structures. [Bou03] A. K. Bousfield. Cosim plicial resolutions a nd homotopy spectral se- quences in model categories. Geom. T opol. , 7:1001 –105 3 (e lectronic), 200 3. [BK72] A. K. Bousfield and D. M. Kan. Ho motopy limits, c ompletions and localiza- tions . Springer-V erla g, Be rlin, 1 972. Lecture Notes in Mathematics, V ol. 304. [CDI02] W . Chach ´ olski, W . G. Dwyer , and M. Intermont. The A - complexity of a space. J. London Math. Soc. (2) , 65(1 ) :204– 222, 2002 . [DD77] E. Dror a nd W . G. Dwyer . A long homology localization tower . Comment. Math. Helv . , 5 2(2): 185–2 10, 1977. [DF87] Emmanuel Dror Farjoun. Homotopy and homology of d iagrams of spaces. In Algebraic topolog y ( Seattle, W ash., 1985 ) , volume 1 286 of Lec- ture Notes in Math. , pa ges 93–1 34. Springer , Berlin, 1987 . [EM65] Samuel Eilenberg and John C. Moore. Adjoint functors and triples. Illi- nois J. Math . , 9:381– 398, 1965. [GJ99] Paul G. Goerss and John F . Jardine. Simp licial h omotopy theory . Birkh ¨ auser V erlag, Basel, 1999. [HV92] J. Hollender and R. M. V ogt. M odules of topological spaces, applications to homotopy limits and E ∞ structures. Arch. Math . (Basel) , 59(2) :115– 129, 1992. [Hov99] Mark Hovey . Model categories , volume 63 of Mathematical Surveys and Monographs . America n Mathematical S ociety , Providence, RI, 199 9. [Kel0 5 ] G. M . Kelly . Basic concepts of enriched category theory . Repr . Theory Appl. Categ. , 1 0:vi+137 pp. ( electronic), 2005. Reprint of the 198 2 original. [Lew78] L. G. Lewis, Jr . The stable cat egory and generalized Thom spectra . PhD thesis, University of Chicago, 19 78. [Lib03 ] Assaf Libman. Homoto py limits of triples. T opology Ap pl. , 130(2) :133– 157, 2003. [ML98] Saunders Mac Lane. Categories for t h e working mathematician , volume 5 of Graduate T exts in Mathema t ics . Springer-V erlag, New Y ork, second edi- tion, 1 998. [May72] Peter Ma y . The Geometry of It erated Loop Spaces . Number 27 1 in LNM. Springer V erlag, 1 9 72. [MT78] J. P . Ma y and R. Thomason. The uniqueness of infinite loop space ma- chines. T opolog y , 17(3) :205– 224, 19 78. [Spa66 ] Edwin H. Spanier . Algebraic topology . McGraw-Hill Book Co., Ne w Y ork, 1966. [Ste67 ] N. E. Steenrod. A convenient category of topological spa ces. Michig an Math. J. , 14 :133– 152, 1967. Journal of Homotop y and Related St ructures, vol. ??(? ?), ???? 23 T ilman Bauer tilman @few. vu.nl Faculteit der exacte wetenschappe n V rije Universiteit A msterdam De Boelelaan 1081A 1081HV Amsterdam, The Netherlands Assaf Libman assa f@mat hs.abd n.ac.uk Department of M athematical Sciences, King’s C ollege University of Aberdeen Aberdeen AB24 3UE , Scotland, U.K.