HHIGHER CYCLIC OPERADS
PHILIP HACKNEY, MARCY ROBERTSON, AND DONALD YAU
Abstract.
We introduce a convenient definition for weak cyclic operads,which is based on unrooted trees and Segal conditions. More specifically,we introduce a category Ξ of trees, which carries a tight relationship to theMoerdijk–Weiss category of rooted trees Ω. We prove a nerve theorem exhibit-ing colored cyclic operads as presheaves on Ξ which satisfy a Segal condition.Finally, we produce a Quillen model category whose fibrant objects satisfy aweak Segal condition, and we consider these objects as an up-to-homotopygeneralization of the concept of cyclic operad.
For certain operads, such as the moduli space of Riemann spheres with labeledpunctures or the endomorphism operad of a vector space V equipped with a non-degenerate bilinear form, there is not really a qualitative difference between thenotion of input and output. Indeed, in the former case, the ‘output’ of a givenelement arises solely from our choice of labels and not the underlying geometry,while in the latter case we have natural isomorphismsEnd V ( n ) = hom( V ⊗ n , V ) = hom( V ⊗ n , V ∗ ) = hom( V ⊗ n +1 , k ) . This consideration leads directly to the notion of cyclic operad , introduced byGetzler and Kapranov in [22] (although we add the axiom due to van der Laan, see[30, §
11] and [35, § II.5.1]). A cyclic operad is an operad O with extra structure,namely an action of the cyclic group C n +1 = (cid:104) τ (cid:105) on the space O ( n ). Applied to anelement f ∈ O ( n ), we should regard f τ ∈ O ( n ) as f with the first input changed tothe output and the output changed to the last input, as in Figure 1. To get a feelfor how this cyclic operator should act on compositions, one should look at treeswith several vertices like the one in Figure 2. Given a cyclic operad, one can begintalking about graph homology (see [29] as well as the generalizations of Conant& Vogtmann [17]), whereas algebras over cyclic operads admit a cyclic homologytheory [22]. Figure 1. f and f τ a r X i v : . [ m a t h . A T ] A ug P. HACKNEY, M. ROBERTSON, AND D. YAU = fg h fτgτ h fτgτh Figure 2.
A composition of f, g, h , and the action of τ on this compositionFurther examples of cyclic operads include the associative, Lie, and commutativeoperads, (certain models for) the framed little n -disks operad [11, 27], the A ∞ operad [22], and also any monoid with involution [31, 43] (that is, the involution x (cid:55)→ x † satisfies x † y † = ( yx ) † ) regarded as an operad concentrated in degree 1. Thelast of these is useful for giving small examples (see Example 8.9 and Proposition8.10), but also gives a connection with another interesting class of mathematicalobjects.A dagger category [41, Definition 2.2] is a category C together with an involutivefunctor † : C op → C which is the identity on objects. In other words, a daggerstructure on C is an assignment ( f : X → Y ) (cid:55)→ ( f † : Y → X ) satisfying f †† = f and f † g † = ( gf ) † ; this is the many-objects version of a monoid with involution.As Baez argued eloquently in [2], to understand the similarities between generalrelativity and quantum theory, one should begin by considering the natural daggerstructure on the category of n -cobordisms and on the category of Hilbert spaces,respectively. Other important examples of dagger categories include any groupoid,categories of relations, and categories of correspondences.Colored cyclic operads are a simultaneous generalization of cyclic operads (whichwe might term ‘monochrome cyclic operads’) and of dagger categories. There areadditional examples in the literature (e.g., [16, § § O is a C -colored operad and n ≥
0, then the object O n = (cid:96) c,c ,...,c n ∈ C O ( c , . . . , c n ; c )admits a right action by Σ n = Aut { , . . . , n } compatible with operadic composition.Write Σ + n = Aut { , , . . . , n } and identify Σ n as the isotropy group of 0. Definition 0.1. A cyclic structure on a C -colored operad O is a collection of maps − · σ : O ( c , . . . , c n ; c ) → O ( c σ (1) , . . . , c σ ( n ) ; c σ (0) ) for σ ∈ Σ + n and c i ∈ C . Theseshould satisfy two conditions. First, they assemble into a right Σ + n action on O n ,which agrees with the existing Σ n action. For the second condition, let τ n +1 ∈ Σ + n be the element with τ n +1 ( n ) = 0 and τ n +1 ( i ) = i + 1 for 0 ≤ i < n . We insist that IGHER CYCLIC OPERADS 3 if g ∈ O ( c , . . . , c k ; c ) and f ∈ O ( d , . . . , d (cid:96) ; c i ) are composable at position i , then (1) ( g ◦ i f ) · τ k + (cid:96) = (cid:40) ( g · τ k +1 ) ◦ i − f if 2 ≤ i ≤ k ( f · τ (cid:96) +1 ) ◦ (cid:96) ( g · τ k +1 ) if i = 1 and (cid:96) (cid:54) = 0.Informally, when O is equipped with a cyclic structure, we will say that O is a C -colored cyclic operad. The up-to-homotopy cyclic operads that we develop in this paper are a variationon ‘dendroidal models’ for ∞ -operads (cf. [9, 13, 14, 15, 37]). The dendroidalcategory Ω is a category of rooted trees [36]; each such rooted tree T (with edge setEd( T )) can be regarded as a free object in the category of Ed( T )-colored operads.The dendroidal category is then defined to be the full subcategory of the categoryof all colored operads whose objects are the rooted trees. Not only is Ω defined as asubcategory of colored operads, but it turns out that there is a model structure onthe category of presheaves of Ω (see [13, Theorem 2.4]) that is Quillen equivalentto a model structure on the category of simplicially-enriched colored operads (see[15]). This is an extension of the equivalence between the Joyal model structure onsimplicial sets and the Bergner model structure on simplicially-enriched categories(see [8] for references).It would be a very ambitious project to attempt to do all of the above for coloredcyclic operads, and we are skeptical that the full Cisinski–Moerdijk program canbe carried out in the cyclic case. A key difficulty is that the adjunction betweencategories and dagger categories is badly behaved, in particular with respect toequivalences. Thus, in the present paper we limit what is said about colored cyclicoperads. It is true that every unrooted tree S freely generates an Ed( S )-coloredcyclic operad C ( S ) (see Section 5), but we do not ever consider the full-subcategoryof colored cyclic operads spanned by the unrooted trees. The cyclic operad C ( S )is nearly always infinite, even when S is a linear tree, and arbitrary maps C ( S ) → C ( R ) do not admit decompositions into cofaces and codegeneracies, as they do inthe dendroidal setting. Instead, we directly construct a category Ξ of unrootedtrees that is reminiscent of Ω. The assignment S (cid:55)→ C ( S ) gives a faithful, non-fullfunctor from Ξ to Cyc (Theorem 5.6, Example 5.7). We use this to prove a nervetheorem for colored cyclic operads (Theorem 6.7).Our main goal is to propose a model for weak monochrome cyclic operads. Theseare called
Segal cyclic operads in Section 8, and they are certain reduced presheavessatisfying a Segal condition. The Segal cyclic operads are patterned after the Segaloperads appearing in the work of Bergner and the first author [9], which havebecome important in current work of Boavida, Horel, and the second author onprofinite completions of the framed little disks operad.The profinite completion of a product of spaces is weakly equivalent, but ingeneral not isomorphic, to the product of the profinite completions. For this reason, We exclude the case i = 1 , (cid:96) = 0 from (1), as the formula ( g ◦ f ) · τ = ( g · τ ) ◦ k f followsfrom the first case. Indeed,[( g · τ k +1 ) ◦ k f ] · τ k − k = ( g · τ k − k +1 ) ◦ k − ( k − f = ( g · τ k +1 k +1 ) ◦ f = g ◦ f. We should note that our definition is not a symmetric version of the ‘cyclic multicategories’ ofCheng, Gurski, and Riehl [12]. A non-symmetric colored cyclic operad is a non-symmetric coloredoperad O together with an action of the subgroup (cid:104) τ n +1 (cid:105) ≤ Σ + n on O n , so that (1) holds. Theseform a reflective subcategory of the category of cyclic multicategories. P. HACKNEY, M. ROBERTSON, AND D. YAU the profinite completion of an operad does not yield an operad on the nose butrather an ∞ -operad. This fact has played a crucial role in work of Horel [25]when he generalized work of Fresse [20] and computed a profinite version of theGrothendieck–Teichm¨uller group (cid:100) GT ∼ = π End h ( (cid:99) D ), where D is the little 2-disks operad. In work by Boavida, Horel and the second author, they show thatconsidering the framed little 2 disks as an operad, they recover exactly the sameresult, i.e., (cid:100) GT ∼ = π End h ( (cid:99) D ) ∼ = π End h ( (cid:100) f D ). Considering f D as a cyclicoperad would necessarily result in a smaller set of endomorphisms and conjecturallywould provide refinement on these computations; of course one would expect theprofinite completion of a cyclic operad to be some type of infinity cyclic operad.Providing a good foundation for this project is one of the major motivations forthe present paper. Overview.
We give a brief outline of the paper. Each section begins with a moresubstantial summary of its contents.The first section is dedicated to the construction of the category Ξ of unrootedtrees. In the second section, we examine exactly how close Ξ is to the category Ωof rooted trees. The third and fourth sections are devoted to two structures on thecategory Ξ: a generalized Reedy structure and an active / inert (or generic / free)weak factorization system.The next two sections deal with the relationship of Ξ to colored cyclic operads.In the fifth section we construct the functor Ξ → Cyc , and in the sixth we prove anerve theorem for colored cyclic operads.The final two sections are devoted to model-categorical matters. The penulti-mate section is about the model structure on diagrams indexed by a generalizedReedy category, and at the beginning of the section we show that this model struc-ture usually has properties which ensure that Bousfield localizations exist. We thenrestrict to the case when the base category is the category of simplicial sets. In Sec-tion 7.1 we discuss certain cases when categories of reduced presheaves of simplicialsets admit model structures. In Section 7.2 we show that these model categoriesare in fact simplicial model categories.In the last section we prove the existence of a model structure on reduced Ξ-presheaves in simplicial sets whose fibrant objects, the Segal cyclic operads, satisfya Segal condition. We show that there is a Quillen adjunction (which is not aQuillen equivalence) between this model structure and the model structure forSegal operads from [9].Finally, in an appendix, we discuss certain additional (co)tensorings by Σ -simplicial sets, which exists for Ξ-presheaves which vanish on non-linear trees. Notational conventions. If C is a category, we will write C ( x, y ) or hom( x, y )for the set of morphisms from x to y , depending on if the name of our categoryis short (e.g., C = Ξ) or long (e.g., C = sSet Ξ op ∗ ). We will write Iso C ( x, y ) for theisomorphisms from x to y , Aut C ( x ) := Iso C ( x, x ) for the invertible self-maps of x ,and Iso( C ) for the wide subcategory of C consisting of all of the isomorphisms. Inall adjunctions C (cid:29) D , the top arrow denotes the left adjoint.Throughout this paper we use freely the language of Quillen model categoriesand take the book of Hirschhorn [24] as our standard reference. IGHER CYCLIC OPERADS 5
Acknowledgments.
We are grateful to the Hausdorff Research Institute for Math-ematics and the Max Planck Institute for Mathematics for their hospitality duringthe fall of 2016.The authors have had many interesting discussions about parts of this paper sinceits conception, but we would like especially to thank Clark Barwick, Julie Bergner,and Richard Garner for some helpful insights that came at precisely the right time.We would also like to thank the participants of the workshop ‘Interactions betweenoperads and motives’ at HIM for useful feedback and questions. Finally, we wouldlike to express our appreciation to the anonymous referees of this paper for theirgenerous and extensive feedback.1.
The unrooted tree category
ΞThe main goal of this section is to define a category of unrooted trees Ξ. Wewill begin with a formalism for general graphs, before defining the objects of Ξin Definition 1.3. We give two distinct descriptions of the morphisms of Ξ inDefinition 1.12 and Definition 1.13. Each has its own advantage: morphisms in theformer sense (here called complete ) immediately form a category, while morphismsin the latter sense are specified by a smaller set of data, and are easier to work within most situations. We then embark on a sustained study of the nature of thesemorphisms; key tools are the notions of distance and a (minimal) path in a tree.Along the way, we recover the Moerdijk–Weiss dendroidal category Ω. Finally, inProposition 1.32, we show that the two definitions of morphisms coincide.At the heart of this work is the notion of ‘graph with legs’. One can chooseseveral formalisms; for concreteness, let us say that an undirected graph with legs consists of two finite sets E and V and a function Nbhd : V → P ( E ) (the set ofsubsets of E ). This data should satisfy one axiom, namely that, for each e ∈ E , |{ v ∈ V | e ∈ Nbhd( v ) }| ≤ . We will package the triple (
E, V,
Nbhd) into a single symbol G , and write Ed( G ) = E and Vt( G ) = V . Edges actually come in two types, namely interior edgesInt( G ) = { e ∈ E | e ∈ Nbhd( v ) ∩ Nbhd( w ) for some v (cid:54) = w } and the set of legs Legs( G ) = Ed( G ) \ Int( G )which are edges incident to at most one vertex. If v is a vertex of G , we also write | v | for the valence of v , or the cardinality of the set Nbhd( v ).Every graph has an underlying topological space, which can be described asfollows. See the left hand side of Figure 3 for an example. Definition 1.1 (Space associated to a graph) . Fix an (cid:15) with 0 < (cid:15) <
1, which wecan use to scale the closed unit disc D in the complex plain C . Define | ☆ | = (cid:15) D = { re iθ | ≤ r ≤ (cid:15), ≤ θ ≤ π } (cid:40) D (cid:40) C If e ∈ Ed( G ) is not incident to any vertex, then one should really think that e appears twicein Legs( G ). Since we are only concerned with connected graphs for the bulk of this paper, onlyone graph (see Example 1.4) has an edge with this property, so we will just systematically singleout that special case. P. HACKNEY, M. ROBERTSON, AND D. YAU
Figure 3.
A graph with legs and its corresponding rwb graphand, for n > | ☆ n | = (cid:15) D ∪ n − (cid:91) k =0 { re kn πi | ≤ r ≤ } , considered as a subspace in the closed unit disc of the complex plane. If G is anundirected graph with legs, fix bijections κ v : Nbhd( v ) ∼ = → { e k | v | πi } = S ∩ (cid:12)(cid:12) ☆ | v | (cid:12)(cid:12) and define | G | = (cid:96) v ∈ Vt( G ) (cid:12)(cid:12) ☆ | v | (cid:12)(cid:12) κ v ( e ) ∼ κ w ( e ) (cid:113) Q × [0 , e ∈ Nbhd( v ) ∩ Nbhd( w ) and Q = Ed( G ) \ (cid:83) v ∈ Vt( G ) Nbhd( v ).Notice that the homeomorphism type of | G | determines the isomorphism typeof G . This would not be the case if we did not add some thickness at the centersof | ☆ n | by using the (cid:15) D . Indeed, a variation of realization with (cid:15) = 0 produces theclosed unit interval [0 ,
1] on both the graph G with one edge and no vertices andon the graph G with one vertex v , one edge e , and Nbhd( v ) = { e } .The following is an alternative, equivalent formalism for graph with legs. Definition 1.2 (Red-white-black formalism) . An rwb graph is an ordinary undi-rected graph (see, for example [18, § • red vertices are univalent, • white vertices are bivalent and are only adjacent to black vertices, and • a black vertex is not adjacent to any other black vertex.From a graph with legs, we can form an rwb graph by coloring all vertices black,adding a white vertex on each interior edge, and adding a red vertex to the looseend of each leg. Each rwb graph determines a graph with legs by deleting the white IGHER CYCLIC OPERADS 7 u vw u vwab c f d e
12 0 0 1 23001 23
Figure 4.
Underlying graph (left), unpinned and pinned tree data (right)vertices and joining the edges on either side and deleting all of the red vertices. SeeFigure 3 for an illustration of this correspondence.1.1.
Trees.
The category Ξ governing cyclic dendroidal sets has ‘unrooted’ or‘cyclic’ trees as objects.
Definition 1.3. An unpinned tree S is an undirected graph with legs which iscontractible, has at least one leg, and is equipped with bijectionsord v : { , , . . . , n v } ∼ = → Nbhd( v ) , where Nbhd( v ) ⊆ Ed( S ) is the set of vertices adjacent to v . A pinned tree , or just tree , has, in addition, a mapord : { , , . . . , n } → Legs( S ) , where Legs( S ) ⊆ Ed( S ) is the set of legs of S which is a bijection if S contains avertex and is otherwise the unique map from { , } to the single edge.A typical example of such a graph is found in Figure 4. In pictures of graphs,we will always draw the ordered set of legs Nbhd( v ) in a counterclockwise fashion.Using this convention, to specify the unpinned structure we only need to mark theedges { ord v (0) } v in the figures. Example 1.4.
Let us fix several foundational examples of trees (Figure 5). • The graph with one edge and no vertices, which we write as η . • For each n >
0, the graph ☆ n . This graph has a single vertex v and n edges { , , . . . , n − } (and take ord v = ord = id). • For n ≥
0, the linear graph L n with n distinct vertices { v , . . . , v n } , n +1 distinct edges { e , . . . , e n } , Nbhd( v i ) = { e i − , e i } , so that ord v i ( t ) = e i − t , ord(0) = e , and ord(1) = e n . Note that L ∼ = η . • We will call any tree with all vertices bivalent a linear graph.
Remark 1.5.
A directed tree is a tree S where each edge has an orientation (seeFigure 7); another way to say this when S (cid:54) = η is to say that there are partitions Nbhd( v ) = out (cid:48) ( v ) (cid:113) in( v )Legs( S ) = out (cid:48) ( S ) (cid:113) in( S ) Note the shift in index compared with [35, p. 250]: they use the notation ∗ n for what we call ☆ n +1 . Outside of this remark, we will only consider directed trees which are rooted, hence we willconsider out( S ) and out( v ) as single edges, rather than the one element sets which contain them;cf., Definition 1.20. P. HACKNEY, M. ROBERTSON, AND D. YAU
Figure 5.
The trees η = L , ☆ , and L so that out (cid:48) ( v ) ∩ out (cid:48) ( w ) = ∅ = in( v ) ∩ in( w ) for v (cid:54) = w and out (cid:48) ( v ) ∩ in( S ) = ∅ = in( v ) ∩ out (cid:48) ( S ). This description actually has a little bit more informationfloating around than we would like; namely a ( | in( S ) | , | out (cid:48) ( S ) | )-shuffle and, foreach v ∈ Vt( S ), a ( | in( v ) | , | out (cid:48) ( v ) | )-shuffle. This is simply because for a directedgraph we only need separate orderings on the inputs and outputs, not orderings onthe entire neighborhoods.Making a choice for the ( p, q )-shuffles above, every directed tree determines atree. We will use the convention that the total order on Nbhd( v ) is determined bythat on out (cid:48) ( v ) and in( v ) by insisting, for e ∈ out (cid:48) ( v ) and e (cid:48) ∈ in( v ), that e < e (cid:48) .Similarly, we get an order on Legs( S ) by saying out (cid:48) ( S ) < in( S ) (unless S consistsof a single edge). This convention makes it so that for a rooted tree, the downwardedge is always labeled by ‘0’. This gives the map Ob(Ω) → Ob(Ξ); we will actuallydefine a variant of Ω in Definition 1.20.1.2.
Morphisms of Ξ . When discussing subgraphs of trees, we will always assumethat they are nonempty, connected, and contain all edges incident to any of theirvertices.
Definition 1.6. A subgraph of a tree S consists of a pair of subsets V ⊆ Vt( S ) E ⊆ Ed( S )so that • if v ∈ V , then Nbhd( v ) ⊆ E (which means that R = ( V, E,
Nbhd) consti-tutes the structure of an undirected graph without orderings), • the underlying space of the graph R = ( V, E,
Nbhd) is contractible.Write Sbgph( S ) for the set of subgraphs of S . Remark 1.7.
Subgraphs of S are naturally unpinned trees. The orderings ord v ateach vertex v are inherited from those in S . Example 1.8.
Each edge e ∈ S constitutes a subgraph with E = { e } and V = ∅ .We will write this subgraph as | e . Example 1.9.
For each v ∈ Vt( S ), there is a subgraph ☆ v with V = { v } and E = Nbhd( v ). Thus we have an inclusion ☆ : Vt( S ) (cid:44) → Sbgph( S ). Notice that ☆ v IGHER CYCLIC OPERADS 9 has a preferred ordering withord v ☆ v = ord ☆ v = ord vS : { , , . . . , n } ∼ = → Nbhd( v ) = Legs( ☆ v ) . Proposition 1.10. If R and R (cid:48) are subgraphs of S and R ∩ R (cid:48) (cid:54) = ∅ , then R ∪ R (cid:48) is also a subgraph of S .Proof. Write R = ( V, E ) and R (cid:48) = ( V (cid:48) , E (cid:48) ). The first condition we need to checkfor R ∪ R (cid:48) = ( V ∪ V (cid:48) , E ∪ E (cid:48) ) is immediate, and does not require the hypothesis.The hypothesis R ∩ R (cid:48) (cid:54) = ∅ means ( V ∩ V (cid:48) ) ∪ ( E ∩ E (cid:48) ) (cid:54) = ∅ , which implies thatthe underlying space of R ∪ R (cid:48) is connected (since it is the union of the underlyingspaces of R and R (cid:48) ). Thus it is a connected subspace of a contractible graph, henceis contractible as well. (cid:3) Definition 1.11 (Boundary of a subgraph) . Suppose that S is a tree. • If X is a set, let M X = (cid:96) n ≥ X × n / Σ n be the free commutative unitalmonoid on X (that is, the set of unordered lists of elements of X ). • There is a function ð : Sbgph( S ) → M (Ed( S )) with ð ( R ) = e if R = | e (cid:81) e ∈ Legs( R ) e otherwise.We say that ð ( R ) is the boundary of the subgraph R .If R, T ∈ Sbgph( S ), then the graph R ∩ T is either empty or it is also in Sbgph( S ).Further, R ∪ T ∈ Sbgph( S ) if and only if R ∩ T is nonempty. We will say that R and T overlap if R ∩ T is nonempty (equivalently, if R ∪ T is connected). Definition 1.12.
Suppose that S and R are two trees. A complete morphism R → S consists of two functions • α : Ed( R ) → Ed( S ) • α : Sbgph( R ) → Sbgph( S )that satisfy the following conditions.(1) The equation ð ◦ α = ( M α ) ◦ ð holds.(2) If T, T (cid:48) ∈ Sbgph( R ) overlap, then so do α ( T ) and α ( T (cid:48) ). Furthermore,(a) α ( T ∩ T (cid:48) ) = α ( T ) ∩ α ( T (cid:48) ) and(b) α ( T ∪ T (cid:48) ) = α ( T ) ∪ α ( T (cid:48) ).The set Sbgph( R ) is actually a partial lattice and the second condition juststates that α is a map of partial lattices. As the properties above are closed underfunction composition, there is a category of trees whose morphisms are completemorphisms.Notice in particular that the existence of the function ☆ : Vt( R ) (cid:44) → Sbgph( R )means that every complete morphism has an associated function Vt( R ) → Sbgph( S ). Definition 1.13.
Suppose that R and S are trees. • A morphism φ : R → S is defined to be a pair of maps φ : Ed( R ) → Ed( S ) φ : Vt( R ) → Sbgph( S )satisfying the following:(1) If v is not bivalent (that is, | Nbhd( v ) | (cid:54) = 2), then φ | Nbhd( v ) is injective. (2) For each vertex v , φ (Nbhd( v )) = Legs( φ ( v )) (as unordered sets).(3) Vt( φ ( v )) ∩ Vt( φ ( w )) = ∅ for v (cid:54) = w . • The identity map id R : R → R is given by letting (id R ) = id Ed( R ) andletting (id R ) be the inclusion ☆ : Vt( R ) → Sbgph( R ). • More generally, a morphism φ : R → S is an isomorphism if φ is a bijectionand if φ factors through Vt( S ) as φ = ☆ (cid:101) φ with (cid:101) φ is a bijection.Vt( R ) Sbgph( S )Vt( S ) φ (cid:101) φ ∼ = ☆ In Proposition 1.32, we show that precomposition with ☆ constitutes a bijectionbetween complete morphisms and morphisms. We will also transfer the compositionof complete morphisms back to morphisms in Definition 1.33, after which the readermay wish to verify that this definition of isomorphism is correct from a categoricalstandpoint. Example 1.14. If R = R (cid:48) except for orderings, then there is a unique isomorphism φ : R → R (cid:48) with φ = id. We first note that since Definition 1.13 does not mentionorderings, the pair (id Ed( R ) , ☆ ) constitutes a morphism R → R (cid:48) of Ξ.We now show that the only automorphism φ of R which fixes the edges is id R .For this, we induct on the number of vertices n of R ; the case n = 0 is clear sincethere is only one map η → η . Suppose that uniqueness has been established for all m < n ; pick any e ∈ Legs( R ). There is a unique v ∈ Vt( R ) with e ∈ Nbhd( v ).By assumption φ ( e ) = e , hence φ ( v ) = ☆ v . For each e ∈ Nbhd( v ) \ e ,there is a subgraph R e of R consisting of all vertices and edges on all paths notcontaining v but beginning at e . By the induction hypothesis, φ | R e = id R e isuniquely determined by the fact that φ fixes the edges. It follows that φ ( v ) = ☆ v for all v ∈ Vt( R ), so φ = id R . Remark 1.15.
The argument for uniqueness in the previous example fails if we al-low graphs without legs. Indeed, the graph • — • admits two distinct automorphisms φ with φ = id. Definition 1.16 (Cofaces and codegeneracies) . We describe three basic types ofmorphisms of Ξ. Throughout, S and R will denote objects of Ξ, and d ( R ) will referto the number of vertices of R . • Suppose that S is a subgraph of R and d ( S ) = d ( R ) −
1. Then we say theinclusion S → R is an outer coface . Up to orderings, the tree S is obtainedfrom R by selecting a pair ( v, e ) with v ∈ Vt( R ) and e ∈ Nbhd( v ) suchthat Nbhd( v ) \ { e } ⊆ Legs( R ), and then deleting v and all of the legs inNbhd( v ) \ { e } . We will write δ v : S → R for such a coface map if d ( R ) > i : η → R for the map that hits ord( i ) when d ( R ) = 1. • A map φ : S → R is an inner coface if d ( S ) = d ( R ) − v so that φ ( v ) has exactly two vertices and φ ( v ) has exactly one vertexfor v ∈ Vt( S ) \ { v } . The subgraph φ ( v ) has exactly one inner edge e ,and we will often write δ e : S → R for such an inner coface map. The tree S is obtained from R by contracting an inner edge. IGHER CYCLIC OPERADS 11 u v w v wwu v x wab c de f c de fab ef dab c d (cid:48) d (cid:48)(cid:48) e f δ u δ c Figure 6.
An outer coface, an inner coface, and a codegeneracy • A map φ : S → R is a codegeneracy if d ( S ) = d ( R ) + 1 and there is avertex v (necessarily with | v | = 2) so that φ ( v ) is an edge and φ ( v )has exactly one vertex for v ∈ Vt( S ) \ { v } .A coface is a map which is either an inner coface or an outer coface.An example of each type of map is given in Figure 6.A path in a graph G is an alternating word in the alphabet Ed( G ) (cid:116) Vt( G ) whichcan only contain a subword ve or ev if e ∈ Nbhd( v ). A path from a vertex v to avertex w is a path of the form P = ve v e . . . e n − v n − e n w while a path from an edge e to an edge e (cid:48) is a path of the form P = ev e v . . . v n − e n − v n e (cid:48) . The length of a path, denoted by | P | , is the length of the word.We can concatenate paths P and P (cid:48) if the last letter in P is the first letter in P (cid:48) or if the last letter in P is adjacent to the first letter in P (cid:48) . In the former case,we will remove the duplicate letter. Definition 1.17.
Let G be a graph and v, w ∈ Vt( G ). Define the distance from v to w by d G ( v, w ) = d ( v, w ) = min P | P |− where P ranges over all paths in G from v to w . Any path realizing the distance is said to be a minimal path (and suchexists as long as d ( v, w ) is defined). Similarly, if e, f ∈ Ed( G ), one defines d G ( e, f )and minimal paths between edges in G .If we consider the rwb graph G rwb associated to a graph G , then Vt( G ) is thesame as the set of black vertices of G rwb and Ed( G ) is the union of the sets of red and white vertices of G rwb ; these distances are then half of the usual graph distancebetween vertices (see, e.g., [18, § G rwb . Proposition 1.18.
Let S be a tree. If v, w ∈ Vt( S ) , then a minimal path from v to w exists and is unique. A similar statement applies to minimal paths betweenedges. Minimal paths are characterized as those containing no repeated entries.Proof. Every vertex and edge of S is a vertex in the corresponding rwb graph S rwb .This statement is then [18, Theorem 1.5.1] applied to the tree S rwb . (cid:3) Existence and uniqueness of minimal paths leads to the following result.
Corollary 1.19. If R is a subgraph of S , then d R = d S | R , i.e., the distance in R is the restriction of the distance in S , for all edges and vertices in R . (cid:3) We now have the necessary tools to define the objects and morphisms of thedendroidal category Ω. Though we do not make substantial use of Ω until Section2, we include this definition here, rather than after Definition 1.33, to indicate theusefulness of the notion of distance.
Definition 1.20 (Dendroidal category) . We now define (a variant of) Ω as a sub-category of Ξ. • A rooted tree is a tree R satisfying the following condition: Suppose r =ord(0) ∈ Legs( R ). If v ∈ Vt( R ) and k >
0, then d (ord v (0) , r ) < d (ord v ( k ) , r ) . If R (cid:54) = η , we setin( v ) = Nbhd( v ) \ ord v (0) out( v ) = ord v (0)in( R ) = Legs( R ) \ ord(0) out( R ) = ord(0) = r while if R = η we set in( R ) = { r } and out( R ) = r . • If R and S are rooted trees and φ : R → S is a map in Ξ, we say that φ is oriented if for each v ∈ Vt( R ) and each k > d ( φ (ord v (0)) , s ) ≤ d ( φ (ord v ( k )) , s ) . • The category Ω is the subcategory of Ξ whose objects are rooted trees andwhose morphisms are the oriented maps between rooted trees. • We write ι : Ω → Ξ for the subcategory inclusion.Notice that if φ is an oriented map, then φ ( v ) is a rooted tree (without orderingof the leaves) with root φ (ord v (0)) = φ (out( v )). Remark 1.21.
This definition of Ω is analogous to the equivalent category Ω (cid:48) from[5, Example 2.8]. A rooted tree in our sense is equivalent to a rooted tree togetherwith a planar structure and an ordering of the input edges, and morphisms do notneed to preserve the planar structure.
Remark 1.22.
In the above definition we were able to recognize rooted treesamong all trees; we cannot do something similar for general directed trees (andhence for the category Θ from [23, Remark 6.55]). Indeed, graphs which are linearas undirected graphs generally possess many directed structures, even controllingfor the number of inputs and outputs. See Figure 7. In short, there is a functorfrom (a legged-variant of) Θ to Ξ, but it is not injective on objects.
IGHER CYCLIC OPERADS 13
Figure 7.
Two different directed structures on the same undi-rected tree
Lemma 1.23.
Let φ : R → S be a morphism of Ξ . Then Int( φ ( v )) ∩ Ed( φ ( w )) = ∅ if v (cid:54) = w .Proof. Suppose that φ ( w ) contains a vertex and e ∈ Int( φ ( v )) ∩ Ed( φ ( w )). Since φ ( w ) (cid:54) = | e , there is a vertex w (cid:48) ∈ Vt( φ ( w )) with e ∈ Nbhd( w (cid:48) ). Since e ∈ Int( φ ( v )), we know that w (cid:48) ∈ Vt( φ ( v )), which implies v = w by Definition 1.13(3).In the general case, we induct on d ( v, w ). Suppose we have vertices v and w with d ( v, w ) = n > ve v e v . . . e n − v n − e n w be the shortest path from v to w . If φ ( w ) contains a vertex then we know Int( φ ( v )) ∩ Ed( φ ( w )) = ∅ bythe first paragraph. Suppose that φ ( w ) = | e is a single edge. If n = 1, then e = φ ( e ) ∈ Legs( φ ( v )), which implies e / ∈ Int( φ ( v )) by Definition 1.13(2).Assume the statement of the lemma is true for vertices of distance equal to n − e = φ ( e n ) ∈ Legs( φ ( v n − )), which implies that e / ∈ Int( φ ( v )) by theinduction hypothesis. (cid:3) Lemma 1.24.
Let φ : R → S be a morphism of Ξ . For each vertex v ∈ Vt( R ) , Int( φ ( v )) ⊆ Ed( S ) \ Im( φ ) .Proof. Suppose that φ ( e ) ∈ Int( φ ( v )). Since the graph has a vertex v and isconnected, every edge is adjacent to at least one vertex. If e is adjacent to v , then φ ( e ) ∈ φ (Nbhd( v )) = Legs( φ ( v )) ⊆ Ed( S ) \ Int( φ ( v )), so we conclude that e isnot adjacent to v . Thus there exists a w (cid:54) = v with e ∈ Nbhd( w ). But now φ ( e ) ∈ Int( φ ( v )) ∩ Legs( φ ( w )) ⊆ Int( φ ( v )) ∩ Ed( φ ( w )) , which is empty by Lemma 1.23. (cid:3) Lemma 1.25.
Let φ : R → S be a morphism of Ξ . If | Legs( φ ( v )) ∩ Legs( φ ( w )) | > , then v = w .Proof. Suppose v (cid:54) = w . Let e, e (cid:48) ∈ Legs( φ ( v )) ∩ Legs( φ ( w )). Let P be the shortestpath in φ ( v ) from e to e (cid:48) and let P (cid:48) be the shortest path in φ ( w ) from e to e (cid:48) .Since P and P (cid:48) are also distance minimizing paths in S , uniqueness implies that P = P (cid:48) . If e (cid:54) = e (cid:48) , this path contains a vertex, hence ∅ (cid:54) = Vt( φ ( v )) ∩ Vt( φ ( w ))and we see that v = w by Definition 1.13(3). (cid:3) Lemma 1.26.
Suppose that φ : R → S is a morphism of Ξ . If φ ( e ) = φ ( e (cid:48) ) ,then e and e (cid:48) lie on a common linear subgraph (which may just mean e = e (cid:48) ), allof whose edges map to a common value. Proof.
Induct on d ( e, e (cid:48) ). If d ( e, e (cid:48) ) = 0 then e = e (cid:48) and the result follows. If d ( e, e (cid:48) ) = 1 and φ ( e ) = φ ( e (cid:48) ), then the vertex adjacent to both e and e (cid:48) must bebivalent by Definition 1.13(1). Assume the result is known for d ( e, e (cid:48) ) < n . Suppose φ ( e ) = φ ( e (cid:48) ) = s ∈ Ed( S ) with d ( e, e (cid:48) ) = n >
1. Let e v e v . . . v n − e n − v n e n be the distance minimizing path in R from e = e to e (cid:48) = e n .For each i , let P i be the shortest path in φ ( v i ) from φ ( e i − ) to φ ( e i ). Thepath P . . . P n − contains no repeated entries by Lemma 1.23 and Definition 1.13(3),hence is the unique length minimizing path (Proposition 1.18) from s to φ ( e n − ).Both of these edges are in φ ( v n ), hence P . . . P n − is a path in φ ( v n ). If s (cid:54) = φ ( e n − ), then P . . . P n − contains a vertex, violating Definition 1.13(3). Thus φ ( e n ) = s = φ ( e n − ), so v n is bivalent. The result now follows from the inductionhypothesis since d ( e, e n − ) < d ( e, e (cid:48) ). (cid:3) We will momentarily (in 1.28) define the image of a map, which is essentiallythe union of all of the subgraphs φ ( v ). We first check that this union actually isa subgraph. Proposition 1.27.
Suppose that φ : R → S is a morphism in Ξ and R (cid:54) = η . Then (cid:91) v ∈ Vt( R ) φ ( v ) is a subgraph of S .Proof. Suppose that R contains a vertex. Let P = v e . . . v n − e n − v n be a pathin R containing all vertices at least once. Then we haveIm( φ ) = φ ( v ) ∪ φ ( v ) ∪ · · · ∪ φ ( v n ) . Use induction. By Lemma 1.10 we know that (cid:16) φ ( v ) ∪ · · · ∪ φ ( v k ) (cid:17) ∪ φ ( v k +1 )is a subgraph since φ ( v ) ∪ · · · ∪ φ ( v k ) and φ ( v k +1 ) are (induction hypothesis)and φ ( e k ) ∈ φ ( v k ) ∩ φ ( v k +1 ) . (cid:3) Definition 1.28.
Let φ : R → S be a map in Ξ. Define the image of φ , denotedIm( φ ), to be the subgraphIm( φ ) = (cid:40) | φ ( e ) R = η (cid:83) v ∈ Vt( R ) φ ( v ) Vt( R ) (cid:54) = ∅ of S . Proposition 1.29.
Suppose that φ : R → S is in Ξ . Then φ (Legs( R )) =Legs(Im( φ )) .Proof. The desired identity is clear when R = η is an edge. We show that thedesired equality holds when R contains a vertex. If s ∈ Ed(Im( φ )) and φ − ( s ) = ∅ ,then s ∈ Int( φ ( v )) ⊆ Int(Im( φ )) for some v , and also s / ∈ φ (Legs( R )).Hence, for the remainder of the proof, we will only consider edges s ∈ Ed(Im( φ ))so that φ − ( s ) is nonempty. We will write L s for the linear subgraph of R , guar-anteed by Lemma 1.26, with Ed( L s ) = φ − ( s ). Note that if Nbhd( v ) ∩ Ed( L s ) isnot empty, then v / ∈ Vt( L s ) if and only if φ ( v ) is not an edge. IGHER CYCLIC OPERADS 15
To show that Legs(Im( φ )) ⊆ φ (Legs( R )), we prove the equivalent statement:if s ∈ Ed(Im( φ )) and φ − ( s ) ⊆ Int( R ), then s ∈ Int(Im( φ )). We have alreadyestablished this when φ − ( s ) = ∅ . If ∅ (cid:54) = φ − ( s ) ⊆ Int( R ), then there existdistinct vertices v , v ∈ Vt( R ) with v i / ∈ Vt( L s ) and Nbhd( v i ) ∩ Ed( L s ) (cid:54) = ∅ . As φ ( v i ) is not equal to | s , it contains a vertex w i adjacent to s ∈ Legs( φ ( v i )). SinceVt( φ ( v )) ∩ Vt( φ ( v )) = ∅ , we know that w (cid:54) = w , hence s ∈ Int(Im( φ )).Let us turn to the reverse inclusion φ (Legs( R )) ⊆ Legs(Im( φ )). Suppose r ∈ Ed( R ) and suppose that s = φ ( r ) ∈ Int(Im( φ )). We must show r ∈ Int( R ). Byassumption, there exist distinct vertices w , w ∈ Vt(Im( φ )) with s ∈ Nbhd( w i ).There exist unique v , v ∈ Vt( R ) with w i ∈ Vt( φ ( v i )). If v were equal to v , thenwe would have s ∈ Int( φ ( v )). This is impossible by Lemma 1.24, hence v (cid:54) = v .Now Nbhd( v i ) ∩ Ed( L s ) (cid:54) = ∅ and v i / ∈ Vt( L s ), hence we have Ed( L s ) ⊆ Int( R ).Since r ∈ Ed( L s ) ⊆ Int( R ), we have completed our proof that φ − (Int(Im( φ )) ⊆ Int( R ). (cid:3) Notice that if T ⊆ T (cid:48) are subgraphs of R , then Im( φ | T ) ⊆ Im( φ | T (cid:48) ). Also, notethat if T, T (cid:48) are two subgraphs of R with Vt( T ) = Vt( T (cid:48) ) (cid:54) = ∅ , then T = T (cid:48) . Lemma 1.30.
Let φ : R → S be a morphism. Suppose that T, T (cid:48) ∈ Sbgph( R ) aretwo subgraphs which overlap. Then the following hold: Im( φ | T ∩ T (cid:48) ) = Im( φ | T ) ∩ Im( φ | T (cid:48) )(2) Im( φ | T ∪ T (cid:48) ) = Im( φ | T ) ∪ Im( φ | T (cid:48) ) . (3) Proof.
We start with the case when T (or T (cid:48) ) is a single edge | e . Since T and T (cid:48) are assumed to overlap, T ⊆ T (cid:48) . Then T ∩ T (cid:48) = | e and T ∪ T (cid:48) = T (cid:48) . Both sides of(2) are | φ ( e ) , while both sides of (3) are Im( φ | T (cid:48) ).It remains to prove the result when T and T (cid:48) each contain at least one vertex.For (3), we haveIm( φ | T ) ∪ Im( φ | T (cid:48) ) = (cid:91) v ∈ Vt( T ) φ ( v ) ∪ (cid:91) w ∈ Vt( T (cid:48) ) φ ( w ) = (cid:91) v ∈ Vt( T ) ∪ Vt( T (cid:48) ) φ ( v ) = Im( φ | T ∪ T (cid:48) ) . Thus (3) holds.Since T ∩ T (cid:48) is contained in T (and in T (cid:48) ), it is automatic that ⊆ of (2) holds.We haveVt(Im( φ | T )) ∩ Vt(Im( φ | T (cid:48) )) = (cid:91) v ∈ Vt( T ) Vt( φ ( v )) ∩ (cid:91) w ∈ Vt( T (cid:48) ) Vt( φ ( w )) = (cid:91) v ∈ Vt( T ) ∩ Vt( T (cid:48) ) Vt( φ ( v ))by Definition 1.13(3). This last set is of course just Vt(Im( φ | T ∩ T (cid:48) )). We have twocases to consider: • If Vt(Im( φ | T ∩ T (cid:48) )) (cid:54) = ∅ , then (2) holds since both sides are subgraphs thathave the same non-empty set of vertices. • If Vt(Im( φ | T ∩ T (cid:48) )) = ∅ , then both sides of (2) are a single edge. But wealready saw that ⊆ of (2) holds, so (2) holds. (cid:3) Lemma 1.31. If ( α , α ) is a complete morphism (Definition 1.12) from R to S ,then ( α , α ◦ ☆ ) is a morphism (Definition 1.13) from R to S .Proof. For concision, we write ˇ α := α ◦ ☆ in this proof and the next. Condi-tions (1) and (2) of Definition 1.13 follow immediately from ð ◦ α = ( M α ) ◦ ð . For (3), suppose that v (cid:54) = w and induct on d ( v, w ). If d ( v, w ) = 1, then ☆ v ∩ ☆ w is an edge, hence α ( ☆ v ∩ ☆ w ) = α ( ☆ v ) ∩ α ( ☆ w ) = ˇ α ( v ) ∩ ˇ α ( w )is an edge, thus (3) holds. Assume the result is known for distances less than n , and suppose ve v e . . . e n − v n − e n w is a minimal path from v to w . Let T v = ☆ v ∪ (cid:83) n − i =1 ☆ v i and T w = ☆ w ∪ (cid:83) n − i =1 ☆ v i ; by the induction hypothesis,Vt(ˇ α ( w )) and Vt(ˇ α ( v )) are both disjoint from (cid:83) n − i =1 Vt(ˇ α ( v i )). On the otherhand, Vt(ˇ α ( v )) ∩ Vt(ˇ α ( w )) ⊆ Vt( α ( T v )) ∩ Vt( α ( T w )) = Vt( α ( T v ∩ T w )) = (cid:83) n − i =1 Vt(ˇ α ( v i )), proving the result. (cid:3) Proposition 1.32.
Morphisms and complete morphisms from R to S are in bijec-tive correspondence. Precisely, the assignment that sends ( α , α ) to ( α , α ◦ ☆ ) constitutes a bijective function from the complete morphisms to the the morphisms.Proof. We saw in the previous lemma that this function is well-defined. We nowconstruct an inverse. Suppose that φ = ( φ , φ ) ∈ Ξ( R, S ). If T ∈ Sbgph( R ), defineˆ φ ( T ) = Im( φ | T ). Let us verify that ( φ , ˆ φ ) is a complete morphism.If T = | e , then ð ( T ) = e , so ð ˆ φ ( | e ) = ð ( | φ ( e ) ) = ( φ ( e )) = ( M φ ) e = ( M φ ) ð ( | e ) . If T contains a vertex, then ð ˆ φ ( T ) = ( M φ ) ð ( T ) by Proposition 1.29 applied to φ | T . Thus the first condition of Definition 1.12 holds for the pair ( φ , ˆ φ ) . Thesecond condition of this definition is guaranteed by Lemma 1.30.We only need to check that φ (cid:55)→ ( φ , ˆ φ ) is inverse to ( α , α ) (cid:55)→ ( α , ˇ α ) =( α , α ◦ ☆ ). But ˇˆ φ ( v ) = ˆ φ ( ☆ v ) = Im( φ | ☆ v ) = φ ( v )so this function is a right inverse. It only remains to check that this function is aleft inverse as well. If T = | e is an edge, we haveˆˇ α ( T ) = Im((Υ α ) | T ) = (cid:40) | α ( e ) T = η (cid:83) v ∈ Vt( T ) ˇ α ( v ) Vt( T ) (cid:54) = ∅ so if T is an edge we are done because α ( | e ) = | α ( e ) . If T contains a vertex, thenˆˇ α ( T ) = (cid:91) v ∈ Vt( T ) α ( ☆ v ) = α (cid:91) v ∈ Vt( T ) ☆ v = α ( T )since α preserves unions. Thus ˆˇ α = α . (cid:3) As the collection of trees together with complete morphisms obviously forms acategory, there is then an induced operation ◦ so that graphs and morphisms form acategory. If one traces through the construction of the inverse in Proposition 1.32,we see that this operation takes the following form. IGHER CYCLIC OPERADS 17
Definition 1.33 (Composition in Ξ) . Let φ : R → S and ψ : S → T be morphismsof Ξ. Define two functions ( ψ ◦ φ ) : Ed( R ) → Ed( T )( ψ ◦ φ ) : Vt( R ) → Sbgph( T )by ( ψ ◦ φ ) = ψ ◦ φ and ( ψ ◦ φ ) ( v ) = Im( ψ | φ ( v ) ) . Cyclic dendroidal sets.
Define the category of cyclic dendroidal sets to bethe presheaf category
Set Ξ op . Precomposition with the functor ι : Ω → Ξ inducesa functor ι ∗ : Set Ξ op → Set Ω op from cyclic dendroidal sets to dendroidal sets. This functor has both adjoints,though we will only need the left adjoint ι ! : Set Ω op → Set Ξ op in this paper.Given a tree S , we will write Ξ[ S ] = hom Ξ ( − , S ) for the object represented by S . Definition 1.34.
Let S ∈ Ξ be a tree. • Suppose that δ is a coface map (Definition 1.16) with codomain S . Thenthe δ -horn of S is the subobjectΛ δ Ξ[ S ] = (cid:91) d : R → Sδ (cid:54)∼ = d d ∗ Ξ[ R ] ⊆ Ξ[ S ]where the union is over all coface maps which are not isomorphic (over S )to δ . This is an inner horn if δ is an inner coface; otherwise it is an outerhorn . • Suppose S (cid:54) = η . The Segal core of Ξ[ S ], denoted Sc[ S ], is defined to be theunion (cid:91) v ∈ Vt( S ) Ξ[ ☆ v ] ⊆ Ξ[ S ] , where ☆ v is regarded as a subobject of S . If S = η , it is convenient to alsodefine Sc[ η ] = Ξ[ η ].Likewise, if T ∈ Ω is a rooted tree, we have horns Λ δ Ω[ T ] ⊆ Ω[ T ] and Segal coresSc[ T ] = (cid:83) Ω[ ☆ v ] ⊆ Ω[ T ].2. Using rooting to orient maps in
ΞIn this section we give a careful comparison of the morphism sets of Ξ andΩ. Morally, the category Ξ is built up from the dendroidal category Ω by addingisomorphisms which rotate trees. Thus, every morphism of Ξ should decomposeinto an oriented map (in Ω) along with some rotation data.In the present section we make this precise. To each tree S and a choice of root s ∈ Legs( S ), there is a rooted tree T ( S, s ) ∈ Ω (see Definition 2.1 and Figure 8).Further, given a morphism φ : R → S we can transform φ into an oriented map(Lemma 2.9) L ( φ ) : T ( R, r φ ) → T ( S, s )for some particular choice of root r φ ∈ Legs( R ) (see Definition 2.4). We show that L respects composition in a certain sense (Proposition 2.11, Remark 2.12). Finally,in Theorem 2.13 (see also Corollary 2.16 and Corollary 2.17) we realize our goal and make clear the idea that (non-constant) maps R → S are just certain maps inΩ along with rooting data for S . Definition 2.1 (Rooting of trees) . Suppose we are given a pair (
S, s ) with S ∈ Ob(Ξ) and s ∈ Legs( S ). • Assume that S (cid:54) = η . We now define a rooted tree T ∈ Ob(Ω) with Ed( T ) =Ed( S ) and Vt( T ) = Vt( S ). For each v ∈ Vt( S ), let ord v ( k v ) = out( v ) ∈ Nbhd( v ) be the element which minimizes the function d ( − , s ) | Nbhd( v ) . Set-ting in( v ) = Nbhd( v ) \ out( v ), we have an induced ordering on in( v ) via { , , . . . , n v } Nbhd( v ) { , , . . . , n v } { out( v ) } (cid:113) in( v ) { , . . . , n v } in( v ) . ord v ∼ =+ k v mod ( n v +1) ∼ = Similarly, we have an induced ordering on in( T ) = Legs( S ) \ { s } via { , , . . . , n } Legs( S ) { , , . . . , n } { s } (cid:113) in( T ) { , . . . , n } in( T ) , ord ∼ =+ k mod ( n +1) ∼ = where k = ord − ( s ). Although ιT may be different from S , since theyhave different total orderings (though the same cyclic orderings [38]) ordand ord v , there is a unique isomorphism f : ιT → S of Ξ with f = id byExample 1.14. • We write T ( S, s ) = ( T, f : ιT ∼ = → S )for this construction. The second component is redundant (since we insistthat f = id), so we will usually abuse notation and just write T ( S, s ) = T .Since the trivial tree η is already rooted, we also set T ( η, e ) = η . • If r ∈ Legs( R ) and s ∈ Legs( S ), define A r ,s to be the compositeΩ( T ( R, r ) , T ( S, s )) Ξ( ι T ( R, r ) , ι T ( S, s )) Ξ( R, S ) . ι ∼ =Ξ( f − R ,f S ) An example is given in Figure 8.
Proposition 2.2.
Let
R, U, S ∈ Ob(Ξ) , r ∈ Legs( R ) , u ∈ Legs( U ) , and s ∈ Legs( S ) . Then the diagram Ω( T ( U, u ) , T ( S, s )) × Ω( T ( R, r ) , T ( U, u )) Ω( T ( R, r ) , T ( S, s ))Ξ( U, S ) × Ξ( R, U ) Ξ(
R, S ) ◦ A u ,s × A r ,u A r ,s ◦ commutes. IGHER CYCLIC OPERADS 19
012 01 23 0
012 30 12 0 201 01 23 0012 12 30 0 120 01 23 0
Figure 8.
A tree S together with its four associated rooted trees Proof.
Let g ∈ Ω( T ( U, u ) , T ( S, s )) and h ∈ Ω( T ( R, r ) , T ( U, u )). The dia-gram ι ( T ( R, r )) ι ( T ( U, u )) ι ( T ( S, s )) R U S ι ( h ) ∼ = f R ι ( g ◦ h ) ∼ = f U ι ( g ) ∼ = f S A r ,u ( h ) A u ,s ( g ) commutes, hence A u ,s ( g ) ◦ A r ,u ( h ) = f S ◦ ι ( g ◦ h ) ◦ f − R = A r ,s ( g ◦ h ) . (cid:3) Lemma 2.3.
Suppose that R is a subgraph of S , s ∈ Legs( S ) , and r ∈ Ed( R ) minimizes d ( − , s ) | Ed( R ) . Then r ∈ Legs( R ) and d ( r, s ) = d ( r, r ) + d ( r , s ) for any r ∈ Ed( R ) . In particular, the minimizing element r is unique.Proof. Let P be the minimal path from r to s . Then P contains no edges of R other than r by assumption. Let P (cid:48) be the minimal path from r to r . Since P (cid:48) is actually a path in R , we have that P and P (cid:48) only have the single edge r incommon. Thus P (cid:48) P has no repeated edges, and thus is a minimal path from r to s by the characterization in Proposition 1.18. Thus | P (cid:48) P | − | P (cid:48) | −
1) + ( | P | − (cid:3) Given
R, S ∈ Ob(Ξ), we write Ξ ( R, S ) ⊆ Ξ( R, S ) for the set of maps whichfactor through the vertex-free graph η . If R is linear then Ξ ( R, S ) ∼ = Ed( S ),otherwise Ξ ( R, S ) = ∅ . A particular case of Lemma 1.26 says that if φ : R → S is a map in Ξ with φ | Legs( R ) not injective, then φ ∈ Ξ ( R, S ). Combining thisfact with φ (Legs( R )) = Legs(Im( φ )) (Proposition 1.29) and Lemma 2.3 applied toIm( φ ) (cid:44) → S gives that the following function is well-defined. Definition 2.4 (‘Find root’ function) . Let s ∈ Legs( S ). Define a function (cid:12) s Ξ( R, S ) \ Ξ ( R, S ) Legs( R ) φ r φ , (cid:12) s where r φ minimizes the function d ( φ ( − ) , s ) | Legs( R ) .Lemma 2.3 also guarantees that (cid:12) s ( φ ) minimizes the function d ( φ ( − ) , s ),though of course there may be internal edges which also minimize this functionsince φ need not be injective. Remark 2.5. If φ ∈ Ξ ( R, S ), then R must be a linear tree. The function d ( φ ( − ) , s ) is constant , hence is minimized by each of the two extremal edgeswhen R ∼ = L n for n > (cid:12) s , in particular with respect to subgraph inclusions and certaincompositions. Remark 2.6.
Let φ : R → S be a map in Ξ which does not factor through η . If s ∈ Legs( S ), then φ ( (cid:12) s ( φ )) = (cid:12) s (Im( φ ) (cid:44) → S ) . In a similar vein, we have the following lemma.
Lemma 2.7.
Let φ : R → S (cid:54) = η be a morphism of Ξ with Im( φ ) = S . Suppose s ∈ Legs( S ) and write r = (cid:12) s ( φ ) (i.e., φ − ( s ) ∩ Legs( R ) = { r } ) . If v ∈ Vt( R ) is such that φ ( v ) is not an edge, then φ (cid:16) (cid:12) s (cid:16) ☆ v (cid:44) → R φ → S (cid:17)(cid:17) = φ ( (cid:12) r ( ☆ v (cid:44) → R )) . In particular, (cid:12) s (cid:16) ☆ v (cid:44) → R φ → S (cid:17) = (cid:12) r ( ☆ v (cid:44) → R ) . Proof.
The second statement follows from the first since φ | Nbhd( v ) is injective when-ever φ ( v ) is not an edge.Suppose that the elements e = (cid:12) s ( ☆ v → R → S ) and e = (cid:12) r ( ☆ v → R ) aredistinct. Let e v e . . . e n − v n e n be the minimal path from e to e n = r = (cid:12) s ( φ ).Since d ( e , r ) < d ( e , r ), the path e v e v . . . v n e n , where v = v , is a minimalpath from e to e n . For i = 1 , . . . , n , let P i be the minimal path from φ ( e i − ) to φ ( e i ); the path P i is a path in φ ( v i ). Since the v i are distinct, P P . . . P n containsno repeated entries, hence is the minimal path from φ ( e ) to φ ( r ) = s . Likewise, P . . . P n is the minimal path from φ ( e ) to s . Since d ( φ ( e ) , s )) ≤ d ( φ ( e ) , s ),the paths P P . . . P n and P . . . P n are equal. This implies that the path P from φ ( e ) to φ ( e ) does not contain a vertex, hence φ ( e ) = φ ( e ). (cid:3) Lemma 2.8.
Suppose that A (cid:54) = η and that A (cid:44) → B (cid:44) → C is a pair of subgraph inclusions. If c ∈ Legs( C ) and b = (cid:12) c ( B (cid:44) → C ) ∈ Legs( B ) ,then (cid:12) b ( A (cid:44) → B ) = (cid:12) c ( A (cid:44) → C ) . IGHER CYCLIC OPERADS 21
Proof.
Write a = (cid:12) b ( A (cid:44) → B ) and apply Lemma 2.3 twice to get d ( a, b ) = d ( a, a ) + d ( a , b )(4) d ( a, c ) = d ( a, b ) + d ( b , c )(5)for any a ∈ Ed( A ) ⊆ Ed( B ). For the particular case when a = a , (5) becomes d ( a , c ) = d ( a , b ) + d ( b , c ). Combining with (4) we have d ( a, b ) = d ( a, a ) + d ( a , c ) − d ( b , c ) , hence d ( a, c ) = d ( a, b ) + d ( b , c ) = d ( a, a ) + d ( a , c ) . Then a is the element of Ed( A ) which minimizes this function, hence a = (cid:12) c ( A (cid:44) → C ) . (cid:3) Orientation of maps.Lemma 2.9.
Let r ∈ Legs( R ) and s ∈ Legs( S ) . There is a (unique) function L = L r ,s so that the following diagram commutes. (cid:12) − s ( r ) Ω( T ( R, r ) , T ( S, s ))Ξ( R, S ) \ Ξ ( R, S ) Ξ(
R, S ) L A r ,s If φ ∈ Ξ( R, S ) \ Ξ ( R, S ), we will use L s ( φ ) as shorthand for L (cid:12) s ( φ ) ,s ( φ ). Proof.
Let φ : R → S be in (cid:12) − s ( r ). To see that the functions φ : Ed( R ) → Ed( S ) φ : Vt( R ) → Sbgph( S )determine a map T ( R, r ) → T ( S, s ) in Ω, we just need to establish that, foreach v ∈ Vt( R ), we have φ (out( v )) = out( φ ( v )). There is nothing to prove in thecase φ ( v ) is a single edge. For concision, write ¯ r for the element (see Remark 2.6)¯ r := φ ( r ) = φ ( (cid:12) s ( φ )) = (cid:12) s (Im( φ ) (cid:44) → S ) ∈ Legs(Im( φ )) = φ (Legs( R )) . In the case when φ ( v ) is not an edge, we haveout( φ ( v )) = (cid:12) s ( φ ( v ) (cid:44) → S )= (cid:12) ¯ r ( φ ( v ) (cid:44) → Im( φ )) Lemma 2.8= φ ( (cid:12) ¯ r ( ☆ v (cid:44) → R → Im( φ )) Remark 2.6= φ ( (cid:12) r ( ☆ v (cid:44) → R )) Lemma 2.7= φ (out( v )) . Definition 2.1 (cid:3)
Lemma 2.10. If r (cid:54) = r ∈ Legs( R ) , then A r ,s (cid:16) Ω( T ( R, r ) , T ( S, s )) (cid:17) ∩ A r ,s (cid:16) Ω( T ( R, r ) , T ( S, s )) (cid:17) ⊆ Ξ ( R, S ) . Proof. If φ ∈ A r i ,s (cid:16) Ω( T ( R, r i ) , T ( S, s )) (cid:17) \ Ξ ( R, S ), then (cid:12) s ( φ ) = r i . (cid:3) The following proposition describes precisely how the functions L r ,s behavewith respect to composition in Ξ. Special cases of the first part have alreadyappeared in Lemma 2.7 and Lemma 2.8. Proposition 2.11.
Consider a composite R ψ → U φ → S in Ξ , and let s ∈ Legs( S ) . Suppose that φ ◦ ψ / ∈ Ξ ( R, S ) . If u = (cid:12) s ( φ ) , then (cid:12) u ( ψ ) = (cid:12) s ( φ ◦ ψ ) and L s ( φ ) ◦ L u ( ψ ) = L s ( φ ◦ ψ ) . Proof.
Set u = (cid:12) s ( φ ) and r = (cid:12) u ( ψ ). Omitting the subsccripts, we know thatthe diagram (cid:12) − s ( u ) × (cid:12) − u ( r )Ω( T ( U, u ) , T ( S, s )) × Ω( T ( R, r ) , T ( U, u )) Ξ( U, S ) × Ξ( R, U )Ω( T ( R, r ) , T ( S, s )) Ξ( R, S ) L × L ◦ A × A ◦ A commutes by Proposition 2.2 and Lemma 2.9. Further, ( φ, ψ ) is an element in theapex. Thus φ ◦ ψ ∈ A r ,s (cid:16) Ω( T ( R, r ) , T ( S, s )) (cid:17) , so (cid:12) s ( φ ◦ ψ ) = r . Thus we have established the first statement.The second statement is now immediate since the underlying maps of L s ( φ ), L u ( ψ ), and L s ( φ ◦ ψ ) are just φ i , ψ i , and ( φ ◦ ψ ) i by the proof of Lemma 2.9. (cid:3) Remark 2.12.
Let S ∈ Ξ, and let
C ⊆ Ξ ↓ S be the full subcategory with objectset (cid:97) R ∈ Ξ Ξ( R, S ) \ Ξ ( R, S ) . A restatement of the last part of the proof of Proposition 2.11 is that for each s ∈ S , there is a functor C → Ω ↓ T ( S, s ) which on objects sends φ to L s ( φ ). Itis possible to extend this functor to the larger full subcategory C ⊆ C (cid:48) ⊆ Ξ ↓ S whichincludes the objects Ξ( η, S ), so that C (cid:48) → Ω ↓ T ( S, s ) ι → Ξ ↓ S is isomorphic tothe inclusion. It is not generally possible to extend the functor to all of Ξ ↓ S .If T and T (cid:48) are rooted trees, let Ω ( T, T (cid:48) ) ⊆ Ω( T, T (cid:48) ) denote the subset oforiented maps which factor through η . Notice that every morphism of Ξ from T to T (cid:48) that factors through η is automatically oriented, so Ω ( T, T (cid:48) ) = Ξ ( T, T (cid:48) ); wemake the distinction in notation only for emphasis.For the remainder of the section, if s ∈ Legs( S ), we will write A s : (cid:97) r ∈ Legs( R ) Ω( T ( R, r ) , T ( S, s )) → Ξ( R, S )for the coproduct of the A r,s . IGHER CYCLIC OPERADS 23
Theorem 2.13.
Suppose that s ∈ Legs( S ) . The function A s restricts to a bijec-tion A ncs : (cid:97) Legs( R ) (cid:16) Ω( T ( R, r ) , T ( S, s )) \ Ω ( T ( R, r ) , T ( S, s )) (cid:17) → Ξ( R, S ) \ Ξ ( R, S )= (cid:97) Legs( R ) (cid:12) − s ( r ) . Proof.
There is a diagram (cid:96)
Legs( R ) (cid:16) Ω( T ( R, r ) , T ( S, s )) \ Ω ( T ( R, r ) , T ( S, s )) (cid:17) Ξ( R, S ) \ Ξ ( R, S ) (cid:96) Legs( R ) Ω( T ( R, r ) , T ( S, s )) Ξ( R, S ) . A ncs L s A s The bottom triangle commutes by Lemma 2.9. Given ψ : R → S , L s ( ψ ) : T ( R, r ) → T ( S, s ) is the unique map of Ω so that A s ( L s ( ψ )) = ψ . If ψ = A s ( φ : T ( R, r ) → T ( S, s )), then certainly φ satisfies the condition tobe L s ( A s ( φ )). Thus the top triangle commutes.Since the left vertical map is injective, so is A ncs . Further, if φ : R → S is notconstant, then L s ( φ ) ∈ Ω( T ( R, (cid:12) s ( φ )) , T ( S, s ))is also not constant, so A ncs is surjective. (cid:3) Corollary 2.14.
We have, for each s ∈ Legs( S )Iso Ξ ( R, S ) ∼ = (cid:97) Legs( R ) Iso Ω ( T ( R, r ) , T ( S, s )) via A s . Specializing to the case R = S , we have Aut Ξ ( S ) ∼ = (cid:97) s ∈ Legs( S ) Iso Ω ( T ( S, s ) , T ( S, s )) . Proof.
The statement is trivial if S (and hence R ) does not have a vertex. Other-wise, isomorphisms are not constant, so this follows from Theorem 2.13 by takingsubsets. (cid:3) Example 2.15.
Consider the tree S with Ed( S ) = Legs( S ) = { e , e , e } andVt( S ) = { v } . We can apply Corollary 2.14 to reveal some of the structure ofAut Ξ ( S ) = Σ , but A e i as i varies do not behave well together. For example, underthe compositeIso Ω (cid:16) T ( S, e ) , T ( S, e ) (cid:17) (cid:110) φ, ψ (cid:111) (cid:96) i =1 Iso Ω (cid:16) T ( S, e i ) , T ( S, e ) (cid:17) Aut Ξ ( S ) (cid:96) j =1 Iso Ω (cid:16) T ( S, e j ) , T ( S, e ) (cid:17) A e ∼ = A − e ∼ =4 P. HACKNEY, M. ROBERTSON, AND D. YAU (where, say, φ ( e ) = e , φ ( e ) = e , φ ( e ) = e , ψ ( e ) = e , ψ ( e ) = e , ψ ( e ) = e ) φ and ψ map to different coproduct summands. The morphism φ lands in the j = 2 component and ψ lands in the j = 3 component.Let us rephrase this. The bottom line of this diagram may be identified with thefollowing: (cid:97) [ π ] ∈ (13) \ Σ [ π ] Σ (cid:97) [ π (cid:48) ] ∈ (23) \ Σ [ π (cid:48) ] ∼ = ∼ = where the coproducts are indexed by right cosets. So the preceding paragraphreflects that A e and A e perform right coset decompositions for different stabilizersubgroups. Corollary 2.16. If R is non-linear, then A s : (cid:97) Legs( R ) Ω( T ( R, r ) , T ( S, s )) → Ξ( R, S ) is an isomorphism, with inverse given by L s .Proof. If R is non-linear, then Ξ ( R, S ) = ∅ . (cid:3) Corollary 2.17. If R (cid:54) = η is a linear graph with at least one vertex, then Legs( R ) = { r , r } and A s : Ω( T ( R, r ) , T ( S, s )) (cid:113) Ω( T ( R, r ) , T ( S, s )) → Ξ( R, S ) satisfies | A − s ( φ ) | = (cid:40) φ ∈ Ξ ( R, S )1 φ / ∈ Ξ ( R, S ) . If R = η = L , then A s is a bijection.Proof. In general, A s splits as a coproduct of A ncs with A cs : (cid:97) Legs( R ) Ω ( T ( R, r ) , T ( S, s )) → Ξ ( R, S ) . If R is linear and has at least one vertex, thenΩ ( T ( R, r ) , T ( S, s )) (cid:113) Ω ( T ( R, r ) , T ( S, s )) Ξ ( R, S )Ed( S ) (cid:113) Ed( S ) Ed( S ) ∼ = A cs ∼ =id (cid:113) id is two-to-one and A ncs is injective by Theorem 2.13. Hence the first statement isproved.For the second statement, if R = η then all maps R → S are constant, Legs( R ) = { r } has one element, and A s = A cs : Ω ( T ( R, r ) , T ( S, s )) → Ξ ( R, S )is isomorphic to the identity map on Ed( S ). (cid:3) IGHER CYCLIC OPERADS 25 A generalized Reedy structure on
ΞThe category ∆ of nonempty finite ordered sets is the prototypical example ofa
Reedy category . The surjective (resp. injective) maps form a wide subcategory(i.e., a subcategory which contains all of the objects of the ambient category) ∆ − (resp. ∆ + ) of morphisms which lower (resp. raise) degrees, such that any map has aunique factorization f = f + f − . Numerous inductive techniques used in the theoryof (co)simplicial objects actually work in diagrams indexed by arbitrary Reedycategories.Generalized Reedy categories were introduced in [5] and capture the dendroidalcategory Ω as an example, highlighting its similarities to ∆. We will return tothe theory of model structures on diagram categories index by a generalized Reedycategory R in Section 7, but for now we show that Ξ admits such a structure. Definition 3.1 ([5, Definition 1.1]) . A generalized Reedy structure on a smallcategory R consists of • wide subcategories R + and R − , and • a degree function d : Ob( R ) → N satisfying the following four axioms.(i) Non-invertible morphisms in R + (resp., R − ) raise (resp., lower) the degree.Isomorphisms in R preserve the degree.(ii) R + ∩ R − = Iso( R ).(iii) Every morphism f of R factors as f = gh with g ∈ R + and h ∈ R − . Thisfactorization is unique up to isomorphism in the sense that if g (cid:48) h (cid:48) is anothersuch factorization, then there is an isomorphism θ so that θh = h (cid:48) and g = g (cid:48) θ .(iv) If θf = f for θ ∈ Iso( R ) and f ∈ R − , then θ is an identity.If, moreover, the condition(iv’) If f θ = f for θ ∈ Iso( R ) and f ∈ R + , then θ is an identityholds, then we call this a generalized dualizable Reedy structure.An ordinary Reedy category is a generalized Reedy category where there are noisomorphisms other than the identity maps.
Definition 3.2.
Consider the following structures on Ξ. • The degree function d : Ob(Ξ) → N with d ( S ) = | Vt( S ) | . • The wide subcategory Ξ + consisting of all maps φ : R → S so that φ : Ed( R ) → Ed( S ) is injective. • The wide subcategory Ξ − consisting of all maps φ : R → S so that φ issurjective and, for each vertex v ∈ Vt( S ), there exists a vertex w ∈ Vt( R )with v ∈ Vt( φ ( w )).This definition is chosen to be compatible with the known generalized Reedystructure on the dendroidal category Ω from Definition 1.20, in the sense that theequalities Ω + = Ω ∩ Ξ + and Ω − = Ω ∩ Ξ − hold, and the degree functions agree.Both inner and outer cofaces from Definition 1.16 are in Ξ + , while codegeneraciesare in Ξ − . In fact, one can show that Ξ + is generated by the cofaces and Ξ − is See [5, Example 2.8] and the minor correction in [10, p. 216]. generated by the codegeneracies, though we do not need this here. Notice that(6) Ξ ( R, S ) ∩ Ξ + ( R, S ) ∼ = (cid:40) Ed( S ) if R = η ∅ otherwiseis nonempty if and only if R = η . The following lemma is immediate. Lemma 3.3.
Given s ∈ Ed( S ) , the map L : Ξ( R, S ) \ Ξ ( R, S ) → (cid:97) r ∈ Legs( R ) Ω( T ( R, r ) , T ( S, s )) restricts to maps Ξ + ( R, S ) \ Ξ ( R, S ) → (cid:97) r ∈ Legs( R ) Ω + ( T ( R, r ) , T ( S, s ))Ξ − ( R, S ) \ Ξ ( R, S ) → (cid:97) r ∈ Legs( R ) Ω − ( T ( R, r ) , T ( S, s )) . Proposition 3.4.
With the structure from Definition 3.2, Ξ is a dualizable gener-alized Reedy category.Proof. For (i): note that isomorphisms preserve degree. If φ : R → S ∈ Ξ + , then { Vt( φ ( v )) } Vt( R ) is a collection of pairwise disjoint, non-empty subsets of Vt( S ).Thus d ( R ) = | Vt( R ) | = (cid:88) Vt( R ) ≤ (cid:88) v ∈ Vt( R ) | Vt( φ ( v )) | ≤ | Vt( S ) | = d ( S ) . If φ : R → S ∈ Ξ − , then surjectivity of φ implies (by Lemma 1.23) that | Vt( φ ( v )) | ≤ v ∈ Vt( R ). Let Vt( S ) → Vt( R ) be the map which sends v ∈ Vt( S ) to the(unique, by Definition 1.13(3)) vertex w with v ∈ Vt( φ ( w )). Since there is at mostone v in a given φ ( w ), the map Vt( S ) → Vt( R ) is injective, hence d ( S ) ≤ d ( R ).For (ii), it is clear that Iso(Ξ) is contained in Ξ + ∩ Ξ − . For the reverse inclusion,suppose that φ : R → S is in both Ξ + and Ξ − . If φ is constant, then by (6) wemust have R = η ; since φ is also in Ξ − , S is also an edge, and φ is an isomorphism.If φ is not constant, choose a root s for S . Then by Lemma 3.3 we know that L ( φ ) is inΩ + ( T ( R, r ) , T ( S, s )) ∩ Ω − ( T ( R, r ) , T ( S, s )) = Iso Ω ( T ( R, r ) , T ( S, s ))where r = (cid:12) s ( φ ). By Corollary 2.14, we thus have that φ is an isomorphism inΞ. Since φ was arbitrary, we have Ξ + ∩ Ξ − ⊆ Iso(Ξ) as well.For (iv), note that if θ is an isomorphism, φ : R → S ∈ Ξ − , and θφ = φ , then θ φ = φ . Since θ is a bijection of sets and φ is a surjection of sets, it followsthat θ is an identity. There is only one isomorphism S → S in Ξ which is theidentity on edges (Example 1.14), hence θ = id S . The proof that Ξ satisfies (iv’)follows similarly.We finally turn to (iii). We first construct a factorization of a given morphism ofΞ. We may assume that φ ∈ Ξ( R, S ) \ Ξ ( R, S ). Pick a root s for S , and consider L ( φ ) ∈ Ω( T ( R, r ) , T ( S, s )) IGHER CYCLIC OPERADS 27 where r = (cid:12) s ( φ ). Then there is a decomposition L ( φ ) = g ◦ h with g ∈ Ω + ( T, T ( S, s )) and h ∈ Ω − ( T ( R, r ) , T ). We have T ( R, r ) T T ( S, s ); L ( φ ) h g apply the functor ι to this diagram to get ι ( T ( R, r )) ιT ι ( T ( S, s )) R S. ∼ = ι ( L ( φ )) ιh ιg ∼ = A ( L ( φ ))= φ Since ιh ∈ Ξ − , ιg ∈ Ξ + , and isomorphisms are in Ξ + ∩ Ξ − , we have provided thedesired decomposition of φ .Suppose that φ ◦ ψ = φ ◦ ψ with φ i ∈ Ξ + ( U i , S ) and ψ i ∈ Ξ − ( R, U i ). Let u i = (cid:12) s ( ψ i ) and r = (cid:12) u ( φ ) . = (cid:12) u ( φ ). We have, by Proposition 2.11, L u ( φ ) ◦ L s ( ψ ) = L u ( φ ) ◦ L s ( ψ ) . Now L u i ( φ i ) ∈ Ω + and L s ( ψ i ) ∈ Ω − , so there exists an isomorphism a makingthe diagram T ( R, r ) T ( U , u ) T ( S, S ) T ( R, r ) T ( U , u ) T ( S, S ) L u ( ψ )id= L s ( φ ) a ∼ = id= L u ( ψ ) L s ( φ ) commute. Applying ι gives the back square of the diagram: ι ( T ( R, r )) ι ( T ( U , u )) ι ( T ( S, S )) R U Sι ( T ( R, r )) ι ( T ( U , u )) ι ( T ( S, S )) R U S ι ( L u ( ψ )) ∼ == ι ( L s ( φ )) ∼ = ι ( a ) ∼ = ∼ == ψ φ ι ( L u ( ψ )) ∼ = ι ( L s ( φ )) ∼ = ∼ = ψ = φ = and there exists a dashed map making the diagram commute. This map is necessar-ily an isomorphism. Thus, the front face establishes uniqueness of decompositionsin Ξ. (cid:3) A stronger statement is true, namely that Ξ is an EZ-category in the sense of[5, Definition 6.7]. We delay a proof of this fact (Theorem 4.9) until the end of thenext section, as we should first learn a bit more about maps in Ξ − . The active / inert weak factorization system on
ΞIn this section we exhibit a weak factorization system on the category Ξ. Givena class I of morphisms in a category C , write I ⧄ for the maps which have the rightlifting property with respect to every element of I . In other words, f : X → Y isin I ⧄ if, and only if, every commutative square A XB Y i f with i ∈ I admits a lift B → X . Similarly, ⧄ I is the class of maps having the leftlifting property with respect to every element of I .A weak factorization system (see [40, Definition 11.2.1]) consists of two classesof maps L and R so that every morphism factors into a map in L followed by onein R , and so that ⧄ R = L and L ⧄ = R . Remark 4.1.
We have actually already encountered one weak factorization systemin this paper, namely the one whose left class is Ξ − and whose right class is Ξ + .In fact, the following is true: if R is a generalized Reedy category, then ( R − , R + ) isan orthogonal factorization system (see, e.g., [21, 2.2]), that is, a weak factorizationsystem in which all of the liftings are unique. This is mentioned in Remark 8.28of [42]; its proof is an exercise using only axioms (iii), (iv), and closure of R + , R − under composition.Any weak factorization system whose right class R is contained in the monomor-phisms will be orthogonal. This will be true, in particular, of the weak factorizationsystem in Proposition 4.7 (using Theorem 4.9 and Remark 4.3). Definition 4.2.
A morphism φ : R → S in Ξ is called active if Im( φ ) = S .It is called inert if φ : Ed( R ) → Ed( S ) is injective and if w ∈ Vt( φ ( v )) thenNbhd( w ) ⊆ Im( φ ).Outer cofaces (see Definition 1.16) are inert maps, while codegeneracies andinner cofaces are active maps. Notice that every map in Ξ − (see Definition 3.2)is an active map. Further, every inert map is contained in Ξ + by the followingremark. Remark 4.3.
Suppose that φ is inert. Then, since φ is injective, we know0 < | Vt( φ ( v )) | for any v ∈ Vt( R ). Further, if Vt( φ ( v )) contains two (adjacent)vertices w and w , connected by an edge e , then e ∈ Int( φ ( v )) ∩ Nbhd( w ) ⊆ Int( φ ( v )) ∩ Im( φ ) = ∅ by Lemma 1.24. Thus | Vt( φ ( v )) | ≤
1. In other words, φ is (isomorphic to) a subgraph inclusion.Let us spell out an alternative characterization of the active maps. The followinglemma will be useful in the proof of Proposition 4.5. Lemma 4.4.
Suppose that R ∈ Sbgph( S ) is not an edge. If Legs( R ) = Legs( S ) ,then R = S .Proof. Suppose that v ∈ Vt( S ) is connected to a different vertex w ∈ Vt( R ) byan edge e . Then e / ∈ Legs( S ) = Legs( R ), so v ∈ Vt( R ). Since S is connectedand Vt( R ) (cid:54) = ∅ , we have Vt( R ) = Vt( S ). The definition of subgraph impliesEd( R ) = Ed( S ) as well. (cid:3) IGHER CYCLIC OPERADS 29
Proposition 4.5.
Suppose φ : R → S is in Ξ .(1) Suppose φ / ∈ Ξ ( R, S ) . Then φ is active if and only if φ (Legs( R )) =Legs( S ) .(2) Suppose φ ∈ Ξ ( R, S ) . Then φ is active if and only if S = η .Proof. The forward direction of (1) follows immediately from Proposition 1.29.For the reverse direction, by hypothesis and Proposition 1.29 we have Legs( S ) =Legs(Im( φ )), so by Lemma 4.4 we have Im( φ ) = S . For (2), simply note that if φ ∈ Ξ ( R, S ), then Im( φ ) is an edge. (cid:3) Lemma 4.6.
The set of active morphisms has the left lifting property with respectto the set of inert morphisms.Proof.
Suppose we are given a commutative diagram in Ξ(7)
R PS Q φ α ψγβ with φ active and ψ inert (so, in particular, P is a subgraph of Q ); we wish toshow that a lift γ : S → P exists. Temporarily write α L : Legs( R ) → Ed( P ) and φ L : Legs( R ) → Legs( S ) for the restrictions of α and φ to legs.If S = η consists of the single edge e , then R is linear. We have a diagramLegs( R ) Ed( P ) { e } Ed( Q ) φ α L ψ β with ψ an injection. Thus Im( α L ) = | p is a single edge, and we define γ : S → P by γ ( e ) = p ∈ Ed( P ).If S contains a vertex, then φ L : Legs( R ) → Legs( S ) is a bijection. Define γ L = α L ◦ ( φ L ) − : Legs( S ) → Ed( P ). We wish to extend this to Int( S ), which wewill do in a moment. Since φ is active, every w ∈ Vt( S ) is in Vt( φ ( v )) for some v ∈ Vt( R ). Notice that (cid:91) w ∈ φ ( v ) β ( w ) = ( β ◦ φ ) ( v ) = ( ψ ◦ α ) ( v ) = (cid:91) t ∈ α ( v ) ψ ( t ) ∈ Sbgph(Im( ψ ))and, since P → Im( ψ ) is an isomorphism, Sbgph(Im( ψ )) ∼ = Sbgph( P ). Thus,for w ∈ Vt( φ ( v )), there is a unique subgraph G w ∈ Sbgph( P ) which maps to β ( w ) ∈ Sbgph(Im( ψ )) ⊆ Sbgph( Q ) under ψ .Suppose that s ∈ Int( S ) is an interior edge. The edge s is adjacent to two distinctvertices w and w (cid:48) . We have β ( w ) ∩ β ( w (cid:48) ) = { β ( s ) } , hence G w ∩ G w (cid:48) = { p } forsome edge p . Set γ ( s ) = p ; by definition we have φ γ ( s ) = β ( s ). Further, since φ is active it sends legs to legs, so if s = φ ( r ), then r is an internal edge betweendistinct vertices v and v (cid:48) , with w ∈ Vt( φ ( v )) and w (cid:48) ∈ Vt( φ ( v (cid:48) )). Since G w ⊆ α ( v ) and G w (cid:48) ⊆ α ( v (cid:48) ), we have { γ ( s ) } = G w ∩ G w (cid:48) ⊆ α ( v ) ∩ α ( v (cid:48) ) = { α ( r ) } .Thus γ φ ( r ) = γ ( s ) = α ( r ). In conclusion, α = γ φ and ψ γ = β .Next, define γ ( w ) = G w ∈ Sbgph( α ( v )). The pair γ = ( γ , γ ) is a morphismof Ξ, which quickly follows from inertness of ψ and the fact that β is a morphism ofΞ. By definition, we have ( ψ ◦ γ ) ( w ) = (cid:83) v ∈ γ ( w ) ψ ( v ) = β ( w ), hence ψ ◦ γ = β . Then ψ ◦ γ ◦ φ = β ◦ φ = ψ ◦ α , hence the injective map ψ : Sbgph( P ) (cid:44) → Sbgph( Q )takes ( γ ◦ φ ) ( v ) and α ( v ) to the same element. It follows that ( γ ◦ φ ) ( v ) = α ( v ),so γ ◦ φ = α . Thus we have shown that (7) always has a lift. (cid:3) Proposition 4.7.
The active and inert morphisms form a weak factorization sys-tem.Proof.
Given a map φ : R → S , we can factor φ as R → Im( φ ) → S . The map R → Im( φ ) is active since Legs(Im( φ )) = φ (Legs( R )), while Im( φ ) → S is asubgraph inclusion, hence inert.For the remainder of the proof, write L for the set of active morphisms and R forthe set of inert morphisms. In Lemma 4.6 we showed that R ⊆ L ⧄ (or, equivalently, L ⊆ ⧄ R ).For the reverse inclusions, suppose that φ : R → S is in L ⧄ . Consider thedecomposition R φ − −−→ T φ + −−→ S coming from the generalized Reedy structure on Ξ. We know that φ − is an activemorphism, hence we can lift in the diagram R RT S φ − id φψφ + which implies ψφ − = id, hence φ − is an isomorphism. Thus φ is injective. Further,the diagram R R
Im( φ ) S ¯ φ id φi admits a lift; if w is a vertex in φ ( v ) ⊆ Im( φ ), thenNbhd( w ) = i (Nbhd( w )) = φ ψ (Nbhd( w )) ⊆ Im( φ ) . Hence φ is inert, and we see that L ⧄ ⊆ R .Now suppose that φ : R → S is in ⧄ R . Then R Im( φ ) S S φ ¯ φ i id ψ admits a lift ψ : S → Im( φ ). Factor ψ = ψ + ψ − with ψ + ∈ Ξ + , ψ − ∈ Ξ − . Sinceid S = ( iψ + ) ψ − with iψ + ∈ Ξ + , we have iψ + , ψ − ∈ Iso(Ξ) by Definition 3.1(iii). Itfollows that i ∈ Ξ + preserves degree, hence is an isomorphism. Thus φ is active,and we find that ⧄ R ⊆ L . (cid:3) The active and inert maps which are oriented form a weak factorization systemon the category Ω. This was proved in [28, Proposition 1.3.13], where inert mapsare called ‘free’ and active maps are called ‘boundary preserving’.
IGHER CYCLIC OPERADS 31
Proposition 4.8.
The inclusion ι : Ω → Ξ respects the weak factorization struc-tures. (cid:3) As promised at the conclusion of the previous section, we have the following.
Theorem 4.9.
The category Ξ , together with the degree function d ( S ) = | Vt( S ) | from Definition 3.2, is an EZ-category in the sense of [5, Definition 6.7] .Proof. As we have already established that Ξ is a dualizable generalized Reedycategory (Proposition 3.4), it is enough to show that(a) Ξ + is the subcategory of monomorphisms,(b) Ξ − is the subcategory of split epimorphisms, and(c) any pair of split epimorphisms with common domain has an absolute pushout[39] (that is, can be extended to a commutative square which becomes a pushoutsquare after applying any functor).For efficiency, we rely on the fact that Ω is an EZ-category [5, Examples 6.8]. Everymap φ : R → S in Ξ is isomorphic (in the arrow category Ξ [1] ) to at least one map ι ( ˜ φ : ˜ R → ˜ S ) by Corollary 2.16, Corollary 2.17, and the fact that Legs( S ) (cid:54) = ∅ .Suppose that φ is a map and we have chosen an isomorphism γ : ι ˜ φ ∼ = φ in Ξ [1] .The following hold, which then imply that (a) and (b) hold by the correspondingproperties of Ω:(i) φ is a monomorphism (resp. split epimorphism) if and only if ˜ φ is a monomor-phism (resp. split epimorphism)(ii) φ is in Ξ + (resp. in Ξ − ) if and only if ˜ φ is in Ω + (resp. in Ω − ).The only point that is perhaps not immediate is that if φ is a split epimorphism, sois ˜ φ . Suppose α is a section of φ . If φ is constant then S = ˜ S = η and the composite η α −→ R γ − −−→ ι ˜ R is automatically oriented and is a section for ˜ φ . If φ is not constant,then ˜ φ is isomorphic in the arrow category Ω [1] to L s ,r ( φ ) : T ( R, r ) → T ( S, s )where s and r are the images under γ of the roots of ˜ S and ˜ R , respectively. Wethen have id T ( S,s ) = L s (id S ) = L s ( φ ◦ α ) = L s ( φ ) ◦ L r ( α )by Proposition 2.11. Since L s ( φ ) admits a section, so does ˜ φ .Let us turn to point (c). Suppose that φ : R → S and ψ : R → U are two maps inΞ − (i.e., two split epimorphisms). Pick r ∈ Legs( R ), and let s = φ ( r ) ∈ Legs( S )and u = ψ ( r ) ∈ Legs( U ) (using that φ and ψ are active). There are maps˜ φ : T ( R, r ) → T ( S, s ) and ˜ ψ : T ( R, r ) → T ( U, u ) and an isomorphism ofdiagrams of shape • ← • → • ι T ( U, u ) ι T ( R, r ) ι T ( S, s ) U R S γ U ∼ = γ R ∼ = ι ˜ ψ ι ˜ φ γ S ∼ = ψ φ in Ξ. Let(8) T ( R, r ) T ( S, s ) T ( U, u ) T ˜ ψ ˜ φ hg be an absolute pushout of ˜ φ, ˜ ψ ∈ Ω − . The diagram (8) remains an absolute pushoutafter applying ι , and the resulting square is isomorphic to the square(9) R SU ιT. ψ φ ( ιh ) γ − S ( ιg ) γ − U Thus (9) is an absolute pushout as well, and (c) is established. (cid:3) A functor from Ξ to the category of cyclic operads In this section, we will show that given a tree R , there is a cyclic (colored) operad C ( R ) ∈ Cyc with Col( C ( R )) = Ed( R ). Further, the assignment R (cid:55)→ C ( R ) is theobject part of a functor C : Ξ → Cyc . This functor is faithful but not full.5.1.
A monadic description.
Fix a color set C . A C -colored tree is a (pinned)tree S together with a function ξ : Ed( S ) → C . If c = c , c , . . . , c n is a profile in C ,we will write tree ( c ) = tree C ( c ) for the groupoid of all C -colored trees S so that { , , . . . , n } Legs( S ) Ed( S ) C ord ξ takes i to c i for 0 ≤ i ≤ n . The isomorphisms ( S, ξ ) → ( S (cid:48) , ξ (cid:48) ) in tree ( c ) are thoseisomorphisms φ : S → S (cid:48) in Ξ so that ξ = ξ (cid:48) ◦ φ and φ ◦ ord S = ord S (cid:48) .Consider the groupoid Σ + C whose objects are finite, non-empty, ordered lists c = c , c , . . . , c n ( n ≥
0) of elements of C , and morphisms are cσ σ → c , where σ ∈ Σ + m = Aut { , , . . . , m } , and cσ = ( c σ (0) , c σ (1) , . . . , c σ ( m ) ). Such a morphism ofΣ + C determines a morphism tree ( c ) → tree ( cσ ) sending S to Sσ , which has all ofthe same structure as S except that the leg ordering ord Sσ is the composite { , , . . . , m } { , , . . . , m } Legs( S ) . σ ord S This is, in fact, a contravariant functor tree ( − ) = tree C ( − ) : Σ + C → Gpd .Every object X ∈ Set Σ + C determines a functor tree C ( c ) → Set , given on anobject S ∈ tree C ( c ) by X [ S ] = (cid:89) v ∈ Vt( S ) X ( ξ (Nbhd( v ))) . Remark 5.1.
There is a monad T + : Set Σ + C → Set Σ + C given on objects by T + ( X )( c ) = colim S ∈ tree ( c ) X [ S ] . IGHER CYCLIC OPERADS 33
The category of algebras for T + is Cyc C , the subcategory of Cyc consisting of thosecyclic operads with color set C and morphisms which are the identity on colors. Ananalogous statement appears in [35, § f : C → D is a map of sets, then restriction along Σ + C → Σ + D induces a functor f ∗ : Cyc D → Cyc C . A map α : P → Q in Cyc is the same thingas a pair ( α , α ), where α : Col( P ) → Col( Q ) is a map of sets and α : P → α ∗ Q is a map in Cyc C .5.2. The functor C : Ξ → Cyc.
Let Prof( C ) be the set of non-empty ordered listsof elements in C ; it is the set of objects of the groupoid Σ + C . There is a forgetfulfunctor Cyc C → Set Σ + C → Set
Prof( C ) . Write F C : Set
Prof( C ) (cid:29) Cyc C : U C for the corresponding adjunction. Definition 5.2.
Let S be a tree and C = Ed( S ). Then S determines an object Z = Z S of Set
Prof( C ) with(10) Z c = (cid:40) { v } Nbhd( v ) = c as ordered lists ∅ otherwise.The cyclic operad C ( S ) is defined as F C ( Z ).Applying the left adjoint of Set Σ + C → Set
Prof( C ) (for C = Ed( S )) to the object Z = Z S gives a new object Z (cid:48) . This object has the property that | Z (cid:48) c | = 1 if andonly if c contains no repetitions and c = Nbhd( v ) as unordered lists for some v .Otherwise, | Z (cid:48) c | = 0.Notice that if T ∈ Ω is a rooted tree and F : Op → Cyc is left adjoint to theforgetful functor, then C ( ι ( T )) = F (Ω( T )) . Remark 5.3. If R ∈ Sbgph( S ) and R is a pinned graph, with { , . . . , n } ord R −−−→ Legs( R ) (cid:44) → Ed( S ) sending i to c i , then R determines an element of C ( S )( c , . . . , c n ; c ).There may, of course, be many elements of this set that do not come from subgraphsof S . Varying the vertex orders ord v on R does not change the element produced inthis way. The cyclic operad C ( S ) is nearly always infinite (see Example 5.7 below).Specifically, C ( S ) is infinite whenever S / ∈ { η, ☆ } , as then S has a vertex v with | v | > Example 5.4.
Let S be the graph from Figure 4. Then C ( S ) is the cyclic operad O generated by three operations u ∈ O ( b, a ; c ), v ∈ O ( d, e, f ; c ), and w ∈ O ( ; d ).We see there is an element v ◦ w ∈ O ( e, f ; c ). Notice that there are no elementsof the form q ◦ i r where r ∈ { u, v } , q ∈ { u, v, w } , as c is not an input for sucha q . But we can first apply the rotation to get, say v · τ ∈ O ( e, f, c ; d ), andthen compose to get ( v · τ ) ◦ u ∈ O ( e, f, b, a ; d ). Finally, there is an element(( v · τ ) ◦ u ) ◦ w ∈ O ( f, b, a ; e ); these are all of the elements given by subgraphsof S . Notice there are many other elements, for example u ◦ ( u · τ ) ∈ O ( a, c, a ; c ).We wish to extend S (cid:55)→ C ( S ) to a functor Ξ → Cyc . Defining maps out of C ( S )is easy, as this object is free in Cyc
Ed( S ) . We utilize Remark 5.3 in order to regardsubgraphs of S as elements in C ( S ). Definition 5.5 ( C as a functor) . Suppose φ ∈ Ξ( R, S ) and write C = Col( C ( R )) =Ed( R ). Set C ( φ ) = φ : C = Ed( R ) → Ed( S ). Since C ( R ) is free in Cyc C , it isenough to define C ( φ ) on generators. Let φ (cid:92) : Z R → U C φ ∗ C ( S ) in Set
Prof( C ) by v (cid:55)→ φ ( v ) ∈ Sbgph( S ), endowed with the ordering { , , . . . , n v } ord v −−−→ Nbhd( v ) φ → Legs( φ ( v )). Define C ( φ ) : C ( R ) → φ ∗ C ( S ) to be the adjoint of φ (cid:92) . Theorem 5.6.
The functor C : Ξ → Cyc is faithful.Proof.
Let φ, ψ ∈ Ξ( R, S ), and suppose that C ( φ ) = C ( ψ ). Write f = φ = ψ for the common function on color sets. By assumption, for each v ∈ Vt( R ), wehave that C ( φ ) ( v ) is equal to C ( ψ ) ( v ). But C ( φ ) ( v ) comes from φ ( v ) (withappropriate choice of pinned structure) as in Remark 5.3, and likewise for C ( ψ ) ( v ).It follows then that the subgraphs φ ( v ) and ψ ( v ) are the same, hence φ = ψ . (cid:3) Example 5.7 ( C : Ξ → Cyc is not full) . Consider the graph L = — • — with edges0 and 1 and vertex v . There are exactly four elements in Ξ( L , L ), correspondingto the four maps of edge sets { , } → { , } . But there are infinitely many mapsin hom Cyc ( C ( L ) , C ( L )). One example which is not in C (Ξ( L , L )) is the map f : C ( L ) → C ( L ) which on color sets is f ( i ) = 0 and on morphisms is specifiedby f ( v ) = v ◦ v : — • — • — . In Section 1.3, we saw several examples of cyclic dendroidal sets. Another classof examples are the nerves of cyclic operads.
Definition 5.8 (Nerve) . There is a functor N c : Cyc → Set Ξ op defined, on anobject O ∈ Cyc , by N c ( O ) = Cyc ( C ( − ) , O ) ∈ Set Ξ op . We will refer to N c as the cyclic dendroidal nerve .Recall there is an analogous dendroidal nerve N d : Op → Set Ω op defined by N d ( O ) T = Op (Ω( T ) , O ), where Ω( T ) is the Ed( T )-colored operad freely generatedby T . This functor is a fully-faithful embedding, and the essential image may becharacterized using inner horns and Segal cores from Definition 1.34.
Theorem 5.9 (Moerdijk–Weiss; Cisinski–Moerdijk) . Let X ∈ Set Ω op be a den-droidal set. The following are equivalent.(i’) X ∼ = N d ( O ) for some operad O .(ii’) X T = hom(Ω[ T ] , X ) → hom(Λ δ Ω[ T ] , X ) is a bijection for every inner cofacemap δ .(iii’) X T = hom(Ω[ T ] , X ) → hom(Sc[ T ] , X ) is a bijection for every rooted tree T .Proof. The equivalence of the first two was established in Proposition 5.3 and The-orem 6.1 of [37], while the third was shown to be equivalent to the first two in [14,Corollary 2.6]. (cid:3)
The next section is dedicated to establishing an analogous theorem for the cyclicdendroidal nerve.
IGHER CYCLIC OPERADS 35
Remark 5.10.
Example 5.7, shows, in particular, that the compositeΞ
Cyc Set Ξ op S N c C ( S )is not the Yoneda embedding S (cid:55)→ Ξ[ S ], since the Yoneda embedding is fully-faithful. 6. Cyclic operads and the nerve theorem
Our goal in this section is to prove Theorem 6.7. The method of proof is toreduce to the dendroidal case and apply Theorem 5.9. To do this, we first need tounderstand the relationship between dendroidal Segal cores (resp. dendroidal innerhorns) and their cyclic dendroidal analogues from Definition 1.34. The other mainingredient is to show that if X is a cyclic dendroidal set so that ι ∗ X is the nerve ofan operad, then X was already the nerve of a cyclic operad. This is Theorem 6.3. Lemma 6.1. If T ∈ Ω is a rooted tree, then ι ! (Sc[ T ] → Ω[ T ]) ∼ = (Sc[ ιT ] → Ξ[ ιT ]) . Proof.
Given S ∈ Ξ, define a category C S with Ob( C S ) = Ed( S ) (cid:113) Vt( S ), andnon-identity maps { e → v | e ∈ Nbhd( v ) } . There is a functor F : C S → Set Ξ op which on objects is given by e (cid:55)→ Ξ[ η ] and v (cid:55)→ Ξ[ ☆ | v | ]. Define F (ord v ( k ) → v )to be the inclusion k ∗ : Ξ[ η ] → Ξ[ ☆ | v | ] which hits k ∈ Ξ[ ☆ | v | ] η = { , , . . . , | v | − } .Then colim C S F ∼ = Sc[ S ]. A similar consideration applies for Sc[ T ] ⊆ Ω[ T ]; namelyif T is a rooted tree in Ω, then there is a functor (using the same domain categoryfrom above) F (cid:48) : C T → Set Ω op with colim C T F (cid:48) ∼ = Sc[ T ] ⊆ Ω[ T ].Notice that ι ! Ω[ T ] ∼ = Ξ[ ιT ] since for any X ∈ sSet Ξ op , they are the same onhom( − , X ):hom( ι ! Ω[ T ] , X ) = hom(Ω[ T ] , ι ∗ X ) = ( ι ∗ X ) T = X ιT = hom(Ξ[ T ] , X ) . In particular, ι ! (Ω[ ☆ v ]) = Ξ[ ☆ v ] and ι ! (Ω[ η ]) = Ξ[ η ]. Since ι ! is a left adjoint, itcommutes with colimits, and we have ι ! (Sc[ T ] → Ω[ T ]) ∼ = (Sc[ ιT ] → Ξ[ ιT ]) . (cid:3) Lemma 6.2.
Let δ : R → T be any coface map in Ω . Then ι ! (Λ δ Ω[ T ] (cid:44) → Ω[ T ]) ∼ = (Λ δ Ξ[ T ] (cid:44) → Ξ[ T ]) . Proof.
Our strategy is the same as in the previous lemma – we write both sides asa colimit of representables and then use that ι ! commutes with colimits and takesrepresentables to representables.Suppose that S ∈ Ξ. Let V S be a skeleton of the comma category Ξ + ↓ S ; thereis an evident functor A S : V S → Set Ξ op which sends φ : R → S to the object Ξ[ R ]and R (cid:48) RS φ (cid:48) α φ to α ∗ : Ξ[ R (cid:48) ] → Ξ[ R ]. Since V S has a terminal object (the one isomorphic to id S ),colim V S A S = Ξ[ S ]. If δ : R → S is a coface map, write V Sδ for the full subcategoryof V S that excludes the two objects isomorphic to id S and δ . Then there is anisomorphism colim V Sδ A S | V Sδ Λ δ Ξ[ S ]colim V S A S Ξ[ S ] . ∼ = ∼ = This isomorphism arises from the fact that Ξ[ R ] → Ξ[ S ] is a monomorphism when-ever R → S is in Ξ + (Theorem 4.9) and the fact that all elements of Ξ[ S ] (in par-ticular, all elements of Λ δ Ξ[ S ]) factor through a minimal face by Definition 3.1(iii).Suppose that T is a rooted tree (with root t ), and let φ : R → T be any mapin Ξ + . If R = η , then φ is automatically oriented. Assume that R has at leastone vertex. Then L t ( φ ) : T ( R, (cid:12) t ( φ )) → T ( T, t ) = T is isomorphic to φ over T and is also in Ξ + . Thus we may as well have assumed that φ was oriented inthe first place. In particular, we may assume that all objects in V T are orientedmaps. With this assumption, all morphisms in V T are also oriented maps, for if φα = φ (cid:48) in Ξ + with φ and φ (cid:48) oriented maps, then α is also oriented by Proposition2.11. Thus we have a functor B T : V T → Set Ω op sending R → T to Ω[ R ], andwith ι ! ◦ B T ∼ = A T . Further, as above, colim V Tδ B T | V Tδ is isomorphic to Λ δ Ω[ T ] overΩ[ T ]. Concluding the proof, we have ι ! Λ δ Ω[ T ] ∼ = ι ! (cid:32) colim V Tδ B T | V Tδ (cid:33) ∼ = colim V Tδ (cid:16) ι ! ◦ B T | V Tδ (cid:17) ∼ = colim V Tδ A T | V Tδ ∼ = Λ δ Ξ[ T ] . (cid:3) Theorem 6.3.
Let X ∈ Set Ξ op be a cyclic dendroidal set and O be a coloredoperad. If α : ι ∗ X ∼ = → N d ( O ) is an isomorphism of dendroidal sets, then there is aunique cyclic structure on O so that α lifts to an isomorphism between X and thecyclic dendroidal nerve of O . In other words, there is a unique object ˜ O ∈ Cyc and an isomorphism ˜ α : X → N c ( ˜ O ) such that U ( ˜ O ) = O and ˜ α ιT = α T : X ιT =( ι ∗ X ) T → N d ( O ) T = N d ( U ˜ O ) T = N c ( ˜ O ) ιT for every rooted tree T ∈ Ω .Further, if β : ι ∗ Y → N d ( P ) is another such isomorphism and f : X → Y is amorphism of Set Ξ op , then there is a (unique) morphism f (cid:48) : ˜ O → ˜ P in Cyc makingthe diagram
X N c ( ˜ O ) Y N c ( ˜ P ) ˜ αf N c ( f (cid:48) )˜ β commute. In order to prove this theorem, we need to delve a bit into how the operadstructure of O is manifested in the dendroidal set N d ( O ).Note that ☆ n +1 from Example 1.4 is a rooted tree (which in the dendroidalsetting is usually called C n ∈ Ω). Let i : η → ☆ n +1 be the (oriented) map withimage the single edge i ; if X ∈ Set Ξ op write P = (cid:81) ni =0 i ∗ : X ☆ n +1 → (cid:81) ni =0 X η . IGHER CYCLIC OPERADS 37 a b a b b ae e e
Figure 9.
The rooted trees Z , , Z , , and Z , If O is a colored operad and W = N d ( O ), then W η = Col( O ) and W ☆ n +1 = (cid:96) c,c ,...,c n O ( c , . . . , c n ; c ) with P retrieving the profile ( c, c , . . . , c n ). Now that wehave the elements of O , let us examine the composition. For that, we will utilizethe following trees. Definition 6.4.
Suppose that m ≥ n ≥
0, and 1 ≤ i ≤ m . Define a (rooted)tree Z im,n with two vertices a, b , Legs( Z im,n ) = 0 , , . . . , m + n − e . We further declare that the vertex neighborhoods are thefollowing ordered setsNbhd( a ) = 0 , , . . . , i − , e, n + i, . . . , m + n − b ) = e, i, i + 1 , . . . , n + i − . See Figure 9 for examples. This graph admits one inner coface map δ e : ☆ n + m → Z im,n with δ e ( k ) = k for all edges k , and two outer cofaces δ a : ☆ n +1 → Z im,n δ a ( k ) = (cid:40) e if k = 0 i + k − δ a ( v ) = ☆ b δ b : ☆ m +1 → Z im,n δ b ( k ) = k if 0 ≤ k < ie if k = in − k if i < k ≤ m . δ b ( v ) = ☆ a . All three of these cofaces are oriented.
Remark 6.5.
Let O be a C -colored operad and W = N d ( O ) its dendroidal nerve.Then the operadic multiplication ◦ i is recovered from O ( c , . . . , c m ; c ) × O ( d , . . . , d n ; c i ) W ☆ m +1 × W η W ☆ n +1 W Z im,n O ( c , . . . , c i − , d , . . . , d n , c i +1 , . . . , c m ; c ) W ☆ m + n ◦ i ∼ =( d b × d a ) − d e where the pullback in the top right is obtained from the maps ☆ m +1 i ← (cid:45) η (cid:44) → ☆ n +1 . Given σ ∈ Σ + n = Aut { , , . . . , n } , there is a morphism ψ σ : ☆ n +1 → ☆ n +1 determined by ( ψ σ ) = σ on edges. The diagram X ☆ n +1 X ☆ n +1 n (cid:81) i =0 X η n (cid:81) i =0 X ηψ ∗ σ P P ( − ) · σ commutes. In particular, ψ ∗ σ restricts to a function P − ( x , . . . , x n ) → P − ( x σ (0) , . . . , x σ ( n ) ) . For the purposes of the next lemma, let us fix some notation. For q ≥
1, let τ q ∈ Aut { , , . . . , q − } be given by τ q ( k ) = k +1 mod q . Let ψ q = ψ τ q : ☆ q → ☆ q be the associated map. If 1 ≤ i ≤ m −
1, define a map φ m,n : Z im,n → Z i +1 m,n by e (cid:55)→ e a (cid:55)→ ☆ a k (cid:55)→ τ m + n ( k ) b (cid:55)→ ☆ b . Likewise, if n ≥
1, define a map φ m,n : Z mm,n → Z n,m by e (cid:55)→ e a (cid:55)→ ☆ b k (cid:55)→ τ m + n ( k ) b (cid:55)→ ☆ a . The following lemma is a diagrammatic repackaging of Definition 0.1(1).
Lemma 6.6. If ≤ i ≤ m − , then the diagram ☆ m +1 ☆ n +1 Z im,n ☆ m + n ☆ m +1 ☆ n +1 Z i +1 m,n ☆ m + nδ b ψ m +1 δ a id φ m,n δ e ψ m + n δ b δ a δ e commutes. If i = m and n ≥ , then the diagram ☆ m +1 ☆ n +1 Z mm,n ☆ m + n ☆ m +1 ☆ n +1 Z n,m ☆ m + nδ b ψ m +1 δ a ψ n +1 φ m,n δ e ψ m + n δ a δ b δ e commutes.Proof. One merely needs to check that the composite functions are identical onedge sets, which is straightforward. (cid:3)
IGHER CYCLIC OPERADS 39
Proof of Theorem 6.3.
We begin by defining the cyclic structure on O . Let c , c , . . . , c n ∈ Col( O ), and write x i := α − ( c i ) ∈ X η . Then α determines the isomorphism P − ( x , . . . , x n ) X ☆ n +1 n (cid:81) i =0 X η O ( c , . . . , c n ; c ) N d ( O ) C n n (cid:81) i =0 N d ( O ) ηα ∼ = α ∼ = P α ∼ = P on the left. We define, for σ ∈ Σ + n ,( − ) · σ : O ( c , . . . , c n ; c ) → O ( c σ (1) , . . . , c σ ( n ) ; c σ (0) )as the restriction of the endomorphism α − ◦ ψ ∗ σ ◦ α of N d ( O ) ☆ n +1 . When σ fixes0, this is precisely the usual symmetric group action.Now suppose that we are working with τ n +1 and consider( − ) · τ n +1 : O ( c , . . . , c n ; c ) → O ( c , . . . , c n , c ; c ) . which is the restriction of α − ◦ ψ n +1 ◦ α . By Theorem 5.9, we simply need to showthat { ( − ) · τ n +1 } are compatible with operadic multiplication; using the notationfrom Remark 6.5, we have that ◦ i = d e ◦ ( d b × d a ) − on W = N d ( O ).In the diagrams in Figure 10 and Figure 11, we have W η = Col( O ) and W ☆ m +1 = O m = (cid:97) W × m +1 η O ( c , . . . , c m ; c )since W = N d ( O ). By Lemma 6.6, the diagram in Figure 10 commutes for 1 ≤ i ≤ W ☆ m +1 × W η W ☆ n +1 W ☆ m +1 × W η W ☆ n +1 X ☆ m +1 × X η X ☆ n +1 X ☆ m +1 × X η X ☆ n +1 X Z i +1 m,n X Z im,n W ☆ m + n W ☆ m + n X ☆ m + n X ☆ m + n τ m +1 × ◦ i +1 ◦ i ψ m +1 × id α ∼ = α ∼ = d e ∼ = d b × d a φ m,n d e ∼ = d b × d a τ m + n ψ m + n α ∼ = α ∼ = Figure 10.
Compatibility with operadic composition m − i = m (where σ interchanges thetwo factors). This shows that the proposed Σ + n actions really turn O into a cyclicoperad. Uniqueness of the structure is clear from the requirement that α lifts toan isomorphism of cyclic dendroidal sets. Finally, the existence of a unique f (cid:48) inthe second part is just a characterization of cyclic operad maps as operad mapswhich respect the extra symmetry, the fact that N d is fully-faithful, and that the f (cid:48) guaranteed by fullness of N d preserves the extra symmetry by inspection. (cid:3) W ☆ n +1 × W η W ☆ m +1 W ☆ m +1 × W η W ☆ n +1 X ☆ n +1 × X η X ☆ m +1 X ☆ m +1 × X η X ☆ n +1 X Z n,m X Z mm,n W ☆ m + n W ☆ m + n X ☆ m + n X ☆ m + n ( τ m +1 × τ n +1 ) ◦ σ ◦ ◦ m ( ψ m +1 × ψ n +1 ) ◦ σα ∼ = α ∼ = d e ∼ = d b × d a φ m,n d e ∼ = d b × d a τ m + n ψ m + n α ∼ = α ∼ = Figure 11.
Compatibility with operadic compositionWe are now ready to prove the main theorem about the cyclic dendroidal nerve.Note that our nerve theorem does not fit into the general monadic framework fornerve theorems from [45, 3], as, for the reasons outlined in the introduction, we havechosen a non-full subcategory Ξ of
Cyc as our indexing category (contrast with theparagraph after [3, Definition 2.3], where the indexing category Θ T is always a fullsubcategory). Theorem 6.7.
The functor N c : Cyc → Set Ξ op is fully-faithful. Further, if X ∈ Set Ξ op is a cyclic dendroidal set, then the following are equivalent.(i) X ∼ = N c ( O ) for some cyclic operad O .(ii) X S = hom(Ξ[ S ] , X ) → hom(Λ δ Ξ[ S ] , X ) is a bijection for every inner cofacemap δ .(iii) X S = hom(Ξ[ S ] , X ) → hom(Sc[ S ] , X ) is a bijection for every rooted tree S .Proof. That N c is fully-faithful follows from the existence and uniqueness of f (cid:48) inthe second part of Theorem 6.3.The remainder of the proof relies on Theorem 5.9, and goes by comparing prop-erties of X to properties of ι ∗ X . Notice, for instance, that Theorem 6.3 impliesthat ι ∗ X satisfies (i’) if and only if X satisfies (i).Since every tree S ∈ Ξ is isomorphic to a rooted tree T ∈ Ω, it suffices to examinethe remaining condition only for rooted trees T . Likewise, in (ii), we may assumethat the inner coface map δ : R → T is an oriented map. So, to see that (ii) holdsfor X if and only if (ii’) holds for ι ∗ X , we merely need to note that the commutativediagram hom(Ω[ T ] , ι ∗ X ) hom(Λ δ Ω[ T ] , ι ∗ X )hom( ι ! Ω[ T ] , X ) hom( ι ! Λ δ Ω[ T ] , X )hom(Ξ[ T ] , X ) hom(Λ δ Ξ[ T ] , X ) ∼ = ∼ = ∼ = ∼ = has the indicated isomorphisms by Lemma 6.2. A similar argument shows that (iii)holds for X if and only if (iii’) holds for ι ∗ X , this time applying Lemma 6.1. (cid:3) IGHER CYCLIC OPERADS 41 Berger–Moerdijk Reedy model structure
In this section, we first recall the Berger–Moerdijk Reedy model structure oncategories of diagrams indexed by a generalized Reedy category R (Definition 3.1)and investigate properties of this model structure. The basic reference is the paper[5], while many of these results in the case of classical Reedy categories may be foundin [24, Chapter 15]. We then turn to the special case of simplicial R -presheaves, andconsider the full subcategory sSet R op ∗ ⊆ sSet R op of diagrams which are a point indegree zero. For certain R , this category admits a model structure (Theorem 7.16)which is simplicial (Theorem 7.24), left proper, and cellular (Proposition 7.18).The results of this section are likely well-known among experts, at least in certainspecial cases.A model category M is called R -projective if, for every r ∈ R , the category of(right) Aut( r )-equivariant maps M Aut( r ) admits the model structure where weakequivalences and fibrations are detected in M . This occurs, for instance, if M iscofibrantly generated (see [24, 11.6.1]) or if Aut( r ) = { e } for all r ∈ R .If r ∈ R , let R + ( r ) be the category whose objects are non-invertible maps in R + with codomain r , and whose morphisms α → β are commutative triangles s s (cid:48) r σα β in R + . This is a full subcategory of R + ↓ r . Similarly, for each r ∈ R , there is afull subcategory R − ( r ) ⊆ r ↓ R − whose objects are non-invertible morphisms withdomain r .If Z ∈ M R , define, for each r ∈ R , latching and matching objects L r Z = colim R + ( r ) α : s → r Z s M r Z = lim R − ( r ) α : r → s Z s which come equipped with maps L r Z → Z r → M r Z in M Aut( r ) . Theorem 7.1 (Theorem 1.6, [5]) . If R is a generalized Reedy category and M is an R -projective model category, then the diagram category M R admits a modelstructure where f : X → Y is a • weak equivalence if, for each r ∈ R , f r : X r → Y r is a weak equivalence in M ; • cofibration if, for each r ∈ R , X r (cid:113) L r X L r Y → Y r is a cofibration in M Aut( r ) ; and • fibration if, for each r ∈ R , X r → M r X × M r Y Y r is a fibration in M . We will call this model structure the
Berger–Moerdijk Reedy model structure on M R . It would be convenient to know when this model structure is left proper and cellular, so that we can guarantee the existence of left Bousfield localizations [24,4.1.1]. Theorem 7.2. If R is a generalized Reedy category and M is left proper and R -projective, then the Berger–Moerdijk Reedy model structure on M R is left proper. A key component of the proof is that Reedy cofibrations are levelwise cofibrations(Lemma 7.4); in establishing this lemma, we will utilize the fact that M G → M preserves cofibrations when G is a finite group. Remark 7.3.
Suppose that G is any discrete group and M is a category so thatproducts indexed by G exist. The forgetful functor f : M G → M has a rightadjoint r so that the composite M r −→ M G f −→ M takes an object X to (cid:81) g ∈ G X ,and similarly for morphisms. If M is a model category, then the composite f r takes acyclic fibrations to acyclic fibrations. Suppose further that M G admits theprojective model structure, that is, the model structure so that f creates fibrationsand weak equivalences. In this case, r preserves acyclic fibrations, hence the leftadjoint f : M G → M preserves cofibrations. Lemma 7.4. If R is a generalized Reedy category, M is R -projective, and f : A → B is a Reedy cofibration, then for each r ∈ R the map f r : A r → B r is a cofibrationin M .Proof. A variation on [5, Lemma 5.3] shows that, for each r ∈ R , the map L r A → L r B is a cofibration in M . As it is a bit simpler than that lemma, let us give asketch of the argument. The category S = R + ( r ) is again generalized Reedy, with S = S + and the domain functor ϕ : S → R is a morphism of generalized Reedycategories. For any s → r in S , the diagram L s → r ( ϕ ∗ A ) L s ( A )( ϕ ∗ A ) s → r A s ∼ == commutes where the top isomorphism is a consequence of [5, Lemma 4.4(i)]. Thus, ϕ ∗ ( f ) is a cofibration in M S . Further, since colim S ( ϕ ∗ A ) = L r ( A ), we have (cid:16) L r A L r f −−→ L r B (cid:17) = colim S (cid:18) ϕ ∗ A ϕ ∗ ( f ) −−−−→ ϕ ∗ B (cid:19) which is a cofibration in M by [5, Corollary 1.7].The map A r → A r (cid:113) L r A L r B is a pushout of a cofibration L r A A r L r B A r (cid:113) L r A L r B hence is a cofibration [24, Proposition 7.2.12(a)]. Since f is a Reedy cofibration, A r (cid:113) L r A L r B → B r is a cofibration in M Aut( r ) . By Remark 7.3, it is also a cofibration in M . Now f r : A r → B r is the composite of two cofibrations in M A r → A r (cid:113) L r A L r B → B r , IGHER CYCLIC OPERADS 43 hence is also a cofibration in M . (cid:3) Proof of Theorem 7.2.
Suppose we have a pushout diagram(11)
A BX X (cid:113) A B (cid:39) in M R ; we wish to show that B → X (cid:113) A B is a weak equivalence. Evaluating (11)at r ∈ R gives the pushout square A r B r X r X r (cid:113) A r B r . (cid:39) By Lemma 7.4, A r → B r is a cofibration in M ; since M is left proper, this impliesthat B r → X r (cid:113) A r B r is a weak equivalence. Since r was arbitrary, B → X (cid:113) A B is a weak equivalence. (cid:3) Proposition 7.5. If M is cellular, then so is M R .Proof. The proof given in [24, 15.7] goes through, with the caveat that in the proofof [24, 15.7.1], one should use Lemma 7.4 in place of [24, 15.3.11]. (cid:3)
Remark 7.6.
Implicit in the explanation for Proposition 7.5 is the fact that thegenerating (acyclic) cofibrations of M R are described as in [24, Definition 15.6.23].Indeed, the proof of cofibrant generation in the first paragraph of [5, Theorem 7.6]only relies on R being a dualizable generalized Reedy category and M being cofi-brantly generated. We will need this explicit description of the generating (acyclic)cofibrations for the proof of Theorem 7.16 (in the special case when M = sSet ).7.1. Reduced presheaves in simplicial sets.
Let R be a generalized dualizableReedy category. We assume that R has a unique object η of degree zero and, forany r ∈ R , the set R ( r, η ) is either empty or has exactly one element. Further, weassume that if R ( r, η ) = { f } , then R ( η, r ) (cid:54) = ∅ as well; any inhabitant of R ( η, r ) isautomatically a section of f . In symbols, this says(12) | R ( r, η ) | ≤ min(1 , | R ( η, r ) | ) ∀ r ∈ R which we take as a standing assumption for all that follows. Examples of suchcategories include ∆, Ω, and Ξ.Let sSet R op ∗ be the full subcategory of sSet R op consisting of those X so that X η = ∆[0]. As this category has been thoroughly analyzed in [7] in the case of R = ∆ and [9] in the case of R = Ω, many of the arguments and constructions inthe remainder of this section should look familiar. Write I : sSet R op ∗ (cid:44) → sSet R op for the inclusion. This functor admits a left adjoint R : sSet R op → sSet R op ∗ which we now describe explicitly. Definition 7.7.
We define the reduction of an object X ∈ sSet R op . • A map R [ η ] × X η → X is given in degree r by either X η f ∗ −→ X r if R ( r, η ) = { f } or else the unique map ∅ → X r if R ( r, η ) = ∅ . Since such an f admitsa section, this map is a monomorphism, and we write X (0) ⊆ X for itsimage. Notice that there is a unique map X (0) → R [ η ] and that X (0) r ∼ = (cid:40) X η if R ( r, η ) = ∗ ∅ if R ( r, η ) = ∅ . • Define R ( X ) as the pushout X (0) X R [ η ] R ( X ) . Suppose X ∈ sSet R op . If there is not a map r → η , then R ( X ) r = X r . Suppose Z ∈ sSet R op ∗ . If there is a map from r to η then it is unique, R ( r, η ) = { f } ; thisimplies that Z r has a natural basepoint given by f ∗ ( t ), where t ∈ Z η is the uniqueelement. Remark 7.8.
Many results of this section hold with a weaker assumption than(12), namely one could just assume | R ( r, η ) | ≤ r ∈ R . In this case, R [ η ] × X η → X may not be a monomorphism, and we should define X (0) to be R [ η ] × X η instead of its image in X . We refrain from giving further details, as ourmain application is to the categories Ξ and Ω (which do satisfy (12)).The category sSet R op ∗ is bicomplete: Limits and directed colimits are computedin the larger category sSet R op , while finite coproducts are given by the formula(13) ( X (cid:113) Y ) r = (cid:40) X r (cid:96) Y r R ( r, η ) = ∅ X r (cid:87) Y r R ( r, η ) = ∗ . Equivalently, X (cid:113) Y = R ( I ( X ) (cid:113) I ( Y )). Though I does not preserve coproducts,it does preserve pushouts. Lemma 7.9.
The inclusion functor I : sSet R op ∗ → sSet R op preserves connectedcolimits.Proof. Suppose F : C → sSet R op ∗ is a functor from a connected category, and A = colim C I F . Then A η = (colim c ∈C I F ( c )) η = colim c ∈C ( I F ( c ) η ) = colim c ∈C ∆[0] = ∆[0] . Since A is already in sSet R op ∗ and I is fully-faithful, A is also the colimit of F in sSet R op ∗ . (cid:3) Given an object r ∈ R , recall that the boundary of r is the subobject ∂ R [ r ] ⊆ R [ r ] = R ( − , r ) so that ∂ R [ r ] s ⊆ R ( s, r ) consists of those maps s → r which factorthrough an object of degree less than d ( r ). In particular, if d ( s ) < d ( r ), ∂ R [ r ] s = R [ r ] s . IGHER CYCLIC OPERADS 45
Definition 7.10.
We define two sets of maps in sSet R op . The set I consists of allinclusions ( ∂ R [ r ] × ∆[ n ]) (cid:91) ∂ R [ r ] × ∂ ∆[ n ] ( R [ r ] × ∂ ∆[ n ]) → R [ r ] × ∆[ n ]for n ≥ r ∈ R . The set J consists of all inclusions( ∂ R [ r ] × ∆[ n ]) (cid:91) ∂ R [ r ] × Λ k [ n ] ( R [ r ] × Λ k [ n ]) → R [ r ] × ∆[ n ] . The set I (resp. J ) is a set of generating (acyclic) cofibrations for the Berger–Moerdijk Reedy model structure on sSet R op (see Remark 7.6).We now make a further restriction on our generalized Reedy category R , whichwill imply that all (co)domains of elements of I ∪ J are small relative to the wholecategory. Lemma 7.11.
All objects of sSet R op and sSet R op ∗ are small [24, Definition 10.4.1] .Proof. In any locally presentable category, every object is small in this sense. Anydiagram category
Set C (with C small) is locally presentable [1, Example 1.12],hence sSet R op ∼ = Set ∆ op × R op is. Finally, sSet R op ∗ is a full, reflective subcategory of sSet R op which is further closed under λ -directed colimits ( λ a regular cardinal) byLemma 7.9, hence is locally presentable by [1, Representation Theorem 1.46]. (cid:3) Lemma 7.12.
Let A be a domain or codomain of an element of I ∪ J , with A ⊆ R [ r ] × ∆[ n ] . If d ( r ) > , then A (0) = R [ r ] (0) × ∆[ n ] ∼ = (cid:96) R ( η,r ) R [ η ] × ∆[ n ] . If r = η ,then A (0) = A , hence R ( A ) = R [ η ] .Proof. The important point is that( ∂ R [ r ]) (0) = (cid:40) R [ r ] (0) d ( r ) > ∅ d ( r ) = 0 . If d ( r ) >
0, then R [ r ] (0) × ∆[ n ] = ( ∂ R [ r ] × ∆[ n ]) (0) ⊆ A (0) ⊆ ( R [ r ] × ∆[ n ]) (0) = R [ r ] (0) × ∆[ n ] . If r = η , then A is one of R [ η ] × ∂ ∆[ n ], R [ η ] × Λ k [ n ], or R [ η ] × ∆[ n ], hence A (0) = A . (cid:3) Proposition 7.13.
Suppose that j : A → R [ r ] × ∆[ n ] is in I or J and d ( r ) > . If j ∈ I , then R ( j ) is a cofibration in sSet R op , while if j ∈ J , then R ( j ) is an acycliccofibration in sSet R op .Proof. The cube with front and back squares pushouts A (0) A ( R [ r ] × ∆[ n ]) (0) R [ r ] × ∆[ n ] R [ η ] R ( A ) R [ η ] R ( R [ r ] × ∆[ n ]) =7 .
12 =6 P. HACKNEY, M. ROBERTSON, AND D. YAU reduces to a rectangle A (0) A R [ r ] × ∆[ n ] R [ η ] R ( A ) R ( R [ r ] × ∆[ n ])where the big rectangle and the left square are pushouts, hence so is the rightsquare.The result now follows since the class of (acyclic) cofibrations is closed underpushouts. (cid:3) Proposition 7.14.
Suppose that j : A → R [ η ] × ∆[ n ] is in I or J . Then R ( j ) isthe identity map on R [ η ] .Proof. This is a direct consequence of the second part of Lemma 7.12 and the factthat hom( R [ η ] , R [ η ]) = R ( η, η ) = ∗ . (cid:3) Lemma 7.15. If j ∈ J , then R ( j ) s is an acyclic cofibration in sSet for every s ∈ R . If j ∈ I , then R ( j ) s is a cofibration in sSet for every s ∈ R .Proof. By the previous two propositions, we know that R ( j ) is an (acyclic) cofibra-tion in the Berger–Moerdijk Reedy model structure on sSet R op . The result thenfollows from Lemma 7.4. (cid:3) Theorem 7.16.
Suppose that R is a generalized Reedy category with unique element η in degree zero and | R ( r, η ) | ≤ min(1 , | R ( η, r ) | ) for all r ∈ R . Then the Berger–Moerdijk Reedy model structure lifts along the adjunction R : sSet R op (cid:29) sSet R op ∗ : I . In other words, sSet R op ∗ admits a cofibrantly-generated model structure with weakequivalences (resp. fibrations) those maps which are weak equivalences (resp. fibra-tions) in the larger category sSet R op .Proof. We apply Kan’s lifting theorem [24, 11.3.2]. We know that R I and R J satisfy the small object argument by Lemma 7.11, hence condition (1) is satisfied.For condition (2) we know that R ( j ) s is an acyclic cofibration in sSet for any s ∈ R by Lemma 7.15. Then given any relative R J -cell complex X → Y in sSet R op ∗ , wealso have that X s → Y s is an acyclic cofibration for each s ∈ R . Thus I X → I Y is a weak equivalence, and condition (2) is established. (cid:3) Notice that the inclusion functor does not admit a right adjoint (as it does notpreserve finite coproducts), hence cannot be a left Quillen functor. Nevertheless,we have the following, which hinges on the fact that I preserves pushouts andtransfinite composition (Lemma 7.9). Proposition 7.17.
The inclusion functor I : sSet R op ∗ → sSet R op preserves (acyclic)cofibrations.Proof. The inclusion functor preserves all weak equivalences, so it is enough toshow that it preserves cofibrations. By Proposition 7.13 and Proposition 7.14, weknow that I takes generating cofibrations in R I to cofibrations in sSet R op . Theinclusion functor preserves pushouts (Lemma 7.9) and transfinite composition, sotakes relative R I -cell complexes to cofibrations. Since every cofibration in sSet R op ∗ IGHER CYCLIC OPERADS 47 is a retract of a relative R I -cell complex, every cofibration in sSet R op ∗ is again acofibration in sSet R op . (cid:3) The following has precursors elsewhere in special cases, for instance in the proofof [9, 4.3].
Proposition 7.18.
With the hypotheses of Theorem 7.16, the model structure on sSet R op ∗ is left proper and cellular.Proof. For left properness, it is enough to show that the pushout of a weak equiv-alence along a generating cofibration R ( i ) ∈ R I is again a weak equivalence; thisfollows from Lemma 7.15, left properness of sSet , and the fact that weak equiva-lences are levelwise weak equivalences.We now turn to cellularity, and aim to verify the three conditions from [24,12.1.1]. We know that (2) holds by Lemma 7.11. Recall from Proposition 7.5that sSet R op is a cellular model category. Cofibrations in sSet R op ∗ are also cofibra-tions (Proposition 7.17) in the cellular model category sSet R op , hence are effectivemonomorphisms. Thus (3) holds.It remains to show (1). All elements of the form R [ r ] × ∆[ n ] are compact relativeto I since they are codomains of elements of I and sSet R op is cellular. If A is adomain or a codomain of an element of I , then A (0) is compact by Lemma 7.12. Itfollows that R ( A ) is compact (relative to I ) by [24, 10.8.8], since A , A (0) and R [ η ] = R [ η ] × ∆[0] are compact. This shows that the set R ( I ) is a set of cofibrations of sSet R op with compact domains. By [24, 11.4.9], if W ∈ sSet R op is compact relativeto I , then W is compact relative to R ( I ). In particular, R ( A ) is compact relativeto R ( I ). Presented relative R ( I )-cell complexes in sSet R op with X ∈ sSet R op ∗ are the same thing as presented relative R ( I )-cell complexes in sSet R op ∗ , hence alldomains and codomains of elements in R ( I ) ∪ R ( J ) are compact in sSet R op ∗ . Thus[24, 12.1.1(1)] holds, and we conclude that sSet R op ∗ is a cellular model category. (cid:3) Simplicial model structures.
As in the case of an ordinary Reedy cate-gory, the Berger–Moerdijk Reedy model structure on sSet R op is a simplicial modelstructure (see [24, Ch. 9]). The structure is given as follows: Definition 7.19.
Suppose that
X, Y ∈ sSet R op and K ∈ sSet . • The object X ⊗ K is defined on objects by( X ⊗ K ) r = X r × K ∈ sSet . • The object Y K is defined on objects by( Y K ) r = map sSet ( K, Y r ) ∈ sSet where map sSet is the simplicial mapping space, that is, map sSet ( A, B ) n =hom sSet ( A × ∆[ n ] , B ). • The (simplicial) mapping spaces map(
X, Y ) are defined in degree n bymap( X, Y ) n = hom( X ⊗ ∆[ n ] , Y ) . The simplicially-enriched category sSet R op satisfies the two conditions to be asimplicial model category [24, 9.1.6], namely M6:
For every two objects
X, Y and every K ∈ sSet map( X ⊗ K, Y ) ∼ = map sSet ( K, map( X, Y )) ∼ = map( X, Y K )(with isomorphisms natural in the three variables). M7: If A → B is a cofibration and X → Y is a fibration, then(14) map( B, X ) → map( A, X ) × map( A,Y ) map( B, Y )is a fibration of simplicial sets. If either map is a weak equivalence, then(14) is also a weak equivalence.Assuming M6, condition M7 is equivalent (see, e.g. [33, Remark A.3.1.6] or [19, § M7 (cid:48) : If X → Y is a fibration in our model category and K → L is a cofibrationof simplicial sets, then(15) X L → X K × Y K Y L is a fibration. If either map is a weak equivalence, then so is (15).Any full subcategory of a simplicially-enriched category is simplicially-enriched;we will now work towards Theorem 7.24, where we show that sSet R op ∗ is a simplicialmodel category. Notation 7.20.
For the remainder of this section, we will write M = sSet R op N = sSet R op ∗ for these two model categories. For X, Y ∈ M , K ∈ sSet , we will write X ⊗ M K := X ⊗ K and map M ( X, Y ) := map(
X, Y ).Recall that we have a Quillen adjunction R : M (cid:29) N : I . Definition 7.21.
Suppose that
X, Y ∈ N = sSet R op ∗ and K ∈ sSet . • The object X ⊗ N K is defined to be X ⊗ N K = R ( I ( X ) ⊗ M K ). • The object Y K is defined as Y K = R (cid:2) ( I Y ) K (cid:3) . • We define map N ( X, Y ) = map M ( I X, I Y ). Remark 7.22.
The object Z = ( I Y ) K already has Z η = ∆[0], which explainswhy we’ve elected not to distinguish between the exponential in the two categories.Indeed, Z η = (( I Y ) K ) η = map sSet ( K, Y η ) = map sSet ( K, ∆[0]) = ∆[0] . Lemma 7.23.
There is an isomorphism map N ( R ( − ) , − ) ∼ = map M ( − , I ( − )) of bifunctors N × M → sSet . IGHER CYCLIC OPERADS 49
Proof.
Let Z ∈ N , Y ∈ M , and n ≥
0. We havemap N ( R ( Z ) , Y ) n = map M ( I R ( Z ) , I Y ) n = hom M ( I R ( Z ) ⊗ M ∆[ n ] , I Y )= hom M ( I R ( Z ) , ( I Y ) ∆[ n ] )= hom M ( I R ( Z ) , I R ( I Y ) ∆[ n ] ) Remark 7.22= hom N ( R ( Z ) , R ( I Y ) ∆[ n ] )= hom M ( Z, I R ( I Y ) ∆[ n ] )= hom M ( Z, ( I Y ) ∆[ n ] ) = map M ( Z, I Y ) n . with all isomorphisms natural in Z, Y , and n . (cid:3) Theorem 7.24.
With the structure from Definition 7.21, sSet R op ∗ is a simplicialmodel category.Proof. By Remark 7.22, the fact that M7 (cid:48) holds for M = sSet R op , and the factthat I creates (acyclic) fibrations, M7 (cid:48) holds for N = sSet R op ∗ . Thus it is enoughto check M6.Let X, Y ∈ N and K ∈ sSet . First,map N ( X ⊗ N K, Y ) = map N ( R ( I ( X ) ⊗ M K ) , Y ) = map M ( I ( X ) ⊗ M K, I Y )by Lemma 7.23. Thus, using M6 for M , the simplicial set map N ( X ⊗ N K, Y ) isisomorphic to, on the one hand,map M ( I ( X ) , ( I Y ) K ) = map M ( I X, I R ( I Y ) K ) = map N ( X, Y K )and on the other tomap sSet ( K, map M ( I X, I Y )) = map sSet ( K, map N ( X, Y )) . (cid:3) Remark 7.25.
We are grateful to an anonymous referee for observing that theabove results reflect a general pattern. The adjunction I (cid:97) R is monadic since R is conservative and preserves connected colimits; thus sSet R op ∗ is equivalent to thecategory of I R -algebras. Remark 7.22 should come as no surprise, as the simpli-cial cotensorings are an enriched limit, hence should be computed in the groundcategory sSet R op . The rest of the structure (simplicial tensorings and simplicialhoms) are then forced by the two-variable adjunctions of M6. Provided these exist,the simplicial structure is guaranteed since M7 is equivalent to M7 (cid:48) .8. Segal cyclic operads
In this section we define Segal cyclic operads as certain fibrant objects in sSet Ξ op ∗ which satisfy a Segal condition (Definition 8.8). The Segal cyclic operads may beidentified as the fibrant objects after we have localized the model structure on sSet Ξ op ∗ with respect to Segal core inclusions.We begin this section by specializing the work of Section 7 to the cases R = Ωor Ξ. We show that the Quillen adjunctions from Theorem 7.16 fit into a diagram(18). Afterward, we discuss the left Bousfield localizations, and show that afterlocalization we still have a diagram of Quillen adjunctions. Finally, we check in Proposition 8.10 that the homotopy theory for Segal cyclic operads is distinct fromthat for Segal operads, and speculate on the possibility of rigidification theorems.
Proposition 8.1.
Using the Berger–Moerdijk Reedy model structures, the adjunc-tion ι ! : sSet Ω op (cid:29) sSet Ξ op : ι ∗ is a Quillen adjunction.Proof. The map ι ∗ preserves weak equivalences since those are defined levelwise,hence it is enough to show that ι ∗ preserves fibrations. If T is a rooted tree, we willshow that(16) (Ω op ) − ( T ) (Ξ op ) − ( ιT )(Ω + ( T )) op (Ξ + ( ιT )) op is an initial functor (see [34, § IX.3]). This implies that the natural map M ιT X = lim (Ξ op ) − ( ιT ) X S → lim (Ω op ) − ( T ) ( ι ∗ X ) T (cid:48) = M T ( ι ∗ X )is an isomorphism. Hence if X → Y is a map in sSet Ξ op and T ∈ Ω, the map onthe right of the commutative diagram( ι ∗ X ) T M T ( ι ∗ X ) × M T ( ι ∗ Y ) ( ι ∗ Y ) T X ιT M ιT ( X ) × M ιT ( Y ) Y ιT is an isomorphism. In particular, if X → Y is a fibration, then so is ι ∗ ( X → Y ).As promised, we will now show that (16) is an initial functor; this is equivalentto the induced functor(17) ι T : Ω + ( T ) → Ξ + ( ιT )being final. Suppose that S φ → ιT is an object of Ξ + ( ιT ), that is, φ is an element ofΞ + ( S, ιT ) \ Iso(Ξ). Our goal is to show that φ ↓ ι T is nonempty and connected. Wefirst explain the case when φ is not constant (that is, when S (cid:54) = η ), and indicatelater the changes for the simpler case when S = η . Write s = (cid:12) t ( φ ), where t isthe root of T ; we have a morphism L t ( φ ) : T ( S, s ) → T ( ιT, t ) = T. Using the structure map f : ι T ( S, s ) ∼ = → S from Definition 2.1, the commutativediagram S ι T ( S, s ) ιT φ f − ι L t ( φ ) in Ξ constitutes an object φ f − −−→ ι T ( L t ( φ ))in φ ↓ ι T . To show that this category is connected, suppose that we have anarbitrary object φ γ −→ ι T (cid:16) R α → T (cid:17) IGHER CYCLIC OPERADS 51 of φ ↓ ι T , where α ∈ Ω + ( T ). Such an object corresponds to a commutative diagram S ιRιT φ γ ια with γ ∈ Ξ + . Lifting all maps to Ω, we have the diagram T ( S, s ) T ( ιR, r ) R T ( ιT, t ) T L t ( φ ) L r ( γ ) L t ( ια ) α in Ω, which commutes by Proposition 2.11. Commutativity of the diagram S ι T ( S, s ) ιRιT φ γf − ι ( L t ( φ )) ι ( L r ( γ )) ι ( α ) in Ξ shows that ι ( L r ( γ )) constitutes a morphism f − → γ in φ ↓ ι T ; thus thiscategory is connected.A similar proof holds when φ is constant, that is, when S = η . In this case, T ( S, s ) should be replaced by η , f should be taken to be the identity map, and L t ( φ ) (resp. L r ( γ )) should be replaced by the unique lift η → T of φ (resp. theunique lift η → R of γ ). Since φ ↓ ι T is connected for every object φ ∈ Ξ + ( ιT ),(17) is a final functor. (cid:3) Corollary 8.2.
The adjunction ι ! : sSet Ω op ∗ (cid:29) sSet Ξ op ∗ : ι ∗ is a Quillen adjunction.Proof. We have a commutative diagram sSet Ω op sSet Ξ op sSet Ω op ∗ sSet Ξ op ∗ ι ∗ I ι ∗ I of right adjoints where all but the bottom map ι ∗ are known to be right Quillenfunctors. Suppose that X → Y is an (acyclic) fibration in sSet Ξ op ∗ , which implies I ι ∗ ( X → Y ) = ι ∗ I ( X → Y ) is an (acyclic) fibration in sSet Ω op . Since I : sSet Ω op ∗ → sSet Ω op detects fibrations and weak equivalences we known that ι ∗ ( X → Y ) is an (acyclic) fibration, implying ι ∗ : sSet Ξ op ∗ → sSet Ω op ∗ is a right Quillenfunctor. (cid:3) We now have a diagram of Quillen adjunctions(18) sSet Ω op sSet Ξ op sSet Ω op ∗ sSet Ξ op ∗ ι ! R R ι ∗ ι ! I ι ∗ I and we wish to localize each of these model structures.8.1. Localizations.
Roughly speaking, a left localization of a model category M at a set of maps C is a left Quillen functor from M which is initial among allleft Quillen functors which take elements of C to weak equivalences. Recall from[24, 3.1.4] that an object W of M is called C -local if W is fibrant and for each f : A → B which is an element of C , the map map h ( B, W ) → map h ( A, W ) is aweak equivalence of simplicial sets. A map g : X → Y is called a C -local equivalenceif map h ( Y, W ) → map h ( X, W ) is a weak equivalence for every C -local W . The leftBousfield localization of M with respect to C [24, 3.3.1], denoted L C M , is then amodel structure (which may or may not exist) on M with weak equivalences the C -local equivalences and cofibrations the ordinary cofibrations in M . The fibrantobjects in this model structure (if it exists) are precisely the C -local objects, andthe identity functor M → L C M is a left localization of M .In order to show that the diagram (18) remains a diagram of Quillen adjunctionsafter localization, we will apply the following lemma several times. Lemma 8.3.
Let L : M (cid:29) N : R be a Quillen adjunction. Suppose that C ⊆ M and D ⊆ N are sets of maps with the domain and codomain of each element of C cofibrant. Suppose further that the left Bousfield localizations L C M and L D N exist. If, for each c ∈ C , the map L ( c ) ∈ N is isomorphic to some d ∈ D , then L : L C M (cid:29) L D N : R is a Quillen adjunction.Proof. There is a left Quillen functor F : M → N → L D N . If c ∈ C , then c is a cofibrant approximation to itself. Further, F ( c ) ∼ = d ∈ D is a weak equivalence, hence F takes any cofibrant approximation of c to a weakequivalence by [24, 8.1.24(1)]. By [24, 3.3.18(1)], the functor F is then a left Quillenfunctor when regarded as a functor L C M → L D N . (cid:3) Let R be either Ω or Ξ. Define S R to be the set of Segal core inclusions S R = { Sc[ r ] → R [ r ] } r (cid:54) = η Remark 8.4.
According to § sSet R op and sSet R op ∗ are simplicial model cate-gories . Thus, if A is cofibrant and X is fibrant, we may use the simplicial mappingspace map( A, X ) as a model for the homotopy mapping space map h ( A, X ) (by,for example, [24, Example 17.2.4]). Since Sc[ r ] , R [ r ] , R (Sc[ r ]), and R ( R [ r ]) are allcofibrant, it suffices to work with map rather than map h when discussing S R or R ( S R ) locality. To establish cofibrancy of Ξ[ S ], simply pick a rooted tree with ιT = S . Then since Sc[ T ] iscofibrant [13, Corollary 1.7], so is ι ! (Sc[ T ]) ∼ = Sc[ ιT ] ∼ = Sc[ S ] by Lemma 6.1 and Proposition 8.1. IGHER CYCLIC OPERADS 53
Since sSet R op is left proper and cellular by Theorem 7.2 and Proposition 7.5,the left Bousfield localization L S R sSet R op exists by [24, 4.1.1]. Since sSet R op ∗ is left proper and cellular by Proposition 7.18,we can take the left Bousfield localization with respect to the set of maps R ( S R ).We call the resulting model structure L R ( S R ) sSet R op ∗ . For R = Ω, these two model structures appear in [14, Definition 5.4] and [9, Propo-sition 4.3], respectively. Proposition 8.5.
Let R be either Ω or Ξ . Then the Quillen adjunction R : sSet R op (cid:29) sSet R op ∗ : I induces a Quillen adjunction R : L S R sSet R op (cid:29) L R ( S R ) sSet R op ∗ : I after taking left Bousfield localization.Proof. If r (cid:54) = η is an object of R , then both Sc[ r ] and R [ r ] are cofibrant. ThusLemma 8.3 applies. (cid:3) The following is a variation on Lemma 6.1.
Proposition 8.6. If T ∈ Ω is a rooted tree, then ι ! (cid:16) R (Sc[ T ]) → R (Ω[ T ]) (cid:17) ∼ = (cid:16) R (Sc[ ιT ]) → R (Ξ[ ιT ]) (cid:17) . Proof.
We have ι ! R (Sc[ T ] → Ω[ T ]) = R ι ! (Sc[ T ] → Ω[ T ]) (18) ∼ = R (Sc[ ιT ] → Ξ[ ιT ]) Lemma 6.1. (cid:3) Proposition 8.7.
The diagram (18) gives a diagram L S Ω sSet Ω op L S Ξ sSet Ξ op L R ( S Ω ) sSet Ω op ∗ L R ( S Ξ ) sSet Ξ op ∗ ι ! R R ι ∗ ι ! I ι ∗ I of Quillen adjunctions after localizing.Proof. In light of Proposition 8.5, we only need to show that the horizontal ad-junctions are Quillen adjunctions. The objects Sc[ T ] and Ω[ T ] are cofibrant in sSet Ω op , so the top adjunction is a Quillen adjunction by Lemma 8.3 and Lemma6.1. Since R is a left Quillen functor, R (Sc[ T ]) and R (Ω[ T ]) are cofibrant in sSet Ω op ∗ . Thus the bottom adjunction is a Quillen adjunction by Lemma 8.3 andProposition 8.6. (cid:3) Definition 8.8. A Segal cyclic operad is a fibrant object in the model category L R ( S Ξ ) sSet Ξ op ∗ . Notice that every Segal cyclic operad has an underlying Segal operad via thefunctor ι ∗ .The following example gives two cyclic operad structures on the same underlyingoperad. In fact, this hints at a whole class of examples: if A is an abelian group,then cyclic structures on the operad given by A are in bijection with the order 1and 2 elements of Aut( A ). Example 8.9.
Let M be the group Z / × Z / { , } × { , } , whose elementsare written as 00 , , ,
11. Then M determines a (monochrome) operad O with O ( n ) = (cid:40) { , , , } n = 1 ∅ n (cid:54) = 1with the operadic multiplication given by the the group operation.The operad O admits distinct cyclic structures C and C (cid:48) . For the first, theaction of Σ +1 = Σ interchanges 01 and 10 and fixes 00 and 11, while in the second,the action fixes every element. These two structures are not isomorphic as cyclicoperads because the Σ -sets C (1) = Σ (cid:113) ∗ (cid:113) ∗ and C (cid:48) (1) = ∗ (cid:113) ∗ (cid:113) ∗ (cid:113) ∗ are notisomorphic. Proposition 8.10.
The Quillen adjunction ι ! : L R ( S Ω ) sSet Ω op ∗ (cid:29) L R ( S Ξ ) sSet Ξ op ∗ : ι ∗ does not induce an equivalence of homotopy categories. In particular, this adjunc-tion is not a Quillen equivalence.Proof. Consider the two cyclic operads
C, C (cid:48) in Set from Example 8.9; recall fromthat example that
U C = U C (cid:48) . Let X = N c ( C ) and X (cid:48) = N c ( C (cid:48) ), and note that ι ∗ X = ι ∗ X (cid:48) . Additionally, let A be the operad with A ( n ) = (cid:40) Z / { e, x | x = e } n = 1 ∅ n (cid:54) = 1 . The operad A admits a unique cyclic operad structure where the Σ action on A (1)fixes x . There are exactly two maps of cyclic operads A → C , while there are fourmaps A → C (cid:48) ; we will show that this remains true once we pass to the homotopycategory of L R ( S Ξ ) sSet Ξ op ∗ . Write W = N c ( A ), and note that W S = ∅ if S isnon-linear, while W L m is the set of words of length m in the alphabet e, x . Theobjects W, X, X (cid:48) are fibrant in L R ( S Ξ ) sSet Ξ op ∗ by Theorem 7.16, Theorem 6.7(iii),and the fact that all maps between discrete simplicial sets are Kan fibrations.Let E Σ → ∆[0] be a cofibrant resolution of the terminal object of sSet Σ ;there is a cofibrant resolution of W in the unlocalized model structure sSet Ξ op ∗ that is isomorphic levelwise to the tensor product W ⊗ E Σ from Definition 7.21.The object W ⊗ E Σ does not take into account the Σ -structure on E Σ , so wemust modify the presheaf structure a bit. As this seems interesting in its ownright, we discuss for K ∈ sSet Σ tensoring ( − (cid:126) K ) and cotensoring ( K (cid:116) − ) indetail in Appendix A. We now show that a cofibrant resolution of W is given by (cid:102) W = W (cid:126) E Σ → W . We have ( W (cid:126) E Σ ) S = ( W ⊗ E Σ ) S for every tree S . Tosee that (cid:102) W (cid:39) W , notice that at L m we have (cid:102) W L m = (cid:16) ( W L m \ { e × m } ) × E Σ (cid:17) + (cid:39) (cid:16) ( W L m \ { e × m } ) (cid:17) + = W L m IGHER CYCLIC OPERADS 55 and (cid:102) W S = ∅ if S is non-linear.To see that (cid:102) W is cofibrant, notice that W and (cid:102) W admit a filtration with W ( k ) consisting of words which have x appearing at most k times and (cid:102) W ( k ) = W ( k ) (cid:126) E Σ .Then (cid:102) W ( k ) is the pushout in sSet Ξ op ∗ , R ( ∂ Ξ[ L k ]) (cid:126) E Σ (cid:102) W ( k − R (Ξ[ L k ]) (cid:126) E Σ (cid:102) W ( k ) . By Proposition A.1 and Proposition A.2, (cid:102) W ( k − → (cid:102) W ( k ) is a cofibration in sSet Ξ op ∗ . Since (cid:102) W (0) = W (0) = Ξ[ η ] is the initial object of this category, it fol-lows that (cid:102) W = colim (cid:102) W ( k ) is cofibrant.We now havehom( (cid:102) W , X ) = hom( N c A (cid:126) E Σ , N c C ) = hom( N c A, E Σ (cid:116) N c C );but N c C is levelwise discrete, so E Σ (cid:116) N c C ∼ = N c C . Using Theorem 6.7, we seehom( (cid:102) W , X ) = hom( N c A, N c C ) = hom( A, C )and similarly hom( (cid:102)
W , X (cid:48) ) = hom(
A, C (cid:48) ). But these sets are especially easy to un-derstand. Maps of cyclic operads from A to any other cyclic operad are determinedby where we send x , and we havehom( (cid:102) W , X ) = hom(
A, C ) = { f , f } hom( (cid:102) W , X (cid:48) ) = hom(
A, C (cid:48) ) = { f , f , f , f } where f a ( x ) = a .Since (cid:102) W is cofibrant in the unlocalized model structure and the objects X, X (cid:48) are fibrant in the localized model structure, we can compute the homotopy classesof maps from (cid:102) W to X (or X (cid:48) ) in either the unlocalized or localized model structureand will get the same answer by [24, 3.5.2]. We now work in the unlocalized modelstructure sSet Ξ op ∗ , which is a simplicial model category. Since our objects arelevelwise discrete, (cid:102) W is cofibrant, and X is fibrant, we havehom( (cid:102) W , X ) = π hom( (cid:102) W , X ) = π map( (cid:102) W , X ) = π map h ( (cid:102) W , X ) = π ( (cid:102) W , X )by Remark 8.4, [24, Proposition 9.5.3], and [24, Proposition 9.5.24] (see [24, § π -notation). Similarly, hom( (cid:102) W , X (cid:48) ) = π ( (cid:102) W , X (cid:48) ). We thus have | π ( (cid:102) W , X (cid:48) ) | =4 > | π ( (cid:102) W , X ) | , which shows that X and X (cid:48) are not isomorphic in the homotopycategory of L R ( S Ξ ) sSet Ξ op ∗ . (cid:3) There is a model structure on the category of (monochrome) simplicial cyclicoperads where the weak equivalences and fibrations are defined as those maps whichare levelwise weak equivalences. This follows by considering either the multi-sortedalgebraic theory of cyclic operads or the colored operad controlling cyclic operads(for the latter, see [32, § Conjecture 8.11.
The model structure for simplicial cyclic operads is Quillenequivalent to L R ( S Ξ ) sSet Ξ op ∗ . Analogous results for simplicial monoids and for simplicial operads appear in [7]and [9], respectively.
Appendix A. Tensoring and cotensoring with Σ simplicial sets As inspiration for this section, recall the cartesian closed structure on the cate-gory
Set Σ of involutive sets. The cartesian product X × Y has the diagonal Σ action, and the internal hom is just the set of ordinary functions, together with theconjugation Σ action ( σ · f )( x ) = σ · f ( σ · x ). The fixed points of the action on theinternal hom are precisely the Σ -equivariant functions X → Y .Let ∇ be the full subcategory of Ξ spanned by the objects L n (Example 1.4),so that ∇ is equivalent to Ξ ↓ η . If n >
0, write ϑ : L n → L n for the uniquenon-oriented isomorphism, and if n = 0 write ϑ = id L . Each morphism in ∇ is ofthe form(19) L n ϑ i → L n γ → L m where γ is oriented and i ∈ { , } . Moreover, the data ( i, γ ) for the decomposition(19) of a map is unique if and only if that map is not constant (otherwise there aretwo such decompositions, see Corollary 2.17). Left Kan extension along ∇ (cid:44) → Ξgives a functor sSet ∇ op ∗ → sSet Ξ op ∗ which allows us to identify the former categoryas being equivalent to the full subcategory consisting of those X ∈ sSet Ξ op ∗ so that X R = ∅ whenever R is non-linear. Notice that if X ∈ sSet ∇ op ∗ , then X L n may beconsidered as an object of sSet Σ using the action of ϑ ∗ .The category sSet ∇ op is tensored and cotensored over sSet Σ . For our purposeswe are interested only in the reduced case, so we will reserve notation for that. Wehave, for X ∈ sSet ∇ op ∗ and K ∈ sSet Σ that X (cid:126) K is given on objects as thepushout K = X L × K X L n × K ∆[0] ( X (cid:126) K ) L n in sSet Σ , where the top map is induced from the unique morphism f n : L n → L . The presheaf structure is given for a morphism (19) by the induced maps onpushouts coming from the commutative diagram∆[0] X L × K X L m × K ∆[0] X L × K X L n × K. = f ∗ m × idid × σ i ( γϑ i ) ∗ × σ i f ∗ n × id Note that this is well-defined on constant maps, though the decomposition (19)need not be unique in this case.We write K (cid:116) − for the right adjoint to − (cid:126) K . If Y ∈ sSet ∇ op ∗ then( K (cid:116) Y ) L n = map sSet ( K, Y L n );in particular, if n = 0 this is just ∆[0]. Given a morphism (19), the map ( K (cid:116) Y ) L m → ( K (cid:116) Y ) L n is given bymap( σ i , ( γϑ i ) ∗ ) : map( K, Y L m ) → map( K, Y L n ) . IGHER CYCLIC OPERADS 57
As simplicial sets, we have (natural in
X, Y )( X (cid:126) K ) L n = ( X ⊗ K ) L n ( K (cid:116) Y ) L n = ( Y K ) L n where the objects on the right are those from Definition 7.21 after forgetting theΣ action on K . Note that if γ : L n → L m is an oriented map, then the abovedefinitions also give equalities of the maps γ ∗ :( X (cid:126) K ) γ = ( X ⊗ K ) γ ( K (cid:116) Y ) γ = ( Y K ) γ . Where these functors differ is exactly on the ϑ ∗ . Proposition A.1. If K ∈ sSet Σ , then the functor − (cid:126) K : sSet ∇ op ∗ → sSet ∇ op ∗ preserves Reedy cofibrations.Proof. To prove this statement, it is of course equivalent to prove that the rightadjoint K (cid:116) − preserves acyclic fibrations. Notice by M7 (cid:48) and the fact that everysimplicial set is cofibrant, we have that ( − ) K preserves (acyclic) fibrations. Since( − ) K preserves acyclic fibrations, (( − ) K ) L m ∼ = ( K (cid:116) − ) L m for all m ≥
0, andReedy weak equivalences are levelwise, we see that K (cid:116) − sends acyclic fibrationsto weak equivalences.We will conclude by showing that that K (cid:116) − preserves Reedy fibrations. Recallthe definition of fibration from Theorem 7.1. The key point to check is that wehave an isomorphism of matching objects M L m ( K (cid:116) Y ) ∼ = M L m ( Y K )for every m ≥
0. For an arbitrary presheaf Z , we have M L m Z = lim ( ∇ op ) − ( L m ) α : L m → L n Z L n = lim ∇ + ( L m ) β : L n → L m Z L n . Let ∆ + ( m ) ⊆ ∇ + ( L m ) be the full subcategory whose objects are the oriented maps L n → L m in ∇ + ⊆ Ξ + . Note that if we have a morphism L n L p L mφγ γ (cid:48) with γ, γ (cid:48) ∈ ∆ + ( m ), then φ is an oriented map. Thus every morphism in ∆ + ( m )is an oriented map as well, which explains the choice of notation. (This argumentis much like one appearing in the proof of Lemma 6.2.) Further, we have that∆ + ( m ) ⊆ ∇ + ( L m ) is essentially surjective: a non-oriented map γϑ in ∇ + ( L m ) isisomorphic to γ . Thus the inclusion map is an equivalence of categories.It follows that M L m Z ∼ = lim ∆ + ( m ) β : L n → L m Z L n . Since K (cid:116) Y and Y K are equal when applied to oriented maps, it follows that wehave, for each K , an isomorphism M L m ( K (cid:116) Y ) ∼ = M L m ( Y K ), natural in Y . Infact, we have (cid:16) ( K (cid:116) Y ) L m → M L m ( K (cid:116) Y ) (cid:17) ∼ = (cid:16) ( Y K ) L m → M L m ( Y K ) (cid:17) so the fact that ( − ) K preserves fibrations implies that K (cid:116) − also preserves fibra-tions. (cid:3) Proposition A.2.
Suppose that sSet ∇ op ∗ and sSet Ξ op ∗ are endowed with the modelstructures from Theorem 7.16. Consider the adjunction sSet ∇ op ∗ (cid:29) sSet Ξ op ∗ induced from the full-subcategory inclusion ∇ → Ξ . Then the left adjoint preservesand detects cofibrations.Proof. We identify sSet ∇ op ∗ as the full subcategory whose objects A satisfy A R = ∅ if R is non-linear. By Theorem 7.16, Proposition 7.17, and RI ∼ = id, the functor I : sSet R op ∗ → sSet R op creates cofibrations (where R = ∇ , Ξ). Thus it is enoughto work with Reedy cofibrations in the unreduced categories.Consider the indexing category for the latching object of A . If R is any object ofΞ, then (Ξ op ) + ( R ) = Ξ − ( R ) has objects of the form R → S in Ξ − \ Iso(Ξ). Givena map R → S , if S is linear, so is R . It follows that L Ξ R ( A ) = colim Ξ − ( R ) R → S A S = ∅ when R is a non-linear tree. If R = L m , then L Ξ L m ( A ) = colim Ξ − ( L m ) L m → S A S = colim ∇ − ( L m ) L m → L n A L n = L ∇ L m ( A )since A S = ∅ when S is not linear. Thus a map A → B in sSet ∇ op ∗ is a cofibrationin sSet ∇ op ∗ if and only if it is a cofibration in sSet Ξ op ∗ . (cid:3) References
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