Operads of natural operations I: Lattice paths, braces and Hochschild cochains
aa r X i v : . [ m a t h . A T ] F e b OPERADS OF NATURAL OPERATIONS I:LATTICE PATHS, BRACES AND HOCHSCHILD COCHAINS by Michael Batanin, Clemens Berger & Martin Markl
Abstract . —
In this first paper of a series we study various operads of natural oper-ations on Hochschild cochains and relationships between them.
R´esum´e (Op´erades des op´erations naturelles I: chemins bris´es, op´erationsbrace et cochaˆınes de Hochschild)
Dans ce premier article d’une s´erie nous ´etudions et comparons plusieurs op´eradesmunies d’une action naturelle sur les cochaines de Hochschild d’une alg`ebre associa-tive.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. The lattice path operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Weak equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. Operads of natural operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. Operads of braces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22A. Substitudes, convolution and condensation . . . . . . . . . . . . . . . . . . . . 30References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Mathematics Subject Classification . —
Primary 55U10, secondary 55S05, 18D50.
Key words and phrases . —
Lattice path operad, Hochschild cohomology, natural operation.M. Batanin was supported by Scott Russell Johnson Memorial Fund and Australian Research CouncilGrant DP0558372.C. Berger was supported by the grant OBTH of the French Agence Nationale de Recherche.M. Markl was supported by the grant GA ˇCR 201/08/0397 and by the Academy of Sciences of theCzech Republic, Institutional Research Plan No. AV0Z10190503.
MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL
1. Introduction
This paper continues the efforts of [
14, 3, 2 ] in which we studied operads natu-rally acting on Hochschild cochains of an associative or symmetric Frobenius algebra.A general approach to the operads of natural operations in algebraic categories wasset up in [ ] and the first breakthrough in computing the homotopy type of suchan operad has been achieved in [ ]. In [ ], the same problem was approached froma combinatorial point of view, and a machinery which produces operads acting on theHochschild cochain complex in a general categorical setting was introduced.The constructions of [ ] have some specific features in different categories which areimportant in applications. In this first paper of a series entitled ‘Operads of NaturalOperations’ we begin a detailed study of these special cases.It is very natural to start with the classical Hochschild cochain complex of anassociative algebra. This is, by far, the most studied case. It seems to us, however,that a systematic treatment is missing despite its long history and a vast amountof literature available. One of the motivations of this paper was our wish to relatevarious approaches in literature and to provide a uniform combinatorial language forthis purpose.Here is a short summary of the paper.In section 2 we describe our main combinatorial tool: the lattice path operad L andits condensation in the differential graded setting. This description leads to a carefultreatment of (higher) brace operations and their relationship with lattice paths insection 3.The lattice path operad comes equipped with a filtration by complexity [ ]. Thesecond filtration stage L (2) is the most important for understanding natural operationson the Hochschild cochains. In section 4 we give an alternative description of L (2) interms of trees, closely related to the operad of natural operations from [ ]. Finally,in section 5 we study various suboperads generated by brace operations. The mainresult is that all these operads have the homotopy type of a chain modelof the little disks operad . For sake of completeness we add a brief appendixcontaining an overview of some categorical constructions used in this paper. Convention.
If not stated otherwise, by an operad we mean a classical symmetric(i.e. with the symmetric groups acting on its components) operad in an appropriatesymmetric monoidal category which will be obvious from the context. The same con-vention is applied to coloured operads, substitudes, multitensors and functor-operadsrecalled in the appendix.
Acknowledgement.
We would like to express our thanks to the referee for carefullyreading the paper and many useful remarks and suggestions.
PERADS OF NATURAL OPERATIONS I
2. The lattice path operad
As usual, for a non-negative integer m , [ m ] denotes the ordinal 0 < · · · < m . Wewill use the same symbol also for the category with objects 0 , . . . , m and the uniquemorphism i → j if and only if i ≤ j . The tensor product [ m ] ⊗ [ n ] is the categoryfreely generated by the ( m, n )-grid which is, by definition, the oriented graph withvertices ( i, j ), 0 ≤ i ≤ m , 0 ≤ j ≤ n , and one oriented edge ( i ′ , j ′ ) → ( i ′′ , j ′′ ) if andonly if ( i ′′ , j ′′ ) = ( i ′ + 1 , j ′ ) or ( i ′′ , j ′′ ) = ( i ′ , j ′ + 1).Let us recall, closely following [ ], the lattice path operad and its basic properties.For non-negative integers k , . . . , k n , l and n ∈ N put L ( k , . . . , k n ; l ) := Cat ∗ , ∗ ([ l + 1] , [ k + 1] ⊗ · · · ⊗ [ k n + 1])where ⊗ is the tensor product recalled above and Cat ∗ , ∗ ([ l + 1] , [ k + 1] ⊗ · · ·⊗ [ k n + 1])the set of functors ϕ that preserve the extremal points, by which we mean that(1) ϕ (0) = (0 , . . . ,
0) and ϕ ( l + 1) = ( k + 1 , . . . , k n + 1) . A functor ϕ ∈ L ( k , . . . , k n ; l ) is given by a chain of l + 1 morphisms ϕ (0) → ϕ (1) → · · · → ϕ ( l + 1) in [ k + 1] ⊗ · · · ⊗ [ k n + 1] with ϕ (0) and ϕ ( l + 1) fulfilling (1).Each morphism ϕ ( i ) → ϕ ( i + 1) is determined by a finite oriented edge-path in the( k + 1 , . . . , k n + 1)-grid. For n = 0, L (; l ) consists of the unique functor from [ l + 1]to the terminal category with one object. . — We will use a slight modification of the terminologyof [ ]. For non-negative integers k , . . . , k n ∈ N denote by Q ( k , . . . , k n ) the integralhypercube Q ( k , . . . , k n ) := [ k + 1] × · · · × [ k n + 1] ⊂ Z × n . A lattice path is a sequence p = ( x , . . . , x N ) of N := k + · · · + k n + n + 1 points of Q ( k , . . . , k n ) such that x a +1 is, for each 0 ≤ a < N , given by increasing exactly onecoordinate of x a by 1. A marking of p is a function µ : p → N that assigns to eachpoint x a of p a non-negative number µ a := µ ( x a ) such that P Na =1 µ a = l .We can describe functors in L ( k , . . . , k n ; l ) as marked lattice paths ( p, µ ) in thehypercube Q ( k , . . . , k n ). The marking µ a = µ ( x a ) represents the number of elementsof the interior { , . . . , l } of [ l + 1] that are mapped by ϕ to the a th lattice point x a of p . We call lattice points marked by 0 unmarked points so the set of marked pointsequals ϕ ( { , . . . , l } ). For example, the marked lattice path(2) • ✲ • ✲ • • ✲ • • • ✲ • MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL represents a functor ϕ ∈ L (3 ,
2; 8) with ϕ (0) = (0 , ϕ (1) = ϕ (2) = ϕ (3) = (1 , ϕ (4) = (2 , ϕ (5) = ϕ (6) = (3 ,
1) and ϕ (7) = ϕ (8) = ϕ (9) = (4 , n = 0, so the unique element of L (; l ) is represented by the point marked l ,i.e. by • l . . — Let p ∈ L ( k , . . . , k n ; l ) be a lattice path. A point of p at which p changes its direction is an angle of p . An internal point of p is a point that is notan angle nor an extremal point of p . We denote by Angl ( p ) (resp. Int ( p )) the set ofall angles (resp. internal points) of p .For instance, the path in (2) has 4 angles, 2 internal points, 4 unmarked pointsand 1 unmarked internal point.Following again [ ] closely, we denote, for 1 ≤ i < j ≤ n , by p ij the projectionof the path p ∈ L ( k , . . . , k n ; l ) to the face [ k i + 1] × [ k j + 1] of Q ( k , . . . , k n ); let c ij := Angl ( p ij ) be the number of its angles. The maximum c ( p ) := max { c ij } iscalled the complexity of p . Let us finally denote by L ( c ) ( k , . . . , k n ; l ) ⊂ L ( k , . . . , k n ; l )the subset of marked lattice paths of complexity ≤ c . The case c = 2 is particularlyinteresting, because L (2) ( k , . . . , k n ; l ) is, by [ , Proposition 2.14], isomorphic to thespace of unlabeled ( l ; k , . . . , k n )-trees recalled on page 20. For convenience of thereader we recall this isomorphism on page 21.As shown in [ ], the sets L ( k , . . . , k n ; l ) and their subsets L ( c ) ( k , . . . , k n ; l ), c ≥ N -coloured operad L and its sub-operads L ( c ) . To simplify formulations, wewill allow c = ∞ , putting L ( ∞ ) := L . . — Since we aim to work in the category of abelian groups, wewill make no notational difference between the sets L ( c ) ( k , . . . , k n ; l ) and their linearspans.The underlying category of the coloured operad L (which coincides with the un-derlying category of L ( c ) for any c ≥
0) is, by definition, the category whose objectsare non-negative integers and morphism n → m are elements of L ( n, m ), i.e. non-decreasing maps ϕ : [ m + 1] → [ n + 1] preserving the endpoints.By Joyal’s duality [ ], this category is isomorphic to the (skeletal) category ∆of finite ordered sets, i.e. L ( n, m ) = ∆( n, m ). The operadic composition makes thecollection L ( c ) ( • , . . . , • n ; • ) (with c = ∞ allowed) a functor (∆ op ) × n × ∆ → Abel,i.e. n -times simplicial 1-time cosimplicial Abelian group.Morphisms in the category ∆ are generated by the cofaces d i : [ m − → [ m ] givenby the non-decreasing map that misses i , and the codegeneracies s i : [ m + 1] → [ m ]given by the non-decreasing map that hits i twice. In both cases, 0 ≤ i ≤ m . Let usinspect how these generating maps act on the pieces of the operad L ( c ) . PERADS OF NATURAL OPERATIONS I . — We describe the induced r th (1 ≤ r ≤ n ) simplicialmaps ∂ ri : L ( c ) ( k , . . . , k r − , m, k r +1 , . . . , k n ; l ) → L ( c ) ( k , . . . , k r − , m − , k r +1 , . . . , k n ; l ) , where m ≥
1, 0 ≤ i ≤ m , and σ ri : L ( c ) ( k , . . . , k r − , m, k r +1 , . . . , k n ; l ) → L ( c ) ( k , . . . , k r − , m +1 , k r +1 , . . . , k n ; l ) , where 0 ≤ i ≤ m . To this end, we define, for each m ≥ ≤ i ≤ m , theepimorphism of the hypercubes D ri : Q ( k , . . . , k r − , m, k r +1 , . . . , k n ) ։ Q ( k , . . . , k r − , m − , k r +1 , . . . , k n )by D ri ( a , . . . , a r , . . . , a n ) := (cid:26) ( a , . . . , a r , . . . , a n ) , if a r ≤ i , and( a , . . . , a r − , . . . , a n ) , if a r > i ,where ( a , . . . , a r , . . . , a n ) ∈ Q ( k , . . . , k r − , m, k r +1 , . . . , k n ) is an arbitrary point. Ina similar fashion, the monomorphism S ri : Q ( k , . . . , k r − , m, k r +1 , . . . , k n ) ֒ → Q ( k , . . . , k r − , m +1 , k r +1 , . . . , k n )is, for 0 ≤ i ≤ m , given by S ri ( a , . . . , a r , . . . , a n ) := (cid:26) ( a , . . . , a r , . . . , a n ) , if a r ≤ i , and( a , . . . , a r + 1 , . . . , a n ) , if a r > i .Let ( p, µ ) be a marked lattice path in Q ( k , . . . , k r − , m, k r +1 , . . . , k n ) representinga functor ϕ ∈ L ( c ) ( k , . . . , k r − , m, k r +1 , . . . , k n ; l ). Then ∂ ri ( ϕ ) is represented by themarked path ( ∂ ri ( p ) , ∂ ri ( µ )), where ∂ ri ( p ) is the image D ri ( p ) of p in Q ( k , . . . , k r − , m − , k r +1 , . . . , k n , l ). The marking ∂ ri ( µ ) is given by ∂ ri ( µ )( D ri ( x )) := P ˜ x µ (˜ x ), with thesum taken over all ˜ x ∈ p such that D ri (˜ x ) = D ri ( x ). A less formal description of thismarking is the following.There are precisely two different points of p , say x ′ and x ′′ , such that D ri ( x ′ ) = D ri ( x ′′ ); let us call the remaining points of p regular. The marking of D ri ( x ) is thesame as the marking of x if x is regular. If x ′ and x ′′ are the two non-regular points,then the marking of the common value D ri ( x ′ ) = D ri ( x ′′ ) is µ ( x ′ )+ µ ( x ′′ ). See Figure 1in which the operator ∂ contracts the column denoted D and decorates the pointobtained by identifying the point (1 ,
0) marked 3 with the point (2 ,
0) marked 1 by3 + 1 = 4. The remaining operators act in the similar fashion.To define the marked lattice path ( σ ri ( p ) , σ ri ( µ )) representing the degeneracy σ ri ( ϕ ),we need to observe that the image S ri ( p ) is not a lattice path in Q ( k , . . . , k r − , m +1 , k r +1 , . . . , k n ), but that it can be made one by adding a unique ‘missing’ lattice pointˆ x . The resulting lattice path is σ ri ( p ). The marking σ ri ( µ ) is given by σ ri ( µ )( S ti ( x )) := µ ( x ) for x ∈ p while σ ri ( µ )(ˆ x ) := 0, i.e. the newly added point ˆ x is unmarked.See Figure 2 in which the new point ˆ x is denoted . Observe that ˆ x is alwaysan internal point. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL • ✲ • ✲ • • ✲ • • • ✲ • D D D • ✲ • ✲ • • • ✲ • • ✲ • ✲ • • ✲ • • ✻ • ✲ • ✲ ✲ • • • ✲ • ✁✁✁✁☛ ∂ ❆❆❆❆❯ ∂ ✲ ∂ Figure 1.
The simplicial boundaries acting on the element of (2). • ✲ • ✲ • • ✲ • • • ✲ • S • ✲ • ✲ • • ✲ • • • ✲ • ✲ σ Figure 2.
The operator σ acting on the element of (2). . — We describe, for l ≥ ≤ i ≤ l , theboundaries δ i : L ( c ) ( k , . . . , k n ; l − → L ( c ) ( k , . . . , k n ; l )and, for 0 ≤ i ≤ l , the degeneracies s i : L ( c ) ( k , . . . , k n , l + 1) → L ( c ) ( k , . . . , k n ; l ) , of the induced cosimplicial structure. Let ( p, µ ) be a marked path in Q ( k , . . . , k n ; l ∓ ϕ ∈ L ( c ) ( k , . . . , k n ; l ∓ δ i nor s i changes theunderlying path, so δ i ( ϕ ) is represented by ( p, δ i ( µ )) and s i ( ϕ ) by ( p, s i ( µ )).Let ˆ x := ϕ ( i ). Then the markings δ i ( µ ) and s i ( µ ) are defined by δ i ( µ )( x ) = s i ( µ )( x ) = µ ( x ) for x = ˆ x , while δ i ( µ )(ˆ x ) := µ (ˆ x ) + 1 and s i ( µ )(ˆ x ) := µ (ˆ x ) −
3. Weak equivalences . — Given an n -simplicial cosimplicial abeliangroup, i.e. a functor X : ∆ op × n × ∆ → Abel, denote by X •∗ = Tot( X ( • , . . . , • n ; • ))the simplicial totalization. It is a cosimplicial dg-abelian group with components(3) X •∗ := M ∗ = − ( k + ··· + k n ) X ( k , . . . , k n ; • ) PERADS OF NATURAL OPERATIONS I bearing the degree +1 differential ∂ = ∂ + · · · + ∂ n , where each ∂ r is induced from theboundaries of the r th simplicial structure in the standard manner. We also denoteby | X | ∗ = Tot(Tot( X ( • , . . . , • n ; • ))) the cosimplicial totalization of the cosimplicialdg-abelian group X •∗ . It is a dg-abelian group with components | X | ∗ = Y ∗ = l − ( k + ··· + k n ) X ( k , . . . , k n ; l ) = Y l ≥ M l −∗ = k + ··· + k n X ( k , . . . , k n ; l )and the degree +1 differential d = δ + ∂ , where ∂ is as above and δ is the standardalternating sum of the cosimplicial boundary operators.According to Appendix A, the dg-abelian groups | L ( c ) | ( n ) := | L ( c ) ( • , . . . , • n ; • ) | are the result of condensation and, therefore, assemble, for each c ≥
0, into a dg-operad | L ( c ) | = {| L ( c ) | ( n ) } n ≥ . Observe that | L (2) | is isomorphic to the Tamarkin-Tsyganoperad T recalled on page 20. (1) Let us denote, for each n, c ≥
0, by B r ( c ) ( n ) the simplicial totalization of the n -times simplicial abelian group L ( c ) ( • , . . . , • n ; 0), that is, B r ∗ ( c ) ( n ) := M ∗ = − ( k + ··· + k n ) L ( c ) ( k , . . . , k n ; 0) , with the induced differential ∂ = ∂ + · · · + ∂ n . Elements of B r ( c ) ( n ) are representedby marked lattice paths ( p,
0) with the trivial marking µ = 0 (all points of p areunmarked). Since the trivial marking bears no information, we will discard it fromthe notation. The whiskering w : B r ( c ) ( n ) → | L ( c ) | ( n ) is defined as(4) w ( p ) := Y s ≥ w s ( p ) , where w s ( p ) ∈ | L ( c ) | ( n ) is the sum of all marked paths, taken with appropriate signs,obtained from p by inserting precisely s new distinct internal lattice points marked 1.The origin of the signs is explained in Proposition 3.2 below. The action of thewhiskering is illustrated in Figure 3.For p ′ ∈ L ( c ) ( a , . . . , a n ; 0), p ′′ ∈ L ( c ) ( b , . . . , b m ; 0) and 1 ≤ i ≤ n define(5) p ′ ◦ i p ′′ := p ′ ◦ i w a i ( p ′′ ) ∈ M b ′ + ··· + b ′ m = b + ··· + b m + a i L ( c ) ( a , ..., a i − , b ′ , ..., b ′ m , a i +1 , ..., a n ; 0)where w a i ( p ′′ ) is the whiskering of the lattice path p ′′ by a i points and ◦ i in the righthand side is the operadic composition in the coloured operad L ( c ) . By linearity, (5)extends to the operation ◦ i : B r ( c ) ( n ) ⊗ B r ( c ) ( m ) → B r ( c ) ( m + n − . — Operations ◦ i above make the collection B r ( c ) = { B r ( c ) ( n ) } n ≥ a dg-operad. The signs in (4) can be chosen such that the map w : B r ( c ) ֒ → | L ( c ) | isan inclusion of dg-operads. (1) Whenever we refer to sections 4 or 5, we shall keep in mind that Convention 4.2 is used in thesesections.
MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL • ••• •• • w Figure 3.
An element p ∈ B r ( c ) (2) (left) and one of the terms in w ( p ) ∈| L ( c ) | (2) (right). The newly added internal points are marked by 1. Proof . — The first part of the proposition can be verified directly. There is an in-ductive procedure to fix the signs in (4), but we decided not to include this clumsyand lengthy calculation here. A conceptual way to get the signs is to embed the dg-operad | L ( c ) | into the coendomorphism operad of chains on the standard simplex ofa sufficiently large dimension, cf. [ , Remark 2.20], and to require that the whiskering w : B r ( c ) → | L ( c ) | induces, via the isomorphism of Proposition 3.4 below, the actionof the surjection operad, with the sign convention of [ , Section 2.2]. Remark.
One of the main advantages of the ‘operadic’ sign convention (see 4.1)which we use in sections 4 or 5 is that in the corresponding whiskering formula (17)all terms, quite miracously, appear with the +1-signs.So the operad structure of B r ( c ) is induced by the operad structure of | L ( c ) | andthe whiskering map. Notice that B r (2) is the brace operad B r recalled on page 22 andthe map w : B r (2) → | L (2) | the whiskering defined in (17). Proposition 3.2 thereforegeneralizes Proposition 5.7. . — Let X ( • , . . . , • n ; • ) be an n -simplicial cosimpli-cial abelian group as in 3.1. We will need also the traditional n -simplicial normalized totalization, or simplicial normalization for short, denoted X •∗ = Nor( X ( • , . . . , • n ; • )),obtained from the un-normalized totalization (3) by modding out the images of sim-plicial degeneracies. We then denote by | X | ∗ = Nor(Nor( X ( • , . . . , • n ; • ))) the nor-malized cosimplicial totalization of the cosimplicial dg-abelian group X •∗ . It is theintersection of the kernels of cosimplicial degeneracies in the un-normalized cosimpli-cial totalization of X •∗ . As argued in [ ], the n -simplicial cosimplicial normalization | L ( c ) | of the lattice path operad L ( c ) is a dg-operad.Let us denote, for each n, c ≥
0, by Nor( B r ( c ) )( n ) = Nor( L ( c ) ( • , . . . , • n ; 0)) thesimplicial normalization of the n -simplicial abelian group L ( c ) ( • , . . . , • n ; 0), with theinduced differential. The explicit description of the simplicial structure in 2.4 makes PERADS OF NATURAL OPERATIONS I it obvious that elements of Nor( B r ( c ) )( n ) are represented by (unmarked) lattice pathswith no internal points.One defines the operadic composition on Nor( B r ( c ) ) = { Nor( B r ( c ) )( n ) } n ≥ andthe whiskering w : Nor( B r ( c ) ) ֒ → | L ( c ) | by the same formulas as in the un-normalizedcase. The operad Nor( B r (2) ) is the normalized brace operad Nor( B r ) recalled onpage 23. We leave as an exercise to verify that Nor( B r (1) ) is the operad for unitalassociative algebras and Nor( B r (0) ) the operad whose ‘algebras’ are abelian groupswith a distinguished point. . — The operads
Nor( B r ( c ) ) are isomorphic to the suboperads F c X of the surjection operad X introduced in [ , 1.6.2] , resp. the suboperads S c of the sequence operad S introduced in [ , Definition 3.2] .Proof . — We rely on the terminology of [ , 1.6.2]. A non-degenerate surjection u : { , . . . , m } → { , . . . , n } , m ≥ n , in F c X ( n ) induces a lattice path ϕ u representingan element of Nor( B r ( c ) )( n ) as follows. For 1 ≤ i ≤ n denote by d i ∈ Z × n the vector(0 , . . . , , . . . ,
0) with 1 at the i th position, and k i := u − ( i ) −
1. Then ϕ u is thepath in the grid [ k + 1] ⊗ · · · ⊗ [ k n + 1] that starts at the ‘lower left corner’ (0 , . . . , d u (1) , then by d u (2) , etc., and finally by d u ( m ) . It is obvious that thecorrespondence u ϕ u is one-to-one.The following statement follows from [ , Examples 3.10(c)] and [ , Section 1.2]. . — The whiskering w : Nor( B r ( c ) ) ֒ → | L ( c ) | is an inclusion of dg-operads. We will need also the following statement. . —
The natural projection π : B r ( c ) ։ Nor( B r ( c ) ) to the normaliza-tion is an epimorphism of dg-operads for each c ≥ .Proof . — It is almost obvious that the operadic composition in B r ( c ) preserves thenumber of internal points, that is, if p ′ (resp. p ′′ ) is a lattice path with a ′ (resp. a ′′ )internal points, then p ′ ◦ i p ′′ is, for each i for which this expression makes sense,a linear combination of lattice paths with a ′ + a ′′ internal points. This implies thatthe degenerate subspace Dgn ( B r ( c ) ) of B r ( c ) which is the subcollection spanned bylattice paths with at least one internal point, form a dg-operadic ideal in B r ( c ) , sothe projection π : B r ( c ) ։ B r ( c ) / Dgn ( B r ( c ) ) = Nor( B r ( c ) ) is an operad map. The factthat π commutes with the differentials follows from the standard properties of thesimplicial normalizations.Let c B r ( c ) = { c B r ( c ) ( n ) } n ≥ be the subcollection of B r ( c ) such that c B r ( c ) ( n ) ⊂ B r ( c ) ( n ) is spanned by paths with no internal points, for n ≥
1, and c B r ( c ) (0) := 0. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL . —
The collection c B r ( c ) is a (non-dg) suboperad of B r ( c ) for any c ≥ . It is dg-closed if and only if c ≤ .Proof . — It follows from the property stated in the proof of Proposition 3.6 thatthe subcollection c B r ( c ) is closed under the operad structure of B r ( c ) for an arbitrary c ≥
0. It remains to prove that c B r ( c ) is closed under the action of the differential ifand only if c ≤
2. Let us prove first that it is dg-closed for c ≤ c = 2 this follows from the fact that c B r (2) = c B r is a dg-suboperad of B r (2) = B r , see Proposition 5.2 and the description of the dg-operad structures of B r and c B r in terms of trees following that proposition, or [ ]. For c = 0 ,
1, the propositionis obvious.If c ≥
3, the differential may create internal points, as shown in the followingpicture where the piece ∂ of the differential creates the internal point : •• •• • D •• • ✲ ∂ So c B r ( c ) is not dg-closed if c ≥ . — For each n, c ≥
0, one may also consider the collec-tion | ˙ L ( c ) | := {| ˙ L ( c ) | ( n ) } n ≥ defined by | ˙ L ( c ) | ( n ) := Tot(Nor( L ( c ) ( • , . . . , • n ; • )))i.e. as the n -simplicial normalization followed by the un-normalized cosimplicial to-talization.Observe that there is a natural projection π : | L ( c ) | ։ | ˙ L ( c ) | of collections. Weemphasize that, for c ≥
3, the collection | ˙ L ( c ) | has no natural dg-operad structurealthough it will still play an important auxiliary role in this section. We, however, have . — For c ≤ , the collection | ˙ L ( c ) | has a natural operad structuresuch that the projection π : | L ( c ) | ։ | ˙ L ( c ) | is a map of dg-operads.Proof . — The proof uses the fact that | ˙ L (2) | is the normalized Tamarkin-Tsyganoperad Nor( T ) which is a quotient of T = | L (2) | , see 4.9. This proves the propositionfor c = 2. For c = 0 , . — For each c ≥ , there is the following chain of weak equivalencesof dg-operads: | L ( c ) | w ←− B r ( c ) π −→ Nor( B r ( c ) ) w −→ | L ( c ) | , in which the maps w are the whiskerings of Propositions 3.2 and 3.5, and π is thenormalization projection of Proposition 3.6. PERADS OF NATURAL OPERATIONS I ......... · · ·· · ·· · ·· · ·· · · ✻✻✻ ✻✻✻ ✻✻✻ ✻✻✻ ✻✻✻ ✲✲✲✲ ✲✲✲✲ ✲✲✲✲✲ ✲✲✲ ✲✲✲ L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) − L ( c ) ( n ) Figure 4.
The structure of the dg-operad | L ( c ) | . Proof . — The map π : B r ( c ) ( n ) → Nor( B r ( c ) )( n ) is a homology isomorphism for each n, c ≥ n -simplicial abelian group, so π isa weak equivalence of dg operads.Let us analyze the un-normalized whiskering w : B r ( c ) ( n ) ֒ → | L ( c ) | ( n ). The arity n piece of the dg-operad | L ( c ) | can be organized into the bicomplex of Figure 4 in whichthe l th column L ( c ) ( n ) l ∗ , l ≥
0, is the simplicial totalization Tot( L ( c ) ( • , . . . , • n ; l ))and the horizontal differentials are induced from the cosimplicial structure. Thedg-abelian group | L ( c ) | ( n ) is then the corresponding Tot Q -total complex (see [ ,Section 5.6] for the terminology).The dg-abelian group B r ( c ) ( n ) appears as the leftmost column of Figure 4, soone has the projection proj : | L ( c ) | ( n ) → B r ( c ) ( n ) of dg-abelian groups which is theidentity on the leftmost column and sends the remaining columns to 0. Since clearly proj ◦ w = id, it is enough to prove that proj is a homology isomorphism.We interpret proj : | L ( c ) | ( n ) → B r ( c ) ( n ) as a map of bicomplexes, with B r ( c ) ( n )consisting of one column, and we prove that proj induces an isomorphism of the E -terms of the spectral sequences induced by the column filtrations. These filtra-tions are complete and exhaustive, thus the Eilenberg-Moore comparison theorem [ ,Theorem 5.5.1] implies that proj is a homology isomorphism.Let ( E ∗∗ , d ) be the 0th term of column spectral sequence for | L ( c ) | ( n ). Thismeans that ( E l, ∗ , d ) = (Tot( L ( c ) ( • , . . . , • n ; l )) ∗ , ∂ ), the l th column of the bicomplexin Figure 4 with the simplicial differential.To calculate E l ∗ := H ∗ ( E l ∗ , d ), we recall the explicit description of the simplicialstructures given in 2.4 and observe that the vertical differential d = ∂ does notincrease the number of angles of lattice paths. We therefore have, for each fixed l ≥ MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL another spectral sequence ( E r ∗∗ , d r ) induced by the filtration of L ( c ) ( n ) l ∗ by the numberof angles. The piece E uv of the initial sheet of this spectral sequence is spanned bymarked paths ( p, µ ) ∈ L ( c ) ( k , . . . , k n ; l ) with − u angles and v = − u − ( k + · · · + k n ).With this degree convention, the total degree of an element of E ∗∗ is the same asthe degree of the corresponding element in E l ∗ . By simple combinatorics, ( E r ∗∗ , d r )is a spectral sequence concentrated at the region { ( u, v ); u ≤ − n, u − v ≥ − n } of the ( u, v )-plane, thus no convergence problems occur. One easily sees that, asdg-abelian groups,(6) ( E u ∗ , d ) ∼ = (cid:18) M p ∈ Nor( B r ( c ))( n ) Angl ( p )= − u M i + ··· + i u +1 = n − −∗ { B i ⊗ Z [ x ] · · · ⊗ Z [ x ] B i u +1 | {z } − u + 1 factors } l , d B (cid:19) , where B ∗ = B ∗ ( Z [ x ] , Z [ x ] , Z [ x ]) is the un-normalized two-sided bar construction ofthe polynomial algebra Z [ x ] and the differential d B is induced in the standard mannerfrom the bar differential. The subscript l in (6) denotes the l -homogeneous part withrespect to the grading induced by the number of instances of x . The factors of thedirect sum are indexed by unmarked paths with no internal points representing a basisof Nor( B r ( c ) )( n ). The isomorphism (6) is best explained by looking at the markedpath ••••••••••••• ❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ with 4 angles which is an element of E − , − represented, via the isomorphism (6), bythe element x ⊗ [ x | x ] ⊗ x ⊗ [ x ] ⊗ x ⊗ [ x | x | x ] ⊗ x ⊗ [ ] ⊗ x ⊗ [ x ] ⊗ x in B ⊗ Z [ x ] B ⊗ Z [ x ] B ⊗ Z [ x ] B ⊗ Z [ x ] B . It is a standard result of homological algebrathat ( B ∗ ⊗ Z [ x ] · · · ⊗ Z [ x ] B ∗ , d B ) is acyclic in positive dimensions, thus the cohomologyof the right hand side of (6) is spanned by cycles of the form(7) x l ⊗ [ ] ⊗ · · · ⊗ [ ] ∈ E − Angl ( p ) ,n − . At this point we need to observe that the differential ∂ decreases the number ofangles of lattice paths p with no internal points representing elements of Nor( B r ( c ) )( n )by one. Indeed, it is easy to see that a simplicial boundary operator described in 2.4may either decrease the number of angles of p by 1 or by 2. When it decreases it by2 it creates an internal point, so the contributions of all simplicial boundaries thatdecrease the number of angles by 2 sum up to 0, by the standard property of thesimplicial normalization. We conclude that ( L ∗ = u + v E uv , d ) ∼ = (Nor( B r ( c ) ) ∗ ( n ) , ∂ )as dg-abelian groups and that ( E r ∗∗ , d r ) collapses at this level.Let us return to the column spectral sequence ( E r ∗∗ , d r ) for the bicomplex in Fig-ure 4. It follows from the above calculation that the l th column E l ∗ of the first term PERADS OF NATURAL OPERATIONS I ( E ∗∗ , d ) equals H ∗ (Nor( B r ( c ) )( n )) for each l ≥
0. It remains to describe the differ-ential d : E l ∗ → E l +1) ∗ . To this end, one needs to observe that the expressions (7)representing elements of E l ∗ = H ∗ (Nor( B r ( c ) )( n )) correspond to marked lattice pathswithout internal points, whose only marked point is the initial one, marked by l . Fromthe description of the cosimplicial structure given in 2.5 one easily obtains that d : E l ∗ → E l +1) ∗ = (cid:26) , if l is even andid , if l is odd.We conclude that E ∗∗ := H ∗ ( E ∗∗ , d ) is concentrated at the leftmost column whichequals H ∗ (Nor( B r ( c ) )( n )) and that, from the obvious degree reasons, the columnspectral sequence collapses at this stage. Since we already know that the projection B r ( c ) π ։ Nor( B r ( c ) ) is a weak equivalence i.e., in particular, that H ∗ ( B r ( c ) ( n )) ∼ = H ∗ (Nor( B r ( c ) )( n )), the above facts imply that proj : | L ( c ) | ( n ) → B r ( c ) ( n ) induces anisomorphism of the E -terms of the column spectral sequences, so it is a homologyisomorphism and w is a homology isomorphism, too.Let us finally prove that the normalized whiskering w : Nor( B r ( c ) )( n ) ֒ → | L ( c ) | ( n )is a weak equivalence. We have the composition(8) Nor( B r ( c ) )( n ) w ֒ → | L ( c ) | ( n ) ˙ ι ֒ → | ˙ L ( c ) | ( n )in which the obvious inclusion ˙ ι is a homology isomorphism by a simple lemma for-mulated below. As in the un-normalized case, the dg-abelian group Nor( B r ( c ) )( n ) isthe first column of the semi-normalized version of the bicomplex in Figure 4, so thereis a natural projection proj : | ˙ L ( c ) | ( n ) → Nor( B r ( c ) )( n ). This proj is a homologyisomorphism by the same arguments as in the un-normalized case, only using in (6)the normalized bar construction instead. The proof is finished by observing that proj is the left inverse of the composition (8).In the proof of Theorem 3.10 we used the following . — The inclusion ˙ ι : | L ( c ) | ( n ) ֒ → | ˙ L ( c ) | ( n ) is a homology isomorphismfor each n, c ≥ .Proof . — The lemma follows from the fact that | L ( c ) | ( n ) is the cosimplicial normal-ization of the dg-cosimplicial group | ˙ L ( c ) | ( n ).– – – – – MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL
In the following two sections we consider several operads. To simplify the naviga-tion, we give a glossary of notation. B , big operad of all natural operations, page 15Nor( B ) , normalized big operad, page 20 b B , non-unital big operad, page 19 T , Tamarkin-Tsygan suboperad of B , page 20Nor( T ) , normalized Tamarkin-Tsygan operad, page 21 b T , non-unital Tamarkin-Tsygan operad, page 21 B r, brace operad, page 22Nor( B r ) , normalized brace operad, page 23 c B r, non-unital brace operad, page 23The operads mentioned in the list and their maps are organized in Figure 10 onpage 29.
4. Operads of natural operations
In the previous sections we studied versions of the lattice path operad and its sub-operads. We only briefly mentioned that some of these operads act on the Hochschildcochain complex of an associative algebra. The present and the following sections willbe devoted to this action. It turns out that, in order to retain some nice features ofthe constructions in the previous section, namely the ‘whiskering’ formula (4) withoutsigns, on one hand, and to have simple rules for the signs in formulas for natural oper-ations on the other hand, one needs to use the ‘operadic’ degree convention, recalledin the next subsection. . — There are two conventions in defining the Hoch-schild cohomology of an associative algebra A . The classical one used for instancein [ ] is based on the chain complex C ∗ cl ( A ; A ) = L n ≥ C n cl ( A ; A ), where C n cl ( A ; A ) := Lin ( A ⊗ n , A ) (the subscript cl refers to “classical”). Another appropriate name wouldbe the (co)simplicial convention, because C ∗ cl ( A ; A ) is a natural cosimplicial abeliangroup. With this convention, the cup product ∪ is a degree 0 operation and the Ger-stenhaber bracket [ − , − ] has degree −
1, see [ , Section 7] for the ‘classical’ definitionsof these operations.On the other hand, it is typical for this part of mathematics that signs are difficultto handle. A systematic way to control them is the Koszul sign rule requiring thatwhenever we interchange two “things” of odd degrees, we multiply the sign by − ]. Now the underlying chain complex is(9) C ∗ ( A ; A ) := Lin ( T ( ↓ A ) , ↓ A ) ∗ , PERADS OF NATURAL OPERATIONS I where ↓ denotes the desuspension of a (graded) vector space and T ( ↓ A ) the tensoralgebra generated by A placed in degree −
1. Explicitly, C ∗ ( A ; A ) = L n ≥− C n ( A ; A ),where C n ( A ; A ) := Lin ( A ⊗ n +1 , A ), so C n ( A ; A ) = C n +1 cl ( A ; A ) for n ≥ −
1. With thisconvention, the cup product has degree +1 and the Gerstenhaber bracket degree 0.Depending on the choice of the convention, there are two definitions of the ‘big’operad of natural operations, see 4.3 below. The classical one introduces B cl as a cer-tain suboperad of the endomorphism operad E nd C ∗ cl ( A ; A ) of the graded vector space C ∗ cl ( A ; A ), and the operadic one introduces B as a suboperad of the endomorphismoperad E nd C ∗ ( A ; A ) . Here A is a generic , in the sense of Definition 4.6, unital associa-tive algebra. The difference between B cl and B is merely conventional; the operad B cl is the operadic suspension s B of the operad B [ , Definition II.3.15] while, of course, E nd C ∗ cl ( A ; A ) ∼ = s E nd C ∗ ( A ; A ) . . — In sections 4 and 5 we accept the operadic convention becausewe want to rely on the Koszul sign rule. As explained above, the operads B and B cl differ from each other only by the regrading and sign factors. . — Recall the dg-operad B = {B ( n ) } n ≥ of all natural multilinear operations on the (operadic) Hochschild cochain complex (9)of a generic associative algebra A (see Definition 4.6) with coefficients in itself intro-duced in [ ] (but notice that we are using here the operadic degree convention,see 4.2, while [ ] uses the classical one).Let A be a unital associative algebra. A natural operation in the sense of [ ] isa linear combination of compositions of the following ‘elementary’ operations:(a) The insertion ◦ i : C k ( A ; A ) ⊗ C l ( A ; A ) → C k + l ( A ; A ) given, for k, l ≥ − ≤ i ≤ k , by the formula ◦ i ( f, g )( a , . . . , a k + l ) := ( − il f ( a , . . . , a i − , g ( a i , . . . , a i + l ) , a i + l +1 , . . . , a k + l ) , for a , . . . , a k + l − ∈ A – the sign is determined by the Koszul rule!(b) Let µ : A ⊗ A → A be the associative product, id : A → A the identity map and1 ∈ A the unit. Then elementary operations are also the ‘constants’ µ ∈ C ( A ; A ),id ∈ C ( A ; A ) and 1 ∈ C − ( A ; A ).(c) The assignment f sgn( σ ) · f σ permuting the inputs of a cochain f ∈ C k ( A ; A )according to a permutation σ ∈ Σ k +1 and multiplying by the signature of σ is anelementary operation.Let B ( A ) lk ,...,k n denote, for l, k , . . . , k n ≥
0, the abelian group of all natural, inthe above sense, operations(10) O : C k − ( A ; A ) ⊗ · · · ⊗ C k n − ( A ; A ) → C l − ( A ; A ) . The regrading in the above equation guarantees that the super- and subscripts of B ( A ) lk ,...,k n will all be non-negative integers. Moreover, with this definition thespaces B ( A ) lk ,...,k n agree with the ones introduced in [ ]. The system B ( A ) lk ,...,k n MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL clearly forms an N -coloured suboperad B ( A ) of the endomorphism operad of the N -coloured collection { C n − ( A ; A ) } n ≥ .Recall that the Hochschild differential d H : C n − ( A ; A ) → C n ( A ; A ) is, for n ≥ d H f ( a ⊗ . . . ⊗ a n ) := ( − n +1 a f ( a ⊗ . . . ⊗ a n ) + f ( a ⊗ . . . ⊗ a n − ) a n + n − X i =0 ( − i + n f ( a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a n ) , for a i ∈ A . Apparently, d H is a natural operation belonging to B ( A ) n +1 n . Therefore,if O ∈ B ( A ) lk ,...,k n is as in (10), one may define δO ∈ B ( A ) l +1 k ,...,k n and, for 1 ≤ i ≤ k ,also ∂ i O ∈ B ( A ) lk ,...,k i − ,k i − ,k i +1 ,...,k n by(11) δO ( f , . . . , f n ) := d H O ( f , . . . , f n ) and ∂ i O ( f , . . . , f n ) := ( − k i + ··· + k n + l + n + i · O ( f , . . . , f i − , d H f i , f i +1 , . . . , f n ) . The sign in the second line of the above display equals ( − deg( f )+ ··· +deg( f i − ) · ( − deg( O ) as dictated by the Koszul rule.It follows from definition that elements of B ( A ) lk ,...,k n can be represented by linearcombinations of ( l ; k , . . . , k n )-trees in the sense of the following definition in which,as usual, the arity of a vertex of a rooted tree is the number of its input edges andthe legs are the input edges of a tree, see [ , II.1.5] for the terminology. . — Let l, k , . . . , k n be non-negative integers. An ( l ; k , . . . , k n ) -tree is a planar rooted tree with legs labeled by 1 , . . . , l and three types of vertices:(a) ‘white’ vertices of arities k , . . . , k n labeled by 1 , . . . , n ,(b) ‘black’ vertices of arities ≥ n = 0 we allow also the exceptional trees and • with no vertices.We call an internal edge whose initial vertex is special a stub (also called, in [ ],a tail ). It follows from definition that the terminal vertex of a stub is white; theexceptional tree • is not a stub. An example of an ( l ; k , . . . , k n )-tree is given inFigure 5.Each ( l ; k , . . . , k n )-tree T as in Definition 4.4 has its signature σ ( T ) = ± T is planar, its white vertices are naturally linearly ordered bywalking around the tree counterclockwise, starting at the root. The first white vertexwhich one meets is the first one in this linear order, the next white vertex differentfrom the first one is the second in this linear order, etc. For instance, the labels of PERADS OF NATURAL OPERATIONS I •• •• ◦ ◦ ◦ ◦ ❏❏❏❏✂✂✂✂❩❩❩❩❩❅❅❅❆❆❆(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) root Figure 5.
An (8; 3 , , , B , , , . Ithas 4 white vertices, 2 black vertices and 2 stubs. We use the conventionthat directed edges point upwards so the root is always on the top. the tree in Figure 5 agree with the ones given by the natural order, which of courseneed not always be the case.One is therefore given a function w p ( w ) that assigns to each white vertex w of the tree T its position p ( w ) ∈ { , . . . , n } in the linear order described above.This defines a permutation σ ∈ Σ n by σ ( i ) := p ( w i ), where w i is the white vertexlabelled by i , 1 ≤ i ≤ n . Let, finally, σ ( T ) be the Koszul sign of σ permuting n variables v , . . . , v n of degrees k − , . . . , k n −
1, respectively. In other words, σ ( T )is determined by(12) σ ( T ) · v ∧ · · · ∧ v n = v σ (1) ∧ · · · ∧ v σ ( n ) , satisfied in the free graded commutative associative algebra generated by v , . . . , v n .An ( l ; k , . . . , k n )-tree T determines the natural operation O T ∈ B ( A ) lk ,...,k n givenby decorating, for each 1 ≤ i ≤ n , the i th white vertex by f i ∈ C k i − ( A ; A ), theblack vertices by the iterated multiplication, the special vertices by the unit 1 ∈ A ,and performing the composition along the tree. The result is then multiplied by thesignature σ ( T ) defined above.When evaluating on concrete elements, we apply the Koszul sign rule and use the‘desuspended’ degrees, that is f : A ⊗ n → A is assigned degree n − a ∈ A degree −
1, see 4.2. For instance, the tree in Figure 5 represents the operation O ( f , f , f , f )( a , . . . , a ) := − a f ( f ( a a , , a ) , a , f ( a )) f ( a , , a ) ,a , . . . , a ∈ A , where, as usual, we omit the symbol for the iteration of the associativemultiplication µ . The minus sign in the right hand side follows from the Koszul ruleexplained above. The exceptional (1; )-tree represents the identity id ∈ C ( A ; A ). Notation . — For each l, k , . . . , k n ≥ B lk ,...,k n the free abelian groupspanned by all ( l, k , . . . , k n )-trees. The correspondence T O T defines, for eachassociative algebra A , a linear epimorphism ω A : B lk ,...,k n ։ B ( A ) lk ,...,k n . MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL
Let T ′ be an ( l ′ ; k ′ , . . . , k ′ n )-tree, T ′′ an ( l ′′ ; k ′′ , . . . , k ′′ m )-tree and assume that l ′′ = k ′ i for some 1 ≤ i ≤ n . The i th vertex insertion assigns to T ′ and T ′′ the tree T ′ ◦ i T ′′ obtained by replacing the white vertex of T ′ labelled i by T ′′ . It may happen thatthis replacement creates edges connecting black vertices. In that case it is followedby collapsing these edges. The above construction extends into a linear operation ◦ i : B l ′ k ′ ,...,k ′ n ⊗ B l ′′ k ′′ ,...,k ′′ m → B l ′ k ′ ,...,k ′ i − ,k ′′ ,...,k ′′ m ,k ′ i +1 ,...,k ′ n , ≤ i ≤ n, l ′′ = k ′ i . Recall the following: ( [ ] ) . — The spaces B lk ,...,k n assemble into an N -coloured operad B with the operadic composition given by the vertex insertion and the symmetric grouprelabelling the white vertices. With this structure, the maps ω A : B lk ,...,k n ։ B ( A ) lk ,...,k n form an epimorphism ω A : B ։ B ( A ) of N -coloured operads. In [ ] we formulated the following important: . — A unital associative algebra A is generic if the map ω A : B ։ B ( A )is an isomorphism.In [ ] we also proved that generic algebras exist; the free associative unital algebra U := T ( x , x , x , . . . ) generated by countably many generators x , x , x , . . . is anexample. We may therefore define the operad B alternatively as the operad of naturaloperations on the Hochschild cochain complex of a generic algebra .The differentials (11) clearly translate, for a generic A , to the tree language of theoperad B as follows. The component ∂ i , 1 ≤ i ≤ n , of the differential ∂ = ∂ + · · · + ∂ n replaces the white vertex of an ( l ; k , . . . , k n )-tree T labelled i with k i ≥ i • · · · ◦ ❅❅✁✁(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅ + • i · · · ◦ ❅❅✁✁(cid:0)(cid:0) ❅❅❅(cid:0)(cid:0) + ( − k i +1 X ≤ s ≤ k i − ◦ is (cid:0)(cid:0)(cid:0) ✓✓✓ ❅❅❅ • ❆❆✁✁ · · ·· · · in which the white vertex has k i − i . The result is thenmultiplied by the overall sign in the second line of (11). In the summation of (13),the black binary vertex is inserted into the s th input of the white vertex. If the i thwhite vertex of T has no inputs then ∂ i ( T ) = 0.The differential δ replaces an ( l ; k , . . . , k n )-tree symbolized by the triangle ❅❅(cid:0)(cid:0) · · · with l inputs by the linear combination • ❅❅(cid:0)(cid:0) · · · (cid:0)(cid:0)(cid:0) ❅❅ + • ❅❅(cid:0)(cid:0) · · · ❅❅❅(cid:0)(cid:0) + ( − l X ≤ s ≤ l ❅❅(cid:0)(cid:0)✁✁✁ ✁✁✁ ❆❆❆ s • ❆❆✁✁ · · ·· · · If a replacement above creates an edge connecting black vertices, it is followed bycollapsing these edges.
PERADS OF NATURAL OPERATIONS I ......... · · ·· · ·· · ·· · ·· · · ✻✻✻ ✻✻✻ ✻✻✻ ✻✻✻ ✻✻✻ ✲✲✲✲ ✲✲✲✲ ✲✲✲✲✲ ✲✲✲ ✲✲✲ B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) B ( n ) Figure 6.
The structure of the operad B . In the above diagram, B ( n ) mk := Q k + ··· + k n = k B mk ,...,k n . The vertical arrows are the simplicialdifferentials ∂ and the horizontal arrows are the cosimplicial differentials δ . We finally define the arity n piece of the operad of natural operations as B ∗ ( n ) := Y l − ( k + ··· + k n )+ n − ∗ B lk ,...,k n , with the degree +1 differential d : B ∗ → B ∗ +1 defined by d := ( ∂ + · · · + ∂ n ) − δ . It isevident that the collection B = {B ∗ ( n ) } n ≥ , with the operadic composition inheritedfrom the inclusion B ⊂ E nd C ∗ ( A ; A ) for A generic, is a dg-operad.The structure of the operad B is visualized in Figure 6. We emphasize that thedegree m -piece of B ( n ) is the direct product , not the direct sum, of elements on thediagonal p + q = m − n + 1 in the ( p, q )-plane. It follows from our definitions thatthe Hochschild complex C ∗ ( A ; A ) of an arbitrary unital associative A is a natural B -algebra. . — From now on, we will assume that A is a generic algebra inthe sense of Definition 4.6 and make no distinction between natural operations on theHochschild complex of A and the corresponding linear combinations of trees. . — An important suboperad of B is the suboperad b B generated by trees without stubs and without • . The operad b B is the operad of all natural multilinearoperations on the Hochschild complex of a non-unital generic associative algebra. Itis generated by natural operations (a)–(c) above but without the unit 1 ∈ C − ( A ; A )in (b). Let us denote by b B lk ,...,k n the space of all operations (10) of this restrictedtype. An important feature of the operad b B is that it is, in a certain sense, bounded.Indeed, one may easily prove that b B lk ,...,k n = 0 if k + · · · + k n − l ≥ n , see Figure 7. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL ......... · · ·· · ·· · ·· · ·· · ·· · ·· · · ✻✻✻ ✻✻✻ ✻✻ ✻✻✻✻ ✻✻✻ ✻✻✻ ✲✲✲ ✲✲✲ ✲✲✲✲ ✲✲✲✲✲ ✲✲✲ ✲✲✲ b B ( n ) n +1 b B ( n ) n b B ( n ) n − b B ( n ) b B ( n ) b B ( n ) n b B ( n ) n − b B ( n ) b B ( n ) b B ( n ) n − b B ( n ) b B ( n ) Figure 7.
The structure of the non-unital operad b B . In the diagram, b B ( n ) mk := Q k + ··· + k n = k b B mk ,...,k n . One also has the quotient Nor( B ) of the collection B modulo the trees with stubs.As explained in [ ], Nor( B ) forms an operad which is in fact the componentwisesimplicial normalization of B . The operad Nor( B ) acts on the normalized Hochschildcomplex of a unital algebra. One has the diagram of operad maps(14) b B ι ֒ → B π ։ Nor( B ) , in which the projection π is a weak equivalence and the components πι ( n ) of thecomposition πι are isomorphisms for each n ≥
1. If U denotes the functor thatreplaces the arity zero component of a dg-operad by the trivial abelian group, then U ( πι ) is a dg-operad isomorphism U ( b B ) ∼ = U (Nor( B )). . — There is also a suboperad T of B generated byelementary operations of types (a) and (b) only, without the use of permutationsin (c). Its arity- n piece equals T ∗ ( n ) := Y l − ( k + ··· + k n )+ n − ∗ T lk ,...,k n , where operations in T lk ,...,k n are represented by linear combinations of unlabeled ( l ; k , . . . , k n )-trees, that is, planar trees as in Definition 4.4 but without the labelsof the legs. The inclusion T lk ,...,k n ֒ → B lk ,...,k n is realized by labeling the legs ofan unlabeled tree from the left to the right in the orientation given by the planar PERADS OF NATURAL OPERATIONS I embedding. The groups T lk ,...,k n form a coloured operad T and the inclusion aboveis the inclusion of operads T ֒ → B .The operad T is the condensation of T and it is a chain version of the operadconsidered in [ , Section 3]. There is also the operad b T := b B ∩ T generated byunlabeled trees without stubs and without • . It is clear that b T is bounded in thesame way as b B . We finally have the normalized Tamarkin-Tsygan operad Nor( T )defined as the image of T under the canonical projection π : B ։ Nor( B ). One has thediagram b T ι ֒ → T π ։ Nor( T ) with the properties analogous to that of (14).Summing up, we have the following N -coloured operads:- the operad B whose piece B lk ,...,k n equals the span of the set of all ( l ; k , . . . , k n )-trees,- the operad b B whose piece b B lk ,...,k n is the span of the set of all ( l ; k , . . . , k n )-treeswithout stubs and without • if n = l = 0,- the operad T whose piece T lk ,...,k n equals the span of the set of all unlabeled( l ; k , . . . , k n )-trees, and- the operad b T = T ∩ b B whose piece b T lk ,...,k n is the span of the set of all unlabeled( l ; k , . . . , k n )-trees without stubs and without • if n = l = 0.We close this section by recalling the isomorphism between the set of unlabeled( l ; k , . . . , k n )-trees and L (2) ( k , . . . , k n ; l ) constructed in the proof of [ , Proposi-tion 2.14]. Let T be an unlabeled ( l ; k , . . . , k n )-tree. We run around T coun-terclockwise via the unique edge-path that begins and ends at the root and goesthrough each edge of T exactly twice (in opposite directions). The lattice path ϕ T : [ l + 1] → [ k + 1] ⊗ · · · ⊗ [ k n + 1] corresponding to T starts at the ‘lowerleft’ corner with coordinates (0 , . . . ,
0) and advances according the following rules:- when the edge-path hits the white vertex labeled i , 1 ≤ i ≤ n , we advance ϕ T inthe direction of the vector d i := (0 , . . . , , . . . ,
0) (1 at the i th place),- when the edge-path hits the leg, we do not move but increase the marking of ourposition by one.The correspondence T ϕ T is illustrated in Figure 8. ( [ ] , Proposition 2.14) . — The above correspondence inducesan isomorphism of coloured operads T and L (2) , and hence, the isomorphism between s T and | L (2) | . (2) More conceptually, the difference between the N -coloured operads T and B andthe corresponding operads T and B can be explained as follows. Let O and O be thecategories of operads and of nonsymmetric operads in the category of chain complexes C hain correspondingly. There is the forgetful functor Des : O → O which forgetthe symmetric group actions. Let M be the nonsymmetric operad for unital monoids. (2) The operadic suspension s applied to T is a consequence of Convention 4.2. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL root T : ◦ ◦• ✂✂ •• • (cid:0)(cid:0) ❅❅✁✁❅❅❆❆(cid:0)(cid:0)(cid:0) ϕ T : • ✻ • ✻ • ✻ • • ✲ • • ✻ Figure 8.
An unlabeled (6; 2 , T and the corresponding latticepath ϕ T ∈ L ( c ) (2 ,
2; 6). . — The category of multiplicative nonsymmetric operads is thecomma-category M /O , see [ ]. The category of multiplicative operads is the comma-category M / Des .So, a multiplicative operad is an operad A equipped with a structure morphism p : M →
Des ( A ). Equivalently, by adjunction, a structure morphism can be replacedby a morphism U A ss → A, where U A ss is the operad for unital associative algebras.The description in [ , 1.5.6] of the coloured operad whose algebras are symmetricoperads, readily implies the following proposition which illuminates the main resultof [ ]. . — The category of algebras over the coloured operad T is iso-morphic to the category of multiplicative nonsymmetric operads. The category ofalgebras of the coloured operad B is isomorphic to the category of multiplicative op-erads. Under this identification, the inclusion T ֒ → B induces the forgetful functorfrom multiplicative operads to nonsymmetric multiplicative operads.
5. Operads of braces
Throughout this section we use Convention 4.7. There is another very importantsuboperad B r of B generated by braces, cup-products and the unit whose normalizedversion was introduced in [ , Section 1] under the notation H . Let us recall itsdefinition. The operad B r is the suboperad of the operad B generated by the followingoperations.(a) The cup product − ∪ − : C ∗ ( A ; A ) ⊗ C ∗ ( A ; A ) → C ∗ ( A ; A ) defined by f ∪ g := µ ( f, g ).(b) The constant 1 ∈ C − ( A ; A ).(c) The braces −{− , . . . , −} : C ∗ ( A ; A ) ⊗ n → C ∗ ( A ; A ), n ≥
2, given by(15) f { g , . . . , g n } := X f (id , . . . , id , g , id , . . . , id , g n , id , . . . , id) , where id is the identity map of A and the summation runs over all possible substitu-tions of g , . . . , g n (in that order) into f . PERADS OF NATURAL OPERATIONS I Notice that, for f ∈ C k ( A ; A ) and g ∈ C l ( A ; A ), the cup product f ∪ g ∈ C k + l +1 ( A ; A ) evaluated at a , . . . , a k + l +1 ∈ A equals(16) ( f ∪ g )( a , . . . , a k + l +1 ) = ( − ( k +1) l f ( a , . . . , a k ) g ( a k +1 , . . . , a k + l +1 ) , with the sign dictated by the Koszul rule. This formula differs from the originalone [ , Section 7] due to a different degree convention used here, see 4.2. We leaveas an exercise to write a similar explicit formula for the brace.The brace operad has also its non-unital version c B r := b B ∩ B r generated by el-ementary operations (a) and (c). One can verify that both B r and c B r are indeeddg-suboperads of B , see [ ]. We also denote by Nor( B r ) ⊂ Nor( B ) the image of B r under the projection B ։ Nor( B ). One has again an analog c B r ι ֒ → B r π ։ Nor( B r )of (14).Let us describe the operad B r , its suboperad c B r and its quotient Nor( B ) in termsof trees. . — Let k , . . . , k n be integers. An amputated ( k , . . . , k n ) -tree is an(0; k , . . . , k n )-tree in the sense of Definition 4.4. We denote by A k ,...,k n the (finite)set of all amputated ( k , . . . , k n )-trees, by Nor( A ) k ,...,k n its subset consisting of ampu-tated ( k , . . . , k n )-trees without stubs and b A k ,...,k n the set that equals Nor( A ) k ,...,k n for n ≥ ∅ for n = 0. . — For each n ≥ and d ≤ n − , there is a natural isomorphism w : Span( { A k ,...,k n ; n − − ( k + · · · + k n ) = d } ) ∼ = B r d ( n ) which restricts to the isomorphism (denoted by the same symbol) w : Span( { b A k ,...,k n ; n − − ( k + · · · + k n ) = d } ) ∼ = c B r d ( n ) . and projects into the isomorphism (denoted again by the same symbol) w : Span( { Nor( A ) k ,...,k n ; n − − ( k + · · · + k n ) = d } ) ∼ = Nor( B r ) d ( n ) . The map w is defined in formula (17) below. From the reasons apparent later wecall it the whiskering . The proof of the proposition is postponed to page 28. Before wegive the definition of w , we illustrate the notion of amputated trees in the following: . — The space B r ∗ (0) is concentrated in degree − B r ∗ (0) = B r − (0) = Span( • ) , while c B r ∗ (0) = 0 = Span( ∅ ). The space B r − d (1) is, for d ≥
0, the span of the singleelement | {z } d -times · · ·••• ◦ ❅❅✁✁(cid:0)(cid:0) MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL while c B r ∗ (1) = c B r (1) = Span( ◦ ). Similarly B r (2) = c B r (2) = Span (cid:18) ◦◦ • ❅(cid:0) , ◦◦ • ❅(cid:0) (cid:19) , c B r (2) = Span (cid:18) ◦◦ , ◦◦ (cid:19) and B r (2) = c B r (2) ⊕ Span (cid:18) ◦◦ • ❅(cid:0) • , ◦◦ • ❅(cid:0) • , ◦◦ • ❅(cid:0) • , ◦◦ • ❅(cid:0) • (cid:19) . . — We call an unlabeled ( l ; k , . . . , k n )-tree amputable if all terminalvertices of its legs are white. For such a tree T we denote by amp ( T ) the amputated( k , . . . , k n )-tree obtained from T by removing all its legs. . — The (1; 1 , ◦◦ • ❅(cid:0) • is amputable, and amp (cid:18) ◦◦ • ❅(cid:0) • (cid:19) = ◦◦ • ❅(cid:0) • . The (2; 1)-tree ◦• ❅(cid:0) is not amputable.For each amputated ( k , . . . , k n )-tree S we define the whiskering to be the product(17) w ( S ) := Y ( T ; T is an amputable tree such that amp ( T ) = S ) . Recall that, by Convention 4.7, we interpret the unlabeled trees in the right hand sideas operations in T d ( n ) ⊂ B d ( n ), d = n − − ( k + · · · + k n ), via the correspondence T ↔ O T introduced on page 17. An equivalent definition in terms of the whiskeredinsertion into a corolla is given in (20). . — Of course, w ( • ) = • represents the unit 1 ∈ C − ( A ; A ). Theelement given by the whiskering of ◦ , w ( ◦ ) = Y d ≥ | {z } d -times · · ·◦ ❅❅✁✁(cid:0)(cid:0) ∈ b T ⊂ b B (1) , is the identity f f , i.e., the unit of the operad B . The whiskering of ◦◦ • ❅(cid:0) , w ( ◦◦ • ❅(cid:0) ) = ◦◦ • ❅(cid:0) ⊔ ◦◦ • ❅(cid:0) ⊔ ◦◦ • ❅(cid:0) ⊔ ◦◦ • ❅(cid:0) ⊔ ◦◦ • ❅(cid:0) ⊔ ◦◦ • ❅(cid:0) ⊔ · · · , gives the cup product (16). The whiskering of the element · · · ◦◦◦ ◦ ❅❅✁✁(cid:0)(cid:0)
12 3 n PERADS OF NATURAL OPERATIONS I ✟✟✟ ❍❍❍ ❅❅(cid:0)(cid:0)❅❅(cid:0)(cid:0) (cid:0) ❏❏❅(cid:0) •• ◦◦ ◦• ◦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Figure 9.
Angles of a tree symbolized by ∗ , . . . , ∗ . Their linear order,indicated by the subscripts, is given by walking around the tree counter-clockwise, starting at the root. Unlike [ , Section 5.2], black vertices donot have angles. The labels of white vertices are not shown. gives the brace (15). In particular, ◦◦ gives Gerstenhaber’s ◦ -product and ◦◦ − ◦◦ the Gerstenhaber bracket. Observe that the whiskering of the tree(18) ◦◦ • ❅(cid:0) is the operation that assigns to f ∈ C m ( A ; A ) and g ∈ C n ( A ; A ) the expression( − mn g ∪ f . The sign comes from the tree signature factor (12) in the definitionof the operation O T , because the order of the white vertices of the tree (18) and itswhiskerings does not agree with the natural planar one.We are going to define operations ∂ and ◦ i acting on amputated trees that trans-late, via the whiskering (17), into the dg-operad structure of B r . For an amputated( k , . . . , k n )-tree S as in Definition 5.1 denote ∂ ( S ) := ∂ ( S ) + · · · + ∂ n ( S ), where ∂ i ( S ) is, for k i ≥
1, the linear combination of amputated trees obtained by replacingthe i th white vertex of S by (13) followed by the contraction of edges connecting blackvertices if necessary. For k i = 0 we put ∂ i ( S ) = 0.The description of the ◦ i -operations is more delicate. Following [ , Section 5.2],define the set of angles of an amputated ( k , . . . , k n )-tree S to be the disjoint union Angl ( S ) := G ≤ i ≤ n { , . . . , k i } . Angles come with a natural linear order whose definition is clear from Figure 9borrowed from [ ]. Now, for an amputated ( k ′ , . . . , k ′ n )-tree S ′ , an amputated( k ′′ , . . . , k ′′ m )-tree S ′′ and 1 ≤ i ≤ n , define S ′ ◦ i S ′′ to be the linear combination(19) S ′ ◦ i S ′′ := X β ( S ′ ◦ i S ′′ ) β , where the sum runs over all (non-strictly) monotonic maps β : In ( w ′ i ) → Angl ( S ′′ )from the set of incoming edges of the vertex w ′ i of S ′ labelled i , to the set of anglesof S ′′ . In the sum, ( S ′ ◦ i S ′′ ) β is the tree obtained by removing the vertex w ′ i from MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL S ′ and replacing it by S ′′ , with the incoming edges of w ′ i glued into the angles of S ′′ following β . An important particular case is k ′ i = 0 when w ′ i has no input edges.Then S ′ ◦ i S ′′ is defined as the tree obtained by amputating w ′ i from S ′ and graftingthe root of S ′′ at the place of w ′ i .We call the operation ◦ i the whiskered insertion . A similar operation defines in [ ]the structure of the operad for pre-Lie algebras. As observed in [ ], the whiskeringof Proposition 5.2 can also be expressed as the product(20) w ( S ) = Y d ≥ | {z } d -times · · · ◦ ❅❅✁✁(cid:0)(cid:0) ◦ S. The following proposition can be verified directly. . —
With ∂ and ◦ i as defined above, the whiskering of Proposi-tion 5.2 satisfies w ( ∂S ) = d ( w ( S )) and w ( S ′ ◦ i S ′′ ) = w ( S ′ ) ◦ i w ( S ′′ ) , for all amputated trees S , S ′ , S ′′ and for all i for which the second equation makessense. . — We show how the classical calculations of [ ] can be conciselyperformed in the language of amputated trees (but recall that we are using a differentsign and degree convention, see 4.2). Let us start by calculating the differentials oftrees representing the cap product, the circle product and the Gerstenhaber bracket.By definition, one has(21) ∂ (cid:16) ◦◦ • ❅(cid:0) (cid:17) = 0 . Since (13) replaces ◦ by ◦• ❅(cid:0) + ◦ • ❅(cid:0) , one gets(22) ∂ (cid:16) ◦◦ (cid:17) = ◦◦ • ❅(cid:0) + ◦◦ • ❅(cid:0) which implies that(23) ∂ (cid:16) ◦◦ − ◦◦ (cid:17) = ◦◦ • ❅(cid:0) + ◦◦ • ❅(cid:0) − ◦◦ • ❅(cid:0) − ◦◦ • ❅(cid:0) = 0 . We want to interpret these equations in terms of operations. To save the space,let us agree that in the rest of this example f will be an element of C m ( A ; A ), g an element of C n ( A ; A ) and h an element of C k ( A ; A ), m, n, k ≥ − ∪ - recalled in (16)and considered as an element of B (2) is zero, d (- ∪ -) = 0, which, by the definition (11)of the differential in B means that − d H ( f ∪ g ) = d H f ∪ g + ( − m f ∪ d H g. PERADS OF NATURAL OPERATIONS I We recognize [ , Eqn. (20)] saying that - ∪ - is a chain operation. Since ◦◦ representsthe ◦ -product, (22) means that f ∪ g + ( − mn g ∪ f = d H f ◦ g + ( − m f ◦ d H g − d H ( f ◦ g ) , which is the graded commutativity (3) of the cup product up to the homotopy - ◦ -proved in [ , Theorem 3]. The origin of the sign factor at the second term in theright hand side is explained in Example 5.6. The meaning of (23) is that d H [ f, g ] = [ d H f, g ] + ( − m [ f, d H g ] , so the bracket [ − , − ] is a chain operation.Let us investigate the compatibility between the cup product and the bracket.Since, in B (3), [ − ∪ − , − ] = [ − , − ] ◦ ( − ∪ − ), the description of the ◦ i -operations interms of amputated trees gives that [ f ∪ g, h ] is represented by gf h ◦◦ • ❅(cid:0) ◦ + f gh ◦ ◦• (cid:0)❅ ◦ − gf h ◦◦ ◦• ❅(cid:0) where we, for ease of reading, replaced the labels of white vertices by the correspondingcochains. Similarly, since −∪ [ − , − ] = ( −∪− ) ◦ [ − , − ] in B (3), f ∪ [ g, h ] is representedby f gh ◦ ◦• (cid:0)❅ ◦ − f hg ◦ ◦• (cid:0)❅ ◦ and, by the same reason, [ f, h ] ∪ g is represented by gf h ◦◦ • ❅(cid:0) ◦ − gh f ◦◦ • ❅(cid:0) ◦ . Combining the above, one concludes that the expression [ f ∪ g, h ] − f ∪ [ g, h ] − [ f, h ] ∪ g is represented by(24) f hg ◦ ◦• (cid:0)❅ ◦ + gh f ◦◦ • ❅(cid:0) ◦ − gf h ◦◦ ◦• ❅(cid:0) . Because, by (13), ∂ replaces h ◦ ❅(cid:0) by h ◦• ❅ ❅(cid:0) + h ◦ • ❅(cid:0)(cid:0) − h ◦• ❅(cid:0) , (3) Since we use the convention in which the cup product has degree +1, its commutativity is the anti symmetry. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL the expression in (24) equals ∂ (cid:18) gf h ◦◦ ◦ ❅(cid:0) (cid:19) . The meaning of the above calculations is that the bracket and the cup product arecompatible up to the homotopy given by the brace −{− , −} . Proof of Proposition 5.2 . — It follows from Proposition 5.7 that the image of w con-tains B r . Indeed, Im ( w ) is a suboperad of B which, by Example 5.6, contains thegenerators of B r , i.e. the cup product, braces and 1. The map w is clearly a monomor-phism, since each amputated ( k , . . . , k n )-tree S equals the amputated part (i.e. thecomponent belonging to Q B k ,...,k n ) of its whiskering w ( S ).Therefore it remains to prove that Im ( w ) ⊂ B r or, more specifically, that w ( S ) ∈ B r ( n ) for each amputated ( k , . . . , k n )-tree S and n ≥
0. We need to show that eachsuch S is build up, by the iterated whiskered insertions ◦ i of (19) and relabelings ofwhite vertices, from the ‘atoms’(25) ◦ , • , ∪ := ◦◦ • ❅(cid:0) and br d := · · · ◦◦◦ ◦ ❅❅✁✁(cid:0)(cid:0)
12 3 d +1 , d ≥ , representing the generators of B r . Since the whiskering w is an operad homomorphismand the atoms are mapped to B r , this would indeed imply that Im ( w ) ⊂ B r .The first step is to get rid of the stubs. If S has s ≥ S thetree S with each stub replaced by ◦ . Let us label these new white vertices of S by n + 1 , . . . , n + s . Then clearly S = ± ( · · · (( S ◦ n +1 • ) ◦ n +2 • ) · · · ) ◦ n + s • . The sign in the above expression, not important for our purposes, is a consequenceof the Koszul sign rule, since • represents 1 ∈ A placed in degree −
1. So we maysuppose that S has no stubs and proceed by induction on the number of internaledges. Assume that S has e internal edges. If e ≤ S is either ◦ or br , so wemay assume that e ≥
2. We distinguish two cases.
Case 1.
The root vertex (i.e. the vertex adjacent to the root edge) is white; assumeit has d ≥ S looks as: · · ·◦ ◗◗◗◗✁✁✁✑✑✑✑❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S d where S , . . . , S d are suitable amputated trees. It is then clear that S can be obtainedfrom ( · · · (( br d ◦ S ) ◦ S ) · · · ) ◦ d S d , PERADS OF NATURAL OPERATIONS I b B b T c B r T B B r Nor( T )Nor( B )Nor( B r ) ✘✘✿✘✘✘✘✘✘✘ ✘✘✿✘✘✘✘✘✘✘PPPPPPPPPqPPPPPPPqPPPPPPPPPq✘✘✘✘✘✘✘✘✘✘✿✻ ✻ ✻✻✻ ✻✻ ✻✻✁✁✁✁✁✕✁✁✁✁✁✕✁✁✁✁✁✕ π ππι ιι Figure 10.
Operads of natural operations and their maps; see also theglossary on page 14. The horizontal maps are inclusions. where br d is the tree in (25), by relabeling the white vertices and changing the sign ifnecessary. Clearly, each S , . . . , S d has less than e internal edges, and the inductiongoes on. Case 2.
The root vertex is black, with d ≥ d = 2, we argue as inCase 1, only using ∪ instead of br . If d ≥
3, we use the equality · · ·• ◗◗◗◗✁✁✁✑✑✑✑❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S d = · · ·• ◗◗◗◗◗◗ • ✑✑✑✑❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S ❅❅(cid:0)(cid:0) · · · S d and argue as if d = 2. This finishes the proof.We finish this section by completing the proof of the following theorem of [ ]. . — The operads introduced above can be organized into the diagramin Figure 10. In this diagram: Operads in the two upper triangles have the chain homotopy type of the operadC −∗ ( D ) of singular chains on the little disks operad D with the inverted grading.In particular, the big operad B of all natural operations has the homotopy typeof C −∗ ( D ) , all morphisms between vertices of the two upper triangles are weak equivalences, operads in the bottom triangle of Figure 10 have the chain homotopy type ofthe operad C −∗ ( D ) with the component of arity replaced by the trivial abeliangroup, and MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL all morphisms in Figure 10 become weak equivalences after the application ofthe functor U that replaces the component of arity of a dg-operad by the trivialabelian group.Proof . — The only piece of information that was missing in [ ] and for which we hadto refer to this paper was that the whiskering w : B r → T is a weak equivalence.This fact follows from Theorem 3.10, the identification s T ∼ = | L (2) | established inProposition 4.10, and the induced identification s B r ∼ = B r (2) of suboperads. . — Theorem 5.9 shows that, up to homotopy, there is no differencebetween actions on the Hochschild cochains of the operads B , T and B r , resp. b B , b T and c B r in the nonunital case, resp. Nor( B ), Nor( T ) and Nor( B r ) in the normalizedcase. ASubstitudes, convolution and condensation
In this appendix we briefly remind the reader of some categorical definitions andconstructions we use in the paper. Most of the material is contained in [ ],[ ],[ ]and [ ].Let V be a symmetric monoidal closed category. Let A be a small V -category andlet [ A, V ] be the V -category of V -functors from A to V. The enriched
Hom -functor
Nat A ( F, G ) is given by the end:
Nat A ( F, G ) := Z X ∈ A V ( F ( X ) , G ( X )) . We also define the tensor product of the V -functors F : A op → V and G : A → V bythe coend F ⊗ A G := Z X ∈ A F ( X ) ⊗ G ( X ) . A.1 Definition . — A V -substitude ( P, A ) is a small V -category A together with asequence of V -functors: P n : A op ⊗ · · · ⊗ A op | {z } n − times ⊗ A → V, n ≥ ,P n ( X , . . . , X n ; X ) = P XX ,...,X n equipped with – a V -natural family of substitution operations µ : P XX ,...,X n ⊗ P X X , ··· ,X m ⊗ · · · ⊗ P X n X n ,...,X nmn → P XX ,...,X nmn – a V -natural family of morphisms (unit of substitude) η : A ( X, Y ) → P ( X ; Y ) = P YX PERADS OF NATURAL OPERATIONS I – for each permutation σ ∈ S n a V -natural family of isomorphisms γ σ : P XX ,...,X n → P XX σ (1) ,...,X σ ( n ) , satisfying some associativity, unitality and equivariancy conditions [ ].Notice that P is a V -monad on A in the bicategory of V -bimodules ( V -profunctorsor V -distributors). The Kleisli category of this monad is called the underlying categoryof P. The concept of substitude generalizes operads and symmetric lax-monoidal cate-gories. Indeed, any coloured operad P in V with the set of colours S is naturally asubstitude ( P, U ( P )) with U ( P ) equal the V -category with the set of objects S andthe object of morphisms U ( P )( X, Y ) = P ( X ; Y ) ∈ V . The substitution operationin the coloured operad P makes the assignment P n ( X , . . . , X n ; X ) = P XX ,...,X n afunctor P n : U ( P ) op ⊗ · · · ⊗ U ( P ) op | {z } n − times ⊗ U ( P ) → V, n ≥ , and the sequence of these functors form a substitude. The category U ( P ) is the un-derlying category of this substitude also called the underlying category of the colouredoperad P. In fact, a substitude is a coloured operad P together with a small V -category A and a V -functor η : A → U ( P ) [ , Prop. 6.3]. A.2 Definition . — [
1, 7 ] A symmetric lax-monoidal structure or a multitensor ona V -category C is a sequence of V -functors E n : C ⊗ · · · ⊗ C | {z } n − times → C equipped with – a family of V -natural transformations: µ : E n ( E m , . . . , E m k ) → E m + ··· + m k ; – A V -natural transformation (unit) Id → E ; – an action of symmetric group γ σ : E n ( X , . . . , X n ) → E n ( X σ − (1) , . . . , X σ − ( n ) ) , satisfying some natural associativity, unitarity and equivariance conditions. A.3 Definition . — [ ] A multitensor is called a functor-operad if its unit is anisomorphism. MICHAEL BATANIN, CLEMENS BERGER & MARTIN MARKL
McClure and Smith observed in [ ] that functor-operads can be used to defineoperads. Their observation works also for multitensors. Let δ ∈ C be an object of C then the coendomorphism operad of δ with respect to a multitensor E is given by acollection of objects in V Coend E ( δ )( n ) = C ( δ, E n ( δ, . . . , δ )) . Substitudes and multitensors are related by the following convolution operation[
8, 7 ]. A.4 Definition . — Let (
P, A ) be a substitude. We define a multitensor E P on C = [ A, V ] as follows:(26) E Pn ( φ , . . . , φ n )( X ) = P X − ,..., − ⊗ A φ ( − ) ⊗ A · · · ⊗ A φ n ( − ) . A special case of this construction is when A is equal to the underlying categoryof P. In this case the convolution operation produces a functor-operad.Let (
P, A ) be a substitude and let δ : A → V be a V -functor. A.5 Definition . — By a δ -condensation of the substitude ( P, A ) we mean the op-erad C ( P,A ) ( δ ) = Coend E P ( δ ). So, as a collection it is given by C ( P,A ) ( δ )( n ) = Nat A ( δ, E Pn ( δ, . . . , δ )) . The operad C ( P,A ) ( δ ) naturally acts on the objects of the form Tot δ ( φ ) = Nat A ( δ, φ )for an arbitrary V -functor φ : A → V ( δ -totalization of φ ) [
17, 2 ].Let i : B → A and δ : B → V be two V -functors. Let Lan i ( δ ) be a ( V -enriched)left Kan extension of δ along i. Then
Tot
Lan i ( δ ) ( φ ) = Nat A ( Lan i ( δ ) , φ ) = Nat B ( δ, i ∗ ( φ )) = Tot δ ( i ∗ ( φ )) , where i ∗ is the restriction functor induced by i. There is a similar formula which expresses the condensation with respect to
Lan i ( δ ) . Let (
P, A ) be a substitude and let i ∗ ,..., ∗ ( P ) be a sequence of functors i ∗ ,..., ∗ ( P ) n : B op ⊗ · · · ⊗ B op ⊗ A → V,i ∗ ,..., ∗ ( P ) AB , ··· ,B n = P Ai ( B ) ,...,i ( B n ) . We define a sequence of functors E i ∗ ,..., ∗ ( P ) n : [ B, V ] ⊗ · · · ⊗ [ B, V ] → [ A, V ]by the formula similar to formula (26). We also define i ∗ P as the substitude ( i ∗ P, B )obtained from P by restricting P n along i. PERADS OF NATURAL OPERATIONS I A.6 Proposition . —
For the functors φ , . . . φ n ∈ [ B, V ] the following V -naturalisomorphisms hold: E Pn ( Lan i ( φ ) , . . . , Lan i ( φ n )) = E i ∗ ,..., ∗ ( P ) n ( φ , . . . , φ n ) . In particular, C ( P,A ) ( Lan i ( δ ))( n ) = Tot δ ( i ∗ E i ∗ ,..., ∗ ( P ) n ( δ, . . . , δ ))= Tot δ ( E i ∗ ( P ) n ( δ, . . . , δ )) = C ( i ∗ ( P ) ,B ) ( δ )( n ) . This result allows to see many of the operads in this paper as the result of δ -condensation of some substitudes. For us V will be the category of chain complexes Ch . Our category A will be the category of nonempty ordinals ∆ (linearized) or thecrossed interval category ( IS ) op [ ] (also linearized). B can be ∆ or its subcategoryof injective order preserving maps ∆ in . These categories are related by the canonicalinclusions: ∆ in i −→ ∆ j −→ ( IS ) op . Let δ : ∆ → Ch be the cosimplicial chain complex of normalized chains on standardsimplices. It is classical that the totalization of a cosimplicial chain complex X • withrespect to δ is the normalized cosimplicial totalization Nor( X • ) and the tensor product X • ⊗ ∆ δ for a simplicial chain complex X • is the normalized simplicial realizationNor( X • ) . Hence, the condensation of the lattice path operad L ( c ) with respect to δ isprecisely the n -simplicial cosimplicial normalization | L ( c ) | = Nor(Nor( L ( c ) ( • , . . . , • n ; • ))) = C ( L ( c ) , ∆) ( δ ) . Proposition A.6 shows that the condensation of the lattice path operad L ( c ) withrespect to Lan i ( i ∗ ( δ )) is the unnormalized n -simplicial cosimplicial totalization | L ( c ) | = Tot(Tot( L ( c ) ( • , . . . , • n ; • ))) = C ( i ∗ ( L ( c ) ) , ∆ in ) ( i ∗ ( δ )) = C ( L ( c ) , ∆) ( Lan i ( i ∗ ( δ )))Analogously, for the operad of natural operations on the Hochschild cochains weuse the condensation with respect to Lan j ( δ ) for the normalized version and withrespect to Lan ji ( i ∗ δ ) for the unnormalized version that is B = C ( B, ( IS ) op ) ( Lan i ( i ∗ ( δ ))) . In [ ] similar calculations were applied to the cyclic version of the lattice path operad. References [1]
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Michael Batanin , Macquarie University, North Ryde, 2109, NSW, Australia
E-mail : [email protected]
Clemens Berger , Universit´e de Nice, Lab. J.-A. Dieudonn´e, Parc Valrose, F-06108 Nice, France
E-mail : [email protected]