A simplicial A ∞ -operad acting on R -resolutions
aa r X i v : . [ m a t h . A T ] J a n A SIMPLICIAL A ∞ -OPERAD ACTING ON R -RESOLUTIONS TILMAN BAUER AND ASSAF LIBMANA
BSTRACT . We construct a combinatorial model of an A ∞ -operad which acts sim-plicially on the cobar resolution (not just its total space) of a simplicial set withrespect to a ring R .
1. I
NTRODUCTION
Let K be a commutative ring and X a simplicial set (henceforth, simply called“space”). Bousfield and Kan [BK72] defined the K-resolution of X to be (a variantof) the cosimplicial space R K ( X ) n = K ( K ( . . . ( K | {z } n + ( X ) . . . ) ,where K ( X ) denotes the degreewise free K -vector space on X . The K -completionof X is derived from the cosimplicial space R K and can be defined in at least threedifferent ways, taking values in different categories that each come with a specificnotion of homotopy.Completion functor taking values in with homotopy defined by X ˆ K = Tot R K ( X ) spaces (simplicial) homotopy X pro K = { Tot s R K ( X ) } s ≥ towers of spaces pro-homotopy R K cosimplicial spaces external homotopyFor more precise information on pro-homotopical structures, see [Isa04, FI07].There are natural transformations R K → ( − ) pro K → ( − ) ˆ K under which homotopiesmap to homotopies, but the three notions of homotopy are strictly contained inone another. To illustrate this, consider the Bousfield (unstable Adams) spectralsequence associated to R K . If the cosimplicial space R K ( X ) is externally homotopyequivalent [Qui67, Bou03] to R K ( Y ) , the spectral sequences agree from E on. Ifthe towers are pro-weakly equivalent, then for every ( p , q ) there is an R such thatthe E rp , q -terms are isomorphic for r ≥ R . If we merely know that X ˆ K ≃ Y ˆ K , even ifthis is induced by a map X → Y , the spectral sequences might not be isomorphicanywhere at any term.The functors ( − ) pro K and R K can be made into endofunctors by enlarging thesource category to towers or cosimplicial spaces, respectively, applying the com-pletion functor levelwise, and taking the diagonal.In [BK72, I.5.6], Bousfield and Kan claimed that ( − ) pro K is a monad (levelwise,not just pro-). In fact, they try to prove that R K is a monad. Both claims werenot proved and later retracted by the authors. In the companion paper [BL07] Date : December 8, 2008.2000
Mathematics Subject Classification.
Key words and phrases. monad, A ∞ -monad, completion, cobar resolution. e showed that ( − ) ˆ K , while not a strict monad, can be made into a monad up tocoherent homotopy, an A ∞ -monad . This works in great generality: whenever givena monad ( A ∞ or strict), there is a completion functor which is again an A ∞ -monad.In this paper, we address the problem of realizing this structure on the resolu-tion. While related in its results, the methods we use here are completely differentfrom [BL07], and this paper can thus be read independently.Our result is slightly more general than outlined above. Let c E denote the cat-egory of cosimplicial objects in a complete and cocomplete monoidal category ( E , ⋄ , I ) . The category c E is endowed with an “external” simplicial structure (seeSection 3) and inherits a levelwise monoidal structure ⋄ from E . Every object X ∈ c E gives rise to an operad O End ( X ) (of simplicial sets) whose n th spaceis map c E ( X ⋄ n , X ) . As usual [May72], an O -algebra in ( c E , ⋄ , I ) is an object X ∈ c E with a morphism of operads O → O
End ( X ) . For example, if O ( n ) = ∗ is theassociative operad, X is simply a monoid. When the spaces O ( n ) are weakly con-tracible we call X an A ∞ -monoid.Now let ( K , η , µ ) be a monoid in ( E , ⋄ , I ) defined by η : I → K and µ : K ⋄ K → K .Let R K ∈ c E be its cobar construction with R Kn = K ⋄ n + and with coface maps d i = K ⋄ i η K ⋄ n − i and codegenracy maps s i = K ⋄ i µ K ⋄ n − i , cf. [BK72, Ch I, § Definition. A twist map for K is a morphism t : K ⋄ K → K ⋄ K such that(1) µ ◦ t = µ ,(2) t ◦ ( K ⋄ η ) = η ⋄ K and t ◦ ( η ⋄ K ) = K ⋄ η ,(3) t ◦ ( µ ⋄ K ) = ( K ⋄ µ ) ◦ ( t ⋄ K ) ◦ ( K ⋄ t ) and t ◦ ( K ⋄ µ ) = ( µ ⋄ K ) ◦ ( K ⋄ t ) ◦ ( t ⋄ K ) .1.2. Theorem.
There exists an A ∞ -operad of simplicial sets TW such that for everymonoid ( K , η , µ ) with a twist map t in a complete and cocomplete monoidal category E ,R K is a TW -algebra in c E , i.e. there is a morphism of operads TW → O End ( R K ) . As an application, let E be the category of endofunctors of simplicial sets withcomposition as its monoidal structure; one needs to pass to a larger universe inorder for E to make sense. A monad K on the category of spaces is a monoid in E .Let K be the free K -module functor on spaces for a ring K , and t the “twist map”defined by t = ( K ⋄ µ ) − id K ⋄ K +( µ ⋄ K ) (cf. [BK72, p. 28])We obtain as a corollary the motivating example:1.3. Corollary.
For any ring K, R K is a TW -algebra and in particular an A ∞ -monad. We do not know any interesting example of a monad with a twist map thatis not of the free- K -module type. It would be interesting to know if the monad X Ω ∞ ( E ∧ X ) , for an S -algebra E in the sense of [EKMM97] or [HSS00] canbe given a twist map – this would imply that the Bendersky-Thompson resolution[BT00] is an A ∞ -monad. Despite the lack of other examples, we decided to workin this generality as, if anything, it makes the proofs clearer.2. T HE “T WIST MAP O PERAD ” TWIn this section we will give an explicit combinatorial construction of the operadTW of spaces. In the next section we will construct its action on the Bousfield-Kanresolution [BK72]. .1. Informal description.
The k -simplices of the n th space of the operad TW arebraid-like diagrams as illustrated in Figure 1 below. These braid diagrams have n ( k + ) incoming strands labelled ( k k . . . . . . 0 . . . k ) and k + ( k ) in this order from left to right. Two strands can join if theyhave the same label, or they can cross provided the label of the strand on the leftis bigger than the one of the strand on the right. In addition we allow new strandsof any label to emerge.(2.1) Cross ( a > b ) : a a //// bb (cid:15)(cid:15)(cid:15)(cid:15) Join: a ,,, a (cid:18)(cid:18)(cid:18) a Emerge: (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) a Note that a crossing has no orientation, that is, none of the strands is consideredpassing above or below the other. Figure 1 shows an example of such a braid.Two such diagrams are considered equal if they are isotopic in R relative to the0 0 ?????????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ?????????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ?????????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? ????? F IGURE
1. An example of an element of TW ( ) end points; in particular, we do not allow Reidemeister moves. However, we doidentify the following diagrams involving the new strand operation:(2.2) (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) a ??? a (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ∼ aa ∼ (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) a (cid:127)(cid:127)(cid:127) a ?????? , (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) a ???? b b (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ∼ bb (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) a , (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) b (cid:127)(cid:127)(cid:127)(cid:127) aa ?????? ∼ aa (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) b The sequence of { TW ( n ) k } k ≥ forms a simplicial set, where the i th face map d i is obtained by erasing the strands of color i , whereas the i th degeneracy map s i duplicates the i th strand. An illustration of how this works on the “basic buildingblocks” of the braids is depicted by (here a > b )(2.3) s a a ---- a (cid:17)(cid:17)(cid:17)(cid:17) a = a ;;;;; a (cid:3)(cid:3)(cid:3)(cid:3)(cid:3) a + ::::: a + (cid:4)(cid:4)(cid:4)(cid:4)(cid:4) a a + s b a a /////// bb (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) = a + a + JJJJJJJJJJ bb (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) b + b + (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) thus introducing new crossings.Composition in the operad TW is obtained by grafting of these braid-like dia-grams with matching labels.We will now give a more rigid combinatorial description of the operad TWwithout direct reference to the braid-like diagrams. To do this we will constructa simplicial object {B k } k in the category of small monoidal categories. The k -simplices of the space TW ( n ) are the morphisms between certain objects in B k .In fact, each B k is a free monoidal category in a sense which we will now makeprecise. .2. Free monoidal categories and the simplicial monoidal category {B k } k ≥ . Bya graph we mean a pair ( A ⇒ O ) of two sets and two maps (codomain, domain)between them. A unital graph is a graph with a common section O → A of domainand codomain. Every small category C has an underlying unital graph U u ( C ) =( Mor C x ⇒ Obj C ) by forgetting the composition rule on morphisms. By furtherforgetting the unit maps, we obtain a graph U ( C ) . Conversely, every (unital) graph G = ( A ⇒ O ) has an associated category C ( G ) . It is the free category generatedby G in the sense that C is left adjoint to the functor U (resp. U u ) from the categoryof small categories to the category of (unital) graphs. See [ML98, Section II.7, inparticular Theorem 1]. This section is about a version of this free construction formonoidal categories.2.4. Definition. A vertex-monoidal graph is a graph ( A ⇒ O ) , where O has thestructure of a monoid. A (unital) monoidal graph is a graph object in the category of(unital) monoids.2.5. Lemma.
The following forgetful functors all have left adjoints: { small monoidal categories } U um −−→{ unital monoidal graphs } U m −→{ monoidal graphs } U vm −−→ { vertex-monoidal graphs. } In particular, for every vertex-monoidal graph G , there exists a small monoidal category ( M , ⊔ , ∅ ) with a morphism of vertex-monoidal graphs ǫ : G → U ( M ) , such that for anyother monoidal category ( C , ⊗ , I ) and vertex-monoidal graph morphism X : G → U ( C ) ,there is a unique functor X ♭ : M → C of monoidal categories such that ǫ ◦ U ( X ♭ ) = X. We give a proof of this (easy) fact in the appendix.2.6.
Remark.
Fix a set Σ , and let Σ ⊔ be the set of words, i.e. finite sequences in Σ , including the empty sequence () . This is a monoid under concatenation, whichwe denote by ⊔ . If ( A ⇒ O ) is a vertex-monoidal graph whose vertices O = Σ ⊔ form such a free monoid, then a morphism of vertex-monoidal graphs X : ( A ⇒ Σ ⊔ ) → U ( M ) , where M is monoidal, is given by a map X : Σ → Obj ( M ) and, forevery arrow a : ( σ , . . . , σ k ) → ( τ , . . . , τ l ) in A , a morphism X ( σ ) ⊗ · · · ⊗ X ( σ k ) → X ( τ ) ⊗ · · · ⊗ X ( τ l ) .2.7. Lemma.
Let Σ and G = ( A ⇒ Σ ⊔ ) be as in Remark 2.6 and let M denote the freemonoidal category generated by G by Lemma 2.5. Let ∼ be an equivalence relation on themorphism set of M such that if f ∼ f ′ then their domains and codomains are equal.Then there exists a monoidal category M / ∼ together with a quotient functor M →M / ∼ which is the identity on the object sets and such that M / ∼ is the free monoidal cat-egory on ( Σ , Λ ) subject to the relation ∼ . That is, given any monoidal category ( C , ⊗ , I ) and a morphism of monoidal graphs X : G → U ( C ) whose associated monoidal functorX : M → C has the property that X ( f ) = X ( g ) if f ∼ g, there exists a unique monoidalfunctor X : M / ∼ → C which factors X through the quotient M → M / ∼ .Proof. Consider the equivalence relation ≈ on the morphism set of M which isgenerated by ∼ and which is closed to the property that ( f ⊔ g ) ≈ ( f ′ ⊔ g ′ ) and f ◦ g ≈ f ′ ◦ g ′ if f ≈ f ′ and g ≈ g ′ (for the composition, whenever it makes sense).One easily checks that the domain and codomain of ≈ -equivalent morphisms areequal. We can now apply [ML98, § II.8, Proposition 1] to obtain M / ≈ which ismonoidal due to the requirements on ≈ . (cid:3) he definition of the category B k is modelled on the ideas in 2.1.2.8. Definition.
We use Lemma 2.7 to define B k as the free monoidal categorygenerated by the set of symbols [ k ] = {
0, . . . , k } and morphisms (cf. (2.1)) () u a −→ ( a ) and ( a , a ) m a −→ ( a ) for all a =
0, . . . , k ( a , b ) t a , b −→ ( b , a ) for all 0 ≤ b < a ≤ k subject to the relations (cf. 2.2) ( R ) m a ◦ ( id ( a ) ⊔ u a ) = id ( a ) = m a ◦ ( u a ⊔ id ( a ) )( R ) i t c , d ◦ ( id ( c ) ⊔ u d ) = u d ⊔ id ( c ) ( R ) ii t c , d ◦ ( u c ⊔ id ( d ) ) = id ( d ) ⊔ u c where 0 ≤ a , c , d ≤ k and d < c .For an object σ = ( a , · · · , a n ) and a morphism f in B k we will write σ ⊔ f forid σ ⊔ f . The morphism in B k between sequences ( a , . . . , a s ) → ( b , . . . , b t ) can bedepicted by braid-like diagrams with s incoming strands labelled by the integers a , . . . , a s and r outgoing strands labelled by ( b , . . . , b t ) . For example, the diagram1 1 /////// (cid:0)(cid:7)(cid:1)(cid:6)(cid:2)(cid:5)(cid:3)(cid:4) rrrrrrrrrr /////// /////// tttttt describes a morphism ( ) → ( ) in B which is the composition [ m ⊔ ( )] ◦ [( ) ⊔ t ⊔ ( )] ◦ [( ) ⊔ m ⊔ t ] ◦ [( ) ⊔ t ⊔ u ] .Most of this section is devoted to the proof of the following result.2.9. Proposition.
There are face and degeneracy operators d i : B k → B k − and s i : B k →B k + which make {B k } k ≥ a simplicial object of small monoidal categories. On objects, d i and s i act by deleting (resp. duplicating) the symbol i ∈ [ k ] . A more elaborate description of the simplicial operators is given in Defini-tion 2.14. In what follows we will write [ k ] for the object (
01 . . . k ) in B k use [ k ] n todenote the n -fold concatenation of [ k ] with itself, i.e. the object ( k k . . . ) .2.10. Definition.
Define TW ( n ) k = Hom B k ([ k ] n , [ k ]) . By Proposition 2.9, this col-lection of sets, for all k , forms a simplicial set.The composition law in the operad TW ( n ) × ∏ ni = TW ( m i ) −→ TW ( ∑ ni = m i ) isdefined using the monoidal structure of B k : B k ([ k ] n , [ k ]) × n ∏ i = B k ([ k ] m i , [ k ]) id ×⊔ −−−−−−→ B k ([ k ] n , [ k ]) × B k ([ k ] ∑ i m i , [ k ] n ) ◦ −−−−→ B k ([ k ] ∑ i m i , [ k ]) .Unitality and associativity of the monoidal structure in B k descend to this compo-sition law.The goal of this section is to prove2.11. Theorem.
The sequence TW ( n ) is a (nonsymmetric) A ∞ -operad of simplicial sets. ere are some easy results on the categories B k (Definition 2.8.) An object σ ∈B k is a sequence of symbols ( s , . . . , s n ) and we write u σ for u s ⊔ · · · ⊔ u s n .2.12. Lemma.
Fix an object τ = ( t < · · · < t n ) in B k . Then given an object σ ∈ B k ,any morphism ϕ : τ → σ has the form (2.13) ϕ = u σ ⊔ ( t ) ⊔ u σ ⊔ · · · ⊔ u σ n − ⊔ ( t n ) ⊔ u σ n where σ i are subsequences of σ such that σ = σ ⊔ ( t ) ⊔ σ ⊔ · · · ⊔ σ n − ⊔ ( t n ) ⊔ σ n . Inparticular, the object () is initial in B k and End B k ( τ ) = { id } .Proof. By Lemma 2.7, the morphisms in B k are compositions of morphisms of theform x ⊔ u a ⊔ y , x ⊔ m a ⊔ y , and x ⊔ t ab ⊔ y , where x , y are objects of B k .Let ϕ be as in the statement of the lemma. We prove the result by induction onthe “generation length” s of ϕ . If s = ϕ = id. If ϕ satisfies (2.13) and isfollowed by some x ⊔ u a ⊔ y , then the resulting morphism still has the form (2.13).If ϕ is followed by some x ⊔ t ab ⊔ y then ( a , b ) cannot be contained in τ because a > b , hence one of a or b belongs to σ i and we can apply the relation (R2) toshow that the composition is again of the form (2.13). Finally, if ϕ is followed by x ⊔ m a ⊔ y then the domain ( a , a ) of m a cannot be contained in τ , and relation (R1)guarantees that the composition is of the form (2.13). (cid:3) Definition.
We turn the categories B k into a simplicial monoidal categoryas follows. To do this it is convenient to replace [ k ] with a totally ordered finiteset S of cardinality k +
1. Then monotonic functions S → T are generated by thefollowing functions η a : ( S − { a } ) inclusion −−−−−→ S for some a ∈ S ǫ a : ( S − { a } ) ⊔ { a ′ , a ′′ } collapse { a ′ , a ′′ }→{ a } −−−−−−−−−−−−→ S where a ′ < a ′′ are inserted to S instead of a ∈ S . In this notation the simplicialidentities become(2.15) η a ◦ η b = η b ◦ η a , ǫ b ◦ η a = η a ◦ ǫ b , ǫ a ◦ ǫ b = ǫ b ◦ ǫ a ( a = b ∈ S ) ǫ a ◦ ǫ a ′ = ǫ a ◦ ǫ a ′′ , ǫ a ◦ η a ′ = ǫ a ◦ η a ′′ = Id ( a ∈ S ) We define face operators d a = ( η a ) ∗ : B S → B T where T = S − { a } and degeneracyoperators s a = ( ǫ a ) ∗ : B T → B S where T = S − { a } ⊔ { a ′ < a ′′ } as follows. Onobjects we define d a (( b )) = ( b ) = s a (( b )) if b = a ; d a (( a )) = () ; s a (( a )) = ( a ′ , a ′′ ) Extend s a and d a to all the objects of B k by monoidality with respect to ⊔ .On morphisms we define d a and s a by defining them on generators: d a ( m a ) = id () , s a ( m a ) = ( m a ′ ⊔ m a ′′ ) ◦ (( a ′ ) ⊔ t a ′′ , a ′ ⊔ ( a ′′ )) cf. (2.3) d a ( m b ) = m b , s a ( m b ) = m b if b = ad a ( u a ) = id () , s a ( u a ) = u a ′ ⊔ u a ′′ d a ( u b ) = u b , s a ( u b ) = u b if b = ad a ( t cd ) = t cd , s a ( t cd ) = t cd if c , d = ad a ( t cd ) = id ( d ) , s a ( t cd ) = ( t a ′ d ⊔ ( a ′′ )) ◦ (( a ′ ) ⊔ t a ′′ d ) if c = ad a ( t cd ) = id ( c ) , s a ( t cd ) = (( a ′ ) ⊔ t ca ′′ ) ◦ ( t ca ′ ⊔ ( a ′′ )) if d = a , cf. (2.3) o see that these assignments define functors we need to check that they respectthe relations (R1) and (R2). For (R1), we compute ( a = b ): s b ( m a ◦ (( a ) ⊔ u a )) = m a ◦ (( a ) ⊔ u a ) = id ( a ) = s b ( id ( a ) ) s a ( m a ◦ (( a ) ⊔ u a )) = ( s a ′ ⊔ s a ′′ ) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ )) ◦ (( a ′ a ′′ ) ⊔ u a ′ ⊔ u a ′′ )= (R2) ( s a ′ ⊔ s a ′′ ) ◦ (( a ′ ) ⊔ u a ′ ⊔ ( a ′′ ) ⊔ u a ′′ )= (R1) id ( a ′ ) ⊔ id ( a ′′ ) = id ( a ′ a ′′ ) = s a ( id ( a ) ) .Similarly one checks that d a and d b carry the relation (R1) to equal morphisms.The second relation of (R1) is checked similarly.For (R2), assume that c > d and consider some a . If a = c , d then it is clearthat applying d a and s a to the relations (R2) leaves them unaffected. If a = c thenapplying s a to the relation (R2) i gives s a ( t ad ◦ (( a ) ⊔ u d )) = ( t a ′ d ⊔ ( a ′′ )) ◦ (( a ′ ) ⊔ t a ′ d ) ◦ (( a ′ a ′′ ) ⊔ u d ) = ( R ) ( t a ′ d ⊔ ( a ′′ )) ◦ (( a ′ ) ⊔ u d ⊔ ( a ′′ )) = ( R ) u d ⊔ id ( a ′ a ′′ ) = s a ( u d ⊔ id ( a ) ) When a = d a similar calculation shows that s a ( t ca ◦ ( id ( c ) ⊔ u a )) = s a ( u a ⊔ id ( c ) ) .Checking that s a respects the relation ( R ) ii is analogous, and we leave it to thereader to check that d a respects (R2) as well. (cid:3) Proof of Proposition 2.9.
We will use the “coordinate free” notation of Definition 2.14and prove that the functors d a and s a render {B k } k ≥ a simplicial monoidal cate-gory. The functors d a and s a are monoidal, so it only remains to prove that theysatisfy the simplicial identities dual to the identities (2.15), i.e. d b d a = d a d b ( a = b ∈ S ) (2.16) d b s a = s a d b ( a = b ∈ S ) (2.17) s b s a = s a s b ( a = b ∈ S ) (2.18) s a ′ s a = s a ′′ s a ( a ∈ S ) (2.19) d a ′ s a = d a ′′ s a = id ( a ∈ S ) (2.20)This is obviously the case on the object sets and it only remains to check the sim-plicial identities on the generating morphisms u c , m c , t cd . If c , d ∈ S − { a , b } thenby definition d a , d b as well as s a and s b map u c , m c and t cd to themselves, hence allthe identities hold. Thus, it remains to check the simplicial identities when { c , d } nd { a , b } are not disjoint. For (2.16) and (2.18), the only non-immediate cases are s b s a ( m a ) = s b ( m a ′ ⊔ m a ′′ ) ◦ s b (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ ))= ( m a ′ ⊔ m a ′′ ) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ )) = s a s b ( m a ) s b s a ( t ab ) = s b ( t a ′ b ⊔ ( a ′′ )) ◦ s b (( a ′ ) ⊔ t a ′′ b )= (( b ′ ) ⊔ t a ′ b ′′ ⊔ ( a ′′ )) ◦ ( t a ′ b ′ ⊔ ( b ′′ ) ⊔ ( a ′′ )) ◦ (( a ′ ) ⊔ ( b ′ ) ⊔ t a ′′ b ′′ ) ◦ (( a ′ ) ⊔ t a ′′ b ′ ⊔ ( b ′′ ))= (( b ′ ) ⊔ t a ′ b ′′ ⊔ ( a ′′ )) ◦ ( t a ′ b ′ ⊔ t a ′′ b ′′ ) ◦ (( a ′ ) ⊔ t a ′′ b ′ ⊔ ( b ′′ ))= (( b ′ ) ⊔ t a ′ b ′′ ⊔ ( a ′′ )) ◦ (( b ′ ) ⊔ ( a ′ ) ⊔ t a ′′ b ′′ ) ◦ ( t a ′ b ′ ⊔ ( a ′′ ) ⊔ ( b ′′ )) ◦ (( a ′ ) ⊔ t a ′′ b ′ ⊔ ( b ′′ ))= s a (( b ′ ) ⊔ t ab ′′ ) ◦ s a ( t ab ′ ⊔ ( b ′′ )) = s a s b ( t ab ) The remaining cases, for example d b d a ( m a ) = d a d b ( m a ) or when { a , b } 6 = { c , d } ,are straightforward.For (2.17), we check d b s a ( m a ) = d b ( m a ′ ⊔ m a ′′ ) ◦ d b (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ ))= ( m a ′ ⊔ m a ′′ ) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ )) = s a d b ( m a ) d b s a ( t ac ) = d b ( t a ′ c ⊔ ( a ′′ )) ◦ d b (( a ′ ) ⊔ t a ′′ c )= ( t a ′ c ⊔ ( a ′′ )) ◦ (( a ′ ) ⊔ t a ′′ c ) = s a d b ( t ac ) and leave the rest to the reader.For (2.19), we use the convention that if a ∈ S then the domain of ǫ a : S ′ → S is formed from S by replacing a with a ′ < a ′′ . The morphisms ǫ a ′ , ǫ a ′′ : S ′′ → S ′ have S ′′ = S − { a } ⊔ { a ′ , a ′′ , a ′′′ } and s a ′ ( a ′ ) = ( a ′ , a ′′ ) , s a ′ ( a ′′ ) = a ′′′ whereas s a ′′ ( a ′ ) = a ′ , and s a ′′ ( a ′′ ) = ( a ′′ , a ′′′ ) . We now have the following calculation,which exploits the monoidal structure of B k : s a ′ s a ( m a ) = s a ′ ( m a ′ ⊔ m a ′′ ) ◦ s a ′ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ ))= { [( m a ′ ⊔ m a ′′ ) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ ))] ⊔ m a ′′′ }◦{ ( a ′ a ′′ ) ⊔ [(( a ′ ) ⊔ t a ′′′ a ′′ ) ◦ ( t a ′′′ a ′ ⊔ ( a ′′ ))] ⊔ ( a ′′′ ) } = ( m a ′ ⊔ m a ′′ ⊔ m a ′′′ ) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ a ′′′ a ′′′ )) ◦ (( a ′ a ′′ a ′ ) ⊔ t a ′′′ a ′′ ⊔ ( a ′′′ )) ◦ (( a ′ a ′′ ) ⊔ t a ′′′ a ′ ⊔ ( a ′′ a ′′′ ))= ( m a ′ ⊔ m a ′′ ⊔ m a ′′′ ) ◦ (( a ′ a ′ a ′′ ) ⊔ t a ′′′ a ′′ ⊔ ( a ′′′ )) ◦ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′′ a ′′ a ′′′ )) ◦ (( a ′ a ′′ ) ⊔ t a ′′′ a ′ ⊔ ( a ′′ a ′′′ ))= { m a ′ ⊔ [( m a ′′ ⊔ m a ′′′ ) ◦ (( a ′′ ) ⊔ t a ′′′ a ′′ ⊔ ( a ′′′ ))] }◦{ ( a ′ ) ⊔ [( t a ′′ a ′ ⊔ ( a ′′′ )) ◦ (( a ′′ ) ⊔ t a ′′′ a ′ )] ⊔ ( a ′′ a ′′′ ) } = s a ′′ ( m a ′ ⊔ m a ′′ ) ◦ s a ′′ (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ )) = s a ′′ s a ( m a ) This is the most subtle calculation in verifying (2.17). The remaining cases, e.g. s a ′ s a ( u a ) = s a ′′ s a ( u a ) , are proved similarly and left to the reader, as is the proof of(2.20). (cid:3) .21. Proposition.
The spaces TW ( ) and TW ( ) are points.Proof. This follows immediately from Lemma 2.12. (cid:3)
Remark.
The categories B k also enjoy the structure of a “cofacial monoidalcategory”, namely a cosimplicial object in the category of small monoidal cate-gories without codegeneracy maps. In the notation of Definition 2.14 it is inducedby the inclusion of symbols T ⊆ S where T and S are totally ordered sets. We willnot need this structure but we will make use of the operator d : B k → B k + which has the effect of shifting indices by 1. That is, d ( i ) = i + i ∈ [ k ] and on morphisms d ( u a ) = u a + , d ( m a ) = m a + and d ( t ab ) = t a + b + . It isobvious that the relations (R1) and (R2) are respected by these assignments. The“coface operator” d also has the following relations with the face and degeneracyoperators:(2.23) d d = id, s d = d d d i d = d d i − , s i d = d s i − for all i > σ is some sequence, we use the power notation σ n for the n -fold concatenationof σ with itself. We observe that d a ([ k + ] n ) = [ k ] n = s a ([ k − ] n ) and d [ k ] =(
1, 2, . . . , k + ) .2.24. Definition.
For every n ≥ b n ∈ B denote the object ( n ) . Define β n : [ ] n → b n by induction on n ≥ β = { m ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } , β n = { β n − ⊔ ( ) } ◦ { [ ] n − ⊔ β }
00 0 (cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)
11 11 00 mmmm lllllllllllll · · ·· · ·· · ·
11 0 11 0 11 112.25.
Lemma.
The following hold for all n ≥ . (1) d ( β n ) = id [ ] n ∈ B . (2) s ( β n ) = { ( ) ⊔ d ( β n ) } ◦ s β n (in B ). (3) s d ( β n ) = { ( ) ⊔ d d ( β n ) } ◦ β n (in B ). The content of (2) is displayed as the obvious equivalence of the following“braids”.00 11 22 0 1 22 · · · jjjjjjjjjjjjjjjjjjjjjj jjjjjjjjjjjjjjjjjjjjjj
22 00 11 22 22 · · ·
22 220 1 0 1 0 jjjjjjjjjjjjjjjjjjjjjj jjjjjjjjjjjjjjjjjjjjjj The content of (3) is obtained by deleting all the strands labelled with 2. roof. (1) By inspection of Definitions 2.14 and 2.24, d ( β ) = d ( m ⊔ ( )) ◦ d (( ) ⊔ t ⊔ ( )) = id .The result follows easily using the inductive definition of β n .(2) Using Definition 2.24 and the pictures given there, we first observe that β n = γ n ⊔ ( ) for some γ n : [ ] n − → b n − . It now follows, using the monoidal structurein B , that for every morphism ϕ : [ ] → b in B we have(2.26) { s b n − ⊔ d ϕ } ◦ { s β n ⊔ ( ) } = { s b n − ⊔ d ϕ } ◦ { s γ n ⊔ ( ) } = { s γ n ⊔ ( ) } ◦ { [ ] n − ⊔ ( ) ⊔ d ϕ } = { s β n ⊔ ( ) } ◦ { [ ] n − ⊔ ( ) ⊔ d ϕ } .To complete the proof of (2) we use induction on n . The hardest part is the base ofinduction n =
2. Using Definition 2.24 and the monoidal structure in B we find { ( ) ⊔ d β } ◦ s ( β ) = { ( ) ⊔ m ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } ◦ { m ⊔ ( ) } | {z } ( ) ◦ m = m ◦ ( ) ◦ { ( ) ⊔ s ( t ) ⊔ ( ) } | {z } expand = { m ⊔ m ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } | {z } ◦{ ( ) ⊔ t ⊔ ( ) } = { m ⊔ m ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } ◦ { ( ) ⊔ t ⊔ ( ) } = s ( m ⊔ ( )) ◦ s (( ) ⊔ t ⊔ ( )) = s ( β ) .Now assume that the formula holds for n and we prove it for β n + by using itsinductive definition. Note that ( ) ⊔ d [ ] n = ( ) = s ( b n ) . { ( ) ⊔ d β n + } ◦ s β n )= { ( ) ⊔ d β n ⊔ ( ) } ◦ { s b n − ⊔ d β } ◦ { s β n ⊔ ( ) } | {z } Formula (2.26) ◦{ [ ] n − ⊔ s β } = { ( ) ⊔ d β n ⊔ ( ) } ◦ { s β n ⊔ ( ) } | {z } induction ◦ { [ ] n − ⊔ ( ) ⊔ d β } ◦ { [ ] n − ⊔ s β } | {z } induction = s ( β n ⊔ ( )) ◦ { [ ] n − ⊔ s β } = s ( β n + ) .This completes the proof of (2). Part (3) follows by applying d to (2), using thesimplicial identities and (2.23). (cid:3) Remark.
In Proposition 2.21 we saw that TW ( ) = TW ( ) = ∗ . One can alsocheck that TW ( ) is a point, whereas all TW ( n ) are infinite-dimensional for n ≥ Proof of Theorem 2.11.
We have to show that TW ( n ) is contractible for all n ≥
0. Thecases n =
0, 1 are covered by Proposition 2.21. Consider n ≥
2. Use the morphisms β n , see Definition 2.24, to define a 0-simplex τ ( n ) ∈ TW ( n ) = B ([ ] n , [ ]) by(2.28) τ ( n ) = d ( β n ) .It is depicted by the tree obtained by deleting the strands labelled 1 from the dia-gram describing β n in Definition 2.24.For every k ≥
1, observe that ( ) ⊔ d [ k − ] n = s k − ( b n ) , and define(2.29) Θ n , k = s k − ( β n ) ∈ B k ([ k ] n , ( ) ⊔ d [ k − ] n ) . e picture Θ n , k by duplicating k − k .Augment TW by TW ( n ) − = ∗ . To show that TW ( n ) is contractible, it sufficesto show that there is a left contraction [DMN89] s − : TW ( n ) k → TW ( n ) k + , k ≥ − d s − = id; d i s − = s − d i − ; s i s − = s − s i − for all i ≥ s − : ∗ → TW ( n ) by s − ( ∗ ) = τ ( n ) . For any ϕ ∈ TW ( n ) k , k ≥
0, we let s − ( ϕ ) be the composition depicted in Figure 2; in formulae: [ k + ] n Θ n , k + −−−→ ( ) ⊔ d [ k ] n ( ) ⊔ d ( ϕ ) −−−−−→ ( ) ⊔ d [ k ] = [ k + ] (cf. Remark 2.22).The first identity of (2.30) holds trivially when k = −
1. For k ≥
0, Lemma 2.25(1),0 ????????? ?????????????????????????? · · · k k ?????????????????????????? · · · ????????? ?????????????????????????? · · · k k ?????????????????????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ?????????????????????????? · · · k k ?????????????????????????? . . . 0 ????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) . . . · · · · · · · · · · · · | {z } B ′ F IGURE
2. The image of B under the simplicial contraction. B ′ denotes B with all labels increased by 1.the simplicial identities, and (2.23) imply d ( Θ n , k + ) = d s k ( β n ) = s k + . . . s k d ( β n ) = id .It follows that d s − = id because for any ϕ ∈ TW ( n ) k where k ≥ d s − ( ϕ ) = d (( ) ⊔ d ϕ ) ◦ d ( Θ n , k + ) = d d ( ϕ ) = ϕ .For the second identity in (2.30), note that k ≥ i >
0. If k = i = d d ( ϕ ) = id () for any ϕ ∈ B . Therefore d s − ( ϕ ) = d (( ) ⊔ d ( ϕ )) ◦ d ( Θ n ,1 ) = d ( β n ) = τ ( n ) = s − ( ∗ ) = s − d ( ϕ ) .If k ≥
1, the simplicial identities imply that d i Θ n , k + = d i s k ( β n ) = s k − ( β n ) = Θ n , k .Therefore for any ϕ ∈ TW ( n ) k , we have d i s − ( ϕ ) = d i ( Θ n , k + ) ◦ d i (( ) ⊔ d ( ϕ )) = Θ n , k ◦ { ( ) ⊔ d d i − ( ϕ ) } = s − d i − ( ϕ ) .For the third identity of (2.30), we start with the case i >
0, which forces k ≥ s i ( Θ n , k + ) = s i s k ( β n ) = s k + ( β n ) = Θ n , k + .It follows that for any ϕ ∈ TW ( n ) k , s i s − ( ϕ ) = s i ( Θ n , k + ) ◦ s i (( ) ⊔ d ( ϕ )) = Θ n , k + ◦ { ( ) ⊔ d s i − ( ϕ ) } = s − s i − ( ϕ ) . t remains to check that s s − = s − s − , which is the most subtle identity. When k = − s − s − ( ∗ ) = s s − ( ∗ ) . By the definition of s − onTW ( n ) and by Lemma 2.25(3) s − s − ( ∗ ) = s − τ ( n ) = { ( ) ⊔ d d β n } ◦ β n = s d β n = s τ ( n ) = s s − ( ∗ ) .For k ≥
0, consider some ϕ ∈ TW ( n ) k . By construction, s − ( ϕ ) = { ( ) ⊔ d ( ϕ ) } ◦ Θ n , k + ,and therefore, by (2.23), s s − ( ϕ ) = { ( ) ⊔ s d ( ϕ ) } ◦ s ( Θ n , k + ) = { ( ) ⊔ d d ( ϕ ) } ◦ s ( Θ n , k + ) s − s − ( ϕ ) = { ( ) ⊔ ( ) ⊔ d d ( ϕ ) } ◦ { ( ) ⊔ d ( Θ n , k + ) } ◦ Θ n , k + .We are left to show that s ( Θ n , k + ) = { ( ) ⊔ d ( Θ n , k + ) } ◦ Θ n , k + .This follows by applying s k + . . . s to Lemma 2.25(2) because Θ n , k + = s k + ( β n ) ,and because (2.23) implies s ( Θ n , k + ) = s s k ( β n ) = s k s ( β n ) ; d ( Θ n , k + ) = d s k ( β n ) = s k d ( β n ) . (cid:3)
3. T
HE OPERAD ACTION ON COMPLETION COSIMPLICIAL OBJECTS
Let ( E , ⋄ , I ) be a complete and cocomplete monoidal category. In particular, it istensored and cotensored over sets by E ⊗ E K = ∐ K E and [ K , E ] = ∏ K E .3.1. The simplicial category of cosimplicial objects.
Let c E denote the categoryof cosimplicial objects in E . The monoidal structure in E gives rise to a levelwisemonoidal structure ⋄ in c E .The category c E has a simplicial structure [GJ99, II.2] even if E does not haveone, see [Qui67, II.1.7ff] or [Bou03, § external sim-plicial structure and is given by [ K , X ] = δ ∗ [ K • , X • ] ( K ∈ s Sets , X ∈ c E ) X ⊗ c E K = ( LKan δ X ) ⊗ ∆ K ( K ∈ s Sets , X ∈ c E ) map c E ( X , Y ) = Hom c E ( LKan δ X , Y ) ( X , Y ∈ c E ) ,where δ : ∆ → ∆ × ∆ is the diagonal map, LKan δ is the left Kan extension along δ op and ⊗ ∆ denotes the coend.If ∆ [ k ] denotes the standard k -simplex, the usual adjunctions imply(3.1) map c E ( X , Y ) k = Hom c E ( X ⊗ c E ∆ [ k ] , Y ) = Hom c E ( X , [ ∆ [ k ] , Y ]) ,so we think of the k -simplices of map ( X , Y ) as cosimplicial maps X → [ ∆ [ k ] , Y ] .3.2. Proposition. If ( E , ⋄ , I ) is a complete and cocomplete monoidal category then thedefinitions above make ( c E , ⋄ , I ) a simplicial monoidal category. roof. Quillen shows in [Qui67, II.1] that c E is tensored and cotensored over s Sets .The monoidal structure in c E also gives rise to(3.3) map c E ( X , Y ) × map c E ( X , Y ) ⋄ −→ map c E ( X ⋄ X , Y ⋄ Y ) as follows. Given k -simplices ϕ : X → [ ∆ [ k ] , Y ] and : X → [ ∆ [ k ] , Y ] we define ϕ ⋄ ψ : X ⋄ X → [ ∆ [ k ] , Y ⋄ Y ] in cosimplicial degree j as the composition X j ⋄ X j ∏ σ ϕ j ( σ ) ⋄ ψ j ( σ ) −−−−−−−−→ ∏ σ ∈ ∆ [ k ] j Y j ⋄ Y j .One easily checks that ϕ ⋄ ψ is a cosimplicial map and that the assignment ( ϕ , ψ ) ϕ ⋄ ψ gives rise to an associative simplicial map (3.3) which has the constant cosim-plicial object I as a unit.This monoidal product in c E distributes over the composition. That is, given k simplices ϕ i ∈ map ( X i , Y i ) k and ψ i ∈ map ( Y i , Z i ) ( i =
1, 2), we have ( ψ ⋄ ψ ) ◦ ( ϕ ⋄ ψ ) = ( ψ ◦ ϕ ) ⋄ ( ψ ◦ ϕ ) . (cid:3) Thus, as mentioned in the introduction, every object X ∈ c E gives rise to an op-erad O End ( X ) whose n th space is map c E ( X ⋄ n , X ) . Its composition law is obtainedfrom the composition and monoidal structure in c E .3.2. Monoids and twist maps.
Before proving Theorem 1.2, we show how to de-rive Corollary 1.3 from it.
Proof of Corollary 1.3.
We need to see that t : K ⋄ K → K ⋄ K , given by t = ( K ⋄ µ ) − id K ⋄ K +( µ ⋄ K ) ,is a twist map (Definition 1.1). It is convenient to write t = d s − id K ⋄ K + d s where d , d and s are the coface and codegeneracy maps between K ⋄ K and K inthe cosimplicial object R K .We compute µ ◦ t = s ◦ ( d s − id + d s ) = s − s − s = s = µ , t ◦ ( K ⋄ η ) = ( d s − id + d s ) ◦ d = d − d + d = d = η ⋄ K ( K ⋄ µ ) ◦ ( t ⋄ K ) ◦ ( K ⋄ t ) = s ◦ ( d s − id K ⋄ K + d s ) ◦ ( d s − id K ⋄ K + d s )= d s s − s + d s s = t ◦ s = t ◦ ( µ ⋄ K ) .The identities t ◦ ( η ⋄ K ) = K ⋄ η and t ◦ ( K ⋄ µ ) = ( µ ⋄ K ) ◦ ( K ⋄ t ) ◦ ( t ⋄ K ) follow bysymmetry. (cid:3) Proof of Theorem 1.2.
Let ( K , η , µ ) be a monoid in a complete and cocomplete mo-noidal category ( E , ⋄ , I ) , R K ∈ c E its cobar construction with coface maps d iK andcodegeneracy maps s iK . These extend by monoidality to coface and codegeneracymaps on R ⋄ nK .It follows from Lemma 2.7 and Definition 2.8 of B k that there are unique mo-noidal functors Φ k : B k → E for all k ≥ Φ k (( a )) = K , Φ k ( u a ) = η , Φ k ( m a ) = µ , Φ k ( t cd ) = t . he relations (R1) of Definition 2.8 hold due to the fact that ( K , η , µ ) is a monoid,and the relations (R2) hold by Property (2) of the twist map. On objects, we thushave Φ k ([ l ]) = ( R K ) l = K ⋄ ( l + ) .In Definition 2.8 it was convenient to define the categories B S where S is anyfinite ordered set. In this way, given a ∈ S we defined functors d a and s a in Defini-tion 2.14 which render the categories B k a simplicial monoidal category.We now construct two natural transformations D a and S a of functors B S → E .3.4. Lemma.
For a ∈ S, there is a natural transformation D a : Φ d a ( S ) ◦ d a → Φ S withD a : ( b ) ( I η −→ K , if b = aK id −→ K , if b = a , and extended to all objects by monoidality. Here d a ( S ) = S − { a } as in Definition 2.14.Proof. By construction, B k defined in 2.8 is monoidally generated by the objects ( b ) and the morphisms u b , m b and t cd . To prove that D a is a natural transformation, itsuffices by Lemma 2.7 to show that for all a , b , c , d ∈ S , Φ S ( u b ) ◦ D a (()) = D a (( b )) ◦ Φ d a ( S ) ( d a ( u b )) , Φ S ( m b ) ◦ D a (( bb )) = D a (( b )) ◦ Φ d a ( S ) ( d a ( m b )) , Φ S ( t cd ) ◦ D a (( cd )) = D a (( dc )) ◦ Φ d a ( S ) ( d a ( t cd )) .The first equality hold if a = b because d a ( u a ) = id. If a = b then D a (( b )) = id = D a (()) and the equality holds again since d a ( u b ) = u b .The second equality holds if a = b because D a (( bb )) = id = D a (( b )) and d a ( m b ) = m b . If a = b then d a ( m a ) = id () so by the relations between η and µ , Φ S ( m a ) ◦ D a (( aa )) = µ ◦ ( η ⋄ η ) = η = D a (( a )) .The third equality holds when a = c , d because D a (( cd )) = id K ⋄ K = D a (( dc )) and d a ( t cd ) = t cd . If a = c then d a ( t cd ) = id (cf. Definition 2.8) so Φ S ( t cd ) ◦ D a (( cd )) = t ◦ ( µ ⋄ K ) = K ⋄ η = D a (( dc )) ◦ Φ d a S ( d a ( t cd )) .A similar calculation applies when a = d . (cid:3) Lemma.
For a ∈ S, there is a natural transformation S a : Φ s a ( S ) ◦ s a → Φ S withS a : ( b ) ( K ⋄ K µ −→ K , if b = aK id −→ K , if b = a , and extended to all objects by monoidality. Here s a ( S ) = S − { a } ⊔ { a ′ , a ′′ } as in Defi-nition 2.14.Proof. As in the proof of Lemma 3.4, in order to prove the naturality of S a we haveto show that Φ S ( u b ) ◦ S a () = S a ( b ) ◦ Φ s a S ( s a ( u b )) , Φ S ( m b ) ◦ S a ( bb ) = S a ( b ) ◦ Φ s a S ( s a ( m b )) , Φ S ( t cd ) ◦ S a ( cd ) = S a ( dc ) ◦ Φ s a S ( s a ( t cd )) . he first and second equalities are easy to prove when a = b because S a ( bb ) = id = S a ( b ) and s a ( m b ) = m b . When a = b we see from Definition 2.8 that Φ S ( u b ) ◦ S a () = u = µ ◦ ( η ⋄ η ) = S a ( a ) ◦ Φ s a S ( s a ( u a )) .For the second equality ( a = b ), we use the associativity of µ and Definitions 1.1(1)and 2.14 to deduce that S a ( a ) ◦ Φ s a S ( s a ( m a )) = µ ◦ Φ s a S ( m a ′ ⊔ m a ′′ ) ◦ Φ s a S (( a ′ ) ⊔ t a ′′ a ′ ⊔ ( a ′′ ))= µ ◦ ( µ ⋄ µ ) ◦ ( K ⋄ t ⋄ K ) = µ ◦ ( µ ⋄ µ ) = Φ S ( m a ) ◦ S a ( aa ) .The third equality is again straightforward when a = c , d . If a = c then byDefinitions 2.8, 2.14 and 1.1(3), S a ( da ) ◦ Φ s a S ( s a ( t ad )) = ( K ⋄ µ ) ◦ Φ s a S ( t a ′ d ⊔ ( a ′′ )) ◦ Φ s a S (( a ′ ) ⊔ t a ′′ d )=( K ⋄ µ ) ◦ ( t ⋄ K ) ◦ ( K ⋄ t ) = t ◦ ( µ ⋄ K ) = Φ S ( t ad ) ◦ S a ( ad ) .A similar calculation applies when a = d . (cid:3) When S = [ k ] , we observe that Φ S ([ k ]) = K ⋄ ( k + ) = R kK and that for any i ∈ [ k ] we have D i ([ k ]) = d iR K , S i ([ k ]) = s iR K , and consequently D i ([ k ] n ) = d iR ⋄ nK and S i ([ k ] n ) = s iR ⋄ nK .3.6. Definition.
For integers n , m ≥ ( n , m ) denote the simplicial set whoseset of k -simplices is B k ([ k ] n , [ k ] m ) . In particular, TW ( n , 1 ) = TW ( n ) (cf. Defini-tion 2.10).Here we use the fact that d i : B k → B k − and s i : B k → B k + are monoidalfunctors which carry [ k ] to [ k − ] and [ k + ] respectively, see Proposition 2.9.Composition of morphisms and the monoidal operation ⊔ in B k give rise to “com-position” and “monoidal” products of simplicial setsTW ( n , m ) × TW ( m , l ) ◦ −→ TW ( n , l ) ,(3.7) TW ( n , m ) × TW ( n ′ , m ′ ) ⊔ −→ TW ( n + n ′ , m + m ′ ) .Write X ( n ) = R ⋄ nK and E ( m , n ) = map c E ( X ( m ) , X ( n )) . Thus Φ k ([ k ] n ) = X ( n ) k .By Proposition 2.9, B • is a simplicial monoidal category and thus a morphism α ∈ ∆ ([ i ] , [ j ]) gives rise to a functor α ∗ : B j → B i .For f ∈ TW ( n , m ) k and j ≥
0, we define a morphism X ( n ) j → [ ∆ [ k ] , X ( m )] j by X ( n ) j ∏ α Φ j ( α ∗ ( f )) −−−−−−−−−−→ ∏ α ∈ ∆ [ k ] j X ( m ) j .These maps assemble to a cosimplicial map ρ ( n , m ) k ( f ) : X ( n ) → [ ∆ [ k ] , X ( m )] ,because by Lemma 3.4, for every injective ǫ i : [ j ] → [ j + ] in ∆ and every α ∈ ∆ [ k ] j + , d iX ( m ) ◦ Φ j (( α ◦ ǫ i ) ∗ ( f )) = D i ([ k ] m ) ◦ Φ d i [ j + ] ( d i ( α ∗ ( f ))) = Φ j + ( α ∗ ( f )) ◦ D i ([ k ] n ) = Φ j + ( α ∗ ( f )) ◦ d iX ( n ) . imilarly, by Lemma 3.5, for any surjective η i : [ j ] → [ j − ] and every α ∈ ∆ [ k ] j − , s iX ( m ) ◦ Φ j (( α ◦ η i ) ∗ ( f )) = S i ([ k ] m ) ◦ Φ s i ([ j − ]) ( s i ( α ∗ ( f )) = Φ j − ( α ∗ ( f )) ◦ S i ([ k ] n ) = Φ j − ( α ∗ ( f )) ◦ s iX ( n ) .Thus, ρ ( n , m ) k ( f ) ∈ E ( n , m ) k , see (3.1). They assemble to form a simplicial map ρ ( n , m ) : TW ( n , m ) → E ( n , m ) .because for any γ ∈ ∆ ([ ℓ ] , [ k ]) , α ∈ ∆ [ ℓ ] j , and f ∈ TW ( n , m ) k , we have Φ j (( γ ◦ α ) ∗ ( f )) = Φ j ( α ∗ ( γ ∗ ( f )) ,so ρ ( n , m ) ℓ ( γ ∗ ( f )) = γ ∗ ρ ( n , m ) k ( f ) .The following diagrams of spaces commute:TW ( n , m ) × TW ( m , l ) ρ ( n , m ) × ρ ( m , l ) (cid:15) (cid:15) ◦ / / TW ( n , l ) ρ ( n , l ) (cid:15) (cid:15) E ( n , m ) × E ( m , l ) ◦ / / E ( n , l ) TW ( n , m ) × TW ( n ′ , m ′ ) ρ ( n , m ) × ρ ( n ′ , m ′ ) (cid:15) (cid:15) ⊔ / / TW ( n + n ′ , m + m ′ ) ρ ( n + n ′ , m + m ′ ) (cid:15) (cid:15) E ( n , m ) × E ( n ′ , m ′ ) ⋄ (3.3) / / E ( n + n ′ , m + m ′ ) .This follows by verification in every simplicial degree k using the fact that thefunctors Φ k are monoidal. For example, given f ∈ TW ( n , m ) k and g ∈ TW ( m , l ) l ,the cosimplicial map ρ ( m , l ) k ( g ) ◦ ρ ( n , m ) k ( f ) has the form X ( n ) j ∏ α Φ j ( α ∗ ( g )) ◦ Φ ( α ∗ ( f )) −−−−−−−−−−−−−→ ∏ α ∈ ∆ [ k ] j X ( l ) j .Now Φ j ( α ∗ ( g )) ◦ Φ j ( α ∗ ( f )) = Φ j ( α ∗ ( g ◦ f )) ; which assemble to ρ ( n , l ) k ( g ◦ f ) . Asimilar argument works for the second diagram.Since the composition law in TW (Definition 2.10) is derived from (3.7), thecommutativity of the diagrams above implies that the maps ρ ( n , 1 ) : TW ( n ) → map c E ( X ( n ) , X ( )) form a morphism of operads TW → O End ( R K ) . This completes the proof ofTheorem 1.2. (cid:3) A PPENDIX
A. T
HE FREE MONOIDAL CONSTRUCTION
In this appendix we will prove Lemma 2.5. The result is an exercise in free con-structions, and probably known to anybody who has worked with small monoidalcategories, but we could not find it in the literature.A.1.
Lemma.
The functor U vm : { monoidal graphs } → { vertex-monoidal graphs } has aleft adjoint.Proof. The adjoint is given by F vm ( A ⇒ O ) = ( A ⊔ ⇒ O ) , where A ⊔ is the freemonoid on A and the structure maps are extended uniquely to monoidal maps.The adjointness is obvious. (cid:3) .2. Lemma.
The functor U m : { unital monoidal graphs } → { monoidal graphs } has aleft adjoint.Proof. Let A ⇒ O be a monoidal graph. For every σ ∈ O we introduce a symbol ι ( σ ) / ∈ A and let ˜ ∆ denote A ⊔ { ι ( σ ) } σ ∈O . Extend ∂ , ∂ : A → O to ˜ ∆ by setting ∂ ( ι ( σ )) = ∂ ( ι ( σ )) = σ .This is not quite a unital monoidal graph because ι : O → ˜ ∆ is not monoidal. Weenforce this by setting ∆ = ˜ ∆ / ∼ , where ∼ is the equivalence relation generatedby ι ( O ) ∼ A and ι ( σ ⊔ τ ) ∼ ι ( σ ) ⊔ ι ( τ ) .Since ∂ i ( a ) = ∂ i ( a ′ ) if a ∼ a ′ ∈ ˜ ∆ , they are well-defined on ∆ , and we obtain aunital monoidal graph ( ∆ , O ) . Again, adjointness is easily checked. (cid:3) A.3.
Lemma.
The functor { small monoidal categories } U um −−→ { unital monoidal graphs } has a left adjoint.Proof. Let G = ( ∂ , ∂ : A x ⇒ O ι ) be a unital monoidal graph. Let ˜ M denote C ( G ) ,the free category generated by the graph G . However, ˜ M is not monoidal yet.Let ∼ be the smallest equivalence relation on the morphism set of ˜ M with(i) ( f ⊔ g ) ◦ ( f ′ ⊔ g ′ ) ∼ ( f ◦ f ′ ) ⊔ ( g ◦ g ′ ) for all composable arrows f , f ′ and g , g ′ in ˜ M ,(ii) ∼ is closed under ⊔ , i.e. if f ∼ f ′ and g ∼ g ′ then ( f ⊔ g ) ∼ ( f ′ ⊔ g ′ ) .Again, the equivalence relation ∼ has the property that if f ∼ f ′ then their do-mains and codomains are equal, and thus we obtain a quotient category M = ˜ M / ∼ [ML98, § II.8, Proposition 1]. The monoidal structure ⊔ descends from ˜ M thanks to (ii). It renders M a monoidal category by virtue of relation (i). (cid:3) R EFERENCES[BK72] A. K. Bousfield and D. M. Kan.
Homotopy limits, completions and localizations . Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.[BL07] Tilman Bauer and Assaf Libman. A ∞ –monads and completion. Preprint, 2007.[Bou03] A. K. Bousfield. Cosimplicial resolutions and homotopy spectral sequences in model cate-gories. Geom. Topol. , 7:1001–1053 (electronic), 2003.[BT00] Martin Bendersky and Robert D. Thompson. The Bousfield-Kan spectral sequence for pe-riodic homology theories.
Amer. J. Math. , 122:599–635, 2000.[DMN89] William Dwyer, Haynes Miller, and Joseph Neisendorfer. Fibrewise completion and unsta-ble Adams spectral sequences.
Israel J. Math. , 66(1-3):160–178, 1989.[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May.
Rings, modules, and algebras in stablehomotopy theory , volume 47 of
Mathematical Surveys and Monographs . American Mathemat-ical Society, Providence, RI, 1997. With an appendix by M. Cole.[FI07] Halvard Fausk and Daniel C. Isaksen. Model structures on pro-categories.
Homology, Ho-motopy Appl. , 9(1):367–398 (electronic), 2007.[GJ99] Paul G. Goerss and John F. Jardine.
Simplicial homotopy theory . Birkh¨auser Verlag, Basel,1999.[HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra.
J. Amer. Math. Soc. ,13(1):149–208, 2000.[Isa04] Daniel C. Isaksen. Strict model structures for pro-categories. In
Categorical decompositiontechniques in algebraic topology (Isle of Skye, 2001) , volume 215 of
Progr. Math. , pages 179–198. Birkh¨auser, Basel, 2004.[May72] Peter May.
The Geometry of Iterated Loop Spaces . Number 271 in LNM. Springer Verlag, 1972.[ML98] Saunders Mac Lane.
Categories for the working mathematician , volume 5 of
Graduate Texts inMathematics . Springer-Verlag, New York, second edition, 1998. Qui67] Daniel Quillen.
Homotopical Algebra . Number 43 in LNM. Springer Verlag, 1967.F
ACULTEIT DER EXACTE WETENSCHAPPEN , V
RIJE U NIVERSITEIT A MSTERDAM , DE B OELELAAN
MSTERDAM , T HE N ETHERLANDS
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICAL S CIENCES , K
ING ’ S C OLLEGE , U
NIVERSITY OF A BERDEEN , A B - ERDEEN
AB24 3UE , S
COTLAND , U.K.
E-mail address : [email protected]@maths.abdn.ac.uk