An infinity operad of normalized cacti
Luciana Basualdo Bonatto, Safia Chettih, Abigail Linton, Sophie Raynor, Marcy Robertson, Nathalie Wahl
AAN INFINITY OPERAD OF NORMALIZED CACTI
LUCIANA BASUALDO BONATTO, SAFIA CHETTIH, ABIGAIL LINTON, SOPHIE RAYNOR,MARCY ROBERTSON, AND NATHALIE WAHL
Abstract.
We show that normalized cacti form an ∞ -operad in the form of a dendroidal spacesatisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketedtrees and an enrichment of the dendroidal category Ω. Introduction
Gluing surfaces along their boundaries allows to define composition laws that have been used todefine cobordism categories, as well as operads and props associated to surfaces. These have playedan important role in recent years, for example in constructing topological field theory or computingthe homology of the moduli space of Riemann surfaces. Of particular interest is the cobordismcategory whose morphism spaces are moduli spaces of Riemann surfaces. It has long been knownthat such moduli spaces admit a graph model: they have the homotopy type of spaces of metric fatgraphs [6, 17, 29]. The composition of moduli spaces induced by the gluing of surfaces was modeledusing graphs in [14, Construction 3.29]. Though the resulting composition is associative on theassociated chain complex, it is not associative on the space level, and, at present, it is not knownhow to make it associative, or even coherently homotopy associative [14, Remark 3.31]. In genus 0,this graph model of the cobordism category includes normalized cacti (eg. [35, Remark 2.8]), whosecomposition was also known not to be associative [20, Remark 2.3.19]. The goal of our paper isto show that the composition of normalized cacti is associative up to all higher homotopies, in theprecise sense that normalized cacti form an ∞ -operad in the way detailed below. We expect thatthe technique presented here can be extended to likewise show that the composition in the graphmodel of the cobordism category is also associative up to all higher homotopies. Figure 1.
Spineless cactus with 7 lobes, with its outside the dotted line.A cactus is a treelike configuration of circles (Figure 1). The cactus operad, originally introducedby Voronov [33, Section 2.7], and its spineless version, introduced by Kaufmann [20, Section 2.3],are models for the framed and unframed little disc operads respectively [20, Section 3.2.1]. Operadiccomposition is by insertion: identifying the outside contour of one cactus with the lobe of another
Date : July 6, 2020. a r X i v : . [ m a t h . A T ] J u l L. BASUALDO BONATTO, S. CHETTIH, A. LINTON, S. RAYNOR, M. ROBERTSON, AND N. WAHL cactus and scaling the inserted cactus appropriately. Here we work with the spineless version forsimplicity.A cactus is normalized if each circle in the cactus has circumference of length one. The space of allnormalized cacti with k -lobes is denoted by C act ( k ) and these spaces assemble into the symmetricsequence C act = {C act ( k ) } k ≥ , with each C act ( k ) ⊂ C act( k ) a homotopy equivalent subspace,for C act( k ) the space of all cacti with k lobes. (See [20, Sec 2.3].) Composition of normalized cactiis defined by insertion as for the cactus operad, but instead of scaling the inserted cactus to thesize of the lobe it is inserted in, one scales the lobe to the size of the inserted cactus. Surprisingly,as illustrated in Figure 16, this new composition is not associative ([20, Remark 2.3.19]). So,normalized cacti do not form an operad. This non-associative composition is, however, the onerelevant to the graph model of the cobordism category, as we explain in Remark 5.1.Our main result is that this composition of normalized cacti is part of an ∞ -operad structure. Inthis paper, an ∞ -operad is a dendroidal Segal space in the sense of [9, Definition 8.1]. A dendroidalspace is a space-valued Ω-diagram, where the dendroidal category Ω is the full subcategory of coloredoperads freely generated by trees (Definition 2.7). Dendroidal spaces are closely related to operadssince there is an isomorphism of categories between one-colored topological operads and reduced(i.e. monochromatic) dendroidal spaces satisfying a strict Segal condition. Dendroidal spaces thatsatisfy a weak Segal condition are a model for ∞ -operads, Quillen equivalent to all other knownmodels for ∞ -operads including: dendroidal sets satisfying an inner Kan condition [9, Proposition6.3; Theorem 8.15], Lurie’s ∞ -operads [18, Section 2.5]; [8, Corollary 1.2] and Barwick’s completeSegal operads [8, Theorem 1.1].Operads can be described as algebras over the operad of operads O , an operad whose elementscan be represented by certain trees (Definition 2.9). In Section 3, we define a bracketing of atree and use it to construct a new topological operad B O (Definition 3.9) whose algebras arehomotopy associative versions of operads: Any B O –algebra has an underlying symmetric sequenceand a preferred composition, but the composition is only associative up to coherent homotopy. Theoperad B O is the realization of an operad whose operations lie in the poset of bracketings of thetrees in O . Given a composition on a symmetric sequence, this operad gives a hands-on way to keeptrack of the homotopies required to show that it is coherently homotopy associative. We illustratehow to construct a B O –algebra in practice by showing: Theorem A (Theorem 5.12) . The symmetric sequence {C act ( k ) } k ≥ of normalized cacti, togetherwith the C act composition described above, extends to a B O –algebra structure. In Section 4, we show that this hands-on notion of an operad up to homotopy is related tomore well-known notions of ∞ -operads. We achieve this by showing that any B O –algebra defines adendroidal Segal space. First we construct a topological enrichment (cid:101) Ω of the dendroidal categoryΩ whose objects are trees, as for Ω, but whose morphisms are the realisation of certain posets ofbracketings in trees, defined in a similar fashion to the operad B O . Diagrams over this thickeneddendroidal category (cid:101) Ω are types of homotopy coherent dendroidal spaces. In Proposition 4.10, weshow that a homotopy coherent (cid:101) Ω –diagram can be rectified to a strict Ω–diagram that satisfies theSegal condition if the original diagram did. By defining a nerve functor that takes a B O -algebrato the category of strictly reduced (cid:101) Ω -diagrams that satisfy a strict Segal condition, we prove thefollowing: Theorem B (Theorem 4.8 and Proposition 4.10 ) . There is an isomorphism of categories between B O -algebras and the category of (cid:101) Ω -diagrams that satisfy a strict Segal condition. In particular, aseach (cid:101) Ω -diagram can be rectified, every B O -algebra is an ∞ -operad. By combining Theorem A and Theorem B, normalized cacti are a rare example of an ∞ -operadthat does not arise via the application of a nerve construction to a known (discrete or topological)operad (Corollary 5.13). Indeed, to our knowledge, the only such examples include the weak operad We use a slight variation of the original definition, for details see Definition 2.6 Remark 2.8
N INFINITY OPERAD OF NORMALIZED CACTI 3 of configuration spaces [16, Corollary 5] and examples that arise as a result of completion as in [4,Proposition 5.1].The idea of using a resolution of the operad of operads O to model ∞ -operads is not a new one.A classical way to resolve an operad is to apply the Boardman-Vogt W -construction. Applied tothe operad O , one gets an operad W O whose algebras are also ∞ -operads: there exists a zig-zagof Quillen equivalences between the category of W O -algebras and reduced dendroidal Segal spaces.(For example this can be seen by combining Theorem 4.1 of [1] with either Theorem 1.1 of [3]or Theorem 8.15 of [10].) However, the operad W O is not easy to work with directly. Indeed,its elements are trees (from the W –construction) whose vertices are themselves decorated by trees(from the operad O ), where the first trees compose by grafting and the second trees compose byvertex substitution. In Appendix A, we show that the operad B O is actually isomorphic to aquotient W O of W O : Theorem C (Theorem A.4) . There exists an isomorphism of topological operads W O ∼ = B O . A W O –algebra is an operad up to homotopy, where the symmetric group action, the unit andassociativity relation are all assumed to hold only up to coherent homotopy. ( W O -algebras arecalled a lax operads in the Ph.D. thesis [7].) On the other hand, a W O –algebra (or equivalently B O –algebra), is a homotopy operad where the composition is still only homotopy associative, butwhere the symmetric group action and unit are strict.Theorem B gives the relationship between the operad B O ∼ = W O and the dendroidal category Ω,showing a “bracketed version” of the equivalence between O –algebras and appropriate Ω–diagrams,i.e. replacing O and Ω by bracketed resolutions B O and (cid:101) Ω . The operad W O is a more completeresolution of O . For a category K , there exists a resolution similar to the W –construction, namelythe “explosion” (cid:101) K of the category, as studied by Segal [31, Appendix B] and Leitch [21]. This“explosion” has the property that (cid:101) K –diagrams are coherently homotopy K –diagrams. Applyingthis construction to the category Ω, one could expect that W O –algebras are related to (cid:101) Ω–diagramsin the same way that B O = W O –algebras are related to (cid:101) Ω –diagrams. We show in Theorem B.6that this does not quite hold, proving instead that there is an embedding of the category of W O -algebras as a full subcategory of the category of (cid:101) Ω–diagrams satisfying a strict Segal condition.The results presented in this paper give a detailed infinity operad structure on normalized cacti.The input of the construction is a pre-given composition that we show to be associative up tocoherent homotopy by using the operad B O = W O / ∼ . The homotopies are constructed using thecontractible space of basepoint preserving monotone reparametrizations of the circle (see the proofof Theorem 5.12). To extend the results to the cobordism category of graphs described above, onewould need to replace O by the operad P O , whose algebras are all symmetric properads [36, Section14.1.2], define a resolution “ BP O ”, as the appropriate quotient of the W -construction applied to P O . Our expectation is that these same reparametrisations of the circle will likewise provide allthe necessary homotopies to provide an infinity composition in the cobordism category. Acknowledgements.
This work was done as part of the Women in Topology Workshop in August2019, supported by the Hausdorff Research Institute for Mathematics, NSF grant DMS 1901795, theAWM ADVANCE grant NSF-HRD-1500481, and Foundation Compositio Mathematica. Additionalwork by L.B.B. and M.R. was carried out while in residence at MSRI in 2020. L.B.B. was supportedby CNPq (201780/2017-8). S.R. acknowledges the support of the Centre of Australian CategoryTheory and Australian Research Council grants DP160101519 and FT160100393. N.W. was sup-ported by the Danish National Research Foundation through the Copenhagen Centre for Geometryand Topology (DNRF151) and the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement No. 772960).In addition, we would like to thank Philip Hackney, Gijs Heuts, Muriel Livernet, Claudia Sche-imbauer and Bruno Vallette for helpful conversations, suggested references and comments.
L. BASUALDO BONATTO, S. CHETTIH, A. LINTON, S. RAYNOR, M. ROBERTSON, AND N. WAHL
Contents
1. Introduction 12. Preliminaries on Operads 42.1. Trees 52.2. The dendroidal category Ω 72.3. The operad of operads 82.4. The relationship between operads and dendroidal spaces 103. The operad of brackets BO B O -algebras 144. Thickening the category Ω 154.1. Bracketing Ω and the category (cid:101) Ω M S + that contains C act C act is a B O –algebra 30Appendix A. Relation between the operads B O and W O W -Construction 35A.2. A variant on the W -construction 35A.3. B O -algebras are strictly symmetric lax operads 37A.4. Proof of Theorem A.4 38Appendix B. The explosion category of Ω 42B.1. The explosion of Ω 42B.2. The relationship between (cid:101) Ω and (cid:101) Ω W O –algebras as (cid:101) Ω–diagrams 45References 472.
Preliminaries on Operads A symmetric sequence in a symmetric monoidal category S is a collection P = {P ( k ) } k ≥ ofobjects in S in which each P ( k ) comes equipped with an action of the symmetric group Σ k . In thispaper, our symmetric monoidal category S will either be the discrete category of sets, the categoryof simplicial sets, or the category of topological spaces with their standard Cartesian products.An operad in S is a symmetric sequence P = {P ( k ) } k ≥ together with a distinguished element ι ∈ P (1), called the unit , and a collection of composition maps ◦ i : P ( k ) × P ( j ) P ( k + j − , ≤ i ≤ k , which are associative, unital, and equivariant. For more complete details see, for example,[23, Definition 11]. Given an operad P , a symmetric sequence Q = {Q ( k ) ⊆ P ( k ) } k ≥ is a suboperad of P if the restriction of the composition maps in P induce an operad structure on Q . A morphismof operads f : P → Q is a family of equivariant maps { f ( k ) : P ( k ) → Q ( k ) } k ≥ that are compatible with composition and units. Remark 2.1.
It is equivalent to work with individual compositions ◦ i : P ( k ) × P ( j i ) → P ( k + j i − N INFINITY OPERAD OF NORMALIZED CACTI 5 or with all ◦ i -compositions simultaneously. In the latter case, the simultaneous compositions aredenoted by a map γ P : P ( k ) × P ( j ) × . . . × P ( j k ) → P (Σ ki =1 j i ) . (eg:[23, Proposition 13]).More generally, we will use colored operads. For any non-empty set C , a C - colored symmetricsequence is a family of objects P := {P ( c ; c , . . . , c k ) } k ≥ in S , where ( c ; c , . . . , c k ) ranges overevery list of colors in C together with a map σ (cid:63) : P ( c ; c , . . . , c k ) → P ( c ; c σ (1) , . . . , c σ ( k ) ) for each σ ∈ Σ k . A C - colored operad is a C -colored symmetric sequence P together with a family of partialcomposition maps ◦ i : P ( c ; c , . . . , c k ) × P ( d ; d , . . . , d j ) → P ( c ; c , . . . , c i − , d , . . . , d j , c i +1 . . . , c k )defined only when c i = d , together with an element ι c ∈ P ( c ; c ) for each c ∈ C , which satisfiesunit, equivariance and associativity conditions. For more details see, for example, [1, Definition1.1]. When the color set is C = {∗} , a C –colored operad is a one-colored operad. In this paper wewill refer to both operads and colored operads as “operads”, only mentioning the color set whennecessary.An algebra over a ( C –colored) operad P is a collection of objects { X ( c ) } c ∈ C in S together withevaluation maps α : P ( c ; c , . . . , c k ) × X ( c ) × · · · × X ( c k ) −→ X ( c )satisfying appropriate associativity, unit and equivariance conditions, see e.g. [1, Definition 1.2].The category of P -algebras in S is denoted P− Alg S .Our main example of a colored operad will be the N -colored operad O , whose algebras are the(non-colored) operads, see Definition 2.9. In Section 5, we will also make use of the following operad: Example 2.2.
Let X be a fixed space in S . The coendomorphism operad of X , CoEnd( X ), has anunderlying symmetric sequence with arity k spacesCoEnd( n )( X ) := Map( X, X × k ) . The symmetric groups act by permuting the factors of f = ( f , . . . , f k ) ∈ CoEnd( k ). If f =( f , . . . , f k ) ∈ CoEnd( k )( X ) and g = ( g , . . . , g j ) ∈ CoEnd( j )( X ) the partial compositions ◦ i : CoEnd( k )( X ) × CoEnd( j )( X ) CoEnd( k + j − X )are given by f ◦ i g = ( f , . . . , f i − , g ◦ f i , . . . , g j ◦ f i , f i +1 , . . . , f k ) . Trees.
Throughout this paper, we use trees to model operad compositions and as the basis ofour main constructions. A graph G is a tuple ( V ( G ) , H ( G ) , s, i ) where V ( G ) is a set of vertices, H ( G ) a set of half-edges, s : H ( G ) → V ( G ) is the source map and i : H ( G ) → H ( G ) is an involution.Orbits of the involution i are called edges of G and the set of edges is denoted by E ( G ). An edgerepresented by a pair { h, i ( h ) } with i ( h ) (cid:54) = h is called an internal edge , and the set of internal edgesis denoted iE ( G ). Edges corresponding to orbits of fixed points of the involution are external .A tree is a simply connected graph. All our trees will be rooted , i.e. they come with a distinguished“outgoing” external edge called the root . All other external edges are “incoming” and called leaves .The set of leaves is denoted L ( T ). The arity of T is the number of leaves | L ( T ) | . The root of thetree is denoted R ( T ).Note that a rooted tree can be canonically made into a directed graph by setting all the edgesto point towards the root. Then note that the set of edges incident to a vertex always has a unique outgoing edge , the one closest to the root, and all other edges are incoming edges . The number ofincoming edges of a vertex v is called the arity of the vertex and denoted by | v | , with | v | ≥ η = | , with no vertices and a single edge. The trees with a single vertexand n leaves are called n -corollas and denoted C n . A rooted tree S is a subtree of T if V ( S ) ⊆ V ( T ), H ( S ) ⊆ H ( T ), and the structure maps for S are restrictions of the structure maps for T , defining L. BASUALDO BONATTO, S. CHETTIH, A. LINTON, S. RAYNOR, M. ROBERTSON, AND N. WAHL i ( h ) = h in S if i ( h ) = h (cid:48) in T with h (cid:48) / ∈ H ( S ), so that for every v ∈ V ( S ), the arity of v in S and T is the same. A planar tree is a rooted tree together with a preferred embedding into the plane.Note that for a planar tree, we get an induced canonical ordering of the incoming edges at eachvertex.We use planar trees to model operad compositions via an operation called grafting. Given trees T and T (cid:48) , of arity n and m respectively, and a leaf i ∈ L ( T ), the grafting of T (cid:48) onto T along the leaf i is defined to be the tree T ◦ i T (cid:48) obtained by attaching the root of T (cid:48) to the leaf i of T so that theyform a new internal edge in the grafted tree (Figure 2). Grafting of trees is also used to model thefree operad generated by a symmetric sequence, as we will explain now. To avoid confusion later,when we will have to decorate vertices of trees by other trees, we will use blackboard fonts for thetrees in the free operad construction (and later the associated W -construction in Section A.1), aswe will soon apply this construction to a symmetric sequence of trees, which will give (blackboard)trees of (plain) trees. viT T (cid:48) (a) Example trees T and T (cid:48) v (b) The grafting T ◦ i T (cid:48) Figure 2.
Grafting of trees.
Definition 2.3.
Let P = {P ( c ; c , . . . , c k ) } c i ,c ∈ C be a C -colored symmetric sequence in S . A planartree T is C -colored if it is equipped with a map f : E ( T ) → C , we refer to f ( e ) as the color of theedge e . A C -colored planar tree T is decorated by P if each vertex v ∈ V ( T ) is labeled by anoperation in p v ∈ P ( out ( v ); in ( v )), where out ( v ) is the color of the outgoing edge of v , and in ( v ) isthe list of colors of the incoming edges of v , ordered by the planar structure. The free operad F ( P )on P is the C -colored operad whose k -ary operations are the C -colored, P decorated, planar trees T of arity k with leaves labeled by a bijection λ : { , . . . , k } → L ( T ).Explicitly, for each c, c , . . . , c k ∈ C , F ( P )( c ; c , . . . , c k ) := (cid:16) (cid:97) ( T ,f,λ ) (cid:89) v ∈ V ( T ) P ( out ( v ); in ( v )) (cid:17) / ∼ , where ( T , f, λ ) runs over all isomorphism classes of leaf-labeled C -colored planar trees with k leavessuch that f ( λ ( i )) = c i , f ( R ( T )) = c , and where the equivalence relation is generated by thefollowing:( ∗ ) two labeled trees ( T , f, λ, ( p v ) v ∈ V ( T ) ) and ( T (cid:48) , f (cid:48) , λ (cid:48) , ( p (cid:48) w ) w ∈ V ( T (cid:48) ) ) are equivalent if thereexists a non-planar isomorphism α : T → T (cid:48) such that f ◦ α = f (cid:48) , α ◦ λ = λ (cid:48) , and σ v ( α ) p v = p α ( v ) , for σ v ( α ) the permutation on in ( v ) induced by α .The symmetric group acts on F ( P ) by permuting the labels of the leaves, acting on λ , andcomposition in F ( P ) is given by grafting of trees, with ◦ i grafting at the leaf λ ( i ). For full detailssee, for example, the construction under Corollary 3.3 [1].We now employ the free operad construction to define a class of free operads Ω( T ) generatedby a planar tree T . This will play a fundamental role in the definition of the dendroidal category(Section 2.2), which describes a model for ∞ -operads. Example 2.4.
A planar tree T generates a free colored operad Ω( T ) as follows. The set of colors ofΩ( T ) is the set of edges C = E ( T ). We define a discrete E ( T )-coloured symmetric sequence X ( T ) N INFINITY OPERAD OF NORMALIZED CACTI 7 by X ( T )( e ; e σ (1) , . . . , e σ ( n ) ) = (cid:40) { σv } if ( e ; e , . . . , e n ) = (out( v ); in( v )), for v ∈ V ( T ) , ∅ otherwise,with its built in free symmetric group action. Then Ω( T ) := F X ( T ) is the free operad on thecollection X ( T ). Explicitly, Ω( T ) is an E ( T )-coloured operad withΩ( T )( e ; e σ (1) , . . . , e σ ( n ) ) = (cid:40) σS if ( e ; e , . . . , e n ) = ( R ( S ); L ( S )), S ⊂ T , ∅ otherwise.for S ⊂ T a subtree of T . Composition, as in the free operad, is given by grafting of subtrees. Forfurther details, see Section 2.2 and just above Definition 2.3.1 in [26].2.2. The dendroidal category Ω . The model we use for ∞ -operads is that of dendroidal Segalspaces that satisfy the weak Segal condition . Dendroidal spaces are diagrams of the dendroidalcategory.The dendroidal category Ω is the full subcategory of colored operads whose objects are the freeoperads Ω( T ) generated by trees (as in Example 2.4). In other words, objects of Ω are planarisomorphism classes of planar rooted trees and morphisms in Ω are defined to be operad mapsHom Ω ( S, T ) = Hom Op (Ω( S ) , Ω( T )) . Morphisms in Ω can be described as a composition of four types of elementary morphisms: isomor-phisms, degeneracies, inner and outer face maps. In terms of trees, isomorphisms are non-planartree isomorphisms, inner face maps are of the form ∂ e : T /e → T , where T /e is the tree obtainedfrom T by contracting an inner edge e ∈ iE ( T ). If v is a vertex of T with only one inner edgeattached to it then T /v is the tree obtained from T by chopping off the vertex v and the inclusion ∂ v : T /v → T is an outer face map . A degeneracy is a map s v : T /v → T where T /v is obtainedfrom T by deleting a vertex v , with | v | = 1, in T .In the opposite category Ω op , outer face maps correspond to restriction to certain allowed sub-trees, while inner face maps correspond to edge collapses. For more details and plenty of examplessee [27, 26]. Remark 2.5.
Our definition of Ω differs slightly from the usual definition in that we have chosenour objects to be planar trees. Technically, what we have described here is the equivalent categoryΩ (cid:48) from [26, 2.3.2].
Definition 2.6. A dendroidal space X is an Ω-diagram X : Ω op → S , where S is either the categoryof simplicial sets or topological spaces.The evaluation of X at a tree T is denoted X ( T ). A dendroidal space is called reduced if X ( η ) (cid:39) ∗ ,where η = | . We will write S Ω op for the category of dendroidal spaces. For any vertex v in a tree T ∈ Ω, we have an associated outer face map in Ω C v −→ T taking the unique vertex of the corolla to v ∈ V ( T ), where C v is the corolla with | v | leaves. Likewise,for any internal edge between vertices u and v in T , there is a commuting diagram in Ω η (cid:47) (cid:47) (cid:15) (cid:15) C u (cid:15) (cid:15) C v (cid:47) (cid:47) T. Let Sk ( T ) be the category whose objects are the edges and vertices of T , thought of copies of η and corollas C v , and whose morphisms are associated to edge inclusions in T , as in the top leftcorner of the above diagram. In the literature a dendroidal space is usually called reduced if X ( η ) = ∗ but we vary this slightly and say thata dendroidal space is reduced if X ( η ) is contractible as in [4, Definition 4.1]. L. BASUALDO BONATTO, S. CHETTIH, A. LINTON, S. RAYNOR, M. ROBERTSON, AND N. WAHL
For a dendroidal space X , the Segal map is the unique map from X ( T ) to the limit lim Sk ( T ) op X induced by the corolla inclusions. When X ( η ) = ∗ , this limit becomes a product over the value of X at the corollas, and the Segal map becomes the map χ : X ( T ) (cid:81) v ∈ V ( T ) X ( C v )with components the restriction to the value of X at each corolla.The category of Ω-diagrams admits two Quillen model category structures: the Reedy modelstructure and the projective model structure which are Quillen equivalent (eg:[5, Remark 2.5]).Throughout, we take the projective model structure in which a morphism of Ω-diagrams is a weakequivalence or fibration if it is entrywise a weak equivalence or fibration. Definition 2.7.
A dendroidal space X ∈ S Ω op satisfies a strict Segal condition if the Segal map isan isomorphism for each η (cid:54) = T ∈ Ω. If X is fibrant and the map χ is only a homotopy equivalencefor each η (cid:54) = T ∈ Ω then we say that X satisfies a weak Segal condition.
Remark 2.8.
We briefly comment that in the original definition in [9, Definition 8.1] a dendroidalspaces satisfies the weak Segal condition if the Segal map is a trivial fibration. Our assumption that X is fibrant allows us to only require that the Segal map is a weak equivalence as in [5, Definition3.1] or [4, Definition 4.1].2.3. The operad of operads.
One of the main constructions in this paper is the operad B O . Thisoperad builds on an N -colored operad O called the operad of operads , whose algebras are one-coloredoperads.Let T be a planar tree. For a vertex v ∈ V ( T ) with arity | v | = m and a planar tree T (cid:48) with m leaves, the substitution T • v T (cid:48) is obtained by removing the vertex v from T and identifying theincoming and outgoing edges of v with the leaves and root of T (cid:48) , respectively. An example is shownin Figure 3. viT T (cid:48) (a) Example trees T and T (cid:48) i (b) The substitution T • v T (cid:48) Figure 3.
Tree substitution (Compare with grafting in Figure 2).A labelled planar tree is a triple (
T, σ, τ ), consisting of a planar tree T equipped with bijections σ : | V ( T ) | → V ( T ) and τ : | L ( T ) | → L ( T ). Two such triples ( T, σ, τ ) and ( T (cid:48) , σ (cid:48) , τ (cid:48) ) are isomorphicif there is a planar tree isomorphism T → T (cid:48) that respects the labelling σ, τ . We represent a labelledplanar tree ( T, σ, τ ) by writing above each leaf (cid:96) ∈ L ( T ) the number τ − ( (cid:96) ), and writing by eachvertex v ∈ V ( T ) the number σ − ( v ), as depicted in Figure 4.We also define a tree substitution that is compatible with the labellings of the leaves . Let ( T, σ, τ )and ( T (cid:48) , σ (cid:48) , τ (cid:48) ) be two planar labelled trees with | V ( T ) | = k , | V ( T (cid:48) ) | = l and | L ( T (cid:48) ) | = | σ ( i ) | = m i .The map τ (cid:48) encodes a permutation in the symmetric group with m i elements. We obtain a newplanar tree ( τ (cid:48) σ ( i ) ) T by applying the permutation τ (cid:48) on the m i incoming edges of the vertex σ ( i ) ∈ V ( T ). We then define(2.1) T • σ ( i ) ,τ (cid:48) T (cid:48) = ( τ (cid:48) σi ) T • σ ( i ) T (cid:48) . In particular, V ( T • σ ( i ) ,τ (cid:48) T (cid:48) ) = { V ( T ) − σ ( i ) }(cid:113) V ( T (cid:48) ). The labelling on the vertices of T • σ ( i ) ,τ (cid:48) T (cid:48) is given by the map σ ◦ i σ (cid:48) , which is the induced bijection { , . . . , k + l − } → V ( T • σ ( i ) ,τ (cid:48) T (cid:48) ) j (cid:55)→ σ ( j ) 1 ≤ j < iσ (cid:48) ( j − i + 1) i ≤ j ≤ i + lσ ( j − l + 1) i + l < j ≤ k + l − . N INFINITY OPERAD OF NORMALIZED CACTI 9
124 3
Figure 4.
Example of a labelled planar tree in O (11; 5 , , , τ (cid:48) on the m i incoming • σ (2) ,τ (cid:48) =
124 3
32 5
Figure 5.
Example of composition in O where τ (cid:48) is different to the planar order of in ( σ (2)).edges of σ ( i ) is the same as the order induced by the planar structure, then T • σ ( i ) ,τ (cid:48) T (cid:48) = T • σ ( i ) T (cid:48) . Definition 2.9.
The operad of operads O is the N –colored operad, for which O ( n ; m , . . . , m k )is the discrete space whose elements are isomorphism classes of labelled planar rooted trees ( T, σ, τ )where T is a planar tree with k vertices and n leaves, with bijections σ : | V ( T ) | → V ( T ), τ : | L ( T ) | → L ( T ), such that the vertex σ ( i ) has arity m i for each 1 ≤ i ≤ k . The composition operation O ( n ; m , . . . , m k ) × O ( m i ; b , . . . , b l ) O ( n ; m , . . . , b , . . . , b l , . . . , m k )(( T, σ, τ ) , ( T (cid:48) , σ (cid:48) , τ (cid:48) )) ( T, σ, τ ) ◦ i ( T (cid:48) , σ (cid:48) , τ (cid:48) ) ◦ i is induced by tree substitution that is compatible with the labelling as in (2.1), where( T, σ, τ ) ◦ i ( T (cid:48) , σ (cid:48) , τ (cid:48) ) = ( T • σ ( i ) ,τ (cid:48) T (cid:48) , σ ◦ i σ (cid:48) , τ ) . The unit for this composition, for the color n , is the element of O ( n ; n ) represented by the corolla C n equipped with the canonical left-right labelling. The symmetric group Σ k acts on ( T, σ, τ ) ∈O ( n ; m , . . . , m k ) by precomposition on the labelling σ of the vertices V ( T ).We further observe that, for each m, n ∈ N , O ( m ; n ) ∼ = (cid:26) Σ n for m = n, ∅ when m (cid:54) = n. The isomorphism O ( n ; n ) ∼ = Σ n corresponds to labelling the leaves of a corolla C n in all possibleways. The unique arity 0 operation in O is represented by the special tree η ∈ O (1; ∅ ). An O -algebra, P , is precisely a one-colored operad. That is to say, P has an underlying N -graded object P = {P ( n ) } n ∈ N in S . Moreover, P admits actions O ( n ; n ) × P ( n ) → P ( n ) for all n and thus P hasan underlying symmetric sequence. By definition, we have O ( n ; m , . . . , m k ) × P ( m ) × . . . × P ( m k ) ⊂ F P ( n ) , where F P is the free operad on the symmetric sequence P , and F P ( n ) = (cid:97) k ∈ N (cid:97) ( m ,...,m k ) ∈ N k O ( n ; m , . . . , m k ) × P ( m ) × . . . × P ( m k ) Σ k so the action maps α : O ( n ; m , . . . , m k ) × P ( m ) × . . . × P ( m k ) → P ( n )induce maps F P ( n ) → P ( n ) for all n , and by the algebra axioms, this is precisely the data of asymmetric operad in S (See [1, Example 1.5.6]). Note that, in particular, the ◦ i -compositions of anoperad P are governed by the trees with one internal edge in O ( n ; m , m ), where n = m + m − The relationship between operads and dendroidal spaces.
Reduced dendroidal spacesthat satisfy a strict
Segal condition are closely related to one-colored operads. Explicitly, everyoperad P can be viewed as a dendroidal space via the dendroidal nerve construction that defines afunctor N d ( P )( T ) = Hom Op (Ω( T ) , P )as T ranges over Ω. The nerve of the free operad Ω( T ) is just the representable dendroidal spaceΩ[ T ] := Hom Ω ( − , T ). A dendroidal space X is the nerve of an operad if, and only if, the Segalmap of Definition 2.7 is an isomorphism for all T [10, Lemma 6.4; Proposition 6.5]. To put thisaltogether, there is an isomorphism of categories O− Alg S ∼ = ( S Ω op ) strict where O is the colored operad whose algebras are one-colored operads (Definition 2.9, below) and( S Ω op ) strict denotes the category of reduced dendroidal spaces satisfying the strict Segal condition.We will prove similar statements for “thickened” versions of Ω in Theorem 4.8 and Theorem B.6.3. The operad of brackets BO In this section we introduce a new topological operad called the operad of bracketed trees . In short,the operad B O captures a weak notion of an operad in the sense that a B O -algebra is a symmetricsequence with ◦ i -operations that are only associative up to higher homotopy. The construction of theoperad B O allows one to check with relative ease whether a symmetric sequence with compositionsassembles into an ∞ -operad. In Theorem 5.12, we use this to show that normalized cacti admitsuch a structure. Moreover, we expect that this construction provides a general method that onecan use to construct other examples of ∞ -operads.One could instead use the classical Boardman-Vogt W -construction on the operad O to obtain anoperad W O whose algebras are homotopy operads ( lax operads in the language of [7]). It is knownto experts that bracketings in trees are related to this operad W O , but the precise details aredifficult to find in the literature. (However, see [28, Section 2.3], in particular Theorem 4, togetherwith Remark 3.7 below, for an algebraic version of this in the case of non-symmetric operads.) InAppendix A we will show that B O identifies with a quotient of the operad W O . Bracketings intrees have also appeared elsewhere, see eg. [11, 12], and the parenthesizations of [32, 2.6]. N INFINITY OPERAD OF NORMALIZED CACTI 11
Bracketings of trees.
We define in this section the poset of bracketings of a tree, startingwith the definition of a bracketing:
Definition 3.1.
A tree is called large if it has at least two vertices (or equivalently, at least oneinternal edge). A set { S j } j ∈ J of subtrees of a tree T is nested if, for any i, j ∈ J , the set of commonvertices V ( S i ) ∩ V ( S j ) is either V ( S i ), V ( S j ) or empty. A bracketing B of a tree T is a (possiblyempty) collection B = { S j } j ∈ J of nested large proper subtrees of T .Recall from Section 2.1 that a subtree of T is a tree S whose vertices are a subset of the vertices of T , and whose half-edges are all the half-edges in T attached to such vertices. Therefore, a subtreeis completely determined by its vertices. With this in mind, we will represent bracketings as inFigure 6. Figure 6.
Example of a tree bracketing with 3 nested subtrees.
Definition 3.2.
Bracketings of a tree T form a poset of bracketings B ( T ) with the relation B (cid:48) ≤ B if B (cid:48) ⊆ B .We denote the geometric realisation of the nerve of the poset B ( T ) by |B ( T ) | . A point in |B ( T ) | = (cid:97) r ≥ N r B ( T ) × ∆ r / ∼ is a pair ( B, t ) with B = B ⊂ · · · ⊂ B r a sequence of bracketings and t ∈ ∆ r . Such a pair ( B, t ) canbe interpreted as a weighted bracketing with underlying set of brackets B r = ∪ ri =0 B i and weightsgiven by t = (1 , t , . . . , t r ) ∈ ∆ r = { t ≥ t ≥ · · · ≥ t r ≥ } where we assign the weight t = 1 to all brackets in B , and for each 1 ≤ i ≤ r , the weight t i to all brackets in B i \ B i − . In particular, a weighted bracketing with all brackets having weight1 corresponds to a vertex B = B in the nerve of the poset. Also, the equivalence relation onthe realization implies that a bracket of weight 0 can be discarded. (See also Appendix A and inparticular the proof of Lemma A.7 where this point of view is used to relate B O to the operad W O .) Example 3.3. If T = C n then T does not admit any large subtree, therefore B ( T ) = {∅} only hasthe empty (or trivial) bracketing. Example 3.4.
Let T be the tree , then the space |B ( T ) | is depicted in Figure 7 (left). Note thatthe initial object in the poset is the empty bracket, in the centre of the pentagon.More generally, let T n be a tree with n vertices such that no vertex is connected to more thantwo inner edges. For such trees, the set of vertices can always be given a total ordering, for instanceby constructing a list starting with a vertex v connected to only one internal edge, and defining thenext element of the list to be the vertex sharing an edge with v that has not yet been listed. Then abracket of T n can be immediately identified with a meaningful placement of parentheses on a wordwith n letters where the word is represented by the ordered set of vertices. Therefore, |B ( T n ) | can Figure 7.
Geometric realization of the poset B ( T ) of Examples 3.4 and 3.5.always be identified with the n -th associahedron (see also Remark 3.7 for another approach to thisstatement). Example 3.5.
Consider a tree T with three inner edges all meeting at a single vertex. Note thatthe poset of bracketings depends only on the relative positions of the vertices (or analogously, theinner edges) of the tree T , and is independent of the number of leaves at each vertex. Therefore,the realization poset of bracketings of T is the one depicted in the Figure 7 (right), using as anexample the tree T = . Example 3.6.
Figure 8 depicts the realisation of the poset of bracketings of a tree T with fourinner edges meeting at a single vertex. Note that by fixing a large subtree S of T , the realisationof the subposet of bracketings of T containing S will correspond to a subspace of the boundary of |B ( T ) | . Each boundary face of top dimension is then associated to a subtree S of T , and two suchfaces S , S share a subface if { S , S } is nested. T = Figure 8.
Tree satisfying the conditions of Example 3.6 together with the geo-metric realisation of its poset of bracketings.
Remark 3.7.
The spaces |B ( T ) | are closely related to the abstract polytopes defined in [28]. Infact, we can show that |B ( T ) | identifies with the hypergraph polytope of the edge-graph H T of T ,as defined in [28, Section 2.2.1]. The set of vertices of H T is the set of inner edges of T , and two N INFINITY OPERAD OF NORMALIZED CACTI 13 such share an edge if they have a common vertex. Then a subset S of vertices of H T uniquelydefines a subforest (cid:104) S (cid:105) of T whose internal edges are precisely the elements of S , and each treein this forest is necessarily large because it has an inner edge (see [28, Section 2.2.1, Lemma 3]).Then we have an order reversing bijection b between the abstract polytope of the edge-graph of T and B ( T ), which can be recursively defined as follows: using the notation established in [28], wetake the construct V ( H T ) to the empty bracketing, and if H T \ Y (cid:32) H T , . . . , H T n , we take theconstruct Y { C , . . . , C n } to the bracketing {(cid:104) H T (cid:105) , . . . , (cid:104) H T n (cid:105) , b ( C ) , . . . , b ( C n ) } . The definition ofthe constructs guarantees that these sets are nested and therefore define a bracketing, and it issimple to check that this is an order reversing bijection. Lemma 3.8.
For any tree T , the space |B ( T ) | , is contractible.Proof. The contractiblity of the space |B ( T ) | follows directly from the fact that the poset B ( T ) hasa minimal element, namely the empty bracketing. (cid:3) An operad of bracketings.
We’ll use the bracketings B ( T ) to construct a topological operad.Let the collection B O ( n ; m , . . . , m k ) = (cid:97) ( T,σ,τ ) ∈O ( n ; m ,...,m k ) |B ( T ) | define the N -coloured symmetric sequence B O . So, elements of B O ( n ; m , . . . , m k ) are tuples( T, σ, τ, B, t ) where (
T, σ, τ ) is an element of O ( n ; m , . . . , m k ) (Definition 2.9) and ( B, t ) is aweighted bracketing of T (ie. a point in |B ( T ) | ).To define operadic composition in B O , we use the composition of trees in O and induce abracketing of the resulting tree. Let ( T, σ, τ, B ) and ( T (cid:48) , σ (cid:48) , τ (cid:48) , B (cid:48) ) be labeled trees with bracketings.The composition in O (Definition 2.9) is given by the substitution of T (cid:48) into the vertex σ ( i ) ∈ V ( T ),( T, σ, τ ) ◦ i ( T (cid:48) , σ (cid:48) , τ (cid:48) ) = ( T • σ ( i ) ,τ (cid:48) T (cid:48) , σ ◦ i σ (cid:48) , τ ) . Since T (cid:48) is canonically a subtree of T • σ ( i ) ,τ (cid:48) T (cid:48) , the bracketing B (cid:48) on T (cid:48) defines a nested collectionof subtrees of T • σ ( i ) ,τ (cid:48) T (cid:48) . We also construct a nested collection of subtrees ˜ B = { ˜ S j } j ∈ J on T • σ ( i ) ,τ (cid:48) T (cid:48) that is induced by the bracketing B = { S j } j ∈ J on T . If T (cid:48) (cid:54) = η , then ˜ B ∼ = B is given by(3.1) ˜ S j = (cid:26) S j if σ ( i ) / ∈ V ( S j ) ,S j • σ ( i ) ,τ (cid:48) T (cid:48) if σ ( i ) ∈ V ( S j ) . If T (cid:48) = η , then ˜ B = { ˜ S j } j ∈ J is defined in the same way, unless σ ( i ) ∈ V ( S j ) and S j has two vertices,in which case S j • σ ( i ) ,τ (cid:48) η is a corolla and is discarded as it is not large. That is, we replace J withanother indexing set J (cid:48) ⊂ J , which is the subset of indices j such that S j is large.We define a bracketing of the tree T • σ ( i ) ,τ (cid:48) T (cid:48) by(3.2) B (cid:48)(cid:48) = (cid:26) ˜ B ∪ B (cid:48) ∪ { T (cid:48) } if T (cid:48) is large˜ B else.See Figure 9. This defines a composition of bracketings of trees. This composition is associative asfollows. Suppose S j ⊂ T is a bracket with only two vertices v and w , and T (cid:48) is a tree with at leasttwo vertices. If we first compose η in v and then T (cid:48) in w , the bracket S j is discarded during the firstcomposition, and then replaced by a new bracket T (cid:48) . Reversing the order of these two compositionsyields the same result because first composing T (cid:48) in w will create a new bracket T (cid:48) , and S j will notbe discarded, but composing further η in v will equate S j and T (cid:48) . Otherwise, the associativity ofthe composition follows from the associativity on the composition in O .The composition also respects inclusions and thus is a poset map(3.3) B ( T ) × B ( T (cid:48) ) B ( T • σ ( i ) ,τ (cid:48) T (cid:48) ) . The realization of the poset map (3.3) induces a map between the geometric realisations of thenerve of the posets.Also recall that the unary elements of O , i.e. the elements of O ( n ; n ) for some n , are given bylabeled corollas. Since there are no non-trivial bracketings of corollas, unary elements of B O have • σ (3) ,τ (cid:48) =
21 3 4 Figure 9.
Example of composition in B O with labelling of the vertices omitted for simplicity.the form ( C n , σ, ∗ , ∅ , ∅ ) ∈ B O ( n ; n ) with σ ∈ Σ n . In particular, the n -coloured identity for thecomposition ◦ in B O is given by ( C n , id n , ∗ , ∅ , ∅ ) ∈ B O ( n ; n ). Therefore B O is an operad. Definition 3.9.
The operad of bracketed trees B O is the N -coloured topological operad with un-derlying symmetric sequence B O ( n ; m , . . . , m k ) = (cid:97) ( T,σ,τ ) ∈O ( n ; m ,...,m k ) |B ( T ) | and composition given by combining the composition in O with the map (3.3) described above. Remark 3.10.
The topological operad B O is the realization of an operad in posets. Indeed, thespace B O ( n ; m , . . . , m k ) is the realization of the poset of elements ( T, σ, τ ) of O ( n ; m , . . . , m k )together with a bracketing of T , where two elements are comparable only if they have the sameunderlying element of O . Likewise, the operad structure is defined as the realization of a map onthe level of posets.3.3. B O -algebras. A B O - algebra is an operad whose ◦ i -compositions are associative up to allhigher homotopies. In particular, a B O -algebra P = {P ( n ) } n ∈ N has an underlying symmetricsequence . To see this, we note that the labelling of the leaves of a corolla ( C n , τ, ∗ , ∅ , ∅ ) ∈ B O ( n ; n )identifies with elements of the symmetric group and we have isomorphisms B O ( n ; n ) ∼ = O ( n ; n ) ∼ = Σ n . The action B O ( n ; n ) × P ( n ) P ( n )makes P = {P ( n ) } n ∈ N into a symmetric sequence. B O -algebras also have a notion of operadic ◦ i composition . To see this, recall that such compo-sitions are encoded in the operad O by the trees with exactly two vertices, one attached to the i thincoming edge of the other. As such trees admit no large, proper subtrees, they admit no non-trivialbracketing and we have isomorphisms for any n, m ≥ B O ( m + n − m, n ) | V ( T ) ≤ ∼ = O ( m + n − m, n ) | V ( T ) ≤ between the components of the tuples ( T, σ, τ, ∅ ,
0) (resp. (
T, σ, τ )) with T having at most twovertices. It follows then that P is equipped with operadic ◦ i -compositions.A B O -algebra is not in general an operad, however. The brackets that arise in trees with morethan two vertices capture the different choices one has in iterated compositions of ◦ i operations.More explicitly, if {P ( n ) } n ∈ N is a B O –algebra, then for any collection of elements x i ∈ P ( m i ) thatdecorate the vertices of a tree ( T, σ, τ ) ∈ O ( n ; m , . . . , m k ), we have a chosen composition of thoseelements, namely the one determined by ( T, σ, τ, ∅ , ∅ ) ∈ B O ( n ; m , . . . , m k ). This “unbracketed”tree sits in the middle of a polytope of all possible elements ( T, σ, τ, B, s ) for any bracketing B , asin Figure 7. The corners of this polytope correspond to the possible maximal bracketings of T (the N INFINITY OPERAD OF NORMALIZED CACTI 15 maximal elements of B ( T )). Just like the corners of the Stasheff polytopes give all the possible waysto bracket a k –fold multiplication, these maximal bracketings correspond precisely to the possibleways to bracket the composition of ◦ i operations, which are those defined using trees with exactlytwo vertices. The polytopes arising from the posets of bracketing in trees can be thought of as anoperadic analogue of the Stasheff polytopes. Remark 3.11.
In [20, Definition 1.1.1], a quasi-operad is a symmetric sequence P = {P ( n ) } n ∈ N together with operadic ◦ i -compositions and no further structure. In this way, a B O -algebra is anextension of a quasi-operad. The operad B O is closely related to the W -construction of O , whosealgebras go under the name lax operads , see Appendix A, where we show that B O -algebras can bedescribed as strictly symmetric lax operads.4. Thickening the category
ΩWe have seen that operads are O -algebras. Also recall from Section 2.4 that operads can bedescribed as strict Segal dendroidal spaces. The dendroidal category Ω is defined as a full subcat-egory of the coloured operads generated by trees. To obtain a similar description of B O –algebrasas certain “homotopy dendroidal Segal spaces,” we construct a topological category (cid:101) Ω that is acategory with the same objects as Ω but its spaces of morphisms are built using posets similarto the posets used to define B O . Theorem 4.8 establishes that this category, (cid:101) Ω , has the desiredproperty that strict reduced Segal (cid:101) Ω op –spaces are precisely B O -algebras. In Section 4.3, we thenshow how rectification of diagrams can be used to produce an actual Segal dendroidal space fromsuch a homotopy version of a dendroidal space.Given any category K with a discrete set of objects, Leitch [21] constructed a new category (cid:101) K with the property that (cid:101) K –diagrams are homotopy coherent K –diagrams. A similar enrichment(the explosion category ) was also used by Segal [31, Appendix B] to relate his Γ–space approach toinfinite loop spaces to the operadic approach of Boardman-Vogt and May. Because (cid:101) Ω –diagrams arehomotopy coherent Ω –diagrams, one can expect that the category (cid:101) Ω is related to this constructionof Leitch applied to Ω. In Appendix B, we construct an equivalence between these two categories,and show that strict Segal (cid:101) Ω–spaces are closely related to W O –algebras.4.1. Bracketing Ω and the category (cid:101) Ω . Recall from Section 2.2 that the objects of Ω are planarisomorphism classes of planar rooted trees. Morphisms in Ω are compositions of inner and outerface maps, degeneracies and isomorphisms of trees. Inner face maps ∂ e : T /e → T create inneredges and correspond to operadic composition, while outer face maps are subtree inclusions and areassociated to projection maps. A degeneracy creates a vertex that is adjacent to exactly two edges.The category (cid:101) Ω is a version of Ω with the same set of objects, but with the realization of a posetof bracketings over each composition of inner face maps.We define the morphism spaces of (cid:101) Ω as follows. Let g : S → T be a morphism in Ω. For eachvertex v ∈ V ( S ), let C v ⊂ S denote the corolla of the vertex v that is, C v = i v ( C | in ( v ) | ) where i v : C | in ( v ) | → S is the composition of outer faces in Ω sending the vertex of the corolla C | in ( v ) | to v . Since g is alternatively considered as a map of operads between Ω( S ) and Ω( T ), the image in S of C v under g is a subtree in T , which we denote g ( C v ) ⊂ T. Note that the trees g ( C v ) are precisely the subtrees of T that correspond to expansion of verticesinto subtrees, going from S to T , or collapsed by g op : T → S in the opposite category Ω op . Thesesubtrees correspond to the part of g made out of inner face maps.For a vertex v ∈ V ( S ), let B gv be a bracketing of g ( C v ) as defined in Definition 3.1. We define aposet L g whose objects are tuples ( B gv ) v ∈ V ( S ) of bracketings of the trees g ( C v ). The poset relation iscomponentwise inclusion. Taking the realization of these posets, for each morphism g we associatethe space L g := (cid:89) v ∈ V ( S ) |B ( g ( C v )) | where B ( g ( C v )) is the poset of bracketings of the tree g ( C v ) as defined in Definition 3.2. Note alsothat |B ( g ( C v )) | = ∗ if g ( C v ) admits only the trivial bracketing. Example 4.1.
Consider the morphism f ∈ Hom Ω ( R, S ) of Figure 10. Since the image of eachcorolla under f only admits a trivial bracketing, L f = (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) = ∗ . f = ∂ e −−−−−→ e Figure 10.
Example of map f in Ω and the subtrees f ( C v ). Example 4.2.
Let s be the morphism in Figure 11(a). By Example 3.4, if s ( C v ) has 3 vertices suchthat no vertex is connected to more than two inner edges, then |B ( s ( C v )) | is the 3rd associahedron,which is an interval. As in Example 3.5, when s ( C v ) is a tree whose three internal edges meet at asingle vertex, the realization poset |B ( s ( C v )) | corresponds to a hexagon. Thus L s is identified withthe hexagonal prism of Figure 11(b). s −→ (a) Map s in Ω and the subtrees s ( C v ). (b) Space L s for the map s of Figure 11(a) Figure 11.
A map s in Ω and its corresponding space L s .The space of morphisms between any two objects in (cid:101) Ω isHom (cid:101) Ω ( S, T ) = (cid:97) g ∈ Hom Ω ( S,T ) L g . It remains to define composition in (cid:101) Ω . To do this, we first define a map of posets(4.1) L g × L f L g ◦ f for any two morphisms f : R → S and g : S → T in Ω, then we take the realization of thiscomposition map to get a composition of spaces L g . Let ( B gv ) v ∈ V ( S ) ∈ L g and ( B fw ) w ∈ V ( R ) ∈ L f be N INFINITY OPERAD OF NORMALIZED CACTI 17 two collections of bracketings. So for each v ∈ V ( S ), B gv is a bracketing of g ( C v ) ⊂ T and for each w ∈ V ( R ), B fw is a bracketing of the tree f ( C w ) ⊂ S . To define the image of (4.1), we construct abracketing of the tree ( g ◦ f )( C w ) from the bracketings of f and g .Fix a vertex w ∈ V ( R ). For each v ∈ f ( C w ) ⊂ S , there is the subtree g ( C v ) ⊂ ( g ◦ f )( C w ), aswell as a bracketing B gv of g ( C v ). Also, for each bracket in S i ∈ B fw , the image g ( S i ) is a subtree of( g ◦ f )( C w ). Therefore we have the following collections of subtrees in ( g ◦ f )( C w ):˜ B gf ( w ) = (cid:91) v ∈ f ( C w ) B gv = { S j : S j ∈ B gv and v ∈ f ( C w ) } ˜ B g ◦ fw = { g ( C v ) : v ∈ f ( C w ) and g ( C v ) (cid:40) ( g ◦ f )( C w ) is large } ˜ B fw = { g ( S i ) : S i ∈ B fw and g ( S y ) (cid:40) ( g ◦ f )( C w ) is large } . All of these are collections of proper large subtrees of ( g ◦ f )( C w ). We set the bracketing ˜ B w of( g ◦ f )( C w ) to be the union ˜ B w := ˜ B gf ( w ) ∪ ˜ B g ◦ fw ∪ ˜ B fw . To see that ˜ B w is a bracketing of ( g ◦ f )( C w ), it remains to verify that this collection is comprisedof nested subtrees. First, each B gv ⊂ ˜ B gf ( w ) is a bracketing of g ( C v ) ⊂ ( g ◦ f )( C w ), so it is nested.Moreover, the subtrees g ( C v ) are all disjoint and each tree of ˜ B gf ( w ) is contained in a tree of ˜ B g ◦ fw ,so the union ˜ B gf ( w ) ∪ ˜ B g ◦ fw is nested too. The ˜ B g ◦ fw ∪ ˜ B fw is also nested, since each bracket g ( C v ) inthe first set is included in each g ( S y ) of the second set whenever v ∈ S y and otherwise is disjointfrom it. Hence ˜ B gf ( w ) ∪ ˜ B fw is also nested, and thus ˜ B w is nested.Define the composition ( B fv ) v ∈ V ( S ) ◦ ( B gw ) w ∈ V ( R ) to be the collection( ˜ B w ) w ∈ V ( R ) ∈ L g ◦ f . Associativity of this composition is analogous to the associativity of the B O composition in Sec-tion 3.2. In most cases, the composition is associative because vertex substitution is associative. Ina composition with a degeneracy, a vertex is removed and so a bracket may be discarded if it is nolonger large. Any discarded bracket is recreated in a subsequent composition if it should not havebeen discarded in the total composition.Furthermore, this composition definition respects componentwise inclusion and thus defines theposet map (4.1). The realization of this poset map induces a map(4.2) L g × L f L g ◦ f . This defines a composition on the morphism spaces of ˜Ω . Example 4.3.
Let f : R → S and g : S → T be the morphisms in Figure 12. Then R is a corolla C w = C , and f ( C w ) ⊂ S is the proper subtree of S whose vertices are v , v , v . The images g ( C v ) , g ( C v ) , g ( C v ) ⊂ T are the corollas C u , C u , C u respectively and g ( C v ) is the subtree withvertices u , u , u . The only images of corollas that admit a non-trivial bracketing are f ( C w ) and g ( C v ). If the bracketing of f ( C w ) consists of the bracket B in Figure 13(a) and the bracketing of g ( C v ) consists of B in Figure 13(b), then˜ B gf ( w ) = { B } , ˜ B g ◦ fw = { g ( C v ) } , ˜ B fw = { g ( B ) } . The bracketing ˜ B w ∈ B (( g ◦ f )( C w )) is illustrated in Figure 13(c).By Example 3.4, if T n is a tree with n vertices such that no vertex is connected to more than twoinner edges, then |B ( T n ) | is the n th associahedron. The 3rd associahedron is an interval. Thus, L f = |B ( f ( C w )) | = (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) ∼ = [0 , L g = |B ( g ( C v )) | × (cid:12)(cid:12)(cid:12) B (cid:16) (cid:17)(cid:12)(cid:12)(cid:12) × |B ( g ( C v )) | ∼ = [0 , . g −−−−→ f −−−−→ w v v v u u u u u u v Figure 12.
Morphisms f and g in Ω. v v v (a) Bracketing B of f ( C w ). u u u (b) Bracketing B of g ( C v ). u u u u u (c) Bracketing ˜ B w of g ◦ f ( C w ). Figure 13.
Example of a bracketing induced by f and g in Figure 12.Again by Example 3.4 and since ( g ◦ f )( C w ) is a tree on five vertices, L g ◦ f = |B (( g ◦ f )( C w )) | is the5th associahedron, which is a three dimensional polytope called an enneahedron. Definition 4.4.
The category (cid:101) Ω has the same objects as Ω. Morphism spaces in (cid:101) Ω areHom (cid:101) Ω ( S, T ) = (cid:97) g ∈ Hom Ω ( S,T ) L g = (cid:97) g ∈ Hom Ω ( S,T ) (cid:89) v ∈ V ( S ) |B ( g ( C v )) | with composition (4.2) as described above. Example 4.5.
Suppose T n is a planar tree with ( n + 1) leaves and n vertices, each of which isconnected to at most two inner edges. Let the inner edges of T n be named e , . . . , e n − . Mor-phisms g ∈ Hom Ω ( C n +1 , T n ) are compositions of inner face maps ∂ e , . . . , ∂ e n − but since the orderof the composition does not affect the total composition, there is only one such morphism g . HenceHom (cid:101) Ω ( C n +1 , T n ) = L g = |B ( T n ) | . Thus Hom (cid:101) Ω ( C n +1 , T n ) is the n th associahedron by Exam-ple 3.4; the centre point of the polytope is defined by the empty bracket, which is the initial objectin the poset B ( T n ).Lemma 3.8 tells us that each bracketing space |B ( g ( C v )) | is contractible, which implies that each L g is contractible. Let p : (cid:101) Ω → Ω be the functor that is the identity on objects and projects eachmorphism space L g to g . By considering Ω as a discrete topological category, we have the followingproposition. Proposition 4.6.
The functor p : (cid:101) Ω → Ω induces a homotopy equivalence on morphism spaces. (cid:3) This proposition will allow us to associate an actual dendroidal space to any homotopy dendroidalspace in Section 4.3.
N INFINITY OPERAD OF NORMALIZED CACTI 19
Homotopy dendroidal spaces.
In Section 2.2, we defined a Segal condition for dendroidalspaces X : Ω op → S using the Segal map χ : X ( T ) lim Sk ( T ) op X. We recall that the category Sk ( T ) has the vertices and edges of T as objects, with morphisms givenby edge inclusions ι e : η → C v into the corollas of adjacent vertices. The Segal map χ is the uniquemap to the limit induced by the edge and corolla inclusions ι e : η → T and ι v : C | v | → T. Note that the spaces L ι e and L ι v in (cid:101) Ω op which lie above the morphisms ι e and ι v are always just asingle point, so the Segal map exists unchanged for functors X : (cid:101) Ω op → S . This allows us to makethe following definition: Definition 4.7. A homotopy dendroidal space X is a diagram X : (cid:101) Ω op → S . A homotopy dendroidalspace is reduced if X ( η ) (cid:39) ∗ and strictly reduced if X ( η ) = ∗ . A homotopy dendroidal space satisfiesthe strict Segal condition if the Segal map is an isomorphism for each η (cid:54) = T ∈ Ω and a homotopydendroidal space satisfies a weak
Segal condition if the Segal map is a homotopy equivalence foreach η (cid:54) = T ∈ Ω.Recall from Section 2.4 that one-colored operads are identified with strictly reduced dendroidalSegal spaces via the dendroidal nerve N d : O -Alg → S Ω op . The following theorem is a version of this nerve theorem for homotopy dendroidal spaces. Weconstruct a functor Φ : B O− Alg S −→ S (cid:101) Ω op and show that a homotopy dendroidal space X ∈ S (cid:101) Ω op with X = ∗ is strictly Segal if, and only if, X ∼ = Φ( P ) for some B O -algebra P .Write ( S (cid:101) Ω op ) strict for the full subcategory of (cid:101) Ω -diagrams whose objects are strictly reducedhomotopy dendroidal spaces satisfying the strict Segal condition. Then we have the followingresult: Theorem 4.8.
There exists an isomorphism of categories
Φ : B O− Alg S ( S (cid:101) Ω op ) strict . ∼ = Proof.
Given a B O –algebra P = {P ( n ) } n ≥ with structure maps α P : B O ( n ; m , . . . , m k ) × P ( m ) × · · · × P ( m k ) −→ P ( n )we will define Φ( P ) = Φ( P , α P ) : (cid:101) Ω op −→ S as follows. We set Φ( P )( η ) = ∗ . On objects T (cid:54) = η of (cid:101) Ω , we setΦ( P )( T ) = (cid:89) w ∈ V ( T ) P ( | w | ) . Given a morphism g : S → T in Ω, we need to define mapsΦ( P )( g ) : L g × (cid:89) w ∈ V ( T ) P ( | w | ) −→ (cid:89) v ∈ V ( S ) P ( | v | ) . We proceed one vertex v at a time. As L g = (cid:81) v ∈ V ( S ) |B ( g ( C v )) | , at each v ∈ V ( S ) we haveprojection maps π v : L g × (cid:89) w ∈ V ( T ) P ( | w | ) −→ |B ( g ( C v )) | × (cid:89) w ∈ V ( g ( C v )) P ( | w | ) . An application of the structure map α P defines a map( ∗ ) α v : |B ( g ( C v )) | × (cid:89) w ∈ V ( g ( C v )) P ( | w | ) −→ P ( | v | ) . Indeed, an element of |B ( g ( C v )) | is a weighted bracketing ( B, t ) of the subtree g ( C v ) ⊂ T . Because T is a planar tree, g ( C v ) inherits a planar structure. We consider g ( C v ) as an element of O by pickingan ordering σ of its vertices { w , . . . , w k } , and labeling its leaves via the map τ ordering them ac-cording to its planar structure. This way (( g ( C v ) , σ, τ ) , B, t ) is an element of B O ( | v | ; | w | , . . . , | w k | ).To define the map ( ∗ ), we first order the factors P ( | w | ) for w ∈ V ( g ( C v )), in accordance with ourchosen σ , and then apply α P noting that our choice of ordering does not affect the result by theequivariance of α P . Finally we act on the resulting element of P ( | v | ) by the permutation inducedby g that identifies the inputs of v with the leaves of g ( C v ), comparing the labeling τ from theplanar structure of T to the planar ordering of in ( v ) (which comes from the planar structure of S ).We now set Φ( P )( g ) := ( α v ◦ π v ) v ∈ V ( S ) . The fact that Φ( P ) commutes with composition follows from the fact that composition in (cid:101) Ω is defined exactly as the operadic composition of B O by taking the union of the brackets fromthe first morphism which remain large after applying the second morphisms, the brackets from thesecond morphism, and new “middle brackets”, the images of the middle corollas, if they are large.It follows then that Φ( P ) : (cid:101) Ω → S is a functor. Since Φ( P )( η ) = ∗ , the Segal map is the mapΦ( P )( T ) −→ (cid:89) v ∈ V ( T ) Φ( P )( C v )induced by the inclusions of the corollas. It is an isomorphism by definition of Φ( P ).The data required in the definition of the homotopy dendroidal space Φ( P ) is the underlyingsymmetric sequence P = {P ( m ) } , the B O -algebra structure maps α P and the projection maps π v ,all of which are natural under maps of B O -algebras. Thus, the assignment P (cid:55)→ Φ( P ) defines afunctor Φ : B O− Alg S −→ ( S (cid:101) Ω op ) strict . It remains to show that the functor Φ is an isomorphism of categories. Given two B O -algebras P and Q with Φ( P ) = Φ( Q ), the underlying symmetric sequences {P ( n ) } n ≥ and {Q ( n ) } n ≥ arenecessarily equal, being the value at the corollas C n and the corolla isomorphisms Hom (cid:101) Ω ( C n , C n ) ∼ =Hom Ω ( C n , C n ) ∼ = Σ n . Moreover, the structure maps α P and α Q likewise must agree as they agreewith the evaluation of Φ( P ) = Φ( Q ) at corresponding morphisms in (cid:101) Ω op . It follows that Φ isinjective.On the other hand, given any X ∈ ( S (cid:101) Ω op ) strict , we can construct a B O -algebra P X by setting P X ( n ) = X ( C n ) with a symmetric group action induced by the image under X of the isomorphismsof C n in (cid:101) Ω . The B O -algebra structure maps of P X are defined using the above identification ofthe spaces B O ( n ; m , . . . , m k ) with morphism spaces in (cid:101) Ω . The fact that X is a functor will thengive that P X is a B O –algebra. Thus the functor Φ is surjective. (cid:3) Rectifying homotopy dendroidal spaces.
We have just seen that B O -algebras correspondto homotopy dendroidal spaces satisfying the strict Segal condition. In this section we will showhow to produce, from a B O -algebra, an actual dendroidal space satisfying the weak Segal condition.To do this we will use some elementary facts about the homotopy theory of diagram categories inthe form of Quillen model categories.A commutative diagram in a topological category S is a functor from a discrete category K to S . A homotopy commutative diagram can be similarly described as a functor from a topologicalcategory (cid:101) K to S , with the homotopies encoded as paths in the spaces of morphisms. In this language,a homotopy commutative diagram X : (cid:101) K → S can be rectified , or strictified , to a functor X (cid:48) : K → S precisely when there is an equivalence p : (cid:101) K → K . We briefly recall this rectification of diagrams,
N INFINITY OPERAD OF NORMALIZED CACTI 21 which was used by Segal in [31], and treated in great generality by Dwyer and Kan [13]; see also [34,Sec 2] for a detailed account of what we will use here. Our examples will be K = Ω with (cid:101) K = (cid:101) Ω .Let p : (cid:101) K → K be a functor between categories enriched over topological spaces. There is aninduced functor p ∗ : S K −→ S (cid:101) K defined by precomposition with p . The homotopy left Kan extension defines also a functor p ∗ : S (cid:101) K −→ S K that can be explicitly given as follows: given a diagram Y ∈ S (cid:101) K , its evaluation at an object d of K is the realization of a simplicial space with space of k –simplices( p ∗ Y ( d )) k = (cid:97) c ,...,c k ∈ ob ( (cid:101) K ) Y ( c ) × Hom (cid:101) K ( c , c ) × · · · × Hom (cid:101) K ( c k − , c k ) × Hom K ( c k , d ) . Lemma 4.9. [34, Proposition 2.1]
Let p : (cid:101) K → K be a functor inducing a homotopy equivalence ofmorphism spaces, and let Y : (cid:101) K → S be a diagram, with p ∗ Y : K → S its rectification as definedabove. Then there exists a zig-zag of natural transformations p ∗ p ∗ Y ← p ∗ p ∗ Y → Y , which inducesa homotopy equivalence on objects: p ∗ p ∗ Y ( d ) (cid:39) p ∗ p ∗ Y ( d ) (cid:39) Y ( d ) . In the statement, p ∗ p ∗ Y is an explicit functor from (cid:101) K to S associated to Y given by a two-sidedbar construction (details in the proof of [34, Proposition 2.1]).Proposition 4.6 states that the functor p : (cid:101) Ω → Ω induces a homotopy equivalence of mor-phism spaces. Below, we apply Lemma 4.9 to describe Segal dendroidal spaces which arise as therectification of a homotopy dendroidal Segal space Y ∈ S (cid:101) Ω op .For any small category K , the category of diagrams K op → S admits a projective model structure in which weak equivalences and fibrations are defined entrywise [19, 11.6.1]. In particular, there isa projective model category structure on the category of reduced dendroidal spaces [3, Proposition3.10]. Similarly, there is a projective model category structure on the category of reduced homotopydendroidal spaces. Moreover, an application of [22, Proposition A.3.3.7] implies that the homotopyleft Kan extension p ∗ is the left Quillen functor in a Quillen equivalence S (cid:101) Ω op S Ω op . p ∗ p ∗ We note that a fibrant diagram in either S (cid:101) Ω op or S Ω op is, in particular, entrywise fibrant. When Y is a reduced homotopy dendroidal space and fibrant, then one could identify the limit definingthe Segal map with the homotopy limit. In this case, the Segal map (cid:101) χ can be written Y ( T ) (cid:81) v ∈ V ( T ) Y ( C v ) . (cid:101) χ Proposition 4.10.
Let Y ∈ S (cid:101) Ω op be a fibrant reduced homotopy dendroidal space. Then the recti-fication p ∗ Y ∈ S Ω op is a reduced dendroidal space, and the fibrant replacement ( p ∗ Y ) f satisfies theweak Segal condition if and only if Y does.Proof. Note first that, because p is the identity on objects, we have p ∗ X ( T ) = X ( T ) for any X : Ω op → S and any T ∈ Obj (Ω) ≡ Obj ( (cid:101) Ω ). It follows that p ∗ Y ( η ) (cid:39) ∗ if Y ( η ) (cid:39) ∗ as p ∗ Y ( η ) = p ∗ p ∗ Y ( η ) (cid:39) Y ( η ) (cid:39) ∗ .We are left to show that the Segal map a is weak equivalence for every T (cid:54) = η for p ∗ Y : Ω op → S if and only if it is the case for the original functor Y : (cid:101) Ω op → S . Recall that the Segal maps for Y and p ∗ Y are the maps Y ( T ) (cid:101) χ −→ lim Sk ( T ) op Y ( C v ) p ∗ Y ( T ) χ −→ lim Sk ( T ) op p ∗ Y ( C v )both of which are induced by corolla and edge inclusions in (cid:101) Ω and Ω, respectively. Now, p takesthe map (cid:101) χ , and each map ι e and ι v in (cid:101) Ω used to define the limit, to the corresponding map in Ωused to define χ . Using that p is the identity on objects, for any X : Ω op → S , we have X ( T ) χ (cid:47) (cid:47) lim Sk ( T ) op X ( C v ) p ∗ X ( T ) (cid:101) χ (cid:47) (cid:47) lim Sk ( T ) op p ∗ X ( C v )in which the two horizontal maps describe the exact same map in S .Since p ∗ is the left adjoint in our Quillen pair, it may not the be the case that p ∗ Y is fibrant.However, we do know that p ∗ ( p ∗ Y ) f is fibrant, where ( p ∗ Y ) f denotes the fibrant replacement of p ∗ Y in S Ω op . Similarly, we let ( p ∗ p ∗ Y ) f denote the fibrant replacement of the diagram p ∗ p ∗ Y . Sincelimits commute with homotopy equivalences whenever our diagram is fibrant, the natural equiva-lences of functors p ∗ p ∗ Y ← p ∗ p ∗ Y → Y of Lemma 4.9 give us the vertical homotopy equivalencesin the following commuting diagram in S p ∗ ( p ∗ Y ) f ( T ) (cid:81) v ∈ V ( T ) p ∗ ( p ∗ Y ) f ( C v )( p ∗ p ∗ Y ) f ( T ) (cid:81) v ∈ V ( T ) ( p ∗ p ∗ Y ) f ( C v ) Y ( T ) (cid:81) v ∈ V ( T ) Y ( C v ) . (cid:101) χ (cid:101) χ (cid:39)(cid:39) (cid:39)(cid:39) (cid:101) χ Using the previous remark in the case X = p ∗ Y identifies the top line of the diagram with theSegal map for p ∗ Y . It thus follows that p ∗ Y satisfies the weak Segal condition (i.e. the top map isa weak equivalence) if, and only if, Y satisfies the weak Segal condition (i.e. the bottom map is aweak equivalence). (cid:3) We are now ready to prove Theorem B from the introduction.
Proof of Theorem B.
Let P be a B O -algebra. Applying the functor Φ of Theorem 4.8, we obtaina homotopy dendroidal space X := Φ( P ) ∈ S (cid:101) Ω op , which we know, by the theorem, is a strictlyreduced homotopy dendroidal space satisfying the strict Segal condition. If X is not fibrant, wetake a fibrant replacement ( X ) f .Set Y := ( p ∗ X ) f = ( p ∗ Φ( P )) f ∈ S Ω op which, by Lemma 4.9, has the property that Y ( C w ) =( p ∗ Φ( P )) f ( C w ) (cid:39) Φ( P )( C w ) = P ( | w | ) and the value of Y on inner face maps identifies under thesehomotopy equivalences with the value of X on inner face maps, and hence identifies with the B O –algebra composition. We can now apply Proposition 4.10 to X to conclude that Y is a reduceddendroidal space that satisfies the weak Segal condition. (cid:3) Remark 4.11.
Since the dendroidal category is a generalized Reedy category [2, Example 1.6],there is also a Reedy model structure on the category of reduced dendroidal spaces. Proposition3.3 of [3] says that the identity functor induces a Quillen equivalence between the Reedy modelstructure and the projective model structures on reduced dendroidal spaces.
N INFINITY OPERAD OF NORMALIZED CACTI 23
We use the projective model structure here because, for our purposes, it is not necessary to showthat the category (cid:101) Ω is an enriched generalized Reedy category. If one wished to do so, one wouldneed to put an enriched generalized Reedy model structure on (cid:101) Ω and then repeat the argument inProposition 4.10 with the appropriate Reedy fibrant objects.5. Normalized Cacti as an infinity operad
The first goal of this section is to define an operad
M S + and show that, despite not being anoperad itself, normalized cacti and their composition can be described as elements and compositionsinside M S + . In Section 5.2, we will use M S + to show that normalized cacti and the normalizedcomposition extends to define a B O -algebra structure. Using the results of Sections 3 and 4, thisimplies that we have an explicit construction of an ∞ -operad with underlying sequence the spaces C act ( n ).A cactus is a configuration of circles of various lengths attached to each other in a treelike fashion.In the original definition by Voronov [33, Section 2.7], there is a global basepoint associated to the“outside circle” of the cactus, as well as a basepoint for each circle (or lobe ). A spineless cactus isa variant introduced by Kaufmann [20, Section 2.3], where the basepoint of each lobe is its closestpoint to the global basepoint along the outside circle. See Figure 14 for an example. The space2 11 43 1
13 12
12 1214 23 Figure 14.
Cactus with 8 lobes, its outside circle indicated by the dotted line.of all spineless cacti with k lobes is denoted C act( k ). The symmetric group acts on this spaceby permuting the labels of the lobes. The symmetric sequence C act = {C act( k ) } k ≥ is given acomposition ◦ i : C act( k ) × C act( j ) → C act( k + j − i th lobe of the first cactus and aligning itsglobal basepoint with the basepoint of the i th lobe. The insertion is done by rescaling the secondcactus so that its total length is equal to the length of the i th lobe of the first cactus, then identifyingthe outside circle of the second cactus with the i th lobe of the first cactus. This composition makes C act into an operad, which is equivalent to the little 2-discs operad [20, Section 3.2.1]. A rigorousdefinition of this composition requires close attention to subtleties and we refer to [20, Section 2]for precise definitions.The space of normalized cacti C act ( k ) ⊂ C act( k ) is the subspace of spineless cacti whose lobesall have length equal to 1 ([20, Definition 2.3.1]). They form a symmetric sequence C act = {C act ( k ) } k ≥ . Composition of normalized cacti(5.1) ◦ i : C act ( k ) × C act ( j ) → C act ( k + j − , is defined by reparameterizing the i th lobe of a cactus x ∈ C act ( k ) to have length j , then identifyingthis lobe with the outer circle of the second cactus y ∈ C act ( j ) and aligning their basepoints. Incontrast to C act, the i th lobe of the first cactus is scaled instead of scaling the second cactus to ◦ =
12 12
12 11 2 Figure 15.
A composition of normalized cacti.the length of the i th lobe. See Figure 15 for an example. This composition is not associative [20,Remark 2.3.19], as illustrated in Figure 16. Thus C act is not an operad. ◦ =1 ◦
11 2 ◦ =1 ◦
11 2
11 2 3 Figure 16.
Non-associativity in C act Remark 5.1 (Composition in the graph cobordism category) . This composition of normalized cactiis highly relevant to the graph model of the cobordism category of Riemann surfaces mentioned inthe introduction of the paper. To model the gluing of cobordisms, we use graphs to representsurfaces with potentially many incoming and outgoing boundary components. Normalized cactiare a simple case of this model, representing surfaces of genus zero with potentially many inputsbut always just one output. Two surfaces are glued by attaching the incoming boundaries of thefirst surface to the outgoing boundaries of the second. According to [15] (see also [14, TheoremA]), we may assume that all incoming boundaries of a surface are disjoint embedded circles in thecorresponding graph (like the lobes of the cactus, if they where pulled apart a little bit). Sincethese boundary circles are disjoint in the graph, they can be scaled independently to each matchthe length of an outgoing boundary in the graph of the second surface, just like scaling the i th lobeof the first cactus in C act composition. There is no obvious way to define a “ C act-like” compositionfor such more general graphs, because the outgoing circles of the second surface cannot be assumedto be disjoint, and hence cannot be scaled independently to the appropriate length. (See [14, Section3.3] for more details about this gluing of fat graphs.)5.1. An operad
M S + that contains C act . In their proof of the Deligne conjecture, McClureand Smith [24, 25] introduced an operad
M S equivalent the little 2-discs operad. Later, Salvatore[30, Section 4] used similar methods to show directly that the operad
M S is equivalent to the non-normalized cactus operad C act. Here we will define a variant of M S called
M S + , and, following[30], start by showing that it is an operad by proving that it embeds in CoEnd( S ). We then showthat normalized cacti are a subspace of the underlying symmetric sequence of M S + and that theircomposition can be written in terms of compositions in M S + . The operad MS is denoted C (cid:48) in [24, Section 5]. N INFINITY OPERAD OF NORMALIZED CACTI 25
The space of operations
M S + ( k ) is built from a space F ( k ), which we will show is homeomorphicto C act ( k ). In fact, we can think of an element of F ( k ) as the outer circle of a cactus. Definition 5.2. [30, Definition 4.1] Let S = [0 , / ∼ F ( k ) as the space of partitions x = ( I ( x ) , . . . , I k ( x )) of S into closed 1-manifolds I j ( x ) ⊂ S , each of which have total length k , with pairwise disjoint interiors, and such that( ∗ ) there does not exist a cyclically ordered 4-tuple ( z ; z ; z ; z ) ∈ S with z , z ∈ ˚ I j ( x ) and z , z ∈ ˚ I i ( x ), for j (cid:54) = i .For an example, see Figure 17(a). The topology of F ( k ) is induced by the metric measuring thesize of the overlap between partitions: for x, y ∈ F ( k ), d ( x, y ) = 1 − (cid:80) kj =1 (cid:96) ( I j ( x ) ∩ I j ( y )) for (cid:96) thelength function on submanifolds of S .The symmetric group Σ k acts on F ( k ) by reindexing the labels of the 1-manifolds. I ( x )
16 718 13181518 I ( x ) I ( x ) (a) x ∈ F (3). (b) Maps c ix . Figure 17.
Element of x ∈ F (3) and associated projections Definition 5.3.
Given an element x ∈ F ( k ), we associate to each I j ( x ) a projection map c jx : S → S that takes the quotient of S under the identification of all the points in the same pathcomponent of S \ ˚ I j and then scales this circle by a factor of k . See Figure 17(b) for an example.The cactus map c x : S → ( S ) k is the collection of maps c x := ( c x , . . . , c kx ). Then there is a map c : F ( k ) M ap ( S , ( S ) k ) x c x = ( c x , . . . , c kx ) : S → ( S ) k . For any x ∈ F ( k ), we also use x to denote the configuration of circles in the image of the cactusmap c x : S → ( S ) k . Condition ( ∗ ) in Definition 5.2 guarantees that this configuration is treelike,as it forces the submanifolds I j ( x ) to be nested. The global basepoint of x is the image of thebasepoint of S and a planar structure is induced by the orientation of the source S (see [30,Definition 4.2]). Since each part of a partition x ∈ F ( k ) has equal length, x is a normalized cactusas shown in Figure 18. This is the sketch of the proof for the next lemma. Lemma 5.4. [30, Section 4]
For each k ≥ , the space F ( k ) is homeomorphic to C act ( k ) . Recall the coendomorphism operad CoEnd( S ) from Example 2.2, whose underlying symmetricsequence is a collection of CoEnd( k )( S ) := Map( S , ( S ) k ). We use the map c : C act ( k ) ∼ = F ( k ) (cid:44) → Map( S , ( S ) k ) = CoEnd( S )( k )to define an embedding of symmetric sequences. Lemma 5.5.
The map c : F ( k ) → M ap ( S , ( S ) k ) is a topological embedding.Proof. We first check injectivity. Given a map c x = ( c x , . . . , c kx ) in the image of c , we can completelydetermine x ∈ F ( k ). We know that each c jx is a “step-map” with linear of slope k over its non-constant parts, by the definition of c . (See Figure 17(b).) Then I j ( x ) is precisely the subset of points of S where the derivative ( c jx ) (cid:48) equals k . Continuity of c follows from the fact that the topology inthe mapping space can be defined using the convergence metric, using likewise the metric on S . (cid:3) This embedding of symmetric sequences does not extend to an embedding of operads. As alreadymentioned, C act is not an operad and one can check that the image of c is not a suboperad ofCoEnd( S ). Indeed, if we compose two elements in CoEnd( S ) that came from elements of F , theircomposition will not be in the image of any F ( k ) because all elements in the image of F ( k ) arepiecewise linear graphs of slope 0 or k , and this property is not preserved by the composition inCoEnd( S ). I ( x )
16 718 13181518 I ( x ) I ( x )
12 1223 13
112 3
Figure 18.
An element x of F (3) and the corresponding normalized cactus c x .Here we define the symmetric sequence M S + = { M S + ( k ) } k ≥ , which is built from F ( k ) anda collection M on + ( I, ∂I ) of scaling maps on the interval I . It has the important property that C act ( k ) ⊂ M S + ( k ) for each k ≥ Definition 5.6 ( M S + as a symmetric sequence) . For each k ≥
0, we define the space
M S + ( k ) as M S + (0) = ∗ M S + ( k ) = F ( k ) × M on + ( I, ∂I )where
M on + ( I, ∂I ) is the space of strictly monotone self-maps of I that restrict to the identity on ∂I . We consider M on + ( I, ∂I ) as a subspace of the space of self-maps of S = I/∂I . For each k ,there is an action of the symmetric group Σ k on M S + ( k ) by the reindexing of the labels of the1-manifolds in F ( k ). Remark 5.7.
The operad
M S that appears in [24, 25, 30] has an underlying symmetric sequenceobtained by replacing
M on + ( I, ∂I ) by the larger space
M on ( I, ∂I ) of weakly monotone maps.The inclusion
M S + (cid:44) → M S is a homotopy equivalence as both
M on ( I, ∂I ) and
M on + ( I, ∂I ) arecontractible (in fact, they are both convex).In order to show that
M S + is an operad, we start by showing that each space of operations M S + ( k ) embeds in CoEnd( S )( k ). We also check that the operad composition of CoEnd( S )preserves the image of M S + , and hence is a suitable composition for M S + , thus making M S + asuboperad of CoEnd( S ). Proposition 5.8.
There is a topological embedding φ : M S + ( k ) → CoEnd( S )( k ) that sends ( x, f ) ∈ M S + ( k ) to the composite S f −→ S c x −→ ( S ) k where c x is the cactus map as in Definition 5.3. A version of Proposition 5.8 is stated for the operad
M S in [30, Section 4]. As we rely heavilyon this result we give more complete details here.
Proof.
The fact that φ is continuous follows from Lemma 5.5, so we are left to check that φ isinjective. Let x ∈ F ( k ). Recall that the map c x = ( c x , . . . , c kx ) : S → ( S ) k is a collection of“step-maps” of linear of slope k over its non-constant parts. Each map c jx : S → S identifies N INFINITY OPERAD OF NORMALIZED CACTI 27 points in the same path component of S \ ˚ I j ( x ) and linearly takes I j ( x ) (of length 1 /k ) to a circleof circumference 1. So, these maps satisfy that1 k k (cid:88) j =1 c jx = Id S . In particular, this means that if c x ◦ f = c y ◦ g , then f = ( 1 k k (cid:88) j =1 c jx ) ◦ f = 1 k k (cid:88) j =1 ( c jx ◦ f ) = 1 k k (cid:88) j =1 ( c jx ◦ g ) = ( 1 k k (cid:88) j =1 c jy ) ◦ g = g. Moreover, as f, g are strictly monotone and hence invertible, for each j = 1 , . . . , k , c jx = ( c jx ◦ f ) ◦ f − = ( c jy ◦ g ) ◦ f − = ( c jy ◦ g ) ◦ g − = c jy . This shows that c x = c y and therefore the map is injective. (cid:3) Proposition 5.8 shows that
M S + is a symmetric subsequence of CoEnd and this next lemmashows that the operad structure maps of CoEnd preserve this structure. Lemma 5.9.
The operad structure maps of
CoEnd preserve the symmetric subsequence
M S + .Proof. It suffices to consider the composition operations ◦ i in CoEnd as defined in Example 2.2.Given ( x, f ) and ( y, g ) in M S + , we need to check that the composition(5.2) S f −→ S c x −→ ( S ) k × g × −−−−→ ( S ) k × c y × −−−−−→ ( S ) j + k − is in the image of M S + , where 1 × g × g acts only on the i th circle. Forthis, we will show two things:(i) (1 × g × ◦ c x = c ˜ x ◦ ˜ g , for some ˜ g ∈ M on + ( I, ∂I ) and ˜ x ∈ F ( k ),(ii) (1 × c y × ◦ c x = c z ◦ h x,y for some h x,y ∈ M on + ( I, ∂I ) and z ∈ F ( j + k − × g ×
1) acts only on the i th circle, so in the composition with c x it only affects points in I i ( x ). Recall that we identify S with I/∂I . If I i ( x ) = J (cid:116) · · · (cid:116) J r witheach J s a subinterval of [0 ,
1] and I i ( x ) of total length k , then define I i (˜ x ) = ˜ J (cid:116) · · · (cid:116) ˜ J r for ˜ J s aninterval of length k (cid:96) ( g ( J s )). We obtain ˜ x ∈ F ( k ) from x by replacing each subinterval J s by theinterval ˜ J s and shifting each path component of [0 , \ ˚ I r ( x ) accordingly. This makes sense as, byconstruction, the total length of I i (˜ x ) is again k . Also ˜ g is defined as the canonical identificationof I r ( x ) with I r (˜ x ) for all r ∈ { , . . . , k } . See Figure 19 for an example.0 1 J J c ix ˜ gx J ˜ J c i ˜ x ˜ x J J g J ˜ J Figure 19.
An example of the commutative diagram g ◦ c ix = c i ˜ x ◦ ˜ g For statement (ii), we consider a composition (1 × c y × ◦ c x : S → ( S ) j + k − with c y on the i th position. Such a composition maps the r th partition I r ( x ), for r (cid:54) = i , to the r th (if r < i ) or( r + k − r > i ) component in the target by a slope k map, while I i ( x ) is mapped by slope jk maps to the remaining components. Let h x,y : S → S be the rescaling map that scales each I r ( x ) by a factor kj + k − for r (cid:54) = i , and I i ( x ) by a factor jkj + k − . Then the image under h x,y of each I r ( x ) will be of size j + k − for r (cid:54) = i , while I i ( x ) will have image of total size jj + k − . Note that this gives a well-defined map in M on + ( I, ∂I ) as the sum of the length of the images h x,y ( I r ( x )) is( k − j + k − + jj + k − = 1. Subdividing the image under h x,y of I i ( x ) into j parts as prescribedby y , together with the images of the other I r ( x )’s, then defines z ∈ F ( j + k − × c y × ◦ c x = c z ◦ h x,y holds by construction. (cid:3) Therefore we have shown that
M S + is a suboperad of CoEnd( S ) via the embedding φ inProposition 5.8. Definition 5.10 ( M S + as an operad) . The symmetric sequence
M S + = { M S + ( k ) } k ∈ N becomesan operad with composition(5.3) ( x, f ) • i ( y, g ) := φ − ( φ ( x, f ) ◦ i φ ( y, g ))where ◦ i is the composition in CoEnd( S ) defined in (5.2), and the pre-image exists as a consequenceof Lemma 5.9.We will often use scaling maps in M on + ( I, ∂I ) to encode the scaling of lobes in the compositionof normalized cacti. Given a partition x = ( I ( x ) , . . . , I k ( x )) ∈ F ( k ) ∼ = C act ( k ), and naturalnumbers m , . . . , m k ≥
0, we let(5.4) g = g ( x ; m , . . . , m k ) : S −→ S be the element of M on + ( I, ∂I ) that scales I j ( x ) by the factor km j m + ··· + m k , 1 ≤ j ≤ k . Each I j ( x )has total length k , so the image of I j ( x ) will have length m j m + ··· + m k for each 1 ≤ j ≤ k . Note that g ( x ; 1 , . . . ,
1) = id is just the identity map on S . I ( x )
16 718 13181518 I ( x ) I ( x ) g ( x ;2 , , −−−−−−→ (cid:96) ( I ( x ))= (cid:96) ( I ( x ))= (cid:96) ( I ( x ))= (cid:96) ( g ( I ( x )))= (cid:96) ( g ( I ( x )))= (cid:96) ( g ( I ( x )))= g ( I ( x ))
14 512 23 34 g ( I ( x )) g ( I ( x )) Figure 20.
Map g ( x ; 2 , , x from Figure 18.We will now show that the ◦ i -compositions of normalized cacti from (5.1), ◦ i : C act ( k ) × C act ( j ) C act ( k + j − , and, more generally, the C act –composition maps γ C act : C act ( k ) × C act ( m ) × · · · × C act ( m k ) −→ C act (Σ ki =1 m i ) , are restrictions of the corresponding compositions of appropriately chosen elements of M S + .For a collection of cacti x ∈ C act ( k ) and y j ∈ C act ( m j ), 1 ≤ j ≤ k , the quasi-operad compo-sition γ C act ( x ; y , . . . , y k ) scales each lobe of x so that the i th lobe now has length m i , and theninserts (without any further scaling) each y i in place of the i th scaled lobe.Under the homeomorphism C act ( k ) ∼ = F ( k ) in Lemma 5.4, a normalized cactus x ∈ C act ( k )precisely corresponds to a partition x ∈ F ( k ) of [0 ,
1] into k submanifolds I j ( x ) of equal lengths k ,satisfying the conditions of Definition 5.2. Since the i th lobe of x corresponds to the submanifold I i ( x ), there is a scaling by k in the identification C act ( k ) to take a lobe of length 1 to I i ( x ). In N INFINITY OPERAD OF NORMALIZED CACTI 29 the next lemma we will use x ∈ F ( k ) and y j ∈ F ( m j ) for 1 ≤ j ≤ k to represent a sequence of cactiin C act ( k ) and C act ( m j ) respectively. We will still denote the composition by ◦ i or γ C act . Lemma 5.11.
Let γ MS + and γ C act denote the (quasi-)operad compositions in M S + and C act ,respectively. Then for x ∈ F ( k ) and y j ∈ F ( m j ) , with ≤ j ≤ k , we have γ MS + (( x, g − ( x ; m , . . . , m k )); ( y , id ) , . . . , ( y k , id )) = ( γ C act ( x ; y , . . . , y k ) , id ) in M S + ( (cid:80) m j ) . In particular, ( x, g − ( x ; 1 , . . . , m i , . . . , • i ( y i , id) = ( x ◦ i y i , id) where • i denotes the composition (5.3) of M S + and ◦ i represents the composition (5.1) of C act .Proof. Let F ( m ,...,m k ) ( k ) denote the scaled version of F ( k ) where the i th partition, I i , now haslength m i instead of k . In particular, we have that F ( k ) = F ( k ,..., k )( k ) and C act ( k ) = F (1 ,..., ( k ).Thus the homeomorphism C act ( k ) ∼ = F ( k ) implies that the composition γ C act on C act can beinterpreted as a map in F , written as F ( k ) × ( F ( m ) × · · · × F ( m k )) S −−→ F ( m ,...,m k ) ( k ) × (cid:0) F (1 ,..., ( m ) × · · · × F (1 ,..., ( m k ) (cid:1) γ −−→ F (1 ,..., ( (cid:88) i m i ) N −−→ F (cid:80) mi ,..., (cid:80) mi ( (cid:88) i m i ) = F ( (cid:88) i m i )where S and N are scaling and normalising maps and the map labelled γ is the insertion map.Our task is to write the composition γ C act in terms of the operad M S + . To do this, we will usescaling maps inside M on + ( I, ∂I ). More precisely, define a map F ( k ) × ( F ( m ) × · · · × F ( m k )) G −−−−→ M S + ( k ) × ( M S + ( m ) × · · · × M S + ( m k ))that takes ( x ; y , . . . , y k ) to (( x, g − ); ( y , id) , . . . , ( y k , id)) where g = g ( x ; m , . . . , m k ) ∈ M on + ( I, ∂I )is the map in equation (5.4). The statement we want to prove is that ( γ C act , id) can be written asthe composition F ( k ) × ( F ( m ) × · · · × F ( m k )) G −−−−→ M S + ( k ) × ( M S + ( m ) × · · · × M S + ( m k )) γ MS + −−−−→ M S + ( (cid:88) i m i ) . In particular, we claim that the resulting element of
M S + ( (cid:80) i m i ) is in the image of C act , that is,of the form ( z, id).To prove this, we start by expressing G as a composition G (cid:48) ◦ S , where S is the scaling map inthe description γ C act = N ◦ γ ◦ S given above and G (cid:48) : F ( m ,...,m k ) ( k ) × ( F (1 ,..., ( m ) × · · · × F (1 ,..., ( m k )) −→ M S + ( k ) × ( M S + ( m ) × · · · × M S + ( m k ))is the map that takes a tuple ( x ; y , . . . , y k ) to the tuple (( N ( x ) , g − ); ( N ( y ) , id) , . . . , ( N ( y k ) , id)),with N the normalization also as above. In order to compare γ C act = N ◦ γ ◦ S with γ MS + ◦ G (cid:48) ◦ S ,we have to show that the diagram F ( m ,...,m k ) ( k ) × ( F (1 ,..., ( m ) × · · · × F (1 ,..., ( m k )) G (cid:48) (cid:15) (cid:15) γ (cid:47) (cid:47) F (1 ,..., ( (cid:80) i m i ) ( N,id ) (cid:15) (cid:15) M S + ( k ) × ( M S + ( m ) × · · · × M S + ( m k )) γ MS + (cid:47) (cid:47) M S + ( (cid:80) i m i ) commutes, where the right vertical map takes z to ( N ( z ) , id). To see this, let ( x ; y , . . . , y k ) be anelement in the top left corner of the square. Its image γ MS + ◦ G (cid:48) ( x ; y , . . . , y k ) along the bottomcomposition is the element of M S + ( m + · · · + m k ) given by the following composition: S g − −−→ S c x −→ ( S ) k c y ×···× c yk −−−−−−−−→ ( S ) m + ··· + m k since we consider M S + ( m + · · · + m k ) as a subspace of CoEnd( m + · · · + m k ) and use thecomposition in (5.3). The j th factor S in the above ( S ) k is subdivided into submanifolds I s ( y j )according to c y j .Their inverse image g ◦ ( c x ) − j ( I s ( y j )) in the source S of the composition is thus taken to the( m + · · · + m j − + s )th factor S in ( S ) m + ··· + m k , being first scaled by a factor (cid:80) m i km j (using g − ),then by a factor k (via the j th component of c x ) and finally by a factor m j (via the s th componentof c y j ). So in total the composition takes g ◦ ( c x ) − j ( I s ( y j )) to S = I/∂I linearly by a factor (cid:80) m i ,and is constant on the connected components of the complement of g ◦ ( c x ) − j ( I s ( y j )). In particular, g ◦ ( c x ) − j ( I s ( y j )) has length (cid:80) m i , which is independent of j and s . Thus we see that the resultingelement does indeed live in the image of C act . As the scaling is always independent of s and j , theproportion of each g ◦ ( c x ) − j ( I s ( y j )) inside the source S is always as dictated by c y j , with each g − ( I j ( x )) having total length m j (cid:80) i m i . Hence the composition is the same as following the other sideof the square, which inserts I s ( y j ) inside I j ( x ), scaling each I j ( x ) to length m j , then scales it by (cid:80) m i to be inside M S + . (cid:3) Therefore up to scaling in accordance with the homeomorphism C act ∼ = F in Lemma 5.4, wehave shown that both C act and its composition are contained within the operad M S + , but not asa suboperad.5.2. C act is a B O –algebra. Here we will construct an action of B O on normalized cacti usingthe fact that C act ⊂ M S + , and that its composition can also be described in terms of the com-position in M S + . In Theorem 5.12, we show that the quasi-operad structure on normalized cacti C act = {C act ( k ) } k ≥ is part of a B O -algebra structure. In Corollary 5.13, we conclude that C act determines a dendroidal Segal space X ∈ S Ω op with X ( C v ) = C act ( | v | ).Recall from Section 3.3 that a B O –algebra is a symmetric sequence with ◦ i –operations that arehomotopy associative up to all higher homotopies. Elements from B O are ( T, σ, τ, B, t ) where T isa planar tree equipped with bijections σ : | V ( T ) | → V ( T ) and τ : | L ( T ) | → L ( T ), and ( B, t ) is aweighted bracketing of T .Let(5.5) R : M S + −→ F denote the projection map that forgets the M on + ( I, ∂I ) component, R ( x, f ) = x . This is a mapof symmetric sequences. If we think of elements of M S + as cacti, the map R has the effect of renormalizing , that is, rescaling the lobes so that they all have the same length. Since M S + is anoperad, it is an O -algebra. The O -action λ MS + : O ( k ; m , . . . , m k ) × M S + ( m ) × . . . × M S + ( m k ) −→ M S + ( (cid:88) i m i )takes a sequence of elements(( T, σ, τ ) , ( x , f ) , . . . , ( x k , f k )) ∈ O ( k ; m , . . . , m k ) × M S + ( m ) × . . . × M S + ( m k )to the composition of the elements ( x , f ) , . . . , ( x k , f k ) according to γ MS + , in the order prescribedby the labeled tree ( T, σ ), acting by the permutation τ on the resulting element of M S + ( (cid:80) i m i ).This composition can be depicted by labeling the i th vertex of ( T, σ, τ ) by ( x i , f i ) ∈ M S + ( m i ).This action is compatible with the composition in O because M S + is an operad. We will usethis existing O -algebra structure to define the B O -algebra structure of C act by representing the C act -composition by R ◦ λ MS + . N INFINITY OPERAD OF NORMALIZED CACTI 31
Theorem 5.12.
The C act -composition (5.1) is part of a B O –algebra structure.Proof. In order to construct a B O –algebra structure on the sequence {C act ( n ) } n ≥ , we want todefine a map B O ( k ; m , . . . , m k ) × C act ( m ) × . . . × C act ( m k ) −→ C act ( (cid:88) i m i )that restricts to the Σ n –action on C act ( n ), which permutes the labels on the lobes, and its al-ready defined ◦ i –compositions. Using the homeomorphism C act ∼ = F from Lemma 5.4, we willequivalently construct a map λ : B O ( k ; m , . . . , m k ) × F ( m ) × . . . × F ( m k ) −→ F ( (cid:88) i m i ) . Firstly, the Σ n -action on the n –space of a B O –algebra is encoded by the labeled corollas( C n , , τ, ∅ , ∅ ) ∈ B O ( n ; n ) ∼ = O ( n ; n ) ∼ = Σ n , where τ labels the leaves of the corollas C n , which are thought of as elements of the symmetric groupΣ n , and the identity corresponds to the planar ordering. This fixes the action of such elements of B O as we have already fixed the Σ n –action on C act ( n ).The C act ◦ i -composition is encoded in B O by the trees with exactly two vertices, one attachedto the i th incoming edge of the other. These trees admit no non-trivial bracketings so such elementsof B O have the form ( T, σ, τ, ∅ , ∅ ) ∈ B O ( m + n − m, n )where σ labels the two vertices of T and τ labels its n + m − C act dictates the action of such elements of B O : ( T, σ, τ, ∅ , ∅ )acts on x ∈ C act ( m ) and x ∈ C act ( n ) by taking their ◦ i -composition, as dictated by the tree,and then acting by τ on the lobes of the resulting element of C act ( m + n − (cid:55)−→
14 5325 34 21 ,
Figure 21.
Example of the B O -action on C act .By Lemma 5.11, this C act -composition can be defined in terms of the M S + composition: R ◦ λ MS + (cid:0) ( T, σ, τ ); ( x , g ) , ( x , g ) (cid:1) where R is the projection map (5.5), and g = g ( x , , . . . , k i , . . . ,
1) and g = g ( x , , . . . , l j , . . . , k i = n and l j = 1 if first vertex is the bottom vertex andthe second is attached to its i th input, or k i = 1 and l j = m if the second vertex is the bot-tom vertex with the first attached to its j th input. Let ( y T , f T ) ∈ M S + denote the element λ MS + (cid:0) ( T, σ, τ ); ( x , g ) , ( x , g ) (cid:1) .We will now extend this definition of the B O –action of trees with at most two vertices to an actionof the whole operad. We start by defining an explicit expression for the action of bracketings oftrees ( T, σ, τ, B,
1) with brackets of weight 1, and afterwards extend this definition to the remainingelements of B O , whose brackets have weight strictly between 0 and 1.Let T = ( T, σ, τ, B,
1) be an element of B O ( n ; m , . . . , m k ) with all brackets of weight 1, and let x i ∈ F ( m i ) ∼ = C act ( m i ) for each 1 ≤ i ≤ k . We first construct scaling maps g ∈ M on + ( I, ∂I ) asin (5.4). Recall from Definition 3.1 that a bracketing B = { S j } j ∈ J consists of large, nested propersubtrees of T . Here we allow B to be empty. Recall that σ orders the vertices of T . For a fixed i ∈ { , . . . , k } , let S ∈ B be the smallest bracket that contains the vertex σ ( i ), allowing S = T if there are no such bracket. Recall that in ( σ ( i )) is the set of incoming edges of σ ( i ), and L ( S ) is theset of leaves of the bracket S . We define a map(5.6) ξ : in ( σ ( i )) −→ N by setting(i) ξ ( e ) = 1 if e ∈ L ( S );(ii) ξ ( e ) = | L ( S (cid:48) ) | if e is the root of a bracket S (cid:48) ⊂ S in B , with S (cid:48) ⊂ S the largest such bracket;(iii) ξ ( e ) = | w | if e ∈ iE ( S ) is not the root of any S (cid:48) ∈ B , where | w | denotes the arity of thevertex w ∈ V ( S ) for which e is the outgoing edge. T = S (cid:48) S e e e e e ξ : in ( σ (1)) → N ξ ( e ) = 1 ξ ( e ) = | L ( S ) | = 5 ξ ( e ) = 1 ξ ( e ) = | σ (7) | = 2 ξ ( e ) = 1 Figure 22.
An example of ξ .Figure 22 shows an example of the map ξ . We then set(5.7) g i := g ( x i ; ξ ( e ) , . . . , ξ ( e m i ))for e , . . . , e m the incoming edges of σ ( i ) ordered by the planar ordering of T .We define the action of B O inductively on the size of the bracketing B .If B is empty, then we define λ ( T ; x , . . . , x k ) := R ◦ λ MS + (( T, σ, τ ); ( x , g ) , . . . , ( x k , g k ))and use ( y T , f T ) ∈ M S + to denote the image of λ MS + . Note that when k = 1 or 2, this is the sameas the B O -structure already defined above.If B is not empty, then we define additional scaling maps for each bracket, using the inductivehypothesis that the action has already been defined action on subtrees with fewer brackets.Let T (cid:48) be the tree obtained from T by adding a binary vertex at the root of each bracket S j ∈ B .Extend the order σ of the vertices of T to an order σ (cid:48) of vertices of T (cid:48) by setting the | J | = | B | newvertices last. An example of T (cid:48) is shown in Figure 23. We will use each additional vertex of T (cid:48) toassign a scaling map to the associated bracket.Let w j ∈ V ( T (cid:48) ) \ V ( T ) be the j th vertex of T (cid:48) not in T , according to the chosen order σ (cid:48) . Let S j ∈ B be the bracket associated to w j . Since the number of brackets of B that lie inside S j is lessthan | B | , we have an element ( y S j , f S j ) ∈ M S + defined by the inductive assumption by restricting T = ( T, σ, τ, B,
1) to the subtree S j . Considerthe tree T /S j in which all vertices in S j are identified and internal edges between them are collapsed.The tree T /S j has a vertex [ S j ] associated to the collapsed tree S j . We have an induced bracketing˜ B of T /S j from the bracketing B of T , and thus can define a map ξ j : in ([ S j ]) → N as in (5.6) byreplacing ( T, B ) by (
T /S j , ˜ B ). Then define(5.8) h j := g ( y S j ; ξ j ( e ) , . . . , ξ j ( e l )) ◦ f − S j N INFINITY OPERAD OF NORMALIZED CACTI 33 T (cid:48) = Figure 23.
An example of T (cid:48) for the bracketing of T in Figure 22for e , . . . , e l the incoming edges of [ S j ] in T /S j . We define the action of B O by setting(5.9) λ ( T ; x , . . . , x k ) := R ◦ λ MS + (cid:0) ( T (cid:48) , σ (cid:48) , τ ); ( x , g ) , . . . , ( x k , g k ) , (1 , h ) , . . . , (1 , h | B | ) (cid:1) for the rescaling maps g i and h j defined above.We claim that the formula for the action (5.9) is indeed compatible with composition of brack-etings of trees of weight 1. It is enough to check this for a ◦ i -composition in B O , so consider T = ( T , σ , τ , B ,
1) and T = ( T , σ , τ , B ,
1) in B O . We need to check that(5.10) λ ( T ; x , . . . , x i − , λ ( T ; x i , . . . , x i + l − ) , x i +1 , . . . , x k + l − ) = λ ( T ◦ i T ; x , . . . , x k + l − ) . From the above definition, we have λ ( T ; x , . . . , x i − , λ ( T ; x i , . . . , x i + l − ) , x i +1 , . . . , x k + l − ) = R ◦ λ MS + (cid:0) ( T (cid:48) , σ (cid:48) , τ ); ( x , g ) , . . . , ( x i − , g i − ) , ( y T , g i ) , ( x i + l , g i + l ) , . . . , ( x k , g k ) , (1 , h ) , . . . , (1 , h | B | ) (cid:1) for y T = R ◦ λ MS + (cid:0) ( T (cid:48) , σ (cid:48) , τ ); ( x i , g (cid:48) ) , . . . , ( x i + l − , g (cid:48) l ) , (1 , h (cid:48) ) , . . . , (1 , h (cid:48)| B | ) (cid:1) where the maps g i and h i are those associated to ( T , B ) and the maps g (cid:48) i and h (cid:48) i associated to( T , B ). In the above notation, we also have( y T , f T ) = λ MS + (cid:0) ( T (cid:48) , σ (cid:48) , τ ); ( x i , g (cid:48) i ) , . . . , ( x i + l − , g (cid:48) i + l − ) , (1 , h (cid:48) ) , . . . , (1 , h (cid:48)| B | ) (cid:1) . Note that one can change the
M on + ( I, ∂I ) component of an element of
M S + by doing a ◦ –composition in the operad. In particular,( y T , g i ) = (1 , g i ◦ f − T ) ◦ ( y T , f T )in M S + ane we can rewrite the left hand side of (5.10) as the first component of the M S + –composition λ MS + (cid:0) ( T (cid:48) , σ (cid:48) , τ ) ◦ i ( ¯ T (cid:48) , ¯ σ (cid:48) , τ ); ( x , g ) , . . . , ( x i − , g i − ) , ( x i , g (cid:48) i ) , . . . , ( x i + l − , g (cid:48) i + l − ) , (1 , h (cid:48) ) , . . . , (1 , h (cid:48)| B | ) , (1 , g i ◦ f − T ) , ( x i + l , g i + l ) , . . . , ( x k , g k ) , (1 , h ) , . . . , (1 , h | B | ) (cid:1) where ¯ T (cid:48) has an extra vertex at the bottom of the tree to encode the change of M on + ( I, ∂I )–component for the T composition. If T is large, this extra vertex corresponds exactly to the extrabracket T arising in the B O –composition, and one checks that the corresponding scaling map h defined by the formula (5.8) is precisely the map g i ◦ f − T . As the other labels of the vertices of thecomposed tree agree with those of the right hand side, we see that we recover the right hand sideof (5.10). If T is not large, then there is no such additional bracket in the B O –composition, butin this case f T = id and the left and right hand side agree directly.Recall from Remark 3.10 that we may consider B O as the geometric realization of the simplicialoperad of bracket trees. Then the above definition of λ on bracketings of weight 1 defines theaction of the vertices of B O . We finally extend this action to all bracketings of a tree T by linear interpolation on the rescaling maps g i . For a fixed tree T and a point (( B ⊂ · · · ⊂ B r ) , t ) in therealization of the poset B ( T ), let g i ( T, B j ) denote the definition of the rescaling map g i with respectto the bracketing B j on T in (5.7), and likewise for the maps f j in (5.8). We set g i = t g i ( T, B ) + ... + t r g i ( T, B r ) . This is well-defined as
M on + ( I, ∂I ) is convex. Also note that this is continuous in B O as goingto the l th face of the simplex ( B ⊂ · · · ⊂ B r ) corresponds to t l going to 0, that is, droppingthe bracket B l .Then we define ( y T , f T ) and λ ( T, σ, τ, B, t ) := R ( y T , f T ) as in (5.9) but with thisdefinition of g i instead.This defines the action of B O on C act . It is compatible under composition because the compo-sition in B O is the realization of the composition in the poset operad, and we have already checkedthe compatibility under composition there. (cid:3) Given that normalized cacti, together with the cactus composition (5.1), forms a B O -algebra wecan now use the rectification results from Proposition 4.10 to define an ∞ -operad. Corollary 5.13.
Normalized cacti define dendroidal spaces of the following two flavors:(i) There exists a reduced homotopy dendroidal space X ∈ S (cid:101) Ω op , satisfying the strict Segalcondition, such that X ( C n ) = C act ( n ) and with value on the inner face maps ∂ e given bythe C act –composition.(ii) There exists reduced dendroidal space Y ∈ S Ω op , satisfying the weak Segal condition, suchthat Y ( C n ) (cid:39) C act ( n ) and with value on the inner face maps ∂ e homotopic to the C act –composition.Proof. Theorem 5.12 shows that C act is a B O -algebra. Applying the construction from The-orem 4.8, we define a homotopy dendroidal space X := Φ( C act ) ∈ S (cid:101) Ω op . By construction,Φ( C act )( C n ) = C act ( n ), and by the theorem it is a reduced homotopy dendroidal space satisfyingthe strict Segal condition. The evaluation of Φ( C act ) on an inner edge is the ◦ i composition, asencoded by the B O -structure, which in the present case is the C act –composition by Theorem 5.12.This proves (i) in the statement.For (ii), we set Y := ( p ∗ X ) f = ( p ∗ Φ( C act )) f ∈ S Ω op to be the rectification of X , as constructedin Proposition 4.10. By Lemma 4.9, Y ( C w ) = ( p ∗ Φ( C act )) f ( C w ) (cid:39) Φ( C act )( C w ) = C act ( | w | )and the value of Y on inner face maps is identifies under these homotopy equivalences with the valueof X on inner face maps, and hence identifies with the C act –composition. The (cid:101) Ω -diagram X takesvalues in the category of topological spaces and is therefore fibrant as it is entrywise fibrant. Wecan now apply Proposition 4.10, to get that Y is reduced and satisfies the weak Segal condition. (cid:3) Appendix A. Relation between the operads B O and W O The Boardman-Vogt W -construction is a construction on operads with the property that, forany topological operad P , algebras over W P are “up-to-homotopy” or “weak” P -algebras. A laxoperad [7] is an algebra over the operad W O , the Boardman-Vogt W –construction applied to theoperad of operads O (Definition 2.9), and is a notion of a “weak” or “infinity” operad. It is knownthat there exists a zig-zag of Quillen equivalences between the category of W O -algebras and thecategory of reduced dendroidal spaces by, for example, combining Theorem 4.1 of [1] with eitherTheorem 1.1 of [3] or a restriction of Theorem 8.15 of [10].Here we show how the operad B O can be identified with a variant W of the W -constructionof the operad O of operads (see Theorem A.4). From this, it will follow that B O –algebras arelax operads that are strictly symmetric and with a strict identity (see Example A.2). We start byrecalling the W –construction. N INFINITY OPERAD OF NORMALIZED CACTI 35
A.1.
The W -Construction. The Boardman-Vogt W -construction is an enlargement of the freeoperad construction. Given an operad P , there are canonical morphisms of topological operads F P (cid:44) → W P ∼ −→ P , where the map p : W P → P is a surjective homotopy equivalence. Algebras for W P are up-to-homotopy P -algebras. We briefly recall the construction here and refer the reader to [7, Section 17]or [1, Section 3] for full details. Definition A.1.
Let P be a C –colored (discrete or topological) operad. The operad W P is atopological operad with the same set of colors C , built from the free operad F ( P ) (Definition 2.3)by adding length in [0 ,
1] to the internal edges of the trees that define the elements of F ( P ). Moreprecisely, for each list of colors c ; c , . . . , c k in C , we have W P ( c ; c , . . . , c k ) = (cid:16) (cid:97) ( T ,f,λ ) (cid:0) [0 , | iE ( T ) | × (cid:89) v ∈ V ( T ) P ( out ( v ); in ( v )) (cid:1)(cid:17) / ∼ where the disjoint union, as for the free operad, runs over the isomorphim classes of leaf-labeled C –colored planar trees ( T , f : E ( T ) → C , λ : { , . . . , k } → L ( T ))with k leaves such that f ( λ ( i )) = c i , f ( R ( T )) = c . The equivalence relation is generated by therelation ( ∗ ) in Definition 2.3 in addition to the following additional relations that capture “weak”operadic composition and units:(1) any tree with an internal edge of length of zero is identified with the tree where that edgehas been collapsed and the operations labelling its end vertices composed;(2) any tree that has a vertex with only one input and one output, both colored by c ∈ C , labeledby the identity in ι c ∈ P ( c ; c ), is identified with the tree where that vertex is deleted. Theresulting new edge, if internal, has length the maximum length of the two original internaledges connected to the deleted vertex.See [7, p 75] for a pictorial version of these relations. The symmetric group acts on W P by relabelingthe leaves, as for the free operad. Composition is by grafting, giving length 1 to the newly createdinternal edge.We will denote elements of W P by ( T , f, λ, s, p ), where T is a planar tree, f : E ( T ) → C is themap coloring its edges, λ : { , . . . , k } → L ( T ) is the bijection labeling its leaves, s ∈ [0 , | iE ( T ) | is acollection of weights, and p = ( p v ) v ∈ V ( T ) is a labeling of the vertices by operations in P . An exampleis shown in Figure 24. There is a canonical projection map π : W P → P defined by sending all theedge lengths to 0 and composing the operations of P as dictated by the trees.A.2. A variant on the W -construction. Given a (discrete or topological) C -coloured operad P , the topological operad W P is defined as the quotient of W P by replacing relation (2) inDefinition A.1 by the following stronger relations for arity one vertices, as well as a version for arityzero vertices:(2 (cid:48) ) any tree that has a vertex v with only one input and one output both colored by c , adjacentto at least one other vertex w , with v labeled by any element P ( c ; c ), is identified with thetree where the vertex v is deleted, and the label of v and w are composed in P (Figure 25).If the resulting new edge is internal, then its length is the maximum length of the twooriginal (then necessarily internal) edges adjacent to v .(3’) any tree that has a vertex v with no input, adjacent to another vertex w , with v labeled by any element of P ( c ; ∅ ), is identified with the tree where the vertex v and the edge between v and w are deleted, and the labels of v and w are composed in P (Figure 26).So a W P –algebra is a weak P –algebra ( W P –algebra) for which the nullary and uniary operationare strict. And in particular, one has that W P ( c ; c ) = P ( c ; c ) and W P ( c ; ∅ ) = P ( c ; ∅ ) for any color c . Also, one can always choose representatives of elements of W P using trees with no valence 0 or1 vertices (unless it only has 0 or 1 vertex). In a tree that defines an element of W P , an arity onevertex lying in between two other vertices can be slid up or down to either of its neighboring vertices,
12 710 a nm(cid:96)kjihgfe dcb o P ( f ; a, e ) (cid:51)P ( p ; f, g, h, i, j ) (cid:51)∈ P ( e ; b, c, d ) ∈ P ( j ; k, (cid:96), m, n )1 210 987654 3 p P ( h ; ∅ ) ∈ P ( n ; o ) ∈ Figure 24.
Example of an element of W P ( p ; a, b, c, d, g, i, k, (cid:96), m, o ). p w p w (cid:48) ◦ p v max { s,t } ∼∼ p w p v p w (cid:48) st p v ◦ p w p w (cid:48) ccc max { s,t } c Figure 25.
Local representation of the relation (2 (cid:48) ) on a tree. ∼ p v p w p w ◦ p vc t Figure 26.
Local representation of the relation (3’) on a tree.composing its label with that of the chosen vertex, while an arity zero vertex can be “pushed down”to the vertex it is attached to.
Example A.2.
The example relevant to us here is when we set P = O is the operad of operads. Inthis case, C = N is the natural numbers and an O -algebra is a (monochrome) operad. The nullaryoperations in O (1; ∅ ) encode the identity operation in the O –algebra, while the unary operations N INFINITY OPERAD OF NORMALIZED CACTI 37 in O ( n ; n ) encode the action of the symmetric groups. It follows that a W O –algebra is a strictlysymmetric weak operad with a strict identity.By construction, the canonical projection p : W P → P factors through the quotient map q : W P → W P . Moreover, both W P and W P are homotopy equivalent to P : Proposition A.3.
There are operad maps W P → W P → P , inducing homotopy equivalences W P ( c ; c , . . . , c n ) W P ( c ; c , . . . , c n ) P ( c ; c , . . . , c n ) q ∼ p ∼ for each n ≥ and each c ; c , . . . , c n in C .Proof. For each n ≥ c ; c , . . . , c n the map q : W P ( c ; c , . . . , c n ) → W P ( c ; c , . . . , c n )is the projection on to the quotient. It is an operad map because if elements of W P are equivalentin W P before being composed, they are necessarily also equivalent in W P after composition. Themap p : W P → P contracts the remaining edges in the trees of W P by sending the lengths to 0(and composing the operations in P ). This map is well-defined, as it is compatible with the relations(2’) and (3’), and respects the operad structure. These maps induce homotopy equivalences, withhomotopy inverses given by including P ( c , . . . , c n ; c ) as labelled corollas in W P ( c ; c , . . . , c n ) or W P ( c ; c , . . . , c n ). (cid:3) A.3. B O -algebras are strictly symmetric lax operads. In this section we show that there isan isomorphism of topological operads B O ∼ = W O . In particular, any B O –algebra will receive acanonical W O –structure via the map W O → W O . Combining this with Example A.2, gives adescription of B O –algebras as strictly symmetric lax operads with strict identity.Our main theorem in this appendix is: Theorem A.4.
The operads W O and B O are isomorphic. Combining Theorem 4.8 and Theorem A.4, we immediately get
Corollary A.5.
There exist isomorphisms of categories W O − Alg S ∼ = ( S (cid:101) Ω op ) strict . The proof of the theorem will be given in Section A.4. Though not saying this explicitly, theproof uses the natural association of a bracketing to a clustering tree , which is described for instancein [32, Definition 2.7].Since the W -construction is built out of cubes, to prepare for the proof, we start by giving analternative description of B O in terms of cubes as well. Definition A.6.
We can define a weighted bracketing of a tree T to be a pair ( B, t ) with bracketing B = { S j } j ∈ J of T and t ∈ [0 , J . The j th coordinate t j ∈ t is the weight of S j . The addition ofweights associates to each bracketing a cube [0 , | B | . These cubes fit together to form a space: B ( T ) = (cid:97) B ∈B ( T ) [0 , | B | / ∼ where the equivalence relation is by identifying any bracketings with weights that only differ by abracket of weight 0 (see Figure 27(b)).Recall from Definition 3.2 the poset B ( T ) of bracketings of a tree T under the inclusion relation. Lemma A.7.
Let T be a tree. There is a homeomorphism |B ( T ) | ∼ = B ( T ) , between the realizationof the nerve of the poset B ( T ) and the cubical space B ( T ) .Proof. We consider the topological k –simplex as the space∆ k = { ( s , . . . , s k ) ∈ R k : 1 = s ≥ s ≥ · · · ≥ s k ≥ } . (a) A bracketing B of T with 2 brackets and itsassociated cube [0 , | B | in B ( T ). (b) All bracketings of T and theirassociated cubes assembled to make B ( T ) a hexagon. Figure 27.
A tree T and its corresponding space of bracketings B ( T ).Fix a tree T and let σ denote a k –simplex B ⊆ · · · ⊆ B k of the nerve of the poset B ( T ). To each σ we associate a map χ σ : ∆ k −→ B ( T )where χ σ ( s , . . . , s k ) is the weighted bracketing of T in which all trees of B have weight 1 = s and all trees of B i \ B i − have weight s i for i ≥ χ σ assemble into a continuous map χ : |B ( T ) | = (cid:16) (cid:97) k ≥ B ( T ) k × ∆ k (cid:17) / ∼ −→ B ( T ) = (cid:97) B ∈B ( T ) [0 , | B | / ∼ . This map is a homeomorphism with inverse defined by mapping a cube [0 , | B | in B ( T ) tothe realization of the sub-poset B ≤ B , which is a cube whose dimension is the cardinality | B | ofthe bracketing. Explicitly, given an element ( B, t ) ∈ B ( T ) with B = ( B , . . . , B k ), we order thecoordinates of t = ( t , . . . , t k ) so that they are in decreasing order1 = t σ (1) = · · · = t σ ( r ) > t σ ( r +1) = · · · = t σ ( r + r ) > · · · > t σ ( r + ··· + r l +1 +1) = · · · = t σ ( r + ··· + r l +2 ) = 0 . This defines an l –simplex ¯ B ⊂ ¯ B ⊂ · · · ⊂ ¯ B l by setting¯ B i = B σ (1) ∪ · · · ∪ B σ ( r + ··· + r i +1 ) . (cid:3) A.4.
Proof of Theorem A.4.
In order to prove B O ∼ = W O , we first recall some definitions.Recall that elements ( T , f, λ, s, p ) ∈ W O ( n ; m , . . . , m k ) are represented by a planar tree T with k leaves ordered by the bijection λ : { , . . . , k } → L ( T ) and with an edge colouring f : E ( T ) → N that, in particular, colours the leaves by m , . . . , m k . In addition, T is equipped with a collectionof lengths s ∈ [0 , | iE ( T ) | , and a decoration of the vertices p = ( p v ) v ∈ V ( T ) by operations by p v in O ( out ( v ); in ( v )).We call a representative ( T , f, λ, s, p ) reduced if the tree T has no vertices of arity zero or one,unless such a vertex cannot be removed using the equivalence relation in W O , i.e. if T is the corolla C or C . In particular, every element of W O has a reduced representative, which in general is notunique. It greatly simplifies the proof of Lemma A.8 to work with reduced representative.For a given tree T , and vertices v, w ∈ V ( T ), we say that w is above v if the unique shortest pathbetween w and the root of the tree goes through v . In this case v is below w . Every other vertex of T is above the root vertex v whose outgoing edge is the root of T . Lemma A.8.
There is a map of topological operads
Ψ : W O → B O . The map Ψ is illustrated in Figure 28.
N INFINITY OPERAD OF NORMALIZED CACTI 39 S t t t S S S , t S , t S , t S , t S , t S , t
31 765 4 32 S , t S , t S , t S , t S , t S , t
31 765 4 32 S , t S , t S , t
31 765 4 321 76 5 4 32 τt t t t t t t v v v v T v τ S t t t S S S , t S , t S , t S , t S , t S , t
31 765 4 32 S , t S , t S , t S , t S , t S , t
31 765 4 32 S , t S , t S , t
31 765 4 32
Figure 28.
Element of W O and corresponding element of B O (16; 2 , , , , , , Proof.
Given a reduced element ( T , f, λ, s, p ) ∈ W O ( n ; m , . . . , m k ), we constructΨ( T , f, λ, s, p ) = ( T, σ, τ, B, t ) ∈ B O ( n ; m , . . . , m k ) , where ( B, t ) is a weighted bracketing on the labelled tree(
T, σ, τ ) = p ( T , f, λ, s, p ) ∈ O ( n ; m , . . . , m k )that is the image of ( T , f, λ, s, p ) under the canonical projection p : W O → O .The bracketing B is constructed from the set of vertices of T . If T has at most one vertex, thenset B = ∅ to be the trivial bracketing, in which case there are no weights to chose so t is the emptymap.Otherwise, since ( T , f, λ, s, p ) is reduced, and T is not a corolla, all its vertices have arity ≥ v be the root vertex of T . For each v ∈ V ( T ) \{ v } , let( S v , σ v , τ v ) = p ( T v , f | T v , λ | T v , p | T v ) , where T v is the subtree of T with v as its root vertex, and containing all the vertices above v .Observe, in particular, that, since v (cid:54) = v , the outgoing edge e v of v – that is the root of S v – isinternal in T . Since the vertices of T have arity at least 2, each S v is a large proper subtree of T ,and because composition in O is by substitution, B = { S v : v ∈ V ( T ) \{ v }} is a collection of nested subtrees, and hence a bracketing.To define the weight function t of B , we associate, to each S v the weight t v = s ( e v ), the lengthof e v ∈ iE ( T ). This completes the definition of Ψ( T , f, λ, s, p ).We need to check that the defined bracketing is independent of our choice of (reduced) represen-tative ( T , f, λ, s, p ) ∈ W O ( n ; m , . . . , m k ), and continuous. In particular, we must check that it iscompatible with the relations (1), (2’) and (3’) in Definition A.1 and Section A.2.To prove that Ψ is well defined with respect to relation (1), and hence also continuous, let s j be the length of an internal edge of T with end vertices v, w , where v is above w . Then if s j goesto 0 in W O , the vertices v, w are identified and their labels are composed in O . Applying Ψ, thiswill precisely have the effect of taking the weight of the bracketing S v to 0, which is equivalent tosimply forgetting the bracketing S v in B O .Relation (2’) allows that a vertex v with only 1 input in T , labeled by a permutation α ∈O ( n ; n ) ∼ = Σ n , to be composed to either of the vertices it shares an edge with. So suppose T is thereduction of a tree (cid:101) T with an arity one vertex v attached to two vertices w and w (cid:48) , with w (cid:48) below w . We may assume that w and w (cid:48) both have arity at least two. We let T be the tree obtained from (cid:101) T by collapsing the edge between w and v and let T (cid:48) be the tree obtained from (cid:101) T by collapsingthe edge between v and w (cid:48) . We need to check that the brackets S w and S w (cid:48) are the same ifcomputed using the representative ( T , f, λ, s, p ) associated to T or ( T (cid:48) , f (cid:48) , λ, s (cid:48) , p (cid:48) ) associated to T (cid:48) .This is immediate for the bracket S w (cid:48) because w (cid:48) is below v and thus p ( T w (cid:48) , f | T w (cid:48) , λ | T w (cid:48) , p | T w (cid:48) ) = p ( T (cid:48) w (cid:48) , f (cid:48) | T (cid:48) w (cid:48) , λ | T (cid:48) w (cid:48) , p (cid:48) | T (cid:48) w (cid:48) ). For the vertex w , the two representatives in general do not have thesame image under p , but if p ( T (cid:48) w , f (cid:48) | T (cid:48) w , λ | T (cid:48) w , p (cid:48) | T (cid:48) w ) = ( S (cid:48) w , σ (cid:48) w , τ (cid:48) w ), we still have that S (cid:48) w = S w .In fact, only τ (cid:48) w might differ from τ w as the vertex v is a permutation α ∈ O ( n ; n ) = Σ n that actson a labeling p ∈ O ( n ; k , . . . , k l ) by permuting the leaves of the labeled tree representing p .For relation (3’) in the definition of W , the relation gives a unique way to reduce a tree if anarity zero vertex is attached to another vertex, so the representative with no arity 0 vertices isunique and nothing needs to be checked.Finally, we check that Ψ is a map of operads. Consider a composition ( T , f , λ , s , p ) ◦ i ( T , f , λ , s , p ) of reduced representatives in W O , and let Ψ( T j , f j , λ j , s j , p j ) = ( T j , σ j , τ j , B j , t j )for j = 1 ,
2. Composition in W O is induced by grafting a tree T onto the i th leaf of T , creatinga new internal edge of length 1. If T has at least one vertex of arity 2, this corresponds exactlyunder Ψ to adding a new bracket T of weight 1 in the composed tree T • σ ( i ) ,τ T where thecomposition here is by insertion. If not, then, since ( T , f , λ , s , p ) is reduced, T has either novertices or a single arity 1 vertex, so T is either the exceptional tree η or a corolla C n . In each case,the newly added edge in the composed tree T ◦ i T will be collapsed when going to a reduced tree,corresponding under Ψ to a composition in B O where no extra bracket is added. This finishes theproof. (cid:3) Lemma A.9.
For every ( n ; m , . . . , m k ) the map Ψ : W O ( n ; m , . . . , m k ) −→ B O ( n ; m , . . . , m k ) is a bijection.Proof. We start by checking that Ψ is surjective. So let (
T, σ, τ, B, t ) of B O ( n ; m , . . . , m k ) with B = { S j } j ∈ J and t ∈ [0 , J . We may always choose a representative where all brackets have non-zero weight, so we assume that t j (cid:54) = 0 for any j ∈ J . We will construct an element ( T , f, λ, s, p ) ∈ W O ( n ; m , . . . , m k ) in the preimage of ( T, σ, τ, B, t ).If B = ∅ is the empty bracketing then define T to be the corolla with k leaves, with f coloring itsleaves m , . . . , m k in the ordering given by λ , and the root by n and p labeling the unique vertex by( T, σ, τ ). The weights s are trivial in this case. By definition, Ψ takes this element to ( T, σ, τ, ∅ , B = { S j } j ∈ J is non-empty. To encode the leaf labelling τ on T , it isconvenient to choose a non-reduced representative of its preimage, using a tree T with one valence1 vertex at its root. We define T as follows: we set V ( T ) = { v τ , v T } ∪ { v j } j ∈ J , where the vertex v j corresponds to the bracket S j ∈ B , v T corresponds to an additional “trivial bracket” S T := T , and v τ will be associated to the permutation τ . To construct the tree, we set v i above v j if S i ⊂ S j ,connecting the two vertices by an edge e i if there is no k ∈ J such that S i (cid:40) S k (cid:40) S j , where weallow S j = S T . This edge e i is colored by the number | L ( S i ) | of leaves of the smaller tree S i and wedefine its length by setting s i = t i is the weight of the corresponding bracket. The nesting conditionon the brackets implies that no cycles are formed this way. We also connect v T and v τ by an edgeof length 1, colored by n = | L ( T ) | , which is also the color of the root of the tree.Finally for each vertex v of T , we attach a leaf l v to the vertex v i ∈ V ( T ) if S i is the smallesttree of the bracketing containing v , attaching it to v T if v is contained in no bracket. This leaf iscolored by the arity of v in T . This defines the tree T , with edge lengths s and edge coloring f .We pick some planar structure for T . (Recall that elements of W O are only defined up tonon-planar isomorphism, which is why there is some freedom here.) Note that the leaves of T correspond exactly to the vertices of T . The ordering λ : { , . . . , k } → L ( T ) is determined by σ andthis identification. This defines the tuple ( T , f, λ, s ).All that remains is to define the decoration p of the vertices of T by elements of O . We need tohave that ( T, σ, τ ) is given by the composition of the elements of the vertices of T so to determinethe decorations in T , what we need is to “undo” the compositions in T marked by the bracketings. N INFINITY OPERAD OF NORMALIZED CACTI 41
Let v j ∈ V ( T ). We define p ( v j ) to be the element ( S j / ∼ , σ j , τ j ) ∈ O ( out ( v j ); in ( v j )) where S j / ∼ is the planar tree S j with each subtree S i (cid:40) S j collapsed to a corolla with the same set ofleaves, σ j orders the vertices according to the above chosen planar ordering of T , where we notethat the incoming edges of v j correspond precisely to the vertices of S j / ∼ , and τ j labels the leavesof S j / ∼ , which are also the leaves of S j , in the order given by the planar embedding of T . (Hereit is important that the chosen planar structure of T is compatible with the chosen order σ j of V ( S j / ∼ ). On the other hand, the chosen order τ j of L ( S j / ∼ ) is not important, as we will fix itbelow using the vertex v τ .) This determines p uniquely on all vertices { v j } j ∈ J ∪ { v T } . Finally, thevertex v τ is labeled by the permutation τ ∈ Σ n , considered as an element of O ( n ; n ).This finishes the construction of ( T , f, λ, s, p ). To compute its image under Ψ, we have to passto a reduced representative, which means collapsing the edge between v τ and v T and composingtheir labeling. (The length of that edge is forgotten.) We have that Ψ( T , f, λ, s, p ) = ( T, σ, τ, B, t ),by our choice of p for the tree T and its leaf-labeling τ , our choice of λ for the ordering σ of thevertices, our choice of vertices of T for B , and with a direct correspondence between the length s i of the edge e i and the weight t i of B i .To finish the proof, we check that Ψ is injective. We will check that, up to the equivalencerelations defining W O , there is a unique reduced ( T (cid:48) , f (cid:48) , λ (cid:48) , s (cid:48) , p (cid:48) ) in the preimage of ( T, σ, τ, B, t ).Note that the number of vertices of such a reduced representative is determined by the tree T andthe cardinality of B . We consider first the cases where T (cid:48) has 0 or 1 vertex.If T (cid:48) has no vertices, then T (cid:48) = η representing the identity element in O (1; 1), B = ∅ , and, up tothe equivalence relations of W O , there is only one possibility for ( T (cid:48) , f (cid:48) , λ (cid:48) , s (cid:48) , p (cid:48) ).Suppose now that T (cid:48) = C k has exactly one vertex of arity k . The leaves of T (cid:48) are in one-to-on correspondence with the vertices of T , with λ (cid:48) ordering its leaves, and f (cid:48) coloring them m , . . . , m k , n , with m i the color of λ (cid:48) ( i ). We can choose a representative of ( T (cid:48) , f (cid:48) , λ (cid:48) , s (cid:48) , p (cid:48) ) so thatthe planar structure of T (cid:48) = C k is given by the ordering σ of the vertices of T . Then the labeling p of the vertex is necessarily precisely ( T, σ, τ ). So there is only one possibility for ( T (cid:48) , f (cid:48) , λ (cid:48) , s (cid:48) , p (cid:48) ).Finally, if T (cid:48) has at least two vertices, then it must have precisely | B | + 1 vertices arrangedin a tree according to the nested structure of the bracket, and k leaves, with each leaf attachedto the vertex corresponding to the appropriate bracket. The root vertex of T corresponds to thewhole tree T . The coloring of the edges is determined by the arity of the vertices and brackets in T , and the labeling of the leaves λ is determined by the ordering σ . The vertices are decoratedby tuples ( T j , σ j , τ j ), with T j determined by the bracketing B , σ j determined by the nesting ofthe bracketing once a planar structure for T (cid:48) is chosen. Choosing a different planar structure willgive an equivalent element of W O (in fact also of W O ). The ordering τ j is likewise not uniquelydetermined by the situation, but a different choice that does yield the same tuple ( T, σ, τ, B, s )under Ψ will be equivalent in W O , using relation (2 (cid:48) ). This finishes the proof of injectivity. (cid:3) We are now ready to prove our main result in this appendix, namely that W O and B O areisomorphic as topological operads. Proof of Theorem A.4.
In Lemma A.8 we constructed a map of topological operadsΨ : W O −→ B O . Combining this with Lemma A.9 we know that, for each tuple ( n ; m , . . . , m k ), the mapΨ : W O ( n ; m , . . . , m k ) −→ B O ( n ; m , . . . , m k )is a continuous bijection. As the source of this map is a compact space ( π W O ( n ; m , . . . , m k ) = O ( n ; m , . . . , m k ) is finite and there are finitely many reduced representatives ( T , f, λ, s, p ) defining acube in each component), and the target is a Hausdorff space, Ψ is therefore a local homeomorphismand hence an isomorphism of topological operads. (cid:3) Remark A.10.
A corollary of the result we just proved is that W O is the realization of an operadin posets, namely the operad B O . The operad W O can likewise be seen as the realization ofan operad in posets, namely the poset of elements of the free operad F O , with poset structure generated by edge collapses. The map of operads q : W O → W O is the realization of a map ofposets. Indeed, the map q : W O → W O ∼ = B O respects the poset structure because collapsingan edge in T , which defines the poset structure underlying W O , corresponds under the map q toforgetting a bracket, which defines the poset structure underlying W O = B O . Appendix B. The explosion category of
ΩIn Section 4.1 we introduced an enriched version of the dendroidal category (cid:101) Ω which is closelyrelated to the category of B O -algebras. As mentioned in the introduction of Section 4, the idea ofthe category (cid:101) Ω is to encode homotopy coherent Ω–diagram, and hence (cid:101) Ω should be connected tothe explosion category of Ω, as defined by Leitch [21] and Segal [31, Appendix B].In this appendix we describe the explosion category of Ω, denoted (cid:101) Ω, and show that our topo-logical category (cid:101) Ω sits between (cid:101) Ω and Ω in the sense that there exist equivalences of topologicalcategories (cid:101) Ω (cid:101) Ω Ω q ˜ p p . The explosion construction and the W –construction are very closely related in spirit. One mightthus expect a relationship between Segal (cid:101) Ω–diagrams and W O –algebras, similar to the relationshipbetween Segal dendroidal spaces (Ω–diagrams) and O –algebras, and between Segal homotopy den-droidal spaces ( (cid:101) Ω –diagrams) and W O – or B O –algebras. Theorem B.6 below will show that sucha relationship exists, but without being as close as in the other cases: W O –algebras identify witha full subcategory of the category of reduced strict Segal (cid:101) Ω–diagrams.B.1.
The explosion of Ω . For each morphism g : S → T in Ω, we define a poset of paths Path Ω ( S, T ) g whose objects are the factorizations of g : S → T in Ω S T . . . T n − T, gg g g n where we identify two factorizations if they differ only by identity morphisms. In particular, eachsuch factorisation ( g , . . . , g n ) has a unique reduced representative containing no identity morphismsunless n = 1 and g is the identity on S . (Such a factorisation can be thought of as a path in thenerve of Ω.) The poset structure is by refinement of factorisation: ( g , . . . , g n ) ≤ ( g (cid:48) , . . . , g (cid:48) m ) if n ≤ m and there is a monotone map α : { , . . . , n } → { , . . . , m } such that α (0) = 0, α ( n ) = ( m ),and g i = g (cid:48) α ( i ) ◦ · · · ◦ g (cid:48) α ( i − for each 1 ≤ i ≤ n .We denote the geometric realization of this poset by K g := | Path Ω ( S, T ) g | . Definition B.1.
The topological category (cid:101)
Ω has the same objects as Ω. Morphism spaces in (cid:101)
Ω aredefined as Hom (cid:101) Ω ( S, T ) = (cid:97) g ∈ Hom Ω ( S,T ) K g = (cid:97) g ∈ Hom Ω ( S,T ) | Path Ω ( S, T ) g | . Composition of morphisms of (cid:101)
Ω is given by concatenation of factorizations.
Example B.2.
Fix a tree T with | L ( T ) | = n leaves and three inner edges: e , e , e . Recall that C n denotes the corolla with n leaves. Let ∂ e , ∂ e , ∂ e denote the inner face maps in Ω associated toeach inner edge, and let g = ∂ e ∂ e ∂ e : C n → T be their composition. Then g admits a factorization C n T T T. g as a composition of three inner face maps for each permutation of { , , } . The elements ofPath Ω ( C n , T ) g that involve only these three inner face maps form a subposet with ([1] , g −→ ) as N INFINITY OPERAD OF NORMALIZED CACTI 43 minimum, and for each permutation σ ∈ Σ the elements([3] , ∂ σ (1) −−−→ ∂ σ (2) −−−→ ∂ σ (3) −−−→ ) ([2] , ∂ σ (1) −−−→ ∂ σ (2) ∂ σ (3) −−−−−−→ ) ([2] , ∂ σ (1) ∂ σ (2) −−−−−−→ ∂ σ (3) −−−→ ) . Each permutation σ this way contributes to a square( ∂ σ (1) ∂ σ (2) −−−−−−→ ∂ σ (3) −−−→ ) ( ∂ σ (1) −−−→ ∂ σ (2) −−−→ ∂ σ (3) −−−→ )( g −→ ) ( ∂ σ (1) −−−→ ∂ σ (2) ∂ σ (3) −−−−−−→ )in this subposet, and the dendroidal identities tell us that these squares together form the followinghexagon inside | Path Ω ( C n , T ) g | :Additional elements of Path Ω ( C n , T ) g can be obtained by inserting tree isomorphisms. This exampleshould be compared to Examples 3.4 and 3.5 which can be interpreted as computing morphismspaces in the category (cid:101) Ω likewise associated to trees with three internal edges, where in one casea pentagon occurs, and in the other it is a hexagon. Lemma B.3.
For each g ∈ Hom Ω ( S, T ) the space K g = | Path Ω ( S, T ) g | is contractible.Proof. The poset Path Ω ( S, T ) g has the trivial factorisation S g −→ T as a minimal element. (cid:3) Let ˜ p : (cid:101) Ω → Ω be the functor that is the identity on objects and projects each morphism space K g to g . Considering Ω as a discrete topological category, the lemma immediately gives the followingproposition. Proposition B.4.
The functor ˜ p : (cid:101) Ω → Ω induces a homotopy equivalence on morphism spaces. Note that the proposition identifies Ω with the “path component category” π (cid:101) Ω, which has thesame objects as (cid:101)
Ω and Hom π (cid:101) Ω ( S, T ) := π (Hom (cid:101) Ω ( S, T )).
B.2.
The relationship between (cid:101) Ω and (cid:101) Ω . The category (cid:101) Ω sits between (cid:101) Ω and Ω in the senseof the following proposition.
Proposition B.5.
There is a functor q : (cid:101) Ω → (cid:101) Ω , which is the identity on objects and induces ahomotopy equivalence on each morphism space. Moreover, the composition p ◦ q = ˜ p : (cid:101) Ω → Ω is theprojection functor of Proposition B.4.Proof. Fix two objects
S, T ∈ (cid:101) Ω. Recall from Definition B.1 thatHom (cid:101) Ω ( S, T ) = (cid:97) g ∈ Hom Ω ( S,T ) K g for K g = | Path Ω ( S, T ) g | is the realization of the poset of factorizations of g , and K g is contractible.Likewise by Definition 4.4 Hom (cid:101) Ω ( S, T ) = (cid:97) g ∈ Hom Ω ( S,T ) L g and L g = (cid:81) v ∈ V ( S ) |B ( g ( C v )) | is the realization of the poset L g of bracketings of the trees g ( C v ), with L g likewise contractible. So to prove the proposition, it is enough to produce a functor q which isthe identity on objects and takes K g to L g for each g . We will define the functor by defining a posetmap q g : Path Ω ( S, T ) g → L g and show that it is compatible with composition.Fix a map g : S → T in Ω. An object of Path Ω ( S, T ) g is a factorization ( g , . . . , g n ) of g and tosuch a factorization of g , for each v ∈ V ( S ), we associate a bracketing of g ( C v ) as follows: set B v = { S w = g n ◦ · · · ◦ g i +1 ( C w ) } ≤ i ≤ n − w ∈ V ( g i ◦···◦ g ( C v )) S w (cid:40) g ( C v ) large This is a (possibly empty) bracketing as these sets are by definition nested. We then define q g ( g , . . . , g n ) = ( B v ) v ∈ V ( S ) . Note that this association is a map of posets as refining a factor-ization will correspond under q g to an inclusion of bracketings.We are left to check that the maps q g assemble to define a functor, i.e. that they are compatiblewith composition in (cid:101) Ω and (cid:101) Ω . Let f : R → S be another morphism in Ω. We need to check thatPath Ω ( S, T ) g × Path Ω ( R, S ) fq g × q f (cid:15) (cid:15) (cid:47) (cid:47) Path Ω ( R, T ) g ◦ fq g ◦ f (cid:15) (cid:15) L opg × L opf (cid:47) (cid:47) L opg ◦ f commutes. Because the target is a poset, it is enough to check that it commutes on objects. Let( g , . . . , g n ) and ( f , . . . , f m ) be objects of Path Ω ( S, T ) g and Path Ω ( R, S ) f . By definition, theircomposition is ( f , . . . , f m , g , . . . , g n ) ∈ Path Ω ( R, T ) g ◦ f . We have q f ( f , . . . , f m ) = ( B fx ) x ∈ V ( R ) and q g ( g , . . . , g n ) = ( B gv ) v ∈ V ( S ) with B fx = { S y = f m ◦ · · · ◦ f i +1 ( C y ) } ≤ i ≤ m − y ∈ f i ◦···◦ f ( C x ) S y (cid:40) f ( C x ) large and bracketing of f ( C x ) ⊂ S , and B gv = { S w = g n ◦ · · · ◦ g i +1 ( C w ) } ≤ i ≤ n − w ∈ g i ◦···◦ g ( C v ) S w (cid:40) g ( C v ) large a bracketing of g ( C v ) ⊂ T . By definition, q g ( g , . . . , g n ) ◦ q f ( f , . . . , f m ) is the collection ( ¯ B x ) x ∈ V ( R ) of bracketings of each tree g ◦ f ( C x ) ⊂ T defined by¯ B x = (cid:16) (cid:91) v ∈ f ( C x ) B gv (cid:17) ∪ (cid:16) (cid:91) v ∈ f ( C x ) g ( C v ) (cid:40) g ◦ f ( C x ) large { g ( C v ) } (cid:17) ∪ (cid:16) (cid:91) S y ∈ B fx g ( S y ) large { g ( S y ) } (cid:17) , N INFINITY OPERAD OF NORMALIZED CACTI 45 where B gv is considered as a bracketing of g ◦ f ( C x ) via the inclusion g ( C v ) ⊂ g ◦ f ( C x ). Now we seethat this is exactly the bracketing of g ◦ f ( C x ) defined by the factorization ( f , . . . , f m , g , . . . , g m ),which indeed is the union of the sets { g ( S y ) = g ◦ f m ◦ · · · ◦ f i +1 ( C y ) } ≤ i ≤ m − y ∈ f i ◦···◦ f ( C x ) g ( S y ) (cid:40) g ◦ f ( C x ) large ∪ { S v = g ( C v ) } v ∈ f ( C x ) S v (cid:40) g ◦ f ( C x ) large ∪ { S w = g n ◦ · · · ◦ g i +1 ( C w ) } ≤ i ≤ n − w ∈ g i ◦···◦ g ◦ f ( C x ) S w (cid:40) g ◦ f ( C x ) large . Hence the poset maps q g assemble to define a functor q : (cid:101) Ω → (cid:101) Ω as claimed. Moreover, onereadily checks that the composition with the projection p : (cid:101) Ω → Ω is the canonical projection˜ p : (cid:101) Ω → Ω. (cid:3) B.3. W O –algebras as (cid:101) Ω –diagrams. In Section 4.2 we showed that B O -algebras describe den-droidal Segal spaces. For completeness, we now show how homotopy dendroidal spaces S (cid:101) Ω op arerelated to W O -algebras.We will only need to consider (cid:101) Ω–diagrams X : (cid:101) Ω op → S that are reduced in the strict sense,i.e. such that X ( η ) = ∗ . Recall that in this case, for X : Ω → S , the Segal map becomes the map X ( T ) χ −→ (cid:89) v ∈ V ( T ) X ( C v )induced by the restriction maps T → C v in Ω op . Considering these morphisms as morphisms of (cid:101) Ω,we likewise have a Segal map for X : (cid:101) Ω → S in this strictly reduced case.In analogy to the case of dendroidal and homotopy dendroidal spaces, let ( S (cid:101) Ω op ) strict denote thefull subcategory of S (cid:101) Ω op of (cid:101) Ω–diagrams X : (cid:101) Ω op → S such that X ( η ) = ∗ and such that the Segalmap χ as above is an isomorphism for every T (cid:54) = η . We have the following: Theorem B.6.
There exists a functor
Ψ : W O− Alg → ( S (cid:101) Ω op ) strict that embeds the category of W O -algebras as a full subcategory of the category of strictly reduced (cid:101) Ω –diagrams satisfying the strict Segal condition. As we will see in the proof, (cid:101)
Ω–diagrams are governed by a version of W O where the trees T havean additional level structure, and W O –algebras identify then as the subcategory of diagrams wherethis level structure does not matter. If one wished to describe a category of homotopy dendroidalspaces which is isomorphic to W O -algebras, one could use this observation to take an appropriatequotient of (cid:101) Ω. As this is particularly messy, and not the main focus of this article, we have electednot to include such a construction.
Proof.
The proof is similar to that of Theorem 4.8 treating the case of B O –algebras. We start withthe definition of the functor Ψ. Let P = {P ( n ) } n ≥ be a W O –algebra with structure maps α P : W O ( n ; m , . . . , m k ) × P ( m ) × · · · × P ( m k ) −→ P ( n ) . We associate to this data an (cid:101)
Ω–diagramΨ( P ) = Ψ( P , α P ) : (cid:101) Ω op → S as follows. Set Ψ( P )( η ) = ∗ and, for T (cid:54) = η in (cid:101) Ω, setΨ( P )( T ) = (cid:89) w ∈ V ( T ) P ( | w | ) . For every morphism g : S → T in Ω, we need to define mapsΨ( P )( g ) : K g × (cid:89) w ∈ V ( T ) P ( | w | ) −→ (cid:89) v ∈ V ( S ) P ( | v | ) . As in Theorem 4.8, we do this one vertex of S at a time.Recall that K g is the realisation of the poset Path Ω ( S, T ) g of factorizations S g −→ T g −→ . . . g n − −−−→ T n − g n −→ T of g in Ω. For each v ∈ V ( S ), we consider the restriction of these maps to C v ∈ S :( ∗ ) C v g −→ g ( C v ) g −→ . . . g n − −−−→ g n − ◦ · · · ◦ g ( C v ) g n −→ g ( C v ) ⊂ T. Recall from Remark A.10 that W O is the realization of an operad in posets, whose elements arethose of the free operad F O (identifying elements of F O with elements of W O in which all weightsof internal edges are 1). We will now use the restriction ( ∗ ) of ( g , . . . , g n ) to C v to construct alabeled planar tree ( T , f, λ, p ) ∈ F O ( | v | ; ( | w | ) w ∈ V ( g ( C v )) )by induction on the height of the tree:Starting at the root, we attach a vertex ¯ v of valence | V ( g ( C v )) | . The incoming edges of ¯ v arelabelled in accordance with ( g ( C v ) , σ v , τ v ), where σ v is a chosen ordering of the vertices of the tree g ( C v ), and τ v is induced by the planar structure of g ( C v ) ⊂ T . Specifically, the incoming edgesof ¯ v are labeled by the vertices of g ( C v ) and ordered via the map σ v .For each vertex w ∈ g ( C v ), which is now an incoming edge of ¯ v , we can attach a vertex ¯ w ofvalence | V ( g ( C w )) | . These incoming edges are labeled with the tuple ( g ( C w ) , σ w , τ w ), as in theprevious case.More generally, for vertices with height 2 ≤ i ≤ n , the tree T has a vertex ¯ y for every vertex y in ( g i − ◦ · · · ◦ g )( C v ), attached to the previously constructed vertex ¯ x associated to the vertex x ∈ ( g i − ◦ · · · ◦ g )( C v ) satisfying that y ∈ g i − ( C x ). We label ¯ y by the tuple ( g i (¯ y ) , σ y , τ y ) with τ y induced by the planar structure of T i , giving C ¯ y the planar structure dictated by the chosen σ y .We now set f : E ( T ) → N to be the unique meaningful colouring which makes T an element of F O ( | v | ; | w | , . . . , | w k | ). We set the ordering σ of the vertices w , . . . , w k of g ( C v ) in accordance tothe resulting planar structure on T . As the set of vertices of g ( C v ) is also the set of leaves of thetree T , this also defines λ . We note that the tree constructed this way is in no way reduced andwill, a priori, have many arity one vertices labelled by identities. We can use relations defining F O ,however, to remove such vertices and give an equivalent element in F O .This assignment of the restriction of a factorisation ( ∗ ) to a labelled tree T respects the posetstructure of Path Ω ( S, T ) g and W O as refining the factorisation corresponds to undoing the collapseof edges, namely if ( g , . . . , g n ) ≤ ( g (cid:48) , . . . , g (cid:48) m ), then the image ( T , f, λ, p ) of the first factorisationcan be obtained from the image ( T (cid:48) , f (cid:48) , λ (cid:48) , p (cid:48) ) of the second by collapsing the edges correspondingto the added levels, as collapsing level in the tree correspond in this construction to composingconsecutive maps g i .In this way we can apply the structure map α P one vertex at a time and define a map(B.1) α v : | Path Ω ( S, T ) g | × (cid:89) w i ∈ V ( g ( C v )) P ( | w i | ) −→ P ( | v | )and we can define Ψ( P )( g ) = ( α v ) v ∈ V ( S ) . By construction, the action of Ψ( P ) on morphisms commutes with composition in (cid:101) Ω, and thusΨ( P ) : (cid:101) Ω op → S defines a functor. That the Segal map for Ψ( P ) is an isomorphism for every T (cid:54) = η follows immediately from our definition of Ψ( P ).The assignment P (cid:55)→ Ψ( P ) requires only the data of underlying symmetric sequence of P andthe algebra structure maps α P . This data is natural under maps of W O -algebras and thusΨ : W O− Alg S −→ S (cid:101) Ω op is a functor. N INFINITY OPERAD OF NORMALIZED CACTI 47
It remains to check that Ψ is an embedding of a full subcategory. Injectivity on objects followsfrom the fact that if P and Q satisfy that Ψ( P ) = Ψ( Q ), then we necessarily have that P ( n ) = Q ( n )for each n , as given by the value at the corolla, with agreeing symmetric group actions as givenby the isomorphisms of corollas, and the structure maps α P and α Q likewise must agree as thevalue of the structure map on every element of W O is the value of the functor Ψ( P ) = Ψ( Q ) onan associated morphism of (cid:101) Ω obtained by choosing a level structure on the tree and interpretingthe collapse of each level of the tree as a morphism in Ω. As morphisms of W O –algebras aredetermined by what they do on spaces P ( n ), we see that the functor is faithful. It is also full asnatural transformations between diagrams originating from W O –algebras, will necessarily respectthe W O -algebra structure of their values at the corollas. (cid:3) Remark B.7.
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