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OPERADS OF NATURAL OPERATIONS I: LATTICE PATHS BRACES AND HOCHSCHILD COCHAINS

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Niveau: Supérieur, Doctorat, Bac+8
OPERADS OF NATURAL OPERATIONS I: LATTICE PATHS, BRACES AND HOCHSCHILD COCHAINS MICHAEL BATANIN, CLEMENS BERGER AND MARTIN MARKL Abstract. In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them. Contents 1. Introduction 1 2. The lattice path operad 2 3. Weak equivalences 6 4. Operads of natural operations 13 5. Operads of braces 20 Appendix A. Substitudes, convolution and condensation 26 References 30 1. Introduction This paper continues the efforts of [14, 3, 2] in which we studied operads naturally acting on Hochschild cochains of an associative or symmetric Frobenius algebra. A general approach to the operads of natural operations in algebraic categories was set up in [14] and the first breakthrough in computing the homotopy type of such an operad has been achieved in [3]. In [2], the same problem was approached from a combinatorial point of view, and a machinery which produces operads acting on the Hochschild cochain complex in a general categorical setting was introduced. However, some special instances of the construction of [2] are important in applications and have specific features not present in general. In this first paper of a series entitled ‘Operads of Natural Operations' we begin a detailed study of these special cases.

  • points equals

  • decreasing map

  • operads acting

  • marked lattice

  • natural operations

  • path operad

  • category ∆

  • lattice path


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OPERADS OF NATURAL OPERATIONS I: LATTICE PATHS, BRACES AND HOCHSCHILD COCHAINS
MICHAEL BATANIN, CLEMENS BERGER AND MARTIN MARKL
Abstract.In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them.
Introduction
The lattice path operad
Weak equivalences
Operads of natural operations
Operads of braces
Contents
Appendix A. Substitudes, convolution and condensation
References
1.Introduction
1
2
6
13
20
26
30
This paper continues the efforts of [14, 3, 2] in which we studied operads naturally acting on Hochschild cochains of an associative or symmetric Frobenius algebra. A general approach to the operads of natural operations in algebraic categories was set up in [14] and the first breakthrough in computing the homotopy type of such an operad has been achieved in [3]. In [2], the same problem was approached from a combinatorial point of view, and a machinery which produces operads acting on the Hochschild cochain complex in a general categorical setting was introduced.
However, some special instances of the construction of [2] are important in applications and have specific features not present in general. In this first paper of a series entitled ‘Operads of Natural Operations’ we begin a detailed study of these special cases.
M. Batanin was supported by Scott Russell Johnson Memorial Fund and Australian Research Council Grant DP0558372. C. Berger was supported by the grant OBTH of the French Agence Nationale de Recherche. ˇ M. Markl was supported by the grant GA CR 201/08/0397 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503. 1
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M. BATANIN, C. BERGER AND M. MARKL
It is very natural to start with the classical Hochschild cochain complex of an associative 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 amount of literature available. One of the motivations for this paper was our wish to relate various approaches in literature and to provide a uniform combinatorial language for this purpose. We achieve this goal by first describing the lattice path operadLand its condensation in the differential graded setting in section 2. This description leads to a careful treatment of (higher) brace operations and their relationship with lattice paths in section 3. The lattice path operad comes equipped with a filtration by complexity [2]. The second filtration stageL(2)is the most important for understanding natural operations on the Hochschild cochains. In section 4 we give an alternative description ofL(2)in terms of trees, closely related to the operad of natural operations from [14]. Finally, in section 5 we study various suboperads generated by brace operations. The main result is that all these operads have the same homotopy type, namely that of a chain model of the little disks operad. For sake of completeness we add a brief appendix containing an overview of some categorical constructions used in this paper. Convention.If not stated otherwise, by anoperadwe mean a classical symmetric (i.e. with the symmetric groups acting on its components) operad in an appropriate symmetric monoidal category which will be obvious from the context. The same convention is applied to coloured operads, substitudes, multitensors and functor-operads recalled in the appendix.
2.The lattice path operad
As usual, for a non-negative integerm, [m] denotes the ordinal 0<∙ ∙ ∙< m. We will use the same symbol also for the category with objects 0 . . .  mand the unique morphismijif and only ifij. Thetensor product[m][n] is the category freely generated by the (m n)-grid which is, by definition, the oriented graph with vertices (i j), 0im, 0jn, and one oriented edge (i0 j0)(i00 j00) if and only if (i00 j00) = (i0+ 1 j0) or (i00 j00) = (i0 j0+ 1). Let us recall, closely following [2], thelattice path operad non-and its basic properties. For negative integersk1 . . .  kn landnNput L(k1 . . .  kn;l) :=Cat,([l+ 1][k1+ 1]⊗ ∙ ∙ ∙ ⊗[kn+ 1]) whereis the tensor product recalled above andCat,([l+ 1][k1+ 1]⊗ ∙ ∙ ∙ ⊗[kn the set+ 1]) of functorsϕthat preserve the extremal points, by which we mean that
(1)ϕ(0) = (0 . . . 0) andϕ(l+ 1) = (k1+ 1 . . .  kn+ 1). A functorϕL(k1 . . .  kn;l) is given by a chain ofl+ 1 morphismsϕ(0)ϕ(1) →→ ∙ ∙ ∙ ϕ(l+1) in [k1+1]⊗∙ ∙ ∙⊗[kn+1] withϕ(0) andϕ(l Each morphism+1) fulfilling (1).ϕ(i)ϕ(i+1) is determined by a finite oriented edge-path in the (k1+ 1 . . .  kn+ 1)-grid. [June 17, 2009]
OPERADS OF NATURAL OPERATIONS I
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2.1.Marked lattice paths. ForWe will use a slight modification of the terminology of [2]. non-negative integersk1 . . .  knZ0denote byQ(k1 . . .  kn) the integral hypercube Q(k1 . . .  kn) := [k1+ 1]× ∙ ∙ ∙ ×[kn+ 1]Z×n. Alattice pathis a sequencep= (x1 . . .  xN) ofN:=k1+∙ ∙ ∙+kn+n+ 1 points ofQ(k1 . . .  kn) such thatxa+1is, for each 0a < N, given by increasing exactly one coordinate ofxaby 1. Amarkingofpis a functionµ:pZ0that assigns to each pointxaofpa non-negative numberµa:=µ(xa) such thatPaN=1µa=l. We can describe functors inL(k1 . . .  kn;l) as marked lattice paths (p µ) in the hypercube Q(k1 . . .  kn). The markingµa=µ(xa) represents the number of elements of the interior {1 . . .  l}of [l that are mapped by+ 1]ϕto theath lattice pointxaofp. We call lattice points marked by 0unmarkedthe set of marked points equalspoints so ϕ({1 . . .  l} example, the). For marked lattice path
02 0 (2)2 0031 represents a functorϕL(3 with2; 8)ϕ(0) = (00),ϕ(1) =ϕ(2) =ϕ(3) = (10),ϕ(4) = (20), ϕ(5) =ϕ(6) = (31) andϕ(7) =ϕ(8) =ϕ(9) = (43).
2.2.Definition.LetpL(k1 . . .  kn;l) be a lattice path. A point ofpat whichpchanges its direction is anangleofp. Aninternal pointofpis not an angle nor an extremalis a point that point ofp denote by. WeAngl(p) (resp.Int(p)) the set of all angles (resp. internal points) ofp.
For instance, the path in (2) has 4 angles, 2 internal points, 4 unmarked points and 1 unmarked internal point.
Following again [2] closely, we denote, for 1i < jn, bypijthe projection of the path pL(k1 . . .  kn;l) to the face [ki+ 1]×[kj+ 1] ofQ(k1 . . .  kn); letcij:= #Angl(pij) be the number of its angles. The maximumc(p) := max{cij}is called thecomplexityofp. Let us finally denote byL(c)(k1 . . .  kn;l)L(k1 . . .  kn;l) the subset of marked lattice paths of complexityc. The casec= 2 is particularly interesting, becauseL(2)(k1 . . .  kn;l) is, by [2, Proposition 2.14], isomorphic to the space of unlabeled (l;k1 . . .  kn)-trees recalled on page 18. For convenience of the reader we recall this isomorphism on page 19.
As shown in [2], the setsL(k1 . . .  kn;l) together with the subsetsL(c)(k1 . . .  kn;l),c0, form aZ0-colored operadLand its sub-operadsL(c) simplify formulations, we will allow. To c=, puttingL():=L. [June 17, 2009]