Protoperads I: Combinatorics and definitions

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Protoperads I: Combinatorics and definitions

Johan Leray

To cite this version:

Johan Leray. Protoperads I: Combinatorics and definitions. 2019. �hal-01982314�

PROTOPERADS I: COMBINATORICS AND DEFINITIONS

JOHAN LERAY

Abstract. This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced S-modules. We study the associated monoids, called protoperads, which are a generalization of operads. As operads encode algebraic operations with several inputs and one outputs, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and show that there exists a notion of free pro- toperad. We also show that the monoidal product introduced here is related to Vallette’s one on the category of S-bimodules, via the induction functor.

Introduction

This is the first of two papers in which the author develops the notion of prot- operad, which is a kind of properad (see [Val03, Val07]), and the homotopy theory of these new objects (see [Ler18]). Properad is an algebraic notion which encodes types of bialgebras, i.e. operations with several inputs and several outputs.

The motivation for this work is to determine what is a double Poisson bracket up to homotopy. Double Poisson structure, defined by Van den Bergh in [VdB08a], give a Poisson structure in noncommutative algebraic geometry (see [Gin05, VdB08b]) under the Kontsevich-Rosenberg principle, i.e. if A is a double Poisson algebra, then the family of affine representation schemes Rep _n (A), represented by commutative algebras A n , has a Poisson structure.

Berest, et al define derived representation schemes (see [BKR13, BCER12]) as the left derived functor L (−) n . A natural question is the following: what is a double Poisson structure compatible with the derived side of the Berest’s construction?

What is a double Poisson bracket up to homotopy? Double Poisson structure is properadic in nature. It is encoded by the properad DP ois (see [Ler18]); it gives us a good framework to study what is its version up to homotopy (cf. [Val03, MV09a, MV09b]). The structure of double Poisson algebras up to homotopy is controlled by a cofibrant resolution DP ois ∞ of the properad DP ois (cf. [MV09a, MV09b] for the model structure of the category of properads). Properads are algebraic objects which encode some algebraic structures, there are included in a large family of such objects:

Date: January 15, 2019.

2010 Mathematics Subject Classification. 05E25,18D50,18G35,55U10.

Key words and phrases. Combinatorics, Species, Properad, Protoperad.

This article is the combinatorial part of the PhD thesis of the author, supported by the project

"Nouvelle Équipe", convention n

^◦

2013-10203/10204 between La Région des Pays de Loire and the University of Angers. The author thanks the Centre Henri Lebesgue ANR-11-LABX-0020-01 for its stimulating mathematical research programs. This paper was finished at the University Paris 13, where the author was financed by a postdoctoral allocation given by DIM Math Innov. The author is indebted to G. Powell who has carefully read and corrected the first version of this paper.

1

Associative alg. ⊂ NS-Operads ⊂ Operads ⊂ Dioperads ⊂ Properads ⊂ Props.

Let V be a k-vector space: algebras encode algebraic structures with one input and one output V → V ; non-symmetric and (symmetric) operads encode algebraic structures with several input and one output V ^⊗m → V ; dioperads, properads and props encodes algebraic structures with several input and outputs V ^⊗m → V ^⊗n . We can resume that by expliciting in which categories these objects live and there underlying combinatorics.

Associative

Algebras NS-Operads Operads Dioperads Properads Props Live in the

category Vect k N -mod ^red _k S-mod ^red _k S-bimod ^red _k S-bimod ^red _k S-bimod k

Monoid for

the product ⊗ k ◦ ^ns ◦ ⊠ ^Gan _{c, ∅} ⊠ ^Val _c

Generators

^p p^{· · ·}

1 2 m

· · · p

1 2 m

1· · ·n

· · · p

1 2 m

1· · ·n

· · · p

1 2 m

1· · ·n

· · · p

Composition controlled

by

^p

q

· · ·

· · · p

q

1

i i+n−1

· · ·

i−1m

· · · p

q

connected oriented graphs without genus

connected oriented graphs with genus

oriented graphs with genus Example

of encoded structures

Chain com- plexes

Associative

algebras Lie algebras

Involutive Lie bialge- bras

Lie bialge- bras

Loday in- finitesimal bialgebras A reference [LV12,

Chapter 1]

[LV12, Chapter 5]

[LV12,

Chapter 5] [Gan03] [Val07] [ML65]

For such an object P , e.g. P an operad, a properad, etc... , the notion of P - algebra up to homotopy is given by cofibrant resolution of P (see the Homotopy Tranfert Theorem for operads [LV12]). The homotopy theory of such objects is more or less complicated.

These successive algebraic structures are increasingly more complicate since they encode more and more type of algebras. As a consequence, their homotopy theory is also more and more elaborate. If props encode the largest category of algebraic structures, we do not yet have the same homotopical tools, like the Koszul duality theory, that hold for properads, operads, etc. The chain complex structures are encoded by the algebra of dual numbers. The non-symmetric operad framework is the minimal one to encode associative algebras, symmetric operad framework is the minimal one to encode associative commutative algebras, and so on: dioperads to encode bigebras without genus in the underlying combinatorics, as involutive Lie bialgebras.

In general, it is a hard problem to determine a usable (minimal) cofibrant res-

olution of a properad. However, if a (quadratic) properad P is Koszul, (i.e. the

homology of its bar construction is concentrated in a certain weight), then we have

an explicit minimal resolution of P: Vallette defines the Koszul duality for proper-

ads (see [Val03, Val07, MV09a]). Recall that Koszul duality theory does not exists

for props. Unfortunately, it is also a hard problem to show that a properad is

Koszul. Almost all known examples of Koszul properads come from Koszulness re- sults for operads. For operads, there are technical tools for showing that an operad is Koszul, including rewriting methods, PBW or Gröbner bases and distributive laws (see [Hof10, LV12, DK10]). The only other known example of a Koszul prop- erad which does not come from an operadic result is the Frobenius properad (and its Koszul dual) (see [CMW16]).

A double Poisson bracket on an associative algebra A is a double Lie bracket with some compatibilities with the product of A. The properad DLie, which en- codes double Lie structure, is defined by generators and relations, i.e. DLie = F (V DLie )/hR DJ i, with V DLie concentrated in arity (2, 2):

V DLie =

^{1 2}

1 2

⊗ sgn(S 2 ) x  ^S _S

²op

^{× S}

^op2 2

and the relation

R DJ =

1 2 3

1 2 3

+

2 3 1

2 3 1

+

3 1 2

3 1 2

.

A double Lie bracket on a chain complex A is given by a morphism of properads DLie → End A where End A is the properad of endomorphisms of A (see [Val07] for the definition). The operad Pois ∼ = Com ◦ Lie, which encodes Poisson structures, is Koszul because the operads Com and Lie, which respectively encode commutative and Lie algebras, are Koszul and satisfy a distributive law (see [LV12, Sect. 8.6.3]).

We have an analogous statement for double Poisson structures: as the properad As, which encodes associative algebras, is Koszul (see [LV12]), the properad DPois ∼ = As ⊠ ^Val _c DLie is Koszul if, and only if, the properad DLie is Koszul (see [Ler18, Sect.

4]). As the study of the Koszulness of DLie is still difficult, the idea is to use the diagonal symmetry of the generator and the relation of this properad to simplify the problem. In fact, the S-bimodules (which are families of representations of permutation groups) V DLie and R DJ are respectively induced from representations of the groups S ₂ and S ₃ , by the diagonal morphism

(1) G −→ G × G ^op

g 7−→ g × g ⁻¹ ,

for G in {S 2 , S 3 } respectively. This fundamental observation allows us to reduce the problem to one in the category of S-modules, and to define the minimal frame- work, called protoperad, which encode double Lie algebras. We will see (in [Ler18]) that the homotopy theory of these new objects is more simpler than the homotopy of properads. We resume the most important results of this article in the following theorem.

Theorem (see Def. 2.8, Thm. 4.16, Prop. 5.21). The category S-mod ^red _k of reduced S-modules, i.e. the full sub-category of functors P : Fin ^op → Ch k such that P ( ∅ ) = 0 is monoidal for the connected composition product ⊠ _c . The monoids in this category are called protoperads. There exists the free protoperad functor, denoted by F (−) and the functor

Ind : S-mod ^red _k , ⊠ _c

−→ S-bimod ^red _k , ⊠ ^Val _c

which is exact and satisfies Ind ◦ F = F ^Val ◦ Ind, where F ^Val is the functor of free

properad.

Using the framework of protoperads is successful: we prove in [Ler18] that there is a Koszul duality theory for protoperads which is compatible with that for properads via Ind, and we use it to prove that the properads DLie and hence DPois are Koszul.

Section 1 – Bricks and walls . We develop the combinatorics for protoperads.

The combinatorics is controlled by walls. A wall over a non-empty finite set S is a set of subsets of S , equipped with a particular partial order and such that the union of these subsets is S. We represent a wall diagrammatically as follows:

v₁ v₂ v3

v4 v5

the width of each brick giving us the number of entries and exits of each of the operations v i which is represented. This is encoded by the functor W ^conn and certain subfunctors. Let S be a non-empty finite set and n a natural number, a wall of n bricks over S is the datum of a set W = {W α } α∈A of |A| = n non-empty subsets of S, which are called bricks of W , satisfying:

– the union of all bricks is S;

– for all element s of S, the set of bricks W α which contains the element s is totally ordered

– a compatibility between orders.

We define also the notion of connectedness for a wall and denote by W ^conn (S), the set of connected walls over S.

Section 2 – Products on S -modules . We review two monoidal products on S-mod ^red _k := Func(Fin ^op , Ch k ) ^red which is the full sub-category of functors P : Fin ^op → Ch k from the category Fin of finite sets with bijections, to the category Ch k of k-chain complexes such that P ( ∅ ) = 0. These product are the composition product , also called the Hadamard product, and the concatenation product ⊗ ^conc (see Section 2.2).

We also define the connected composition product on S-mod ^red _k (see Defini- tion 2.8), denoted by ⊠ _c , which encodes algebraic structures which have the same number of inputs and outputs and a diagonal symmetry. It is the bifunctor

− ⊠ _c − : S-mod ^red _k × S-mod ^red _k −→ S-mod ^red _k

defined, for all reduced S-modules P and Q and for all non-empty finite sets S, by:

P ⊠ _c Q(S) := M

(I,J)∈X

^conn

(S)

O

α

P (I α ) ⊗ O

β

Q(J β ).

Protoperads are the monoids for this monoidal structure. For a protoperad P , we can view a homogeneous element p of P (S) as a labelled brick

s1 s₁

s2 s₂

· · ·

· · ·

sm sm

p

,

and the product P ⊠ _c P (S) → P (S) as the composition of two rows of bricks which are connected:

s1

s1 s2

s2

sm

sm

· · · ·

p1 p_j ps

· · ·

p^′₁ p^′_r

7→

s1 s1

s2 s2

· · ·

· · ·

sm sm p

Section 3 – Connected product on S -bimodules . In this section, we also recall three monoidal structures on the category of reduced S-bimodules which are analogous to ones on the S-module category: the concatenation product, the com- position product, and the connected composition product ⊠ ^Val _c , defined by Vallette in [Val03, Val07]. We do this review with an other point of view than the original one: we give an equivalent definition of the connected composition product, which is more adapted to species and the functorial point of view of S-bimodules.

Section 4 – Induction functor . The new product ⊠ _c is the avatar of the prod- uct ⊠ ^Val _c on the category S-bimod ^red _k . The most important property of the prod- uct ⊠ _c is its compatibility with the product of Vallette via the induction functor Ind : S-mod ^red _k → S-bimod ^red _k , defined using Equation (1). We prove the following.

Theorem (see Theorem 4.16). The induction functor Ind : S-mod ^red _k , ⊠ _c

−→ S-bimod ^red _k , ⊠ ^Val _c is monoidal. In particular, it sends protoperads to properads,

Section 5 – Protoperads . In this section, we give an equivalent definition of a protoperad, generalizing the definition of operads in terms of partial compositions.

Proposition (see Proposition 5.8). A protoperad P has canonically a partial com- position system. Conversely, a partial composition system on a S-module P canon- ically extends to a protoperad structure.

Using the work of Vallette on free monoids in abelian monoidal categories (see [Val09]), we show that there exists a free protoperad functor. We also have a combinatorial description of the free protoperad.

Theorem (see Theorem 5.21). Let V be a reduced S-module. The free protoperad functor is the graded functor F ^∗ (−) given, for all finite set S, and for all natural number ρ, by the isomorphism of right Aut(S)-modules

F ^ρ (V )(S) ∼ = M

({W

α

}

α∈A

,6)

∈W

_ρ^conn

(S)

O

α∈A

V (W α ),

with W _ρ ^conn (S ), the set of connected walls with ρ bricks.

Section 6 – Colours on walls . In [Ler18], the author develops the Koszul du- ality for protoperads, which is related to Koszul duality theory of properads (see [Val03, Val07]) via the functor Ind. In this last section, we define the notion of a coloured wall, and we associate to a wall W over a totally ordered set S, the colouring complex, denoted by C ^Col _• (W ). This is motivated by the combinatorial description of the bar construction of the free protoperad (see also [Ler17, Ler18]).

The principal result of this section is the following:

Theorem (see Theorem 6.15). Let S be a finite totally ordered set, and W a wall over S. If the set Succ(W ) (see Section 1.1 for the definition of Succ) is non empty, then the colouring complex C ^Col _• (W ) is acyclic.

This theorem corresponds to the acyclicity result for bar construction of free protoperad (see [Ler18] for the definition of the bar construction of a protoperad).

Contents

Introduction 1

Notation 6

1. Bricks and walls 6

2. Products on S-modules 13

3. Connected product on S-bimodules 18

4. Induction functor 26

5. Protoperads 31

6. Colours on walls 45

References 53

Notation

We use the notation N ^∗ for the set N − {0}. We denote by Fin, the category with finite sets as objects and bijections as morphisms and Set, the category of all sets and all applications. For two integers a and b, we denote by [[a, b]] the set [a, b] ∩ Z , and, for n ∈ N ^∗ , S _n is the automorphism group of [[1, n]], i.e. S _n = Aut ^Fin ([[1, n]]).

We denote by Ch _k the category of Z -graded chain complexes over the field k.

Let (C, ⊙) be a monoidal category: we denote by As(C, ⊙) the category of monoids without unit (e.g. semi-groups) in C and UAs(C, ⊙) the category of unital monoids in C. If (C, ⊙) is symmetric monoidal, we also denote by Com(C, ⊙) the category of commutative monoids without unit and UCom(C, ⊙) the category of commutative unital monoids in C.

A monoidal category (C, ⊙, I ) is an abelian monoidal category if C is also abelian:

we do not suppose any compatibility between the monoidal product ⊙ and the abelian structure.

1. Bricks and walls

We begin by describing the combinatorial framework of this paper. The first sec- tion is about posets and, after that, we introduce the functor of walls. Walls encode the combinatorics of "diagonal properads", as rooted trees govern the combinatorics of operads. In this section, we define two important functors:

X ^conn : Fin ^op → Set ^op and W ^conn : Fin ^op → Set ^op .

The first one, X ^conn , encodes the combinatorics of the new monoidal structure on the category of S-modules, the connected composition product (see Section 2.3).

The second, W ^conn encodes the combinatorics of the free monoid for this monoidal structure (see Theorem 5.21).

Remark 1.1. In this section, we construct (covariant) functors from the opposite category of finite sets to the opposite category of sets, i.e. F : Fin ^op → Set ^op or the category of chain complexes, i.e. F : Fin ^op → Ch k . We choose to consider the opposite category of Fin to get a right action of the automorphism group Aut(S) on F (S). This right action mimics the actions of symmetric groups on the leaves of trees in the operadic case.

1.1. Recollections on posets. Let k and l be two elements of a poset (K, 6 ). We say that k and l are successors if k < l and if there does not exist an element t in K such that k < t < l. We denote by Succ(K), the set of pairs of successors of K.

A chain of a poset K is an increasing sequence of elements of K and the length of

the chain is the number of elements of the chain: we denote the length of a chain k 1 < k 2 < . . . < k r by len(k 1 < k 2 < . . . < k r ). The height of an element k of a poset (K, 6 ) is the element h(k) of N ∪ {∞} defined by

h(k) := max

len(c) ∈ N ^∗ | c = (λ 1 < λ 2 < . . . < λ r−1 < k) .

Proposition 1.2. Let (K, 6 ) be a poset and (k, l) in Succ(K). Then the surjection π _k ^l : K ։ K/ k∼l

induces a partial order on K/ k∼l defined, for all r and s in K, by – [r] 6 [s] if r 6 s and r, s / ∈ {k, l}

– [s] 6 [k ∼ l] (resp. s > [k ∼ l]) if s 6 k or s 6 l (resp. s > k or s > l).

Proof. Left to the reader.

Lemma 1.3. Let (R, 6 _R ) and (S, 6 _S ) be two posets with injections R ֒ → T ← ֓ S.

If, for all a and b in R ∩S, a 6 _R b if and only if a 6 _S b, then R ∪S has a canonical partial order which extends the partial orders 6 _R et 6 _S .

Proof. For any x and y in R ∪S, we have x 6 _R∪S y if and only if one of the following assumption holds:

– x and y are in R and x 6 _R y;

– x and y are in S and x 6 _S y;

– x is in R, y is in S and there exists t in R ∩ S such that x 6 _R t 6 _S y.

1.2. The functors of walls. In the rest of this section, we define some (covariant) functors from the category Fin ^op to the category Set ^op , called functors of walls. Let F be a functor of walls (see below for definitions) and S a finite set with n elements.

An element W of F (S ) should represent a morphism Hom ^C (V ^⊗n , V ^⊗n ), for V a chain complex, with a diagonal action of S _n by permutations inputs and outputs at the same time.

Definition 1.4 (Functor of ordered walls W ^or ). For n in N ^∗ , the covariant functor W _n ^or : Fin ^op −→ Fin ^op is given, for all finite set S, by

W _n ^or (S) :=

 

 

 



W = (W 1 , . . . , W n )

∪ i W i = S; ∀i ∈ [[1, n]], W i 6= ∅ ;

∀s, t ∈ S, Γ ^W _s := {W i |s ∈ W i } is totally ordered (by 6 _s );

∀a, b ∈ Γ ^W _s ∩ Γ ^W _t , a 6 _s b ⇔ a 6 _t b

 

 

 

 where an element W of W _n ^or (S) is a poset for 6 which is the induced partial order (cf. lemma 1.3) on ∪ s∈S Γ ^W _s = {W 1 , . . . , W n }. We denote by (W, 6 ), the elements of W _n ^or (S ). The action of σ, an element of Aut(S), on (W 1 , . . . , W n ), 6

in W _n ^or is induced by the canonical action on S,

(W 1 , . . . , W n ), 6

· σ = (W 1 · σ, . . . , W n · σ), 6 ^σ

where 6 ^σ is induced by the total orders of the sets Γ ^W _s ^·σ := {W i · σ|s ∈ W i · σ}.

The functor W ^or , defined by

W ^or : Fin ^op −→ Sets ^op

S 7−→ `

n∈ N

^∗

W _n ^or (S) .

Remark 1.5. For all non-empty sets S, we have W ₀ ^or (S) = ∅ . For all integers n > 0, the group S _n acts freely on W _n ^or (S) by permuting the position of elements, i.e. for τ in S _n , we have τ · (W 1 , . . . , W n ), 6

= (W τ

⁻¹

(1) , . . . , W τ

⁻¹

(n) ), 6 . The partial order is the same because it doesn’t depend of the W i ’s indexes.

Example 1.6. We consider W a = {1, 2}, W b = {3, 4} and W c = {2, 3}, three subset of S = [[1, 4]]. Then, we have the following four elements of W ₃ ^or (S):

– (W 1 , W 2 , W 3 ), < ¹

with < ¹ given by W 1 < ¹ W 3 and W 2 < ¹ W 3 ; – (W 1 , W 2 , W 3 ), < ²

with < ² given by W 3 < ² W 1 and W 3 < ² W 2 ; – (W 1 , W 2 , W 3 ), < ³

with < ³ given by W 1 < ³ W 3 and W 3 < ³ W 2 ; – (W 1 , W 2 , W 3 ), < ⁴

with < ⁴ given by W 3 < ⁴ W 1 and W 2 < ⁴ W 3 . This four elements of W ₃ ^or (S ) are distinct.

The vertical composition product on W ^or , is the natural transformation:

V : W ^or × W ^or

(−) −→ W ^or (−)

given, for all finite set S, by V n,m,S : W _n ^or (S) × W _m ^or (S) −→ W _m+n ^or (S) which sends the pair (W, 6 _W ), (L, 6 _L )

on R = (W 1 , . . . , W n , L 1 , . . . L m ), 6 ^L _W

where, for all s in S, the total order of the poset Γ ^R _s is induced by the ones of Γ ^W _s and Γ ^L _s and by extension, for all W i in Γ ^W _s and all L j in Γ ^L _s , we have W i 6 ^L _W L j . This product is associative, so, for all finite set S, we have the following commutative diagram:

W ^or (S) × W ^or (S) × W ^or (S) ^V×id / /

id×V

W ^or (S) × W ^or (S)

V

W ^or (S) × W ^or (S)

V / / W ^or (S).

The (horizontal) concatenation product on W ^or , is the natural transformation between bifunctors:

H : W ^or (− 1 ) × W ^or (− 2 ) −→ W ^or (− 1 ∐ − 2 )

given, for all finite sets S and T , by H n,m,S,T : W _n ^or (S) × W _m ^or (T ) −→ W _m+n ^or (S ∐T ) which sends (W, 6 _W ), (L, 6 _L )

to R = (W 1 , . . . , W n , L 1 , . . . L m ), 6 _W,L where, for all s in S and t in T , we have the equalities Γ ^R _s = Γ ^W _s and Γ ^R _t = Γ ^L _t . This product is associative and commutative, so we have the following commutative diagrams:

W ^or (− 1 ) × W ^or (− 2 ) × W ^or (− 3 ) ^H×id / /

id×H

W ^or (− 1 ) × W ^or (− 2 ∐ − 3 )

H

W ^or (− 1 ∐ − 2 ) × W ^or (− 3 )

H / / W ^or (− 1 ∐ − 2 ∐ − 3 ) and

W ^or (− 1 ) × W ^or (− 2 ) ^H / /

∼ =

W ^or (− 1 ∐ − 2 )

∼ =

W ^or (− 2 ) × W ^or (− 1 )

H / / W ^or (− 2 ∐ − 1 ) .

We also have the following commutative diagram of natural transformations, called the interchanging law :

W ^or ×2

(− 1 ) × W ^or ×2

(− 2 ) W ^or (− 1 ) × W ^or (− 2 ) ×2

W ^or (− 1 ) × W ^or (− 2 ) W ^or (− 1 ∐ − 2 ) ×2

W ^or (− 1 ∐ − 2 )

id×σ×id

V×V H×H

H

V

.

Pass to W, the functor of unordered walls:

Definition 1.7 (Functor of walls W). We define the functor W n : Fin ^op → Fin ^op by W n := W _n ^or

S

_n

which is given, for all finite sets S, by W n (S) :=

 

 

 



W = {W α } α∈A , 6

|A| = n; ∀α ∈ A, W α 6= ∅ ; ∪ α W α = S;

∀s ∈ S, Γ ^W _s := {W α |s ∈ W α } is totally ordered (by 6 _s )

∀s, t ∈ S, ∀a, b ∈ Γ s ∩ Γ t , a 6 _s b ⇔ a 6 _t b

 

 

 

 where 6 is the canonical partial order of ∪ s∈S Γ ^W _s = {W α } α∈A . We have the natural projection π : W ^or −→ W. The action of σ, an element of Aut(S), on

{W α } α∈A , 6

∈ W n is induced by the canonical action on S, i.e.

{W α } α∈A , 6

· σ = W = {W α · σ} α∈A , 6 ^σ

where 6 ^σ is induced by total orders of Γ ^W·σ _s·σ := {W α · σ|s · σ ∈ W α · σ}. We also define the functor

W : Fin ^op −→ Sets ^op

S 7−→ `

n∈ N

^∗

W n (S) .

An element W of W(S) is called a wall over S, and an element of a wall W is called a brick of W .

Proposition 1.8 (Products on W). The products V and H on W ^or pass through tue quotient by the actions of the symmetric groups on the indexes of bricks, hence induce natural transformations

V : W × W

(−) −→ W(−) and H : W(− 1 ) × W(− 2 ) −→ W(− 1 ∐ − 2 ), respectively called the composition product and concatenation product on W, such that we have the following commutative diagrams

W ^or × W ^or

(−) W ^or (−)

W × W

(−) W(−)

V

π

^×2

π

V

;

W ^or (− 1 ) × W ^or (− 2 ) W ^or (− 1 ∐ − 2 )

W(− 1 ) × W(− 2 ) W(− 1 ∐ − 2 )

H

π

^×2

π

H

;

W × W

(− 1 ) × W × W

(− 2 ) W(− 1 ) × W(− 2 ) × W(− 1 ) × W(− 2 )

W(− 1 ) × W(− 2 ) W(− 1 ∐ − 2 ) × W(− 1 ∐ − 2 )

W(− 1 ∐ − 2 )

id×σ×id

V×V H×H

H V

.

1.3. Two subfunctors of W. We introduce here two important subfunctors of W. For all finite non-empty sets S and all n in N ^∗ , we define the functor of ordered partitions Y _n ^or : Fin ^op → Fin ^op , by

Y _n ^or (S) := { (K 1 , . . . , K n ) | ∐ ⁿ _i=1 K i = S; ∀i ∈ [[1, n]], K i 6= ∅ }

equipped with the natural injection Y _n ^or ֒ → W _n ^or . By disjoint union, we also define the functor Y ^or by

Y ^or : Fin ^op −→ Set ^op

S 7−→ `

n∈ N

^∗

Y _n ^or (S) .

Via the vertical composition, we have, for all finite sets S and all m and n in N ^∗ , the isomorphism:

Y _m ^or (S) × Y _n ^or (S) ∼ =

 

 

 

 

 

 

 



R = (K 1 , . . . , K m , L 1 , . . . , L n )

∐ i K i = S = ∐ j L j ;

∀i ∈ [[1, m]], K i 6= ∅ ; ∀j ∈ [[1, n]], L j 6= ∅ ;

∀s ∈ S, ∃!i ∈ [[1, m]], ∃!j ∈ [[1, n]]

s.t. Γ ^R _s := {K i , L j } with K i 6 _s L j

∀s, t ∈ S, ∀a, b ∈ Γ ^R _s ∩ Γ ^R _t , a 6 _s b ⇔ a 6 _t b

 

 

 

 

 

 

 

 ,

which gives us the natural injection Y _m ^or (S) × Y _n ^or (S) ֒ → W _m+n ^or (S). Hence, we define, for all non-empty finite sets S, the functor X ^or of ordered pairs of partitions of finite sets, by

X ^or (S) := a

m,n∈ N

^∗

Y _n ^or (S) × Y _m ^or (S).

equipped with the natural injection X ^or ֒ → W ^or . This functor is important: it encodes the combinatorics of our new monoidal product, up to a property of con- nectedness (see Section 1.4)

The natural surjection π : W ^or ։ W gives the following commutative diagrams of natural transformations:

Y ^or W ^or X ^or

Y W X

,

where Y (resp. X) is the quotient of Y ^or (resp. X ^or ) by the action of the sym- metric group on the indexes of bricks. The concatenation product restricts to the subfunctors X and Y:

Y(− 1 ) × Y(− 2 ) W(− 1 ) × W(− 2 )

Y (− 1 ∐ − 2 ) W(− 1 ∐ − 2 )

µ

^conc

H and

X (− 1 ) × X (− 2 ) W(− 1 ) × W(− 2 )

X(− 1 ∐ − 2 ) W(− 1 ∐ − 2 )

µ

^conc

H .

1.4. Connected walls. Now, we introduce the notion of connectedness of a wall.

Let (W = {W α } α∈A , 6 ) be a wall in W(S). We define on W the equivalence relation of connectedness ^conn. ∼ : for two elements a and b of A, we say W a

conn.

∼ W b if there

exist an integer n > 2 and a sequence W 0 , W 1 , . . . , W n−1 , W n of elements of W with W 0 = W a and W n = W b such that, for all i in [[0, n − 1]],

W i ∩ W i+1 6= ∅ and (W i , W i+1 ) ∈ Succ(W ) or (W i+1 , W i ) ∈ Succ(W ).

Definition 1.9 (Projection K). We define the projection K as follows: for a finite set S, we have

K S : W(S) −→ Y(S) ⊂ W(−) W 7−→ n S

B

α

∈π

⁻¹

([B]) B α

[B] ∈ π(W ) o , where π is the projection of W to its quotient by ^conn. ∼ .

We have the natural commutative diagram X  / /

K

W

K

Y  / / W.

Lemma 1.10. The projection K is associative, i.e. the following diagram of natural transformation commutes:

Y ^×3 Y ^×2

Y ^×2 Y

Y×K

K×Y K

K

.

Definition 1.11 (The functor W ^conn ). We define the functor W _n ^conn,or : Fin ^op −→ Fin ^op

by, for all non-empty set S, the fiber of K S : W ^or (S) → Y ^or (S) over the wall with one brick {S }, i.e. the subfunctor of W _n ^or giving by W _n ^conn,or (S) :=

 

 (W, < W ) ∈ W _n ^or

∀α, β ∈ [[1, n]], ∃W α =: W i

0

, . . . , W i

m−1

, W m := W β

s.t. ∀j ∈ [[0, m − 1]], W i

j

∈ W, W i

j

∩ W i

j+1

6= ∅ and (W i

j

, W i

j+1

) or (W i

j+1

, W i

j

) ∈ Succ(W )

 

 . The natural surjection W ^or ։ W gives us the subfunctor

W ^conn,or W ^or

W ^conn W

called the functor of connected walls: an element of W ^conn (S) is called a connected wall on S.

Remark 1.12. By the same arguments as in remark 1.3, we have the natural injection of X ^conn in W ^conn .

Proposition 1.13. Let W be a wall in W(S). Then, there exist n in N and S 1 ∐ . . . ∐ S n a unique non-ordered partition of S such that

W ∈ im H :

Y n i=1

W ^conn (S i ) −→ W(S)

.

Proof. Let S be a finite set and W be in W(S), a wall over S. The partition K(W )

in Y (S) gives the result.

Remark 1.14. About the diagramatic representation of walls. The terminology introduced in definition 1.11 comes from the diagramatic representation of the ele- ments of W(S). Just as the combinatorics of operads is controlled by rooted trees (cf. [LV12, Sect. 5.6]), the combinatorics of protoperads is controlled by a stack of bricks:

or ,

the gray colour indicates a brick that is not connected in this representation. We have some examples.

– First, we consider the set S = [[1, 4]] and the wall W = {W a , W b , W c } in W([[1, 4]]) over S with the three bricks

W a = {1, 2}, W b = {3, 4} et W c = {1, 2}

with the partial order W a < W c . We represent this wall by

(2)

1 2 3 4

a b

c .

This wall is not connected: W is in the image of the product H : W ^conn ([[1, 2]]) × W ^conn ([[3, 4]]) → W([[1, 4]]).

Note that this graphical representation of the wall W depends on the choice of an order on S: so, the diagrams

1 2

3 4

b a

c and

2 1

3 4

a b

b c

represent the same wall W . In the second diagram, the brick W b is not connected. The choice of the natural order on [[1, 4]] corresponds to Equa- tion (2). In cases where there is no ambiguity, we can omit the elements of S and the names of bricks to obtain the following diagram

.

– We consider the connected wall with four bricks W = {W a , W b , W c , W d } in W ^conn ([[1, 4]]) with

W a = {1, 2}, W b = {3, 4}, W c = {2, 3} and W d = {1, 4}

and the partial order W a < W c , W a < W d , W b < W c and W b < W d . We represent this wall by

a b

c

d d

.

2. Products on S -modules

2.1. S-modules. Recall that the category Fin is a groupoid which gives us the equivalence of categories Fin ∼ = Fin ^op by passage to the inverse. One of the key points of the constructions of this section is that (Fin, ∐) is a symmetric monoidal category. We denote by S its skeleton i.e. the category where objects are natural numbers, i.e. Ob S = N and where morphisms are given by Hom S (n, n) = S n for n 6= 0 and Hom(0, 0) = {id}, and which is equivalent to Fin.

Definition 2.1 (S-module, S-bimodule). A (right) S-module is an object of S-mod k

not. := Func(Fin ^op , Ch k ), the category of contravariant functors from Fin to the category of k-chain complexes. A S-bimodule is an object of S-bimod k

not. :=

Func(Fin × Fin ^op , Ch _k ).

Let V be a S-module (respectively W a S-bimodule). We say that V (resp. W ) is locally finite if, for all finite sets S, V (S) has finite total dimension (resp. for all finite sets S, E, the chain complex W (S, E) has finite dimension).

As S is the skeletal category of Fin, we can view an S-module M as a collection M (n)

n∈ N

^∗

of chain complexes indexed by natural numbers, where the the group S _n acts (on the left) on M (n), for n 6= 0. Similarly, an S-bimodule P is a collection of chain complexes P (m, n)

m,n∈N indexed by pairs of integers where P (m, n) has an action of S _m on the left and an action of S _n on the right, or equivalently, has an action of the group S _m × S ^op _n on the left.

Definition 2.2 (Reduced S-(bi)module). A S-module (resp. S-bimodule) P which satisfies P ( ∅ ) = 0 (resp. P( ∅ , S) = 0 and P(S, ∅ ) = 0 for all finite set S) is called reduced. We respectively note by S-mod ^red _k and S-bimod ^red _k , the full subcategories of S-mod k and S-bimod k of reduced S-modules and S-bimodules.

Remark 2.3. We have the equivalence of categories S-mod k ∼ = S ^op -mod k , induced by taking the inverse of elements in symmetric groups. We use this equivalence without mention.

2.2. Composition and concatenation products on S-mod k . In this subsec- tion, we recall the classical constructions of composition and concatenation product of S-modules. The composition product (or vertical product) is the bifunctor

− − : S-mod ^red _k × S-mod ^red _k −→ S-mod ^red _k

defined, for P and Q two reduced S-modules and S a non-empty finite set, by P Q

(S) := P (S) ⊗ Q(S).

This bi-additive bifunctor gives the category S-mod ^red a symmetric monoidal struc- ture, with the identity I , defined, for all non-empty sets S, by I (S) := k concen- trated in degree 0. In the litterature or algebraic operads (cf. [LV12, Sect. 5.1.12]), the composition product of S-modules is also called the Hadamard product. The concatenation product is the bifunctor

− ⊗ ^conc − : S-mod ^red _k × S-mod ^red _k −→ S-mod ^red _k defined, for all finite set S and all reduced S-modules P and Q, by:

P ⊗ ^conc Q

(S) := M

{S

^′

,S

^′′

}∈Y

^or₂

(S)

P(S ^′ ) ⊗ Q(S ^′′ ).

This product has no identity.

Remark 2.4. This product is called the concatenation product because it corre- sponds to a concatenation of operations. It is functorial: it is a particular case of Day’s convolution product. For P and Q two reduced S-modules, we have

P ⊗ ^conc Q(−) :=

Z (S

^′

,S

^′′

)∈Ob ( Fin

^op

)

^×2

k

Hom ^Fin (S ^′ ∐ S ^′′ , −)

⊗ P (S ^′ ) ⊗ Q(S ^′′ ).

Proposition 2.5. The concatenation product is symmetric, i.e. for all reduced S-modules P and Q, we have the following isomorphism of S-modules

P ⊗ ^conc Q ∼ = Q ⊗ ^conc P.

Proof. If (S ^′ , S ^′′ ) is an element of Y ₂ ^or (S), then (S ^′′ , S ^′ ) too. As the monoidal product ⊗ k of the category Ch k is symmetric, we have the natural equivalence τ defined as follows: for all finite sets S,

τ S : P ⊗ ^conc Q

(S) −→ ^{∼ =} Q ⊗ ^conc P (S)

given by τ S (p ⊗ q) = (−1) ^|p||q| q ⊗ p for p ⊗ q in P (S ^′ ) ⊗ ^conc Q(S ^′′ ).

We can extend the concatenation product:

(3) − ⊗ ^conc − : S-mod k × S-mod ^red _k −→ S-mod ^red _k . This extension is induced by the equivalence of categories

S-mod k ∼ = Ch k × S-mod ^red _k ,

by the injection (−) ^S : Ch k ֒ → S-mod k defined, for all chain complexes C and all finite sets S, by

(C) ^S (S ) :=

C if S = ∅ , 0 sinon ;

and by the action of the category Ch k on S-mod ^red _k defined, for all chain complexes C and all finite sets S, by

C ⊗ ^conc V

(S) := C ⊗ V (S).

This extension allows us to define the suspension of a S-module.

Definition 2.6 (Suspension and desuspension of a S-module). Let Σ (respectively Σ ⁻¹ ) be the chain complex k concentrated in degree 1 (resp. in degree −1). For V a reduced S-module, the suspension of V (resp. desuspension of V ) is the reduced S-module ΣV ^not. := Σ ⊗ ^conc V (resp. Σ ⁻¹ V ^not. := Σ ⁻¹ ⊗ ^conc V ).

2.2.1. Free monoids associated to ⊗ ^conc . Recall that the bifunctor −⊗ ^conc −is linear in each of its inputs. We have the functor:

T _⊗ (−) : S-mod ^red _k −→ As(S-mod ^red _k , ⊗ ^conc ) defined, for all finite sets S and all reduced S-modules P , by

T _⊗ P

(S) := M

r∈ N

^∗

T ^r

⊗ P (S) with

T ^r

⊗ P

(S) := P ^⊗

^conc

^r (S) = M

I∈Y

_r^or

(S)

P (I 1 ) ⊗ . . . ⊗ P (I r ).

As ⊗ ^conc is symmetric, we have also the commutative free monoid functor S (−) : S-mod ^red _k −→ Com(S-mod ^red _k , ⊗ ^conc ).

defined, for all reduced S-modules P , by:

S (P) := M

b∈N

^∗

S ^b (P ) with S ^b (P ) := T ^b _⊗ (P )

S

_b

where the action of S b is given by the symmetry τ of the product ⊗ ^conc : explicitly, for p 1 ⊗ · · · ⊗ p b an element of P(I 1 ) ⊗ ^conc · · · ⊗ ^conc P (I b ) which is included in

T ^b _⊗ P

(S), and σ, a permutation in S b , we have

σ · (p 1 ⊗ · · · ⊗ p b ) := (−1) ^|σ(m)| p σ

⁻¹

(1) ⊗ . . . ⊗ p σ

⁻¹

(b)

which lives in P(I _σ

⁻¹

₍₁₎ ) ⊗ ^conc . . . ⊗ ^conc P(I _σ

⁻¹

_(b) ) with (−1) ^|σ(m)| the Koszul sign induced by τ.

Notation 2.7. Let S be a finite set and P be a reduced S-module, as in [LV12, Sect. 5.1.14], we use the following notation

M

I∈Y

r

(S)

O

α∈A

P(I α ) ^not. := M

I∈Y

_r^or

(S)

P(I 1 ) ⊗ . . . ⊗ P (I r )

S

_r

. Let S be a finite set and P and Q be two reduced S-modules. Since

S ^r P

(S) :=

T ^r P (S)

S

_r

∼ = M

I∈Y

_r^or

(S)

P (I 1 ) ⊗ . . . ⊗ P (I r )

S

_r

then, we have

S P

(S) ∼ = M

{I

α

}

α∈A

∈Y(S)

O

α∈A

P (I α ).

We also have the following isomorphism:

S P S Q

(S) ∼ = M

({I

α

}

α∈A

,{J

β

}

β∈B

)∈X(S)

O

α∈A

P (I α ) ⊗ O

β∈B

Q(J β ).

Moreover, as the bifunctor −⊗ ^conc − is biadditive, the functor S has the exponential property:

(4) S (P ⊕ Q) ∼ = S (P ) ⊕ S (Q) ⊕ S (P ) ⊗ ^conc S (Q).

2.3. Connected composition product of S-modules. In this section, we define the new monoidal structure on the category of reduced S-modules, which is called the connected composition product. This monoidal structure is the analogue of the product defined by Vallette in [Val03, Val07].

Definition 2.8 (Connected composition product of S-modules). The connected composition product of reduced S-modules is the bifunctor

− ⊠ _c − : S-mod ^red _k × S-mod ^red _k −→ S-mod ^red _k

defined, for all reduced S-modules P and Q and for all non-empty finite set S, by:

P ⊠ _c Q(S) := M

(I,J)∈X

^conn

(S)

O

α

P (I α ) ⊗ O

β

Q(J β ).

Denote by I ⊠ , the S-module given by I ⊠ (S) ^def :=

k if |S| = 1 0 otherwise , which is the unit of the product ⊠ _c .

Below, we require a description of elements of P ⊠ _c Q(S): we represent by (p 1 ⊗ . . . ⊗ p r ) ⊗ (q 1 ⊗ . . . ⊗ q s )

a class of N

α P (I α ) ⊗ N

β Q(J β ) with (I, J) in X ^conn (S). Recall that we identify (p 1 ⊗ . . . ⊗ p r ) ⊗ (q 1 ⊗ . . . ⊗ q s ) with

(−1) |σ(p)|+|τ(q)| (p σ

⁻¹

(1) ⊗ . . . ⊗ p σ

⁻¹

(r) ) ⊗ (q τ

⁻¹

(1) ⊗ . . . ⊗ q τ

⁻¹

(s) )

for all permutations σ in S r , τ in S s and (−1) ^|σ(p)| , (−1) ^|τ(q)| , the Koszul signs induced by these permutations.

Lemma 2.9. The product ⊠ _c is associative and, for all reduced S-modules A and B, the endofunctor

Φ A,B : S-mod ^red _k −→ S-mod ^red _k X 7−→ A ⊠ _c X ⊠ _c B is split analytic (in the sense of [Val09]).

Proof. The associativity of the product ⊠ _c follow from the associativity of K S : for P ,Q and R three reduced S-modules, and S a non-empty set, we have the following isomorphism of chain complexes:

(P ⊠ _c Q) ⊠ _c R

(S) = M

(I,J)∈X

^conn

(S)

O

α

P ⊠ _c Q

(I α ) ⊗ O

β

R(J β )

= M

(I,J)∈X

^conn

(S)

O

α

M

(K

^α

,L

^α

)

∈X

^conn

(I

α

)

O

γ

P (K _γ ^α ) ⊗ O

δ

Q(L ^α _δ ) ⊗ O

β

R(J β )

∼ = M

(I,J)∈X

^conn

(S)

M

{(K

^α

,L

^α

)}

∈∐

α∈A

X

^conn

(I

α

)

O

γ,α

P(K _γ ^α ) ⊗ O

δ,α

Q(L ^α _δ ) ⊗ O

β

R(J β )

∼ = M

(I,J)∈X

^conn

(S)

M

(K,L)∈Y(S)×Y (S) K

S

(K,L)=I

O

γ,α

P (K _γ ^α ) ⊗ O

δ,α

Q(L ^α _δ ) ⊗ O

β

R(J β )

∼ = M

(K,L,J )∈Y(S)

^×3

K

S

(K

S

(K,L),J)=(S)

O

γ

P (K γ ) ⊗ O

δ

Q(L δ ) ⊗ O

β

R(J β )

∼ =

(1.10)

M

(K,L,J )∈Y(S)

^×3

K

S

(K,K

S

(L,J))=(S)

O

γ

P (K γ ) ⊗ O

δ

Q(L δ ) ⊗ O

β

R(J β )

∼ = M

(K,I)

∈X

^conn

(S)

M

(J,L)∈Y(S)×Y(S) K

S

(J,L)=I

O

γ

P (K γ ) ⊗ O

δ,α

Q(L ^α _δ ) ⊗ O

β,α

R(J _β ^α )

∼ = M

(K,I)∈X

^conn

(S)

O

α

P (K γ ) ⊗ O

α

Q ⊠ _c R (I α )

= P ⊠ _c (Q ⊠ _c R)

(S).

Also, for all reduced S-modules A and B, the endofunctor Φ A,B is well defined by Φ A,B (X) ∼ = M

(K,L,J)∈Y(S)

^×3

K

S

(K

S

(K,L),J)=(S)

O

γ

A(K γ ) ⊗ O

δ

X (L δ ) ⊗ O

β

B(J β )

∼ = M

n∈ N

M

L∈Y

n

(S)

M

(K,J)∈Y(S)

^×2

K

S

(K

S

(K,L),J)=(S)

O

γ

A(K γ ) ⊗ O

δ

X(L δ ) ⊗ O

β

B(J β )

=: M

n∈ N

(Φ A,B ) n (X ).

where (Φ A,B ) n is an homogeneous polynomial functor of degree n; so, for all reduced S-modules A and B, the functor Φ A,B is a split analytic functor (in the sense of

[Val09, Sect. 4]).

Proposition 2.10. The category S-mod ^red , ⊠ _c , τ, I ⊠

is an abelian symmetric monoidal category that preserves reflexive coequalizers and sequential colimits.

Proof. Let P and Q be two reduced S-modules, we have, for all non-empty finite sets S, the isomorphism of S _|S| -modules

P ⊠ _c Q(S) := M

(I,J)∈X

^conn

(S)

O

α

P(I α ) ⊗ O

β

Q(J β )

∼ = M

(I,J)∈X

^conn

(S)

O

β

Q(J β ) ⊗ O

α

P (I α )

by symmetry of ⊗ of the category Ch _k ; since (I, J) is in X (S) if, and only if, (J, I) is in X (S), we have the isomorphism

P ⊠ _c Q(S) ∼ = M

(J,I)∈X

^conn

(S)

O

β

Q(J β ) ⊗ O

α

P (I α ) ∼ = Q ⊠ _c P (S).

We denote this isomorphism τ P,Q (S) : P ⊠ _c Q(S) −→ Q ⊠ _c P(S), which gives us the symmetry of the product. The rest of the proof is similar to [Val09, Prop 13].

We have the following compatibility between these products:

Proposition 2.11. Let P and Q be two reduced S-modules. We have the following natural isomorphism of S-modules:

S (P ⊠ _c Q) ∼ = S P S Q.

Proof. Let S be a finite set, we have S P ⊠ _c Q

(S) = M

Λ∈Y(S)

O

γ

P ⊠ _c Q (Λ γ ) 2.8 =

M

Λ∈Y(S)

O

γ

M

(I

^γ

,J

^γ

)∈X

^conn

(Λ

γ

)

O

α

P (I _α ^γ ) ⊗ O

β

Q(J _β ^γ )

∼ = M

Λ∈Y(S)

M

{(I

^γ

,J

^γ

)}∈ `

γ

X

^conn

(Λ

γ

)

O

α,γ

P (I _α ^γ ) ⊗ O

β,γ

Q(J _β ^γ )

∼ = M

Λ∈Y(S)

M

(e I, J)∈K e

⁻¹_S

(Λ)

O

a

P( I e a ) ⊗ O

b

Q( J e b )

∼ = M

(e I, J)∈X(S) e

O

α

P( I e α ) ⊗ O

β

Q( J e β )

=

S P S Q (S)

which conclude the proof.

3. Connected product on S -bimodules

Just as in the S-modules, we give a description of the three monoidal structures on the category of (reduced) S-bimodules which are the equivalent of the three previous ones defined in S-mod ^red _k . We start with a section on the functors that encode the combinatorics of our monoidal products. We express the combinatorics of the different monoidal structures on the S-bimod k category, using the same formalism as in the previous section.

3.1. Combinatorics of the connected composition of S-bimodules.

Definition 3.1 (Functors Y ^or and Y ). Let S and E be two finite sets. We define (1) the bifunctor Y ^or (−, −) : Fin×Fin ^op → Set given on objects by Y ^or (S, E) =

∐ r∈N

^∗

Y ^or _r (S, E), with Y ^or _r (S, E) := Y _r ^or (S) × Y _r ^or (E) ∼ =

(I, K) = (I j , K j )

j∈[[1,r]]

∐ ^r _j=1 I j = S; ∐ ^r _j=1 K j = E;

∀j ∈ [[1, r]], I j 6= ∅ , K j 6= ∅

, where the elements of Y ^or

r (S, E) are ordered sets of pairs of sets.

(2) Y (−, −) : Fin×Fin ^op → Set the bifunctor given by Y (S, E) = ∐ r∈ N

^∗

Y _r (S, E) with Y _r (S, E) =

 

 {I, K} ^not := {(I α , K α )} α∈A

A ∈ Ob Fin, |A| = r

∐ α∈A I α = S, ∐ α∈A K α = E

∀α ∈ A, I α 6= ∅ , K α 6= ∅

 

 , where the elements of Y _r (S, E) are non ordered sets of pairs of sets.

Note that Y _r (S, E) 6∼ = Y(S) × Y(E). As in the case of Y ^or (−), we have a free action of S _r on Y ^or _r (S, E) given, for all (I, K) in Y ^ord _r (S, E) and all permutation σ in S _r , by

σ · (I, K) = (I _σ

⁻¹

_(j) , K _σ

⁻¹

_(j) )

j∈[[1,r]]

which induces the surjection Y ^or

r (S, E) ։ Y _r (S, E).

As in the case of S-modules, we define a new bifunctor, denoted by X ^conn which encodes connectedness. For r and s in N ^∗ and for S and E two finite sets, the bifunctor X ^n,conn

r,s (S, E) is equal to {I, K ^′ }, {K ^′′ , J }

∈ Y _r (S, [[1, n]]) × Y _s ([[1, n]], E)

(K ^′ , K ^′′ ) ∈ X ^conn ([[1, n]])

S

_n

, where, for all n in N ^∗ , by the functoriality of X ^conn , the quotient by S _n which identifies

{I, K ^′ } · σ, {K ^′′ , J}

∼ {I, K ^′ }, σ ⁻¹ · {K ^′′ , J}

is well defined. We also have the functors X ^n,conn (S, E) := a

r,s∈ N

^∗

X ^n,conn

r,s (S, E) and X ^conn (S, E) := a

n∈ N

^∗

X ^n,conn (S, E).

3.2. Monoidal products of the category S-bimod k . 3.2.1. Composition product of S-bimodules .

Definition 3.2 (The composition product of S-bimodules). The the composition product of S-bimodules is the bifunctor of

− − : S-bimod ^red _k × S-bimod ^red _k −→ S-bimod ^red _k

defined, for all reduced S-bimodules P and Q, and for all finite sets S and E, by P Q

(S, E) := M

n∈N

^∗

M

({I,K

^′

},{K

^′′

,J})∈ X

ⁿ_1,1

(S,E)

P(I, K ^′ ) ⊗ Q(K ^′′ , J) .

S

_n

∼ = M

n∈ N

^∗

P (S, [[1, n]]) ⊗

S

_n

Q([[1, n]], E) ; where the action of S n is induced by the action on [[1, n]].

Remark 3.3. The composition product is defined by a coend. In fact, the tensorial product of the category Ch k gives us the external product

Func(Fin × Fin ^op , Ch k ) ^×2 −→ Func(Fin × Fin ^op × Fin × Fin ^op , Ch k )

(P, Q) 7−→

(S 1 , E 1 , S 2 , E 2 ) 7→ P (S 1 , E 1 ) ⊗ Q(S 2 , E 2 )} , and, by taking the coend of the functors P (S 1 , −) ⊗ Q(−, E 2 ) : Fin ^op × Fin → Fin for S 1 and E 2 two finite sets, we have

− − : Func(Fin × Fin ^op , Ch k ) ^×2 −→ Func(Fin × Fin ^op , Ch k ) (P, Q) 7−→ R S∈ Fin

P (−, S) ⊗ Q(S, −) . As the category S is a skeleton of the category Fin, we finally have

− − : Func(Fin × Fin ^op , Ch k ) ^×2 −→ Func(Fin × Fin ^op , Ch k )

(P, Q) 7−→ M

n∈N P (−, [[1, n]]) ⊗

S

_n

Q([[1, n]], −) . Remark 3.4. Let P and Q be two reduced S-modules. For all m and n in N ^∗ , we have the following isomorphism of S m × S ^op _n -bimodules:

P Q(m, n) ∼ = M

N ∈N

^∗

P(m, N ) ⊗

k[ S

_N

] Q(N, n).

Proposition 3.5. The category S-bimod k , , I

with I (S, E) = k[Aut(S)] for

S ∼ = E and 0 otherwise, is a monoidal category.

Proof. Let P , Q and R be three reduced S-bimodules and S and E be two non- empty finite sets. We note by e the cardinal of the set E. By definition of I , we have the following isomorphisms:

P I (S, E) ∼ = M

n∈ N

P (S, [[1, n]]) ⊗

S

_n

I ([[1, n]], E)

∼ = P (S, [[1, e]]) ⊗

S

_e

I ([[1, e]], E)

∼ = P (S, [[1, e]]) ⊗

S

_e

k[Aut([[1, e]])]

∼ = P (S, E).

By Remark 3.3, we also have the isomorphisms:

P Q

R(S, E) =

Z U∈ Fin

P Q

(S, U ) ⊗ R(U, E)

∼ =

Z U∈ Fin Z V ∈ Fin

P(S, V ) ⊗ Q(V, U ) ⊗ R(U, E)

∼ =

Fubini

Z (U,V )∈ Fin

^×2

P (S, V ) ⊗ Q(V, U ) ⊗ R(U, E)

∼ = P Q R (S, E)

3.2.2. The concatenation product ⊗ ^conc . As in the case of S-modules, we define the concatenation product of two reduced S-bimodules.

Definition 3.6 (Concatenation product of S-bimodules). The concatenation prod- uct is the bifunctor

− ⊗ ^conc − : S-bimod ^red _k × S-bimod ^red _k −→ S-bimod ^red _k ,

defined, for P and Q two reduced S-bimodules and for all finite sets S and E, by P ⊗ ^conc Q

(S, E) := M

(I,K)∈ Y

^ord₂

(S,E)

P (I 1 , K 1 ) ⊗ Q(I 2 , K 2 ).

Remark 3.7. This product is a particular case of a Day convolution product: let P and Q be two reduced S-bimodules, the bifunctor P ⊗ ^conc Q(− 1 , − 2 ) is given by

Z (I

1

,I

2

,J

1

,J

2

)

k

Hom ^Fin

^op

× Fin (I 1 ∐ I 2 , J 1 ∐ J 2 ), (− 1 , − 2 )

⊗ P (I 1 , J 1 ) ⊗ Q(I 2 , J 2 ).

The two product and ⊗ ^conc satisfy the interchanging law.

Proposition 3.8 (Interchanging law). Let A, B, C, D, E and F be reduced S- bimodules. We have the natural injection of S-bimodules

Φ A,B,C,D : (A B) ⊗ ^conc (C D) ֒ → (A ⊗ ^conc C) (B ⊗ ^conc D), which is associative, i.e. we have

Φ A⊗C,B⊗D,E,F ((Φ A,B,C,D ) ⊗ (E F )) = Φ A,B,C⊗E,D⊗F ((A B) ⊗ Φ C,D,E,F ) .

Proof. Let A, B, C and D be reduced S-bimodules. We have (A B) ⊗ ^conc (C D)(−, −)

=

Z (I

1

,I

2

,J

1

,J

2

)∈ Fin

^×4

k[Hom((I

1

∐I

2

,J

1

∐J

2

),(−,−))]

⊗(A B)(I

1

,J

1

)⊗(C D)(I

2

,J

2

)

=

Z (I

1

,I

2

,J

1

,J

2

)∈ Fin

^×4

k[Hom((I

1

∐I

2

,J

1

∐J

2

),(−,−))]

⊗ Z S∈ Fin

A(I

1

,S)⊗B(S,J

1

) ⊗ Z T ∈ Fin

C(I

2

,T )⊗D(T,J

2

)

∼ =

Fubini

Z (I

1

,I

2

,J

1

,J

2

,S,T)∈ Fin

^×6

k[Hom((I

1

∐I

2

,J

1

∐J

2

),(−,−))]

⊗A(I

1

,S)⊗C(I

2

,T)⊗B(S,J

1

)⊗D(T,J

2

) .

The natural injections

k

Hom (I

1

∐I

2

,J

1

∐J

2

),(−,−)

֒ → a

S k

Hom (I

1

∐I

2

,J

1

∐J

2

),(−,S)

⊗k

Hom (I

1

∐I

2

,J

1

∐J

2

),(S,−)

֒ → a

U→S S S→V

k

Hom (I

1

∐I

2

,U),(−,S)

⊗k

Hom (V,J

1

∐J

2

),(S,−)

and

A(I

1

,S)⊗C(I

2

,T)⊗B(S,J

1

)⊗D(T,J

2

) ֒ → a

J

1

,J

2

,K

1

,K

2

A(I

1

,J

1

)⊗C(I

2

,J

2

)⊗B(K

1

,L

1

)⊗D(K

2

,L

2

)

give us the natural injection

(A B) ⊗ ^conc (C D)(−, −)

֒ →

Z (S,I

1

,I

2

,J

1

,J

2

,K

1

,K

2

,L

1

,L

2

) k

Hom (I

1

∐I

2

,J

1

∐J

2

),(−,S)

⊗k

Hom (K

1

∐K

2

,L

1

∐L

2

),(S,−)

⊗A(I

1

,J

1

)⊗C(I

2

,J

2

)⊗B(K

1

,L

1

)⊗D(K

2

,L

2

)

∼ =

Fubini

Z S Z I

i

,J

i

k

Hom (I

1

∐I

2

,J

1

∐J

2

),(−,S)

⊗A(I

1

,J

1

)⊗C(I

2

,J

2

) ⊗ Z K

i

,L

i

k

Hom (K

1

∐K

2

,L

1

∐L

2

),(S,−)

⊗B(K

1

,L

1

)⊗D(K

2

,L

2

)

= Z S

A ⊗ ^conc C

(−, S) ⊗ B ⊗ ^conc D (S, −)

= A ⊗ ^conc C

B ⊗ ^conc D (−, −).

Then we have the injective natural transformation between bifunctors:

Φ A,B,C,D : (A B) ⊗ ^conc (C D) ֒ → (A ⊗ ^conc C) (B ⊗ ^conc D).