Alexandre Quesney

Swiss Cheese type operads and models for relative loop spaces

come

Swiss Cheese type operads and models for relative loop spaces

Introduction

The Swiss Cheese operad SC is a 2-coloured topological operad that mixes, in its m-dimensional part SC m , the m-dimensional and the (m -1)-dimensional parts of the little cubes operad C. It was used by M. Kontsevich [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF] in deformation quantization as a natural way to define C * (C m )-algebras acting on C * (C m-1 )-algebras. In fact, it turns out that the Swiss Cheese operad SC m recognizes actions of m-fold loop spaces on m-fold relative loop spaces as announced in [START_REF] Hoefel | A ∞ -actions and recognition of relative loop spaces[END_REF]. Looking at m = 1 this means that the A ∞ -actions are recognized by SC 1 .

The purpose of this paper is two fold. We first provide a combinatorial model for the Swiss-Cheese operad SC. That is, we construct a (coloured) operad RL in the category of sets and, by mean of condensation and according to a choice of a cosimplicial object, we obtain an operad equivalent to the Swiss Cheese operad SC as well as an operad equivalent to the chains of SC with integers coefficients.

In a second part, we use our newly obtained operads to exhibit algebraic models for actions of 2-fold loop spaces on 2-fold relative loop spaces. We pay close attention to the couple (cobar construction, relative cobar construction).

In [START_REF] Batanin | The lattice path operad and Hochschild cochains[END_REF], Batanin and Berger introduce the notion of condensation of a coloured operad. By applying this condensation to the lattice path operad L they obtain a model for the little cubes operad. More precisely, this means that, in the category of topological spaces Top (resp. the category of chain complexes Ch(Z)), the condensation operad of L is weakly equivalent to the topological (resp. chain) little cubes operad.

We construct a coloured operad RL in the category of sets that may be thought of as a combinatorial model for the Swiss Cheese operad.

Let us fix a closed monoidal symmetric category C with a zero object and a cosimplicial object δ : △ → C. By adapting Batanin-Berger's method, we obtain a functor F : RL-algebra -→ Coend RL (δ)-algebra that sends algebras over RL into algebras over the condensation (2-coloured) operad Coend RL (δ) in C.

The operad RL is filtered by suboperads RL m , m ≥ 1 and we have the corresponding condensed operads Coend RL m (δ), m ≥ 1.

We are interested by two choices for δ: The Swiss Cheese operad that we consider is denoted by SC m , m ≥ 1, and is the augmented (cubical) version of Voronov's Swiss Cheese operad SC vor m defined in [START_REF] Voronov | The swiss-cheese operad[END_REF]. For each m ≥ 1, we construct a cellular decomposition of the Swiss Cheese operad SC m . This provides a recognition principle that we use afterwards to show the following.

Theorem 1.1. Let m ≥ 1. Then, the operad Coend RL m (δ Top ) is weakly equivalent to the topological Swiss-Cheese operad SC m and, the operad Coend RL m (δ Z ) is weakly equivalent to the chain Swiss-Cheese operad C * (SC m ).

For each m ≥ 1, the operad Coend RL m (δ Z ) admits a weakly equivalent suboperad RS m . This operad RS m may be thought of as a Swiss Cheese version (or relative version) of the surjection operad S m studied by McClure-Smith [START_REF] Mcclure | A solution of Deligne's Hochschild cohomology conjecture[END_REF][START_REF] Mcclure | Multivariable cochain operations and little n-cubes[END_REF] and Berger-Fresse [START_REF] Berger | Combinatorial operad actions on cochains[END_REF].

Our cellular decomposition of the Swiss Cheese operad SC m generalizes Berger's cell decomposition of the little m-cubes operad [START_REF] Berger | Combinatorial models for real configuration spaces and en operads[END_REF] and, likewise, gives rise to a recognition principle.

The cells are indexed by a poset operad RK m which is a Swiss Cheese (or relative) version of the extended complete graph operad. Such a decomposition provides a zig-zag of weak equivalences between the Swiss Cheese operad SC m and the classifying operad of RK m . The latter arise as a comparison object: for any topological RK m -cellular operad O (see Definition 3.7) there is a zig-zag of weak equivalences linking O to the classifying operad of RK m . Thus, we prove the following.

Theorem 1.2. Let m ≥ 1. Then any topological RK m -cellular operad is weakly equivalent to the Swiss Cheese operad SC m .

The second objective of this paper is to provide models for relative loop spaces. It is well-known that m-fold loop spaces are recognized by the little m-cubes operad, [START_REF] May | The geometry of iterated loop spaces[END_REF]. A similar pattern is announced in [START_REF] Hoefel | A ∞ -actions and recognition of relative loop spaces[END_REF] for relative loop spaces: couples (m-fold loop space, m-fold relative loop space) are recognized by SC m . Then a "good" model for such a couple should be acted on by a model of SC m .

We focus on the second stage filtration operad RL 2 . This operad encodes couples (M, Z) together with a map M → Z satisfying some properties. In particular, M is a multiplicative operad and Z is an infinitesimal bimodule over the associative operad As. These structures naturally endow M and Z with a cosimplicial structure. Moreover, the functor F sends such a couple (M, Z) to the couple of totalizations (Tot δ M, Tot δ Z). Thus, the condensed operad Coend RL 2 (δ) acts on the couple (Tot δ M, Tot δ Z).

Given a pair of topological spaces (X, Y) pointed at * and such that * ⊂ Y ⊂ X, there exists a cosimplicial space ω(X, Y) such that its totalization Tot δ Top ω(X, Y) is homeomorphic to Ω(X, Y). In particular, ω(X) = ω(X, * ) is a model for the loop space Ω(X, * ) = ΩX. We have the following.

Theorem 1.3. Let (M, N) be a pair of topological monoids pointed at the unit such that N is a submonoid of M. Let δ be δ Top (resp. δ Z ). Then there exists an operad Coend RL 2 (δ) which is weakly equivalent to the topological operad SC 2 (resp. to the chain operad C * (SC 2 )) and which acts on the totalization (Tot δ ω(M), Tot δ ω(M, N)).

Preliminaries

In [BB09] Batanin and Berger introduced the notion of condensation of a coloured operad. It consists of a realization followed by a totalization what "condenses" all the colours into a single one. We consider particular coloured operads that we call SC-split operads. Roughly speaking, the set of colours of an SC-split operad can be split into two subsets that yield two sub-operads. We modify Batanin-Berger's condensation process for the SC-split operads. Our modification consists in condensing separately the colours of each of the two subsets of colours into one colour. This provides, in particular, 2-coloured operads.

SC functor-operads

Let C be a closed symmetric monoidal category. Let A and B be two C-categories (i.e. enriched over C). We denote by A ⊗ B the category with the pairs (a, b) for a ∈ A and b ∈ B as objects and

Hom A⊗B ((a, b), (a ′ , b ′ )) := Hom A (a, a ′ ) ⊗ Hom B (b, b ′ )
as hom-objects, where the tensor on the right hand side is the tensor of C.

Definition 2.1. For a family of C-functors

{ξ A 1 ,...,A k ;A k+1 : A 1 ⊗ • • • ⊗ A k → A k+1 } Ai∈{A,B}
and a permutation σ ∈ Σ k , we denote by

ξ σ A 1 ,...,A k ;A k+1 : A 1 ⊗ • • • ⊗ A k → A k+1 the functor ξ σ A 1 ,...,A k ;A k+1 (X 1 , ..., X k ) = ξ σ A σ -1 (1) ,...,A σ -1 (k) ;A k+1 (X σ -1 (1) , ..., X σ -1 (k) ).

SC functor-operads

A family of C-functors {ξ A 1 ,...,A k ;A k+1 : A 1 ⊗ • • •⊗ A k → A k+1 } A i ∈{A,B} is called twisting symmetric if there exist C-natural transformations φ σ,A 1 ,...,A k ;A k+1 : ξ A 1 ,...,A k ;A k+1 → ξ σ A 1 ,...,A k ;A k+1 for σ ∈ Σ k , such that φ σ 1 σ 2 ,A 1 ,...,A k ;A k+1 = (φ σ 1 ,A 1 ,...,A k ;A k+1 ) σ 2 φ σ 2 ,A 1 ,...,A k ;A k+1
and such that φ id,A 1 ,...,A k ;A k+1 is the identity transformation where id denotes the neutral element of Σ k .

Definition 2.2. An SC functor-operad ξ = {ξ A 1 ,...,A k ;A k+1 } k≥0 over (A, B) is the data, for each k ≥ 0, of twisted symmetric families

ξ A 1 ,...,A k ;A k+1 : A 1 ⊗ • • • ⊗ A k → A k+1 indexed by the (k + 1)-uples (A 1 , ..., A k ; A k+1 ) such that A k+1 = B whenever it exists an 1 ≤ i ≤ k such that A i = B, together with natural transformations µ [A] 1,i 1 ,...,[A] k,i k ;A k+1 : ξ A 1 ,...,A k ;A k+1 • (ξ A 1,1 ,...,A 1,i 1 ;A 1 ⊗ . . . ⊗ ξ A k,1 ,...,A k,i k ;A k ) → ξ A 1,1 ,...,A k,i k ;A k+1 , for i 1 , ..., i k ≥ 0, where [A] a,b = (A a,1 , ..., A a,b ; A a )
. These natural transformations have to satisfy the following three conditions.

1. For A 0 ∈ {A, B}, ξ A 0 ;A 0 is the identity functor and ξ

A k+1 ;A k+1 •ξ A 1 ,...,A k ;A k+1 = ξ A 1 ,...,A k ;A k+1 = ξ A 1 ,...,A k ;A k+1 •(ξ A 1 ;A 1 ⊗. . .⊗ξ A k ;A k )
where the equalities are obtained via µ (A 1 ,...,A k ;A k+1 );A k+1 and µ (A 1 ;A 1 ),...,(A k ;A k );A k+1 respectively.

The natural transformations µ

[A] 1,i 1 ,...,[A] k,i k ;A k+1 are associative.
3. All the diagrams of the following forms commute:

ξ A 1 ,...,A k ;A k+1 • (ξ A 1,1 ,...,A 1,i 1 ;A 1 ⊗ . . . ⊗ ξ A k,1 ,...,A k,i k ;A k ) ξ A 1,1 ,...,A k,i k ;A k+1 ξ σ A 1 ,...,A k ;A k+1 • (ξ σ 1 A 1,1 ,...,A 1,i 1 ;A 1 ⊗ . . . ⊗ ξ σ k A k,1 ,...,A k,i k ;A k ) ξ σ(σ 1 ,...,σ k+j ) A 1,1 ,...,A k,i k ;A k+1 φ σ • (φ σ 1 ⊗ . . . ⊗ φ σ k+j ) µ [A] 1,i 1 ,...,[A] k,i k ;A k+1 φ σ(σ 1 ,...,σ k+j ) µ [A] 1,i 1 ,...,[A] k,i k ;A k+1 Definition 2.3. Let ξ = {ξ A 1 ,...,A k ;A k+1 } k≥0 be an SC functor-operad over (A, B). A ξ-algebra X is a couple (X A , X B ) ∈ A ⊗ B equipped with morphisms in A k+1 α A 1 ,...,A k ;A k+1 : ξ A 1 ,...,A k ;A k+1 (X A 1 , ..., X A k ) → X A k+1 , k ≥ 0,
subject to the following conditions.

1. α A 1 = 1 XA 1 ; 2. α A 1 ,...,A k ;A k+1 • φ σ = α A 1 ,...,A k ;A k+1 , for all σ ∈ Σ k ;
3. all the diagrams of the following form commute

ξ A 1 ,...,A k ;A k+1 • (ξ [A] 1,i 1 (X 1,i 1 ) ⊗ . . . ⊗ ξ [A] k,i k (X k,i k )) ξ A 1,1 ,...,A k,i k ;A k+1 (X A 1,1 , ..., X A k,i k ) ξ A 1 ,...,A k ;A k+1 (X A 1 , ..., X A k+1 ) X A k+1 ξ A 1 ,...,A k ;A k+1 (α [A] 1,i 1 ⊗ . . . ⊗ α [A] k,i k ) µ [A] 1,i 1 ,...,[A] k,i k ;A k+1 α A 1,1 ,...,A k,i k ;A k+1 α A 1 ,...,A k ;A k+1
where X a,b denotes X A a,1 , ..., X A a,b .

SC-split operads

O(-, • • • , - k ; -) : (O op u ) ⊗k ⊗ O u → C, k ≥ 0.
From now, we suppose C endowed with a zero object i.e. an element 0 ∈ C such that 0 ⊗ X = 0 for all X ∈ C. Let us suppose now that O satisfies the following hypothesis.

H1. Col = Col c ⊔ Col o . H2. The collection of the O(n 1 , . . . , n k ; n) for n i , n ∈ Col c , k ≥ 0 forms a sub-operad of O.
H3. The collection of the O(n 1 , . . . , n j ; n) for n i , n ∈ Col o , j ≥ 0 forms a sub-operad of O.

H4. The O(n 1 , . . . , n j ; n) are the zero object for any n ∈ Col c if there exists an

1 ≤ i ≤ j, such that n i ∈ Col o , j ≥ 1.
We call such an operad an SC-split operad.

C O c u (resp. C O o u ) of C-functors from O c u (resp. from O c u ) to C is a C-category.
For k ≥ 0, let (c 1 , ..., c k ; c k+1 ) be a tuple of elements in {c; o} satisfying (2.1)

c k+1 = o if there is an 1 ≤ i ≤ k such that c i = o.
We set A i := C O c i u and we define the C-functor

ξ(O) c 1 ,...,c k ;c k+1 : A 1 ⊗ • • • ⊗ A k → A k+1
as the coend

ξ(O) c 1 ,...,c k ;c k+1 (X c 1 , ..., X c k )(n) = O(-, . . . , - k ; n) ⊗ O c 1 u ⊗•••⊗O c k u X c 1 (-) ⊗ • • • ⊗ X c k (-). An O-algebra X is a family {X(n)} n∈Col of objects X(n) ∈ C equipped with morphisms O(n 1 , ..., n k ; n) ⊗ X(n 1 ) ⊗ . . . ⊗ X(n k ) → X(n), n 1 , ..., n k , n ∈ Col (2.2)
subject to the natural unit, associative and equivariance axioms. In particular, from the hypothesis on O, X can be seen as a pair (X c , X o ) where X c is the sub-family {X c (n)} n∈Colc and X o is the sub-family {X o (n)} n∈Colo . We have an SC analogue to [BB09, Proposition 1.8] or [DS03]:

Proposition 2.4. The functors ξ(O) c 1 ,...,c k ;c k+1 extend to an SC functor-operad ξ(O), such that the category of O-algebras and the category of ξ(O)-algebras are isomorphic.

Condensation

Proof. Let us first show that the family of the ξ(O) c 1 ,...,c k ;c k+1 form an SC functor-operad. We set [c] a,b = c a,1 , ..., c a,b ; c a+1 and we denote by n a,b the list n a,1 , . . . , n a,b of objects n r,s ∈ O cr,s u . The natural transformation µ

[c] 1,i 1 ,...,[c] k,i k ;c k+1 is given by ξ(O) c 1 ,...,c k ;c k+1 • (ξ(O) [c] 1,i 1 (X 1,1 , ..., X 1,i 1 ), ..., ξ(O) [c] k,i k (X k,1 , ..., X k,i k )) = n 1 ,...,n k O(n 1 , ..., n k ; -) ⊗ n 1,i 1 O(n 1,i 1 ; n 1 ) ⊗ X 1,1 (n 1,1 ) ⊗ • • • ⊗ X 1,i 1 (n 1,i 1 ) ⊗ • • • ⊗ n k,i k O(n k,i k ; n k ) ⊗ X k,1 (n k,1 ) ⊗ • • • ⊗ X k,i k (n k,i k ) ∼ = n 1 ,...,n k ,n 1,i 1 ,...,n k,i k O(n 1 , ..., n k ; -)⊗O(n 1,i 1 ; n 1 )⊗• • •⊗O(n k,i k ; n k )⊗X 1,1 (n 1,1 )⊗...⊗X k,i k (n k,i k ) → n 1,i 1 ,...,n k,i k O(n 1,i 1 , ..., n k,i k ; -) ⊗ X 1,1 (n 1,1 ) ⊗ ... ⊗ X k,i k (n k,i k ) =: ξ(O) c 1,1 ,...,c k,i k ;c k+1 (X 1,1 , ..., X k,i k ),
where the last map is induced by the composition map of O. The associativity property of the latter implies that ξ satisfies the associativity axiom. The unit axioms 1 of Definition 2.2 is due to the Yoneda lemma. The twisted symmetric condition is obtained from the equivariance of the operad.

Via the hypothesis H2 and H3, X c and X o can be seen as functors

X c : O c u → C and X o : O o u → C respectively. The maps (2.2) give: O(n 1 , ..., n k ; n k+1 ) ⊗ X c 1 (n 1 ) ⊗ . . . ⊗ X c k (n k ) → X c (n k+1 ), (2.3)
for any n i ∈ Col c i and any (c 1 , ..., c k ; c k+1 ) satisfying (2.1). Since these maps satisfy the associativity and unit axioms, they induce a map

ξ c 1 ,...,c k ;c k+1 (O)(X c 1 , ..., X c k ) = n 1 ,...,n k O(n 1 , ..., n k ; n k+1 ) ⊗ X c 1 (n 1 ) ⊗ . . . ⊗ X c k (n k ) → X c k+1 (n k+1 ).
This way, we obtain

α c 1 ,...,c k ;c k+1 : ξ(O) c 1 ,...,c k ;c k+1 (X c 1 , ..., X c k ) → X c k+1 , k ≥ 0.
We conclude that X is a ξ(O)-algebra because of the unit, associativity and equivariance properties of maps (2.3). Conversely, H4 says that the ξ(O) c 1 ,...,c k ;c k+1 's recover all maps in (2.2).

Condensation

Let δ c : O c u → C and δ o : O o u → C be two functors. We set δ = (δ c , δ o ). We define the coendomorphism operad Coend ξ(O) (δ) as the operad in C with objects:

Coend ξ(O) (δ)(c 1 , ..., c k ; c k+1 ) = Hom C O c k+1 u (δ c k+1 , ξ(O) c 1 ,...,c k ;c k+1 (δ c 1 , ..., δ c k )),
for c 1 , ..., c k ; c k+1 satisfying (2.1).

The composition maps

Coend ξ(O) (δ)(c 1 , ..., c k ; c k+1 )⊗ Coend ξ(O) (δ)(c 1,1 , ..., c 1,i 1 ; c 1 )⊗ • • • ⊗ Coend ξ(O) (δ)(c k,1 , ..., c k,i k ; c k ) → Coend ξ(O) (δ)(c 1,1 , ..., c k,i k ; c k+1 )
are given by sending maps f ⊗ g 1 ⊗ . . . ⊗ g k to the composite

δ c k+1 ξ(O) c 1,1 ,...,c k,i k ;c k+1 (δ c 1,1 , ..., δ c k,i k ) ξ(O) c 1 ,...,c k ;c k+1 (δ c 1 , ..., δ c k ) ξ(O) c 1 ,...,c k ;c k+1 (ξ(O) [c] 1,i 1 (δ), ..., ξ(O) [c] k,i k (δ)). f ξ(O)c 1 ,...,c k ;c k+1 (g 1 , ..., g k ) α c 1 ,...,c k ;c k+1
Given an O-algebra X = (X c , X o ), we denote by

Tot δ c X c := Hom C O c u (δ c , X c ); Tot δ o X o := Hom C O o u (δ o , X o ). In virtue of Proposition 2.4, the couple (Tot δ c X c , Tot δ o X o ) is a Coend ξ(O) (δ)-algebra. The action maps Coend ξ(O) (δ)(c 1 , ..., c k ; c k+1 ) ⊗ Tot δ c 1 X c 1 ⊗ • • • ⊗ Tot δ c k X c k → Tot δ c k+1 X c k+1
are given by sending maps f ⊗ g 1 ⊗ . . . ⊗ g k to the composite

δ c k+1 X c k+1 ξ(O) c 1 ,...,c k ;c k+1 (δ c 1 , ..., δ c k ) ξ(O) c 1 ,...,c k ;c k+1 (X c 1 , ..., X c k ). f ξ(O)c 1 ,...,c k ;c k+1 (g 1 , ..., g k ) α c 1 ,...,c k ;c k+1
Unit, associative and equivariance axioms are deduced from the SC functor-operad properties of ξ(O).

A cellular decomposition of the Swiss Cheese operad

The little cubes operad C has a cellular decomposition indexed by the extended complete graph operad K, see [START_REF] Berger | Combinatorial models for real configuration spaces and en operads[END_REF] and [BFV07, 4.1]. We extend this result to the Swiss Cheese operads SC m , m ≥ 1 what provides a recognition principle for Swiss Cheese type operads. In particular, we construct a poset operad RK m that indexes the cells (SC m ) (α) of SC m . This leads to a zig-zag of weak equivalences of operads

SC m hocolim α∈RKm (SC m ) (α) B RK m , ∼ ∼
between the Swiss Cheese operad SC m and the classifying operad of RK m .

The Swiss Cheese operad

The Swiss Cheese operad that we use is the cubical version of the one defined in [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]. Let m ≥ 1. Let Sym : R m → R m be the reflection Sym(x 1 , ...x m ) = (x 1 , ..., -x m ), and let Half + be the upper half space

Half + = {(x 1 , ..., x m ) ∈ R m |x m > 0}. The standard cube C 0 in R m is C 0 = [-1, 1] ×m . A cube C in the standard cube is of the form C = [x 1 , y 1 ] × [x 2 , y 2 ] × • • • × [x m , y m ] with -1 < x j < y j < 1 for 1 ≤ j ≤ m.
Definition 3.1. For n ≥ 0 and c i , c ∈ {c, o} we define a topological Σ n -space SC m (c 1 , ..., c n ; c) as the empty-set if c = c and there exists 1 ≤ i ≤ n such that c i = o; for the other cases, it is

• the space of the little m-cubes operad C (m) (n) defined in [START_REF] May | The geometry of iterated loop spaces[END_REF] for c = c;

• the empty set if n = 0;

• the one-point space if n = 1;

• in the case s + t = n ≥ 2 with s, t ≥ 0 such that s colours c i are c and t colours c j are o, the space of configuration of 2s

+ t disjoint cubes (C 1 , ..., C 2s+t ) in the standard cube C 0 ∈ R m such that Sym(C i ) = C i+s for 1 ≤ i ≤ s and Sym(C i ) = C i for 2s + 1 ≤ i ≤ 2s + t
c i = i if c i = c; i if c i = o.
A colouring and an orientation on a complete graph on { c 1 , ..., c n } is, for each edge between c i and c j , an orientation σ i,j (that is, c i → c j or c i ← c j ) and a strict positive natural number µ i,j ∈ N >0 as the colour. A monochromatic acyclic orientation of a complete graph is a colouring and orientation such that there exist no oriented cycles with the same colour, i.e. there are no configurations of the form

c i 1 → c i 2 → • • • → c i k → c i 1 with µ i 1 ,i 2 = µ i 2 ,i 3 = • • • = µ i k-1 ,i k = µ i k ,i 1 .
If there exists an i such that c i = o, then we set RK(c 1 , ..., c n ; c) as the empty set. Else, the poset RK(c 1 , ..., c n ; c) is the set of pairs (µ, σ) c of monochromatic acyclic orientations of the complete graph on { c 1 , ..., c n }. The colouring µ is a collection of a colour µ i,j for each pair {i; j} and σ is a collection of an orientation σ i,j for each pair {i; j}, with 1 ≤ i, j ≤ n. This is equivalent to write (µ, σ) as {(µ i,j , σ i,j )} 1≤i<j≤n by setting µ i,j = µ j,i and σ i,j = τ 2 σ j,i for 1 ≤ i < j ≤ n, where τ 2 denotes the non-neutral element of Σ 2 .

The poset structure is given by

(µ, σ) c ≤ (µ ′ , σ ′ ) c ⇔ ∀i < j, either (µ i,j , σ i,j ) = (µ ′ i,j , σ ′ i,j ) or µ i,j < µ ′ i,j .
The filtration (RK m ) m≥1 is as follows.

For RK(c 1 , ..., c n ; c) with c i = c for all i, we set

RK m (c 1 , ..., c n ; c) = {(µ, σ) c ∈ RK(c 1 , ..., c n ; c) | µ i,j ≤ m ∀ i < j}
For RK(c 1 , ..., c n ; o), we set

RK m (c 1 , ..., c n ; o) = {(µ, σ) o ∈ RK(c 1 , ..., c n ; o) | µ i,j ≤ m if c i = c j = c, µ i,j ≤ m -1 if c i = c j = o, µ i,j ≤ m if i → j, (3.1) µ i,j ≤ m -1 if i → j }.
Given a permutation σ ∈ Σ n and an element (µ, τ) c ∈ RK(c 1 , ..., c n ; c), the resulting element σ • (µ, τ) c ∈ RK(c σ -1 (1) , ..., c σ -1 (n) ; c) is given by permuting the numbers i by σ without changing neither the underline nor the orientation nor the colouring. For example, the edges i → j of (µ, τ) c with colours µ i,j become the edges σ(i) → σ(j) with the same colours µ i,j . The compositions

RK(c 1 , ..., c n ; c) × RK(c 1,1 , ..., c 1,k 1 ; c 1 ) × • • • × RK(c n,1 , ..., c n,kn ; c n ) → RK(c 1,1 , ..., c n,kn ; c) send a tuple of RK(c 1 , ..., c n ; c) × RK(c 1,1 , ..., c 1,k 1 ; c 1 ) × • • • × RK(c n,1 , ..., c n,kn ; c n ) to an element in RK(c 1,1 , .
.., c n,kn ; c) obtained as follows. The sub complete graphs with vertices in the same block {c i,1 , ..., c i,k i } is oriented and coloured as in RK(c i,1 , ..., c i,k i ; c i ); the edges with vertices in two different blocks are oriented and coloured as the edges between the corresponding vertices in RK(c 1 , ..., c n ; c).

Remark 3.3. For m = 1 the conditions where µ i,j ≤ m -1 cannot be satisfied. It follows that RK 1 (c 1 , ..., c n ; o) is empty whenever the tuple (c 1 , ..., c n ) has more than one open colour. [START_REF] Berger | Combinatorial models for real configuration spaces and en operads[END_REF]). Let X be a topological space and A be a poset. We say that X admits an A-cellulation if there is a functor Θ : A → Top such that:

Cellular decomposition of SC type operads

1. colim α∈A Θ(α) ∼ = X; 2. β ≤ α ∈ A ⇔ Θ(β) ⊆ Θ(α);
3. the inclusions Θ(β) ⊆ Θ(α) are a closed cofibration;

4. for each α ∈ A, the "cell" Θ(α) is contractible.

Given a cell Θ(α), we denote its boundary β<α Θ(β) by ∂Θ(α); we denote its interior Θ(α) \ ∂Θ(α) by Θ(α). A cell with a non empty interior is called proper. A cell with an empty interior is called improper.

Lemma 3.6. [Ber97, Lemma 1.7] Let X be a topological space with an A-cellulation. Then we have the weak equivalences

X ∼ = colim α∈A Θ(α) hocolim α∈A Θ(α) hocolim α∈A ( * ) ∼ = B A, ∼ ∼
where B A denotes the realization of the nerve of the category A.

Proof. Items 2 and 3 of Definition 3.5 give the left hand equivalence (see [BFSV03, Proposition 6.9] for details); the item 4 gives the right one.

γ O Θ c 1 ,...,cn;c (α) × Θ c 1,1 ,...,c 1,k 1 ;c 1 (α 1 ) × • • • × Θ c n,1 ,...,c n,kn ;cn (α n ) ⊆ Θ c 1,1 ,...,c n,kn ;c (γ RK (α; α 1 , ..., α n )),
for all variables c, c i , c i,j , α, α i , where γ O and γ RK denote the composition map of O and RK respectively.

Definition 3.8. Let m ≥ 1. A topological SC type operad O is called an RK m -cellular operad if there are RK m (c 1 , ..., c n ; c)-cellulations of O(c 1 , ..., c n ; c) Θ c 1 ,...,cn;c : RK m (c 1 , ..., c n ; c) → Top
for each c 1 , ..., c n ; c, n ≥ 0, subject to the following two compatibilities.

1. The cellulations are compatible with the Σ n -action:

Θ c σ -1 (1) ,...,c σ -1 (n) ;c (σ • α) = σ • Θ c 1 ,...,cn;c (α) for all σ ∈ Σ n .
2. The cellulations are compatible with the operadic structure of O.

We have the "Swiss Cheese analogue" to Theorem 1.16 [START_REF] Berger | Combinatorial models for real configuration spaces and en operads[END_REF]:

Theorem 3.9. Let m ≥ 1. Any two topological RK m -cellular operads are weakly equivalent. Moreover, the Swiss Cheese operad SC m has a structure of an RK m -cellular operad.

Proof. Let O be a cellular SC type operad. Analogue to [BFSV03, Lemma 6.11] is the fact that {hocolim α∈RK(c 1 ,...cn;c) Θ c 1 ,...cn;c (α)} c i ,c∈{c;o},n≥0 forms an operad. Moreover, the operad structures are compatible with the weak equivalences of Lemma 3.6. We show that, for each m ≥ 1, the operad SC m has a structure of a cellular SC type operad indexed by RK m . The "closed" part of SC m , that is the little m-cubes operad C m , is already shown to have a structure of a cellular operad (indexed by K m ), cf. [START_REF] Brun | On the multiplicative structure of topological hochschild homology[END_REF][START_REF] Berger | Combinatorial models for real configuration spaces and en operads[END_REF].

We use the description of SC m via cubes and semi-cubes given in Remark 3.2. The number m ≥ 1 is fixed. For C 1 either a cube or a semi-cube and C 2 either a cube or a semi-cube, we note C 1 µ C 2 if there are separated by a hyperplane H i orthogonal to the i-th coordinate axis for some i ≤ µ, such that whenever there is no separating hyperplane H i for i < µ, the left element C 1 lies in the negative side of H µ and C 2 lies in the positive side of H µ .

Note that, whenever C 1 is a semi-cube and C 2 is a cube, if H m exists, then C 1 lies in the negative side of H m .

For

α = (µ, σ) ∈ RK(c 1 , ..., c k ; o), we set SC m (c 1 , ..., c k ; o) (α) the cell {(C 1 , ..., C k ) ∈ SC m (c 1 , ..., c k ; o)|C i µ i,j C j if c i → c j and C j µ i,j C i if c i ← c j }.
To see that SC m (c 1 , ..., c k ; c) is the colimit of its cells, the only delicate point is to show that if

x ∈ SC m (c 1 , ..., c k ; c) (α) ∩ SC m (c 1 , ..., c k ; c) (β) , with neither α ≤ β nor α ≥ β then x ∈ ∂SC m (c 1 , ..., c k ; c) (α) ∩ ∂SC m (c 1 , ..., c k ; c) (β) .
Here ∂SC m (c 1 , ..., c k ; c) (α) denotes the boundary γ<α SC m (c 1 , ..., c k ; c) (γ) . For such an x, we construct γ ∈ RK(c 1 , ..., c k ; c) such that γ ≤ α and γ ≤ β as follows. For each c i and c j we define a colouring and an orientation as the minimum among (µ α i,j , σ α i,j ) and (µ β i,j , σ β i,j ). This minimum exists since α and β represent the same configuration x. This defines an element γ ∈ RK(c 1 , ..., c k ; c).

The compatibility with the operadic structure of SC m is clear. Let us explain how works the contractibility of the cells. Recall that the interior of a cell

SC m (c 1 , ..., c k ; c) (α) is SC m (c 1 , ..., c k ; c) (α) \ ( β<α SC m (c 1 , ..., c k ; c) (β)
). Contractibility of proper cells (cells with a non empty interior) onto an interior point is obtained by coordinate-wise contractions starting from the last coordinate, see [Ber97, Theorem 1.16] for details. For improper cells we remark the following. If a cell SC m (c 1 , ..., c k ; c) (α) is improper then at least three cubes/semi-cubes are involved i.e. k ≥ 3.

Two elements c i and c j of α are said related by a positive (resp. negative) monochromatic path of colour ν if there exist an l ≥ 2 and indices i =: i 0 , i 1 , ..., i l-1 , i l := j such that c ir → c i r+1 (resp. c ir ← c i r+1 ) and µ ir,i r+1 = ν for all 0 ≤ r ≤ l -1. Two elements c i and c j are called an improper pair if there are related by at least one monochromatic path of colour ν such that µ i,j > ν.

Then, an improper cell is exactly a cell indexed by an elements α which contains at least one improper pair.

Let α be such an element indexing an improper cell. To such an improper pair c i and c j of α one assigns the colour µ ′ i,j defined as the minimal ν among all the monochromatic paths of colour ν satisfying µ i,j > ν; also, one assigns the following orientation c i → c j if the1 path of the minimal colour ν = µ ′ i,j is positive, and c i ← c j if this path is negative.

Let β be the element obtained by applying such an assignment for each improper pair of α. Then β is such that β ≤ α and it has no improper pairs, so that its corresponding cell is proper. Moreover, by construction it is unique and maximal for α. Thus, any improper cell has a unique maximal proper cell and then is contractible.

The operad RL

Definition of the operad RL

We describe an SC-split operad RL in the category of sets, Set.

The operad RL has a natural filtration by sub operads RL m for m ≥ 1. For each m ≥ 1, RL m can be thought of as a mix between the sub operads L m and L m-1 of the Lattice paths operad L introduced in [START_REF] Batanin | The lattice path operad and Hochschild cochains[END_REF].

For m = 2, a description of RL 2 using planar trees is given in Section 6.

Definition of RL. Let Cat * , * be the category of bi-pointed small categories and functors preserving the two distinguished objects. An ordinal [i] defines a category freely generated by the linear graph

l i = {0 → 1 → • • • → i}. For two ordinals [i] and [j], the tensor product [i] ⊗ [j]
is the category freely generated by the graph l i ⊗ l j . The category

[i] is bi-pointed in (0, i); the tensor product [i] ⊗ [j] is bi-pointed in ((0, 0), (i, j)).
The set of colours of RL is

Col = Col c ⊔ Col o ,
where Col c is the set N of natural numbers and Col o is the set of natural numbers decorated with an underline. Hence, n ∈ Col c whereas n ∈ Col o . In general a colour in Col is denoted by n, so that it is either n or n.

The set RL( n 1 , ..., n k ; n) is defined as:

RL( n 1 , ..., n k ; n) = ∅ if n = n ∈ Col c and if there is an i such that n i = n i ∈ Col o ; RL( n 1 , ..., n k ; n) = Cat * , * ([ n + 1], [ n 1 + 1] ⊗ [ n 2 + 1] ⊗ . . . ⊗ [ n k + 1])
else.

The substitutions maps are given by tensor and composition in Cat * , * . For instance, an element x ∈ RL(n 1 , n 2 ; n) is a functor

x : [n + 1] → [n 1 + 1] ⊗ [n 2 + 1] (4.1)
that sends (0, n + 1) on ((0, 0), (n 1 + 1, n 2 + 1)) and is determined by the image of the n remaining objects of [n + 1] and the morphisms into the lattice

[n 1 + 1] ⊗ [n 2 + 1].
Example 4.1. The following lattice x belongs to RL(3, 2; 3):

(3, 0) • x(4) • • • x(2) = x(3) x(1) x(0) • (0, 4) 1 2 2 1 1 2 1 Figure 4.1: Lattice paths of (12|211||21) o .
The elements of RL( n 1 , ..., n k ; n) correspond bijectively to a string of (decorated) natural numbers separated by vertical bars. Indeed, let us consider an x ∈ RL( n 1 , ..., n k ; n). The relative lattice x is a path from x(0) to x(n + 1) made of edges in the grid l 1 ⊗ . . . ⊗ l k . By running through x from x(0) to x(n + 1) we construct the integer-string with vertical bars as follows. To each parallel edge to the i-axis of the grid l 1 ⊗ . . . ⊗ l k we assign i if n i = n i or i if n i = n i ; to each object x(s) for 1 ≤ s ≤ n we assign a vertical bar. Additionally, we put an extra labelled according to the nature of the output colour.

Example 4.2.

(121) o ∈ RL(1, 0; 0) whereas (121) ∈ RL(1, 0; 0).

Let us expose the corresponding composition on integer-string representations via an example.

Example 4.3. (12|14231||24) o • 2 (13|213|31) o = (124|1632451||426) o
The composition is at 2 and the second term has 3 outputs. Then one has renumbered the integer-string (12|14231||24) o by increasing by 2 = 3 -1 the numbers greater than 2; one gets (12|16251||26) o . One has increased the numbers of the second integer-string (13|213|31) o by 1 = 2 -1: (24|324|42) o . Finally, one has substituted the three occurrences of 2 by the three sub-sequences 24, 324 and 42.

Definition of the operad RL

We use left action for the symmetric group: for σ ∈ Σ k and x ∈ RL( n 1 , ..., n k ; n), the string-integer representation of σ • x ∈ RL( n σ -1 (1) , ..., n σ -1 (k) ; n) is obtained by permuting the number i (resp. i) of the string-integer representation of x by the number σ(i) (resp. σ(i)).

Example 4.4. For x = (12|3211||21) o and σ(1) = 2, σ(2) = 3, σ(3) = 1 one has:

σ • x = (23|1322||32) o .
The underlying category of RL. Let Cat be the category of small categories. Via Joyal-duality:

Cat * , * ([n + 1], [m + 1]) ∼ = Cat([m], [n]).
The bijection is given by (φ :

[n + 1] → [m + 1]) ↔ (ψ : [m] → [n])
given by ψ(i) + 1 = min{j|φ(j) > i} and φ(j) -1 = max{i|ψ(i) < j}. One has immediately that:

RL(n; m) = Hom △ ([n], [m]); RL(n; m) = Hom △ ([n], [m]); RL(n; m) = Hom △ ([n], [m]),
where △ is the simplicial category. Thus, the underlying category (RL) u of RL is the category with objects

[n] = {0 < 1 < • • • < n} for n ∈ N; and [n] = {0 < 1 < • • • < n} for n ∈ N;
and with hom-sets

Hom([n], [m]) = Hom △ ([n], [m]); Hom([[n], [m]) = Hom △ ([n], [m]); Hom([n], [m]) = Hom △ ([n], [m]); Hom([n], [m]) =∅.
The two sub-categories (RL) c u and (RL) o u are (canonically isomorphic to) the category △. This implies that, for each k ≥ 0, the functor

( n 1 , ..., n k ; n) → RL( n 1 , ..., n k ; n) is a multisimplicial/cosimplicial set.
Filtration by sub operads RL m . Let us define two maps

c i,j , c ′ i,j : RL( n 1 , ..., n k ; n) → N.
For 1 ≤ i < j ≤ k, we denote by

φ ij : RL( n 1 , ..., n k ; n) → RL( n i , n j ; n)
the projection induced by the canonical projection

p ij : [ n 1 + 1] ⊗ • • • ⊗ [ n k + 1] → [ n i + 1] ⊗ [ n j + 1].
For x ∈ RL( n 1 , ..., n k ; n) and 1 ≤ i < j ≤ k, we define c ij (x) as the number of changes of directions in the lattice paths φ ij (x).

The second number c ′ i,j (x) is defined as follows. Since x ∈ RL( n 1 , ..., n k ; n), its integer-string representation is in particular a sequence of numbers (underlined or not) between 1 and k. For 1 ≤ i ≤ k, we set i -(resp. i -) the first occurrence of i (resp. i) in the integer-string representation. Equivalently, i -(resp. i -) is the first edge of the lattice x which is in the i-th direction. We write n -< m -if the element n -precedes m -.

For 1 ≤ i < j ≤ k, we set:

(4.2) c ′ i,j (x) =          c i,j (x) if i -< j -; c i,j (x) + 1 if i -> j -; c i,j (x) + 1 if i -< j -; c i,j (x) if i -> j -.
For m ≥ 1, we define RL m ( n 1 , ..., n k ; n) as the set of elements x ∈ RL m ( n 1 , ..., n k ; n) satisfying the three conditions:

max (i,j) c i,j (x) ≤ m; max (i,j) c i,j (x) ≤ m -1; and, max (i,j) or (i,j) c ′ i,j (x) ≤ m.
Remark 4.5. Changing the filtration defined in (4.2) by:

(4.3) c ′′ i,j (x) =          c i,j (x) + 1 if i -< j -; c i,j (x) if i -> j -; c i,j (x) if i -< j -; c i,j (x) + 1 if i -> j -.
we get another filtration of RL by sub-operads RL ′ m . The operad RL ′ 2 seems to be more adapted for proving the Swiss Cheese version of Deligne's conjecture.

The operad Coend RLm (δ) as a Swiss Cheese operad

We apply the method developed in [BB09, Sections 3.5-3.6]. More precisely, given a functor δ : △ → C where C is a monoidal model category, we construct a zig-zag of weak equivalences of operads

(4.4) Coend RLm (δ) Coend RLm (δ) B δ RK m , ∼ ∼
whenever δ satisfies some conditions. Here, B δ A denote the δ-realization of the nerve of the category A.

The intermediate operad RL m is defined using homotopy colimits in C applied on a decomposition of RL m indexed by RK m . Such a decomposition is a consequence of the following lemma.

Lemma 4.6. There is a morphism q : RL → RK of filtered operads.

Proof. Let us recall that if x ∈ RL m ( n 1 , ..., n k : n) then:

c i,j (x) ≤ m if n i = n i , n j = n j ; c i,j (x) ≤ m -1 if n i = n i , n j = n j ; c i,j (x) ≤ m if i -< j -; c i,j (x) ≤ m -1 if i -> j -; c i,j (x) ≤ m -1 if i -< j -; c i,j (x) ≤ m if i -> j -, for 1 ≤ i < j ≤ k.
The element q(x) = (µ, σ) ∈ RK is defined, for 1 ≤ i < j ≤ n, by:

µ i,j = c i,j (x) c i → c j for i -> j - c i ← c j for i -< j -.
The fact that q preserves the filtration is clear from its definition. Let us sketch the proof that q is a morphism of operad by using similar arguments to [BB09, Proposition 3.4]. It is straightforward to check that the image of q is contained in a sub-operad RK - m of RK m . This operad RK - m consists of colouring and orientation on complete graphs that are acyclic, that is, no (polychromatic) cyclic orientations are allowed. Such acyclic orientations on a complete graph on n elements correspond to the choice of a permutation in Σ n . Then RK - m (c 1 , ..., c n ; c) is the set of pair (µ, σ) of {1, ..., m} ( n 2 ) × Σ n submitted to the same conditions than in (3.1) (i.e. c i → c j corresponds to σ(i) < σ(j) and c i ← c j corresponds to σ(i) > σ(j)). An explicit formula for operadic composition is given in [BB09, Proposition 3.2] for the non Swiss Cheese case; it is the same formula in our context. In particular, permutations are composed as in the (Swiss Cheese version of the) Symmetric operad RΣ, and we have an operadic map RK - m → RΣ. Then, it is sufficient to show that the composite RL m → RK - m → RΣ is a morphism of operads. This follows almost directly from the definition of q. Since q "reverses" the orientations we have to check that the morphisms Rev n : Σ n → Σ n , that sends σ to Rev n • (σ) where Rev n = 1 2 ... n n ... 2 1 , induces a morphism of operads Rev : RΣ → RΣ. The Swiss Cheese symmetric operad RΣ consists of sets RΣ(c 1 , ..., c n ; c) to be either Σ n or empty according to the usual "Swiss-Cheese" condition on the colours; the operadic compositions are defined as in the classical non colour case.

Remark 4.7. Our morphism "reverses" the orientations. This is due to the choice of the cellulation of SC m we have made. However, if RL ′ m is the operad as in Remark 4.5, the similar morphism q ′ : RL ′ m → RK m does not "reverse" the orientations.

Let us recall that for δ = (δ c , δ o ) with δ c , δ o : △ → C, the functor

ξ(RL m ) c 1 ,...,c k ;c (δ) : △ → C denotes the realization of RL m (c 1 , ..., c k ; c) ξ(RL m ) c 1 ,...,c k ;c (δ)(n) = RL m (-, . . . , - k ; n) ⊗ △ k δ c 1 (-) ⊗ • • • ⊗ δ c k (-),
where we use implicitly the strong monoidal functor

Set → C E → e∈E 1 C .
We use the same functor δ c = δ o and we denote it by δ. We fix two functors

δ Top : △ Set △ op Top δ yon | -| and δ Z : △ Set △ op Ch(Z) δ yon C * (-; Z)
where:

• δ yon ([n]) = Hom △ (-, [n]
) is the Yoneda functor;

• | -| : Set △ op → Top is the geometric realization; and,

• C * (-; Z) : Set △ op → Ch(Z) is the normalized chain complex.
For α ∈ RK m (c 1 , ..., c k ; c) and

n i ∈ Col c i , n ∈ Col c , we set (RL m ) α ( n 1 , . . . , n k ; n) = {x ∈ RL m ( n 1 , . . . , n k ; n) | q(x) ≤ α}.
Then we have:

RL m ( n 1 , . . . , n k ; n) = colim RKm(c 1 ,...,c k ;c) (RL m ) α ( n 1 , . . . , n k ; n),
for all (c 1 , ..., c k ; c) and

n i ∈ Col c i , n ∈ Col c . Thus ξ(RL m ) c 1 ,...,c k ;c (δ) = colim RKm(c 1 ,...,c k ;c) ξ(RL m ) α (δ).
We suppose C endowed with a monoidal model structure, cf. [START_REF] Hovey | Model categories[END_REF]. The categories Set △ op and Top are considered with the Quillen model structure; the category Ch(Z) is considered with its projective model structure.

In [BB09, Sections 3.5-3.6] it is proved that, given a standard system of simplices δ, the operad Coend RLm (δ) is weakly equivalent to B δ RK, provided that operad RL m is strongly δ-reductive.

A standard system of simplices δ : △ → C (cf. [BM06, Definition A.6]) provides, in particular, a "monoidal symmetric" cosimplicial frame (-) ⊗ △ δ so that homotopy colimits are compatible with the symmetric monoidal structure of C. Moreover, for such a δ, the realization functor (-) ⊗ △ δ preserves and reflects weak equivalences.

It is proved in [BM06, A.13, A.16] that the functors δ yon , δ Top and δ Z are such standard system of simplicies.

The strong δ-reductivity allows us to show that the zig-zag of weak equivalence (4.4) follows essentially from the weak equivalences ξ(RL m ) c 1 ,...,c k ;c (δ) n → ξ(RL m ) c 1 ,...,c k ;c (δ) 0 for all n, k ≥ 0.

We define the strong δ-reductivity condition for Coend RLm (δ) similarly to [BB09, Definition 3.7] by extending it to the functors

ξ(RL m ) c 1 ,...,c k ;c (δ) n → ξ(RL m ) c 1 ,...,c k ;c (δ) 0 , for all n ≥ 0, k ≥ 0, see next definitions.
Definition 4.8. A weak equivalence in C is called universal if any pullback of it is again a weak equivalence.

Definition 4.9. Let δ be a standard system of simplices in C. The operad RL m is called δ-reductive if for any n ≥ 0 and k ≥ 0 and any colours c i , c ∈ {c; o} satisfying (2.1), the map

ξ(RL m ) c 1 ,...,c k ;c (δ) n → ξ(RL m ) c 1 ,...,c k ;c (δ) 0 is a universal weak equivalence.
The operad RL m is called strongly δ-reductive if in addition the induced maps Coend RLm (δ

)(c 1 , ..., c k ; c) → ξ(RL m ) c 1 ,...,c k ;c (δ) 0 are universal weak equivalence in C.
The proof of [BB09, Theorem 3.8] can be applied mutatis mutandis to the functors ξ(RL m ) c 1 ,...,c k ;c (δ). So, we have almost for free the analogue to [BB09, Theorem 3.8]: Theorem 4.10. Let δ be a standard system of simplices in a model monoidal category C with a zero object. If the operad RL m is strongly δ-reductive, then the operad Coend RL m (δ) is weakly equivalent to B δ RK.

Proof. We outline the proof.

We construct a zig-zag of weak equivalences of operads

(4.5) Coend RLm (δ) Coend RLm (δ) B δ RK m . ∼ ∼
The intermediate operad RL m is defined as

RL m ( n 1 , . . . , n k ; n) = hocolim RKm(c 1 ,...,c k ;c) (RL m ) α ( n 1 , . . . , n k ; n),
for all (c 1 , ..., c k ; c) and n i ∈ Col c i , n ∈ Col c . The properties of the standard system of simplicies δ imply that Coend RLm (δ) is an operad.

We denote by ξ(RL m ) the corresponding SC functor-operad of RL m , so that

ξ(RL m ) c 1 ,...,c k ;c (δ) = hocolim RKm(c 1 ,...,c k ;c) ξ(RL m ) α (δ).
The left hand map in the zig-zag (4.5) is induced by the maps ξ(RL m ) c 1 ,...,c k ;c (δ) → ξ(RL m ) c 1 ,...,c k ;c (δ).

Because of the strong δ-reductivity condition, it is sufficient to show that

ξ(RL m ) c 1 ,...,c k ;c (δ) 0 → ξ(RL m ) c 1 ,...,c k ;c (δ) 0 ,
is a weak equivalence. This follows from general properties of standard system of simplicies.

B δ RK m (c 1 , ..., c k ; c) ∼ = hocolim RKm(c 1 ,...,c k ;c) (ccδ 0 ) ∼ = Hom(δ, ξ(RL m ) c 1 ,...,c k ;c (ccδ 0 ))
where ccδ 0 is the constant cosimplicial object at δ 0 = 1 C . Then it is sufficient to prove that

Hom C △ (δ, ξ(RL m ) c 1 ,...,c k ;c (ccδ 0 )) → Hom C △ (δ, ξ(RL m ) c 1 ,...,c k ;c (ccδ 0 ))
is a weak equivalence. Since Hom C △ (-, -) preserves weak equivalences, this is satisfied if

ξ(RL m ) c 1 ,...,c k ;c (δ) n → ξ(RL m ) c 1 ,...,c k ;c (ccδ 0 ) n
is a weak equivalence for all n ≥ 0. Since we have

ξ(RL m ) c 1 ,...,c k ;c (δ) n ∼ = hocolim RKm(c 1 ,...,c k ;c) ξ(RL m ) α (δ), it remains to show that ξ(RL m ) α (δ) n → ξ(RL m ) α (ccδ 0 ) n = 1 C
is a weak equivalence for all α and n ≥ 0. By δ-reductivity the left vertical arrow in

ξ(RL m ) α (δ) n ξ(RL m ) c 1 ,...,c k ;c (δ) n ξ(RL m ) α (δ) 0 ξ(RL m ) c 1 ,...,c k ;c (δ) 0
is a weak equivalence. Moreover, for each (c 1 , ..., c k ; c) and α ∈ RK m (c 1 , ..., c k ; c), the object ξ(RL m ) α (δ) 0 is weakly contractible. The latter is due to the following fact. Properties of the standard system of simplices imply that: the realization functor (-) ⊗ △ δ preserves and reflect weak equivalences; the two objects ξ(RL m ) α (δ) 0 and ξ(RL m ) α (δ yon ) 0 ⊗ △ δ are weakly equivalent; so that, it is sufficient to prove that for δ = δ Top , the space ξ(RL m ) α (δ Top ) 0 is weakly contractible.

In Theorem 4.10 we have used our version of [MS04, Lemma 14.8] (see also [BB09, Lemma 3.9]):

Lemma 4.11. For each m ≥ 1, (c 1 , ..., c k ; c) and α ∈ RK m (c 1 , ..., c k ; c), the space ξ(RL m ) α (δ Top ) 0 is weakly contractible.

Proof. The proof is quite similar to [MS04, Lemma 14.8]; it is sufficient to check that the arguments are compatible with our operads RK and RL. We outline the proof for convenience.

For each α ∈ RK(c 1 , ..., c k ; c), we construct a retraction ι : ξ(RL m ) α (Cone(δ Top ), ..., Cone(δ Top )) 0 ⇆ ξ(RL m ) α (δ Top , ..., δ Top ) 0 : ρ such that ρ • ι = id. The functor Cone(δ Top ) : △ → Top is defined as follows.

Cone(δ

Top ) n = δ n+1 Top Cone(δ Top )(f : [l] → [n]) = δ Top ( f : [l + 1] → [n + 1]),
where f :

[l + 1] → [n + 1] is defined as f(0) = 0 and f(k) = f(k -1) + 1 for 1 ≤ k ≤ l + 1.
The map ρ sends the class of (x, u 1 , ..., u k ) to the class of (x, d 0 u 1 , ..., d 0 u k ) where d 0 :

δ n Top → δ n+1
Top is the zeroth cosimplicial face operator. For x ∈ RL α ( n 1 , ..., n k ; n), with ( n 1 , ..., n k ; n) according to (c 1 , ..., c k ; c), the map ι assigns to the class of (x, u 1 , ..., u k ) the class of ( x, u 1 , ..., u k ) where x is the lattice obtained from x by doubling the first occurrence of i (or i) for 1 ≤ i ≤ k in its integer-string representation. Thus the lattice x belongs to RL α ( n 1 , ..., n k ; n).

Since which is also universal.

B) The chain complex case δ = δ Z . We will show that the maps

(θ Z ) * c 1 ,...,c k ;c : ξ(RL m ) c 1 ,...,c k ;c (δ Z ) * → ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 (4.9)
are trivial Reedy fibrations in Ch(Z) △ . Indeed, the trivial Reedy fibrations are objectwise trivial fibrations what implies the δ Z -reductivity. Moreover, since δ Z is a standard system of simplicies, the δ Z -totalization functor Hom Ch(Z) (δ Z , -) is a right Quillen functor. Then, the induced maps

(4.10) Coend RL m (δ Z )(c 1 , ..., c k ; c) → ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0
are trivial fibrations in Ch(Z).

Recall that the realization functors | -| δTop and | -| δ Z preserve and reflect the weak equivalences. Then, since (θ Top ) n c 1 ,...,c k ;c is a weak equivalence for each n ≥ 0, we deduce that for n ≥ 0, the maps (θ Z ) n c 1 ,...,c k ;c : ξ(RL m ) c 1 ,...,c k ;c (δ Z ) n → ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 (4.11) so are. It remains to show that the maps (θ Z ) * c 1 ,...,c k ;c are Reedy fibrations in Ch(Z) △ . Let us denote by M n X the matching object of X. Since M n ccξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 = ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 , it is sufficient to show that the maps

j n c 1 ,...,c k ;c : ξ(RL m ) c 1 ,...,c k ;c (δ Z ) n → M n ξ(RL m ) c 1 ,...,c k ;c (δ Z ) * (4.12)
induced by the (θ Z ) n c 1 ,...,c k ;c are surjective for each n ≥ 1. This can be shown by a careful investigation on M n ξ(RL m ) c 1 ,...,c k ;c (δ Z ) * .

The relative surjection operad

For m ≥ 1, we define a {c; o}-coloured operad RS m , sub-operad of Coend RL m (δ Z ). We show that the inclusion RS m ֒→ Coend RL m (δ Z ) is a weak equivalence.

As complexes, we set RS m (c 1 , ..., c k ; c) := ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 . For (c 1 , ..., c k ) where c i ∈ {c, o}, we set

c i = i if c i = c; i if c i = o.
A surjection f : {1, . . . , k+r} → { c 1 , . . . , c k } is called degenerate if there exists an i such that f(i) = f(i+1).

Then, an element in RS m (c 1 , ..., c k ; c) is a sum of non degenerate surjections f : {1, . . . , k + r} → { c 1 , . . . , c k }. Indeed, a generator of the complex

ξ(RL m ) c 1 ,...,c k ;c (δ Z ) 0 = RL m (-, ..., -; 0) ⊗ △ k δ Z ⊗ . . . ⊗ δ Z
is represented by an element in RL m ( n 1 , ..., n k+j ; 0) without repetitions (that is without doubles ii or ii in the integer-string representation). The degree of f : {1, . . . , k + r} → { c 1 , . . . , c k } is the total degree r = n 1 + ... + n k of the corresponding generator in

RL m ( n 1 , ..., n k ; 0) ⊗ (δ n 1 Z ) n 1 ⊗ . . . ⊗ (δ n k Z ) n k .
Following [START_REF] Batanin | The lattice path operad and Hochschild cochains[END_REF] we define maps

ϑ n : ξ c 1 ,...,c k ;c (RL m )(δ Z ) 0 ⊗ δ n Z → ξ c 1 ,...,c k ;c (RL m )(δ Z ) n , n ≥ 0,
f • RS c i g = f • i ϑ n(i) (g ⊗ e n(i) ),
where n(i) denotes the number of occurrences of c i in the surjection f (i.e. n(i) is the cardinal of f -1 ( c i )) and • i denotes the partial composition of Coend RLm (δ Z ). We extend the composition by linearity. Such partial compositions give an operadic structure on RS m .

Example 5.1.

(121

) o • RS 1 (12) = (1312) o + (1232) o .
Proposition 5.2. The inclusion RS m ֒→ Coend RL m (δ Z ) is a weak equivalence of operads.

Proof. The maps {ϑ n } n induce, by adjunction, a map ϑ ′ : RS m ֒→ Coend RLm (δ Z ). Except for signs, the fact that ϑ ′ is compatible with the operadic structures is straightforward from the definition. Such a compatibility implies (and then defines) the signs involved in the definition of ϑ n . In particular, signs can be determined by an induction process on the dimension of the generators of RS m . Moreover, let us denote by π :

Coend RL m (δ Z ) → RS m
the weak equivalence coming from (4.10); it satisfies πϑ ′ = id. Thus, ϑ ′ is a weak equivalence.

Proposition 5.3. As an operad, RS 2 is generated by the following elements

µ c = (12) T k = (1213 • • • 1k1), k ≥ 2 µ o = (12) o T j = (1213 • • • 1j1) o , j ≥ 2 inc = (1) o ,
and the two unit elements id c = (1) and id o = (1) o .

Proof. We suppose by induction on N that any homogeneous element of RS 2 with N ≥ 1 different integers in its integer-string representation is obtained by operadic compositions of elements cited in the statement. The cases N = 1 and N = 2 are trivially verified.

In what follows we abusively do not mark the distinction between integers with under bar and integers without under bar. Let x be a homogeneous element of RS 2 with N + 1 different integers. Because of the filtration condition (4.2), x can be written as a sequence (A 1 • • • A n ) where the A i 's are non empty sequences of integers such that, if j belongs to A i , then j / ∈ A s for s = i. Moreover, because of the symmetric group action, one can suppose that the integers of A i are smaller than the integers of A j whenever i < j. In this case, if n > 1, then x = ((12)

• 1 (A 1 • • • A n-1 )) • max A n-1 +1 ( A n )
where A n is obtained from A n by decreasing each number by max A n-1 . Since the A i 's are not empty, (A 1 • • • A n-1 ) as well as A n have at most N different integers and then satisfy the induction hypotheses. If n = 1 then either x is T k (or T k ) for some k or, x is such that A 1 = jB 1 jB 2 j • • • jB p j with 1 ≤ p < N and for some integer j. Thus there exists at least one B i 0 that contains 2 ≤ q ≤ N -(p -1) and x = (jB 6 The operad RL 2

1 j • • • B i 0 -1 jajB i 0 +1 j • • • jB p j) • a B i 0 for some a, what concludes the proof. • 1 2 µ c id c 1 id o o 1 1 2 3 j -1 j . . . o T k 1 2 3 k -1 k . . .

The operad RL 2 in term of trees

We describe an SC-split operad RL 2 in the category of sets, Set. Given any cocomplete, closed monoidal symmetric category C, one has the strong monoidal functor

Set → C E → e∈E 1 C
where 1 C denotes the unit for the monoidal structure ⊗ of C. This way we can consider algebras over the operad RL 2 that are (family of) objects in Top or Ch(Z).

The operad RL 2 can be thought of as a mix between the Lattice paths operads L 2 and L 1 introduced in [START_REF] Batanin | The lattice path operad and Hochschild cochains[END_REF]. The closed part of RL 2 is the sub-operad L 2 of multiplicative operads. The open part of RL 2 is the sub-operad L 1 of cosimplicial -monoids (cf. [MS04, Definition 2.1]). In Section 6.2 we define the notion of wide and stable wide bimodules over an operad so that a cosimplicial -monoid is a wide bimodule over the non-symmetric operad of associative algebras As.

A multiplicative operad M is a (non-symmetric) operad endowed with an inclusion As ֒→ M of non-symmetric operads.

The operad RL 2 encodes the couples (M, Z) subject to the following conditions. I. M is a (non-symmetric) multiplicative operad.

6.1 The operad RL 2 in term of trees II. Z is a stable wide left-module over M; in particular, using As ֒→ M, it is a wide left-module over As; this is a part of the following structure III.

III. Z is a wide bimodule over As.

A stable wide bimodule is, in particular, an infinitesimal bimodule (called weak bimodule in [Tur10, Definition 4.1]).

Lemma 6.1. [Tur10, Lemma 4.2] The structure of a cosimplicial vector space is equivalent to the structure of an infinitesimal bimodule over As.

Our operad RL 2 is closed to the operad ∅ constructed in [DTT11, section 3.1]. The operad ∅ was constructed in order to obtain an action of a Swiss Cheese operad on the pair (CC * (A, A), A) formed by an associative algebra A and its Hochschild cochain complex CC * (A, A). We enlarge the (non-closed part of the) operad ∅ in order to obtain an action of a Swiss Cheese operad on the couple (Tot δ Z M, Tot δ Z Z) where (M, Z) is a couple as above and Tot δ Z (-) denotes the δ Z -totalization. In particular, our open part takes into account the cosimplicial structure of Z.

Recall from Section 4.1 that the set of colours of RL 2 is

Col = Col c ⊔ Col o ,
where Col c is the set of natural numbers and Col o is the set of natural numbers decorated with an underline.

Definition 6.2. Let T be a planar rooted tree. Let ν be a vertex of T . We denote by T ν the maximal sub-tree of T such that ν is the root vertex of T ν .

For n i , n ∈ Col c , the set RT (n 1 , ..., n k ; n) is the set of equivalence classes of planar rooted trees T satisfying:

• a subset of the set of vertices of the tree T is indexed by the set {1, ..., k} ⊔ {1, ..., n} in such a way that:

the vertices indexed by {1, ..., n} are only terminal vertices,

• the ordered set of edges originating at the vertex indexed by s ∈ {1, ..., k} is identified with

[n i -1]. (We set [-1] = ∅).
The subset of vertices identified with {1, ..., n} benefits of an order given by {1 < ... < n}. We require that this order coincides with the order which is given by turning around the tree in the clockwise direction starting from the root vertex. The equivalence class of equivalence is the same as in [DTT11, 3.2.1]. Explicitly, it is the finest one in which two trees are equivalent if one of them can be obtained from the other by either: -the contraction of an edge with unmarked ends; or, -removing an unmarked vertex with only one edge originating from it and joining the two edges adjacent to this vertex into one edge.

1 2 3 • Figure 6.1: Element in RL 2 (3, 2, 2; 6).
Let ( n 1 , ... n k+j ) be a tuple of colour in Col. We set {s 1 , ..., s k+j } the set such that

s i = i if n i = n i ; i if n i = n i .
The set RT ( n 1 , ... n k+j ; n) is the set of equivalence classes of planar rooted trees T satisfying:

• a subset of the set of vertices of a tree T is indexed by the set {s 1 , ..., s k+j } ⊔ {1, ..., n} in such a way that:

given a vertex ν of T indexed by i ∈ {s 1 , ..., s k+j } then, in the tree T ν there is no vertex different of ν indexed by an element s j ∈ {s 1 , ..., s k+j }, -the vertices indexed by {1, ..., n} are only terminal vertices,

• the ordered set of edges originating at the vertex indexed by s i ∈ {s 1 , ..., s k+j } is identified with

[ n i -1];
• the root is decorated by an o.

The subset of vertices identified with {1, ..., n} benefits of an order given by {1 < ... < n}. We require that this order coincides with the order which is given by turning around the tree in the clockwise direction starting from the root vertex. The equivalence class is the same as the previous one.

• • • • 1 3 2 4 • o Figure 6.2: Element in RT (2, 0, 3, 3; 5). • 1 2 • o Figure 6.3: Element in RT (2, 2; 6).
The composition maps in RT are defined by substitution of trees into marked vertices.

• Proof. Let us take a labelled tree T ∈ RT . One runs through the tree T in clockwise direction starting from the root in such a way that one passes exactly two times on each edges (once per sense). One assigns the number an integer-string by writing down the corresponding label each time one meets a labelled vertex and by writing down a vertical bar each time one meets a terminal vertex. One add an extra label o if the root is decorated by o.

1 2 3 o 1 • 1 2 = • • 1 3 4 2 o = • 1
The inverse construction is given as follows. To an integer-string representation one assigns a tree with: one labelled vertex for each different integer and one terminal vertex for each vertical bar. The 6.2 The algebras over RL 2 labelled vertices have one output less than there are occurrences for the corresponding integer. The corresponding tree is constructed such that its order fits with the reading (from the left to the right) of the integer-string. Note that when two equal integers (or two vertical bars) are adjacent in the integer-string this forces the creation of an unlabelled vertex.

One checks that this two assignments provide an isomorphism of operads.

As an example, the tree from Figure 6.2 corresponds to the integer-string (1|113||3234|4|443) o .

The algebras over RL 2

Let us fix a cocomplete, closed monoidal symmetric category (C, ⊗, 1 C , τ C ).

Definition 6.4. Let M be a (1-coloured) non symmetric operad in C. A family Z = {Z(A)} A∈finite sets of objects in C is a wide left module over M if, for any three finite ordered sets A, A ′ and B such that A ′ ⊂ A and α : B → A \ A ′ a map of ordered sets, there is a map

λ ′ α : M(A) ⊗ a∈A\A ′ Z(α -1 (a)) → Z(B ⊔ A ′ ),
such that all the diagrams of the following form commute

M(A) ⊗ M(B) ⊗ e∈(A∪aB)\(A ′ ⊔B ′ ) Z((α ⊔ β) -1 (e)) M(A) ⊗ e∈A\A ′ Z( α -1 (e)) M(A ∪ a B) ⊗ e∈(A∪aB)\(A ′ ⊔B ′ ) Z((α ⊔ β) -1 (e)) Z(C 1 ⊔ A ′ ⊔ C 2 ⊔ B ′ ), id ⊗ λ ′ β (-• a -) ⊗ id λ α λ ′ α⊔β for A ′ ⊂ A \ {a}, B ′ ⊂ B, α : C 1 → (A \ {a}) \ A ′ , β : C 2 → B \ B ′ , α ⊔ β : C 1 ⊔ C 2 → (A ∪ a B) \ (A ′ ⊔ B ′ )
and where α :

C 1 ⊔ (C 2 ⊔ B ′ ) → A \ A ′ is defined as α(c) = α(c) if c ∈ C 1 ; a if c ∈ C 2 ⊔ B ′ .
The maps λ ′ α are required to be natural in isomorphisms of ordered sets A, A ′ , B and α. Moreover, for all finite set B and α : B → {1}, the map

λ ′ α : Z(B) = 1 M(1) ⊗ Z(B) → Z(B),
is required to be the identity. Proposition 6.5. A wide left module Z over M with actions maps

λ ′ α : M(A) ⊗ a∈A\A ′ Z(α -1 (a)) → Z(B ⊔ A ′ ), for A ′ ⊂ A and α : B → A \ A ′ is, in particular:
1. an infinitesimal left module over M (see [START_REF] Victor Turchin | Hodge-type decomposition in the homology of long knots[END_REF]Definition 4.1]) for the action maps λ α := λ ′ α defined for A ′ = ∅; and, 2. a left module over M for the action maps λ a := λ ′ α defined for A \ A ′ = {a}.

Proof. The diagram from Definition 6.4 gives the required properties for λ α and λ a .

Definition 6.6. A wide left module Z over an operad M is called stable if there are maps in C

ι A : M(A) → Z(A)
for each finite set A, such that all the diagrams of the following form commute

M(A) ⊗ a∈A\A ′ M(α -1 (a)) M(A) ⊗ a∈A\A ′ Z(α -1 (a)) M(B ⊔ A ′ ) Z(B ⊔ A ′ ), id ⊗ b∈B ι α -1 (b) -• α - λ ′ α ι B⊔A ′ where α : B → A \ A ′ .
Definition 6.7. A (resp. stable) wide bimodule over an operad M is a (resp. stable) wide left-module over M and a right-module over M such that the underlying infinitesimal left-module structure together with the right-module structure form an infinitesimal bimodule structure over M.

Definition 6.8. The non-symmetric operad As in C is given as

As(n) = 1 C for n ≥ 0.
Let E be the category with objects the couples (M, Z) ∈ C ⊗ C satisfying the three conditions I, II and III of section 6; and, with morphisms the pair (f, g) : (M, Z) → (M ′ , Z ′ ) where f : M → M ′ is a morphism of multiplicative operads and g : Z → Z ′ is a morphism of stable wide left modules over M and wide bimodules over As. Lemma 6.9. The category of cosimplicial -monoids in C is isomorphic to the category of wide bimodules over As in C. Proposition 6.10. [BB09, Proposition 2.14] The category of L 1 -algebras (resp. of L 2 -algebras) in C is isomorphic to the category of cosimplicial -monoids (resp. of multiplicative operads) in C. Proposition 6.11. Let C be a cocomplete, closed monoidal symmetric category with a zero object. The category of RL 2 -algebras in C is isomorphic to the category E.

Proof. We use the interpretation of RL 2 in terms of planar trees, see Proposition 6.3. Given a tree in RL 2 ( n 1 , ... n k ; n) we recall that {s 1 , ..., s k }, where

s i = i if n i = n i ; i if n i = n i ,
denotes the set that labels "open" and "closed" vertices. The action of RL 2 on an object (M, Z) of E is given as follows. One decorates the closed marked vertex indexed by i with an element x i ∈ M([n i -1]) and one decorates the open marked vertex indexed by j with an element y j ∈ Z([n j -1]); the resulting element in Z({1, ..., n}) is obtained by composing the decorating elements along the tree, using the module structures of M and Z.

Conversely, the vertices of type T1, T2 and T3 give the maps λ ′ α and the vertices of type T4 give the map ι : M → Z. In particular, the vertices of type T2 give the maps λ α for the left module structure; the vertices of type T3 give the maps λ a for the left infinitesimal module structure. The elements of type T5 and other combinations give the diagram of Definition 6.4 using the operadic structure of RL 2 . The diagram of Definition 6.6 is obtained similarly.

Cosimplicial relative loop space

In this section we define a cosimplicial model ω(X, Y) for the relative loop spaces Ω(X, Y), that is, a cosimplicial space such that its δ Top -totalization Tot δ Top ω(X, Y) is homeomorphic to Ω(X, Y). For (M, N) a pair of monoids pointed at the unit * and such that N is a sub-monoid of M, we show 7.1 Cosimplicial relative loop space of monoids that ω(M, N) is endowed with an additional structure. More precisely, we show that there exists an RL 2 -algebra (ω(M), ω(M, N)) associated to (ω(M), ω(M, N)) such that, for δ being δ Top or δ Z , the totalization (Tot δ ω(M), Tot δ ω(M, N)) is an algebra over Coend RL 2 (δ).

Let us start by a definition.

Definition 7.1. Let (X, Y) be two topological spaces pointed at * and such that * ⊂ Y ⊂ X. The relative loop space of (X, Y), Ω(X, Y), is the space of continuous maps γ : [0, 1] → X satisfying γ(0) = * and γ(1) ∈ Y.

Definition 7.2. Let (X, Y) be a pair of topological spaces pointed at * such that * ⊂ Y ⊂ X. The cosimplicial relative loop space ω(X, Y) is the cosimplicial space such that ω(X, Y) 0 = Y, and ω(X, Y) k = X ×k × Y for k ≥ 1, with

d 0 (x 1 , • • • , x k , y) = ( * , x 1 , • • • , x k , y) d i (x 1 , • • • , x k , y) = (x 1 , • • • , x i , x i , • • • , x k , y), 1 ≤ i ≤ k d k+1 (x 1 , • • • , x k , y) = (x 1 , • • • , x k , y, y) s i (x 1 , • • • , x k , y) = (x 1 , • • • , x i , x i+2 , • • • , x k , y), 0 ≤ i ≤ k.
Remark 7.3. For Y = * , the cosimplicial space ω(X, Y) is the cosimplicial space ωX × { * } ∼ = ωX that is a model for the loop space ΩX described in [START_REF] Salvatore | The topological cyclic Deligne conjecture[END_REF].

Proposition 7.4. The maps

Ω(X, Y) × ∆ k → X k × Y (γ, (t 1 , • • • , t k ) → (γ(t 1 ), • • • , γ(t k ), γ(1)), k ∈ N,
induce, by adjunction, a homeomorphism Ω(X, Y) ∼ = Tot(ω(X, Y)).

Cosimplicial relative loop space of monoids

Given a topological monoid M and a sub-monoid N, the totalization of the cosimplicial relative loop space ω(M, N) is homeomorphic to Ω(M, N) ≃ Ω(Ω B M, Ω B N) ∼ = Ω 2 (B M, B N) where B G denotes the classifying space of the monoid G. From [START_REF] Salvatore | The topological cyclic Deligne conjecture[END_REF] we know that ω(M) can be seen as the cosimplicial space coming from a multiplicative operad ω(M). This property implies that the totalization of ω(M) is an E 2 -algebra.

In the same spirit we show that ω(M, N) comes from ω(M, N), which is both a wide bimodule over As and a stable wide left module over the multiplicative operad ω(M). Then, one obtains that the couple of totalizations (Tot δ ω(M), Tot δ ω(M, N)) is a Swiss Cheese algebra for δ = δ Top or δ = δ Z .

The structural map of the operad ω(M) is denoted by

γ : ω(M)(k) × ω(M)(l 1 ) × • • • × ω(M)(l k ) → ω(M)(l 1 + • • • + l k ) (f, g 1 , . . . , g k ) → γ(f; g 1 , . . . , g k ).
Since ω(M) has a unit, γ is equivalent to infinitesimal maps

• i : ω(M)(k) × ω(M)(l) → ω(M)(l 1 + • • • + l k ) (f, g) → f • i g.
Explicitly, if f = (x 1 , . . . , x k ) and g = (y 1 , . . . , y l ), then (7.1) f • i g := (x 1 , . . . , x i-1 , x i y 1 , . . . , x i y l , x i+1 , . . . , x k ), see [START_REF] Salvatore | The topological cyclic Deligne conjecture[END_REF].

The family ω(M, N) = {ω(M, N)(l)} l≥0 has, as underlying spaces, the spaces ω(M, N)(l) := ω(M, N) l . A typical element in ω(M, N)(l) is denoted by a couple (g; n) so that g ∈ M ×l and n ∈ N.

Let us define a stable wide left action. For k ≥ 1, 1 ≤ s ≤ k, l i ≥ 0 and an injective order preserving map β :

{1 < • • • < s} → {1 < • • • < k}, we define (7.2) σ ′ : ω(M)(k) × ω(M, N)(l 1 ) × • • • × ω(M, N)(l s ) → ω(M, N)(l 1 + • • • + l s + k -s)
by σ ′ (f, (g 1 ; n 1 ), . . . , (g s ; n s )) = (γ(f;

1 β(1)-1 , g 1 , n 1 β(2)-β(1)-1 , g 2 ⊲ n 1 , n 2 n 1 β(3)-β(2)-1 , . . . . . . , g s ⊲ n s-1 • • • n 1 , n s • • • n 1 k-β(s)-1 ); n s • • • n 1 ),
where, for n, m ∈ N and j ≥ 0, n j := n, . . . , n j and nm j := nm, . . . , nm j and where g ⊲ n is the diagonal right action i.e. if g = (x 1 , ..., x l ) then g ⊲ n = (x 1 n, ..., x l n).

For s = 1 above, we get an infinitesimal left action

σ i : ω(M)(k) × ω(M, N)(l) → ω(M, N)(k + l -1) (7.3) (f, (g; n)) → (γ(f; 1, . . . , 1 i-1 , g, n, . . . , n); n), for 1 ≤ i ≤ k.
For s = k above, we get a left action

σ : ω(M)(k) × ω(M, N)(l 1 ) × • • • × ω(M, N)(l k ) → ω(M, N)(l 1 + • • • + l k ) (7.4) (f, (g 1 ; n 1 ), ..., (g k ; n k )) → (γ(f; g 1 , g 2 ⊲ n 1 , g 3 ⊲ n 2 n 1 , ..., g k ⊲ n k-1 • • • n 1 ); n k • • • n 1 ),
for all k ≥ 1 and l i ≥ 1.

For s = 0 above, we set the inclusion

ι : ω(M)(k) → ω(M, N)(k) (7.5) f → (f; 1), for all k ≥ 1.
The right action is given by

ρ : ω(M, N)(k) × ω(M)(l 1 ) × • • • × ω(M)(l k ) → ω(M, N)(l 1 + • • • + l k ) (7.6) ((f; n), g 1 , . . . , g k ) → (γ(f; g 1 , . . . , g k ); n),
where γ is the structural map of the operad ω(M).

Lemma 7.5. The maps σ ′ , ι and ρ endow ω(M, N) with a stable wide bimodule structure over ω(M). In particular, using the map As ֒→ ω(M), the module ω(M, N) is a wide bimodule over As. With regard to the infinitesimal bimodule structure over As, the corresponding (via Lemma 6.1) cosimplicial space is ω(M, N).

Proof. The only non obvious property is that of wide left module structure. One has to proves that the following equation σ ′ (f 1 ; 1, ..., (g 1 ; n 1 ), ..., (g s ; n s ), ..., 1 i-1 , σ ′ f 2 ; 1, ..., (g s+1 ; n s+1 ), ..., (g r ; n r ), ..., 1 , ..., (g t ; n t ), ..., 1) = σ ′ (f 1 • i f 2 ; 1, ..., (g 1 ; n 1 ), ..., (g s ; n s ), ..., (g s+1 ; n s+1 ), ..., (g r ; n r ), ..., (g t ; n t ), ..., 1)

holds for all f 1 ∈ ω(M)(k), f 2 ∈ ω(M)(l) and (g j ; n j ) ∈ ω(M, N) and 1 ≤ i ≤ k, 1 ≤ t ≤ k + l -1.
Using the very definition of σ ′ in terms of γ and ⊲, by the associativity of γ, proving the associativity for the left action σ ′ essentially amounts to prove that

(7.7) γ(f; g 1 , ..., g k ) ⊲ n = γ(f; g 1 ⊲ n, ..., g k ⊲ n),
for all f, g 1 , ..., g k ∈ ω(M) and n ∈ N. This results from the following. Let us write x ⊳ (y 1 , ..., y l ) := (xy 1 , ..., xy l ) for x, y i ∈ M. Let f = (x 1 , ...x k ). Then one has

γ(f; g 1 , ..., g k ) ⊲ n = (x 1 ⊳ g 1 , ..., x k ⊳ g k ) ⊲ n = ((x 1 ⊳ g 1 ) ⊲ n, ..., (x k ⊳ g k ) ⊲ n) = (x 1 ⊳ (g 1 ⊲ n), ..., x k ⊳ (g k ⊲ n)) = γ(f; g 1 ⊲ n, ..., g k ⊲ n). 7.2 Action of H * (SC 2 )
In virtue of the above Lemma 7.5, Proposition 6.11, Theorem 4.12 and Section 2. We write down a few operations on (Tot δ Z ω(M), Tot δ Z ω(M, N)) when (M, N) is a pair of monoids pointed at the unit * with N submonoid of M.

The described operations are the sufficient to ensure the existence of an H * (SC 2 )-algebra structure on (H * Tot δ Z C * (M), H * Tot δ Z (C * (M), C * (N))). They form a part of the RS 2 -algebra structure we make explicit in the next section.

Recall that an algebra over the homology operad H * (SC 2 ) is a triple (A, G, f) where A is an associative algebra; G is a Gerstenhaber algebra; and, f : G → A is an algebra morphism such that f(G) belongs to the center of A, for example see [HL13, Proposition 3.2.1].

For a cosimplial set K * , the totalization Tot

δ Z (K) is Π n≥0 Z[K n ]
with the differential usually given as the sum of the cosimplicial face maps. We denote by K(n) the n-th component Z[K n ]; the degree of an element f ∈ K(n) is n and it is denoted by |f|.

Let us recall a consequence of McClure and Smith's work.

Theorem 7.7 ([MS04]

). Let O be a multiplicative operad. Then Tot δ Z O is an E 2 -algebra.

In particular, Tot δ Z O is endowed with a product, ∪, commutative up to a chain homotopy E 1,1 . The product ∪ on Tot δ Z ω(M) is given by

f ∪ g := (µ • 2 g) • 1 f, (7.8)
for f, g ∈ ω(M). We define an associative dg-product ⊔ on Tot δ Z ω(M, N) by u ⊔ v := σ(µ; u, v), (7.9) for u, v ∈ ω(M, N) where σ is the map defined in (7.4).

In other words, for u = (f; m) and v = (g; n) one has

u ⊔ v = (f, g ⊲ m; nm).
The inclusion inc : Tot δ Z ωM ֒→ Tot δ Z ω(M, N) is given on its components by

inc : ω(M)(k) → ω(M, N)(k) (7.10) (a 1 , . . . , a k ) → (a 1 , . . . , a k ; 1)
and it is a chain map.

From this, we easily deduce that

inc(f ∪ g -(-1) |f||g| g ∪ f) ⊔ u = inc(∂E 1,1 (f; g)) ⊔ u = ∂(inc(E 1,1 (f; g)) ⊔ u),
for any homogeneous elements f, g ∈ Tot δ Z ω(M) and u ∈ Tot δ Z ω(M, N). Finally, we define a chain homotopy H such that inc

(f) ⊔ u -(-1) |f||u| u ⊔ inc(f) = ∂H(f, u)
for any two homogeneous elements f ∈ Tot δ Z ω(M) and u ∈ Tot δ Z ω(M, N). We recall the infinitesimal left action σ i defined in (7.3). The homotopy H is defined as

H(f, (g; n)) = 1≤i≤k (-1) i+i|g|+|f||g| σ i (f; (g; n)), (7.11) for any f ∈ Tot δ Z ω(M) of degree k and (g; n) ∈ Tot δ Z ω(M, N).
We have shown, in particular, the following.

Relative cobar construction

For C a 1-reduced (coaugmented, counital, coassociative) dg-coalgebra, we denote by ΩC its cobar construction as defined in [FHT92, p.538].

Definition 8.1. Let (C, N) be a pair of dg-module such that C is a 1-reduced dg-coalgebra (coaugmented, counital, coassociative) and N is a left C-comodule. The relative cobar construction of (C, N) is the free graded ΩC-module on N, ΩC ⊗ N, with the following differential D.

The differential is given by: D = d 0 + d 1 , where d 0 is the Koszul differential of the module ΩC ⊗ N induces by d C and d N ; and d 1 is the quadratic part induces by the quadratic part of the cobar construction ΩC and a twist with the reduced C-comodule structure of N. We write the reduced coproducts ∇

C (c) = c 1 ⊗ c 2 = ∇ C (c) -c ⊗ 1 -1 ⊗ c and ∇ N (n) = z 1 ⊗ n 2 = ∇ N (n) -1 ⊗ n. One has: d 1 (s -1 c) = (-1) c 1 s -1 c 1 ⊗ s -1 c 2 , c ∈ C >1 d 1 (n) = s -1 z 1 ⊗ n 2 , n ∈ N,
where s -1 denotes the desuspension i.e. (s

-1 C) i = C i+1 . The augmentation ǫ : ΩC ⊗ N → k is given by ǫ = ǫ ΩC • ǫ N . The relative cobar construction (ΩC ⊗ N, D, ǫ) is denoted by Ω(C, N).
The action of ΩC is given by concatenation

a Ω(C,N) : ΩC ⊗ Ω(C, N) → Ω(C, N) (s -1 x 1 ⊗ • • • ⊗ s -1 x k ) ⊗ (s -1 x k+1 ⊗ • • • ⊗ s -1 x r ⊗ n) → s -1 x 1 ⊗ • • • ⊗ s -1 x r ⊗ n,
and yields on Ω(C, N) a ΩC-module structure in the category of dg-modules.

Data 8.2. Let us fix a pair of module maps (f : C → A, g : N → M), where:

• C is a 1-connected coaugmented dg-coalgebra;

• A is an augmented dg-algebra;

• N is a left dg-comodule over C with coaction c N : N → C ⊗ N;

• M is a left dg-module over A with action a M : A ⊗ M → M.

We denote by f : ΩC → A the induced algebra morphism of f. Let (8.1) (f, g) : ΩC ⊗ N → M be the map f • g whose the k-th component is

C ⊗k ⊗ N A ⊗ M M. f ⊗ g a M As an immediate consequence, (f, g)a Ω(C,N) = a M (f ⊗ g), so that (f, g) is f-equivariant.
Now, we define a relative version of twisting cochain. We recall first that the set of twisting cochains Tw(C, A) is the set of maps f :

C → A satisfying f ∪ f = ∂f, where f 1 ∪ f 2 = µ A (f 1 ⊗ f 2 )∇ C and ∂ denotes the usual differential in Hom(C, A). Definition 8.3. Let (f : C → A, g : N → M) a pair as in Data 8.2. The pair (f, g) is called relative twisting if f is a twisting cochain and if (8.2) ∂g = a M (f ⊗ g)c N .
Proposition 8.4. Let (f, g) be a pair as in Data 8.2. Suppose f is a twisting cochain. Then the following propositions are equivalent.

• The pair (f, g) is a relative twisting pair.

• The f-equivariant map (f, g) : ΩC ⊗ N → M is a morphism of dg-modules.

it is immediate that Φ is a morphism of dg-coalgebras. Indeed, given a cubical set Q, its cubical chain complex C (Q) is a dg-coalgebra; the coproduct we consider is the Serre diagonal defined using the face operator of Q (see [KS05, 2.4 equation (3)]). The compatibility of φ with the monoidal structures implies that Adams' morphism is a quasi-isomorphism of dg-bialgebras. The construction of φ is realized by an induction process involving another map p : P Sing 1 X → Sing I PX between the cubical path construction and the cubical set of the path space PX. For a simplicial set K, the cubical set P K is defined as the twisted cartesian product ([KS05, Definition 4.2]) (8.5) For a pair (K, L) of 1-reduced simplicial sets such that L is a sub simplicial set of K, we define the relative cubical cobar construction Ω (K, L) as the twisted cartesian product (8.6) Ω (K, L) := Ω K × τ L, where τ : L → Ω K is the inclusion L ⊂ K composed with the universal truncating twisting function τ U : K → Ω K.

P (K) := Ω K × τ U K,
It is easy to show that the concatenation Ω K × Ω (K, L) → Ω (K, L) makes Ω (K, L) into a cubical Ω K-module.

Let (X, Y) be a pair of 1-connected pointed spaces such that Y ⊂ X. We define a map of cubical sets Consequently, we obtain Proposition 8.9. Let (X, Y) be a pair of 1-connected topological spaces pointed at * such that * ⊂ Y ⊂ X. The map ψ : Ω (Sing 1 X, Sing 1 Y) → Sing I Ω M (X, Y) induces a Φ-equivariant morphism Ψ :

C * Ω (Sing 1 X, Sing 1 Y) = Ω(C 1 * X, C 1 * Y) → C * Ω M (X, Y
) that is a quasi-isomorphism of dg-coalgebras.

Unreduced (relative) cobar construction

With regard to its structure, this section is analogous to Section 7.1: we exhibit a couple (M B , Z B,C ) giving rise to the both unreduced cobar and relative cobar constructions over a couple of a coalgebra/comodule (B, C) in the category of algebras; the couple (M B , Z B,C ) is shown to be an algebra over RL 2 .

Let (B, ∇ B ) be an (ungraded) unital/counital bialgebra with counit ǫ (i.e. B is a counital coalgebra in the category of unital algebras) and let (C, ∇ C ) be a left B-comodule in the category of unital algebras.

Let us consider the unreduced cobar construction Ω u B given by ); c k c

(2)

k-1 • • • c (k)
1 ).

For s = 0 above, we set an inclusion

ι B : M B (k) → Z B,C (k) (8.11) f → (f; 1).
The right action is given by Lemma 8.10. The maps λ ′ B , ι B and ρ B endow Z B,C with a stable wide bimodule structure over M B . In particular, using the map As ֒→ M B , the module Z B,C is a wide bimodule over As. With regard to the infinitesimal bimodule structure over As, the δ Z -totalization of the corresponding (via Lemma 6.1) cosimplicial complex of Z B,C is Ω u (B, C).

ρ B : Z B,C (k) ⊗ M B (l 1 ) ⊗ • • • ⊗ M B (l k ) → Z B,C (l 1 + • • • + l k ) (8.
Proof. One has to show that the equation (8.13) λ ′ B (f 1 ; 1, ..., (g 1 ; n 1 ), ..., (g s ; n s ), ..., 1

i-1

, λ ′ B f 2 ; 1, ..., (g s+1 ; n s+1 ), ..., (g r ; n r ), ..., 1 , ..., (g t ; n t ), ..., 1)

= λ ′ B (f 1 • B i f 2 ; 1, ..., (g 1 ; n 1 ), ..., (g s ; n s ), ..., (g s+1 ; n s+1 ), ..., (g r ; n r ), ..., (g t ; n t ), ..., 1) holds for all f 1 ∈ M B (k), f 2 ∈ M B (l) and (g j ; n j ) ∈ Z B,C and 1 ≤ i ≤ k, 1 ≤ t ≤ k + l -1. One easily show that a similar equation to (7.7) holds here. The compatibility of λ ′B with the operadic structure of M B follows from the facts that γ is associative (what is essentially due to the fact that B is a bialgebra) and that C is a B-comodule in the category of unital algebras. More explicitly, the last requirement is used to show that the "coefficients" (i.e. the terms in C) of the two terms from (8.13) coincide: , where w a denotes the distance from (g a ; n a ) to the last term: w a = k + lp a where p a is the position (from the left) of (g a ; n a ) and v a = i + lp a .

Theorem 8.11. Let B be a unital/counital bialgebra and let C be a B-comodule in the category of unital algebras. Then the couple (Ω u B, Ω u (B, C)) is an algebra over Coend RL 2 (δ Z ).

Action of RS 2

Let (M, N) be a pair of 1-connected topological monoids pointed at the unit 1 such that N is a sub-monoid of M. We explain how the following couple (ΩC 1 * M, Ω(C 1 * M, C 1 * N)) can be endowed with an action of RS 2 . The operation are similar to the ones described in the previous section. However, the absence of a unit in the reduced dg-bialgebra (C 1 * M) + constrains us to write down the operations "manually".

Let us fix an unital/counital dg-bialgebra (B, ∇ B ) with counit ǫ, such that B = B + ⊕ Z and with B 1 = 0; and (C, ∇ C ) be a left B-dg-comodule in the category of unital algebras. For example B = C 1 * M and C = C 1 * N. We recall that the cobar construction ΩB is, in particular, given as the free tensor algebra over s -1 B + . We adopt the same notations as in Section 8.2. We denote the reduced coproducts The closed part of RS 2 acts on it, see [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF] for an example with Z 2 as the field of coefficients. The multiplication is the concatenation, and, for k ≥ 1, where f = [a 1 , . . . , a n ] and g i ∈ ΩB.

E
In the same way, we can define a multiplication given by E ′ j (f; (g 1 ; c 1 ), ..., (g j ; c j )) = ±[a 1 , . . . , a i 1 ⊳ g 1 , a i 1 +1 z (1) c 1 , . . . . . . , a i j ⊳ g j ⊲ (z (1) c j-1 • • • z (j-1) c 1 ), . . . , a n (z (1) c j-1 • • • z (j-1)

c 1
)]c j c

(1)

j-1 • • • c (j) 1 ,
for any f = [a 1 , . . . , a n ] and (g i ; c i ) ∈ Ω(B, C).

We claim that, for appropriate signs, the operations (8.14), (8.15), (8.16) and the inclusion of ΩB into Ω(B, C) define an RS 2 -algebra structure on (ΩB, Ω(B, C)).

δ

  Top : △ Set △ op Top δ yon | -| and δ Z : △ Set △ op Ch(Z), δ yon C * (-; Z) 1 Introduction where δ yon ([n]) = Hom △ (-, [n]) is the Yoneda functor. In this manner, the condensation of RL m leads to a topological operad Coend RL m (δ Top ) and a chain operad Coend RL m (δ Z ).

  The sub-operad in H2 is called the closed part of O; the sub-operad in H3 is called the open part of O. The underlying category O u contains two particular categories: • O c u the sub-category of O u with objects the colours in Col c and morphisms the O u (n, m) for n, m ∈ Col c ; • O o u the sub-category of O u with objects the colours in Col o and morphisms the O u (n, m) for n, m ∈ Col o . By H2 and H3 both O c u and O o u are C-categories. The category

  Definition 3.4. Let C be a closed symmetric monoidal model category. A morphism of operads f : O → P is a weak equivalence if each f c 1 ,...,c k ;c : O(c 1 , ..., c k ; c) → P(c 1 , ..., c k ; c) is a weak equivalence in C. Two operads O and P in C are said weakly equivalent if there exists a zig-zag O ← ... → P of weak equivalences.Definition 3.5 ([

  Definition 3.7. A topological 2-coloured operad O with colours {c, o} is called an SC type operad if O(c 1 , ..., c n ; c) is empty whenever there is an i such that c i = o. For such an operad O, suppose we have given an RK(c 1 , ..., c n ; c)-cellulation of O(c 1 , ..., c n ; c) Θ c 1 ,...,cn;c : RK(c 1 , ..., c n ; c) → Top, for each c 1 , ..., c n ; c, n ≥ 0. This families of cellulations Θ c 1 ,...,cn;c is said compatible with the operad structure of O if

  Proof. A) The topological case δ = δ Top . Let us recall from[START_REF] Mcclure | Cosimplicial Objects and little n-cubes[END_REF] Proposition 13.4] that, for all k ≥ 0 and c i = c = c, we have an isomorphism of cosimplicial spaces(θ Top ) * c 1 ,...,c k ;c : ξ(RL m ) c 1 ,...,c k ;c (δ Top ) * → δ * Top × ξ(RL m ) c 1 ,...,c k ;c (δ Top ) 0 . (4.6)The same argument 2 applies to show that, for all k ≥ 0 and all colours (c 1 , ..., c k ; o),(θ Top ) * c 1 ,...,c k ;o : ξ(RL m ) c 1 ,...,c k ;o (δ Top ) * → δ * Top × ξ(RL m ) c 1 ,...,c k ;o (δ Top ) 0 (4.7)is an isomorphism of cosimplicial spaces. Moreover, both the cosimplicial isomorphisms (4.6) and (4.7) are compatible with the projection onto the second factor so that one obtains trivial fibrations (θ Top ) n c 1 ,...,c k ;c for each n ≥ 0 and colours (c 1 , ..., c k ; c) as in (2.1). Moreover θ induces a homeomorphism on Coend Lm (δ Top )(c 1 , ..., c k ; c) ∼ = Hom △ (δ Top , δ Top ) × ξ(RL m ) c 1 ,...,c k ;c (δ Top ) 0 . Using the contractibility of Hom △ (δ Top , δ Top ) one gets a weak equivalence (4.8) Coend RL m (δ Top )(c 1 , ..., c k ; c) → ξ(RL m ) c 1 ,...,c k ;c (δ Top ) 0 ,

  as follows. Let T ∈ RS m (c 1 , ..., c k ; c) = ξ c 1 ,...,c k ;c (RL m )(δ Z ) 0 be a generator. We denote by T # ∈ ξ c 1 ,...,c k ;c (RL m )(δ Z ) n the element represented by the relative lattice obtained by adding n vertical bars in its integer-string representation. Let us denote by e n the generator of(δ n Z ) n = C n (Hom △ (-, [n]); Z), so that (δ n Z ) n ∼ = Z[e n ] for n ≥ 0. Let T ∈ ξ c 1 ,...,c k ;c (RL m )(δ Z ) 0 be agenerator so that T is the class of T ⊗ e n 1 ⊗ . . . ⊗ e n k ; we denote [T ⊗ e n 1 ⊗ . . . ⊗ e n k ] such a class. We define ϑ n (T ⊗ e n ) to be the signed sum of elements [T # ⊗ e n # 1 ⊗ . . . ⊗ e n # k ]; the sum being over all possible relative lattices path of the form T # . We extend ϑ n on ξ c 1 ,...,c k ;c (RL m )(δ Z ) 0 ⊗ δ n Z as a cosimplicial map. Let us described the partial compositions of RS m . For two surjections f ∈ RS m (c 1 , ..., c k ; c) and g ∈ RS m (d 1 , ..., d j ; c i ) we set (5.1)

Figure 5 . 1 :

 51 Figure 5.1: Tree representation of the generators of RS 2 (via the description of Proposition 6.3).

Figure 6 . 4 :

 64 Figure 6.4: Example of composition.

  where τ U : K → Ω K is the universal truncating twisting function [KS05, Section 4]. The maps p and φ satisfy p(σ ′ , σ) = φ(σ ′ ) • p(e, σ) where • stands for the left action Ω M X × P M X → P M X and where e ∈ Ω X is the unit of the monoidal cubical set, see [KS05, Proof of Theorem 5.1].

  ψ : Ω (Sing 1 X, Sing 1 Y) → C * Ω M (X, Y)as follows. The inclusion Y ⊂ X gives rise to an inclusion of cubical sets Ω (Sing 1 X, Sing 1 Y) ⊂ P (Sing I Y). The map ψ is defined as the restriction p| Ω (Sing 1 X,Sing 1 Y) and satisfies ψ(σ ′ , σ) = φ(σ ′ ) • ψ(e, σ) where • stands for the left action induced byΩ M X × Ω M (X, Y) → Ω M (X, Y).The map ψ is then a map of cubical modules. The fact that ψ is a homotopy equivalence follows from the long exact sequences in homotopy induced byΩ M X → Ω M (X,Y) → Y, and |Ω Sing 1 X| → |Ω (Sing 1 X,Sing 1 Y)| → |Sing 1 Y|,using that φ is a homotopy equivalence.By construction (see[START_REF] Kadeishvili | A cubical model for a fibration[END_REF] (4) iii)]), we haveC * Ω (Sing 1 X, Sing 1 Y) = Ω(C 1 * X, C 1 * Y).

Ω

  u B = (TB, D)given byλ i B (a 1 ⊗ . . . ⊗ a k ) ⊗ (b 1 ⊗ . . . ⊗ b l ⊗ c) = (c) (a 1 ⊗ . . . ⊗ a i-1 ⊗ a i ⊳ (b 1 ⊗ . . . ⊗ b l ) ⊗ (a i+1 ⊗ . . . ⊗ a k ) ⊲ z (1) c ⊗ c (2) ), for 1 ≤ i ≤ k.For s = k above, we get a left actionλ B : M B (k) ⊗ Z B,C (l 1 ) ⊗ • • • ⊗ Z B,C (l k ) → Z B,C (l 1 + • • • + l k ) (8.10) defined by λ B (f, (g 1 ; c 1 ), . . . , (g k ; c k )) := (γ(f; g 1 , g 2 ⊲ z (1) c 1 , g 3 ⊲ z (1) c 2 z (2) c 1 , . . . , g k ⊲ z (1) c k-1 • • • z (k-1)c 1

  12) ((f; c), g 1 , . . . , g k ) → (γ B (f; g 1 , . . . , g k ); c), for any (f; c) = b 1 ⊗ . . . ⊗ b k ⊗ c ∈ Z B,C (k) and g i = b i 1 ⊗ . . . ⊗ b i l i ∈ M B (l i ).

  ∇ B (b) = ∇ B (b) -1 ⊗ bb ⊗ 1, ∇ C (c) = ∇ C (c) -1 ⊗ c. We denote [b 1 , . . . , b k ] := (s -1 ) ⊗k (b 1 ⊗ . . . ⊗ b k ).

  µ ′ o : Ω(B, C) ⊗ Ω(B, C) → Ω(B, C) (8.15) [a 1 , . . . , a k ]c ⊗ [b 1 , . . . , b l ]c ′ → (c) ±[a 1 , . . . , a k , (b 1 , . . . , b l ) ⊲ z (1) c ]c ′ c (2) ,where, we recall, for c ∈ C, the element ∇ C (c) is denoted by (c) z(1) c ⊗ c (2) so that 1 ⊗ c is a term of this sum; and, a family of operations E ′ j : (ΩB) ⊗ (Ω(B, C)) ⊗j → Ω(B, C) (8.16)

  SC functor-operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 SC-split operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A cellular decomposition of the Swiss Cheese operad 3.1 The Swiss Cheese operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The SC extended complete graph operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cellular decomposition of SC type operads . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the operad RL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The operad Coend RL m (δ) as a Swiss Cheese operad . . . . . . . . . . . . . . . . . . . . .

	Contents
	1 Introduction
	2 Preliminaries
	2.1 4 The operad RL
	4.1

5 The relative surjection operad 6 The operad RL 2 6.1 The operad RL 2 in term of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The algebras over RL 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cosimplicial relative loop space 7.1 Cosimplicial relative loop space of monoids . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Action of H * (SC 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Action of the whole operad RS 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Relative cobar construction 8.1 Cubical model for relative loop spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Unreduced (relative) cobar construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Action of RS 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Let us fix an arbitrary coloured operad O in C. We denote Col its set of colours, so that O consists of objects O(n 1 , . . . , n satisfying the natural unit, associativity and equivariance axioms. The underlying category of O is the category O u with the colours n ∈ Col as objects and the unary operations O u (m, n) = O(m; n) as morphisms. This way, we have functors

k ; n) ∈ C, for n 1 , ..., n k ∈ Col and k ≥ 0 together with the unit I C → O(n; n) and substitution maps O(n 1 , . . . , n k ; n) ⊗ O(m 1 , . . . , m l ; n i ) → O(n 1 , . . . , n i-1 , m 1 , . . . , m l , n i+1 , . . . , n k ; n)

  and such that all the cubes (C 1 , ..., C s ) are in the upper half space. Remark 3.2. Because of the symmetry conditions imposed by Sym, we may thought of SC m (c 1 , ..., c n ; o) as the configuration space of cubes (C 1 , ..., C s ) and semi-cubes (C s+1 , ..., C s+t ) lying into the standard semi-cube Half + ∩ C 0 .Similarly to the little m-cubes operad C (m) the composition maps• i : SC m (c 1 , ..., c n ; c) × SC m (d 1 , ..., d r ; c i ) → SC m (c 1 , ..., c i-1 ,d 1 , ..., d r , c i+1 , ..., c n ; c) are defined as substitutions of cubes. We denote the resulting 2-coloured operad SC m . We define the SC (or relative) extended complete graph operad RK. It is a 2-coloured poset operad with filtration {RK m } m≥1 . Its closed part is K m , its open part is K m-1 , where {K m } m≥1 denotes the extended complete graph operad defined in [BFV07, Section 4.1]. Given n colours c i ∈ {c, o}, we denote by { c 1 , ..., c n } the set with

	3.2 The SC extended complete graph operad

  Top or δ Z , the operad RL m is strongly δ-reductive. Consequently, the operad Coend RL m (δ) is weakly equivalent to the topological (resp. chain) Swiss Cheese operad SC m (resp. C

Cone(δ Top ) is contractible, it provides a map ξ(RL m ) α (Cone(δ Top ), ..., Cone(δ Top )) 0 → ξ(RL m ) α ( * , ..., * ) 0 = * which is a weak equivalence. Once again, the proof of the following is completely similar to [BB09, Examples 3.10].

5 The relative surjection operad Proposition 4.12. For δ being δ * SC m ) for δ being δ Top (resp. δ Z ).

  3, we haveTheorem 7.6. Let (M, N) be a pair of topological monoids pointed at the unit * such that N is a submonoid of M. Let δ be δ Top or δ Z . Then the operad Coend RL 2 (δ), which is weakly equivalent to the Swiss-Cheese operad SC 2 if δ = δ Top and weakly equivalent to C * (SC 2 ) if δ = δ Z , acts on the couple of totalizations (Tot δ ω(M), Tot δ ω(M, N)).

	7.2 Action of H

* (SC 2 )

  ′ 1,k : ΩB ⊗ (ΩB) ⊗k → ΩB (8.14)is given byE ′ 1,k (f; g 1 , ..., g k ) = ±[a 1 , . . . , a i 1 ⊳ g 1 , . . . , a i k ⊳ g k , . . . , a n ]

More precisely, such a path is not necessarily unique. However, if c i and c j are related by two monochromatic paths with the same colour ν then both have the same direction, either positive or negative.

In Proposition 12.7 and Proposition 13.4 from[START_REF] Mcclure | Cosimplicial Objects and little n-cubes[END_REF] it is sufficient to consider elementsp ⊔ q T S f hwith p ⊔ q instead of k, where p and q stand respectively for the number of closed colours and the number of open colours in (c 1 , ..., c k ; c).

Acknowledgement I would like to warmly thank Muriel Livernet for corrections and suggestions on an earlier version of this paper and Eduardo Hoefel for his constant support. The author was supported by "Bolsista da CAPES à Projeto 88881.030367/2013-01".

Proposition 7.8. Let (M, N) be a pair of 1-connected monoids pointed at the unit and such that N a sub-monoid of M. Then the above operations (7.8), (7.9) and (7.10) induce an H * (SC 2 )-algebra structure on the pair (H * Tot δ Z ω(M), H * Tot δ Z ω(M, N)).

Remark 7.9. For two paths τ, τ ′ : [0, 1] → M, let us denote by τ • τ ′ the path τ • τ ′ (t) = τ(t)τ ′ (t) given by the product of M. For x ∈ M, one denotes by c x the constant path at x. Let us define:

• a (associative and commutative all two up to homotopy) product µ c for Ω(M) as the concatenation µ c (γ, γ ′ ) = γγ ′ ;

• a (associative up to homotopy) product µ o for Ω(M, N) as µ o (γ, γ ′ ) = γ(γ ′ • c γ(1) ); and,

• an inclusion ΩM ֒→ Ω(M, N) as the canonical inclusion of loops.

The above inclusion and multiplication µ o provide a left action l : ΩM × Ω(M, N) → Ω(M, N) given by concatenation l(γ, τ) = γτ. Since, M, N are monoids, it is easy to see that the right action r :

) is homotopic to the left one l.

One can show that these operations induce an H * (SC 2 )-algebra structure on the pair (H * Ω(M), H * Ω(M, N)). It would be interesting to compare this structure with that one of Proposition 7.8.

Action of the whole operad RS 2

The inclusion RS 2 ֒→ Coend RL 2 (δ Z ) implies an action of the operad RS 2 on (Tot δ Z ω(M), Tot δ Z ω(M, N)). We describe this action.

To do that it is sufficient to write down explicitly the operations corresponding to the generators of RS 2 .

The closed part of RS 2 acts as described in [START_REF] Kadeishvili | On the cobar construction of a bialgebra[END_REF]. We recall from Proposition 5.3 that RS 2 is generated by µ c , T k , µ o , T j and inc. By a slight abuse of notation, for an element T ∈ RS 2 (c 1 , ..., c k ; c) d we denote also by T the corresponding operation

where

The multiplication µ c ∈ RS 2 (c, c; c) 0 acts as the product ∪ in (7.8). The trees T k ∈ RS 2 (c, ..., c k ; c) 2k+1 act as follows. For any f ∈ ω(M)(n) and g 1 , . . . , g k-1 ∈ Tot δ Z ω(M, N), one has

The generator µ o ∈ RS 2 (o, o; o) 0 acts as the product ⊔ in (7.9). The generator

; o) 2j+1 acts as follows. For each homogeneous elements f ∈

where σ is the map defined in (7.4). In other words, T j (f; (h 1 ; n 1 ), . . . , (h j-1 ; n j-1 )) is the sum over

Cubical model for relative loop spaces

In this section we define a model for Moore relative loop spaces. Given a pair (X, Y) of topological spaces, pointed at * and such that * ⊂ Y ⊂ X, the Moore relative loop space Ω M (X, Y) is defined as

We denote simply Ω M (X, * ) by Ω M X which is the loop space of X. The Moore path space P M X is defined as

Let us start with a few definitions about cubical sets. Those are extracted from [KS05, Section 2.4].

The product of two cubical sets Q and Q ′ , is

The face and degeneracy operators are induced by those of Q and Q ′ in the obvious way.

For n ≥ 0, let I n be the cube of dimension n given as the cartesian product of the interval I = [0, 1]; then the cube I 0 is a point. Let X be a pointed connected topological space. The cubical set Sing X = {Sing n X} n≥0 is formed by the continuous maps I n → X.

Let Q be a cubical set. We denote by C * Q the normalized chain complex of Q. For a topological space X, C * X denotes the normalized chain complex of Sing I X. Definition 8.6. A monoidal cubical set is a cubical set Q with an associative cubical multiplication µ : Q × Q → Q for which a distinguished element e ∈ Q 0 is the unit.

The chain complex C * Q of a monoidal cubical set is a dg-bialgebra. Definition 8.7. Let Q be a monoidal cubical set Q. A cubical set P is a Q-module if there is an associative cubical map Q × P → P with the unit of Q acting as identity.

The chain complex C

Let X be a 1-connected space pointed at * . Let Sing 1 X = {Sing 1 n X} n≥0 be the simplicial set formed by the singular simplexes σ n : ∆ n → X such that σ n sends the 1-skeleton of ∆ n to the base point * ∈ X. For any topological space Z, let us denote by C 1 * Z the normalized chain complex generated by singular simplexes whose the 1-skeleton is sent to the base point of Z; that is

In [START_REF] Kadeishvili | A cubical model for a fibration[END_REF] the cubical cobar construction Ω (Sing 1 X) on the simplicial set Sing 1 X is constructed. It is a monoidal cubical set; it is shown to be homotopically equivalent to the monoidal cubical set Sing Ω M X, cf. [KS05, Theorem 5.1]. In particular, taking the cubical chain complexes, one obtains Adams' morphism [START_REF] Adams | On the cobar construction[END_REF] Φ :

that is a quasi-isomorphism of dg-algebras. In fact, since Φ comes from a map of cubical sets

where D is the differential

In the same fashion, we define Ω u (B, C) to be the unreduced relative cobar construction

Let us fix some notations. For k ≥ 0, B ⊗k is an algebra for the product (a 1 ⊗.

where the dot • stands for the multiplication in B ⊗k ; and, a right B-module

, and more generally, (∇

Let us define M B to be the following multiplicative operad (for instance see [START_REF] Menichi | Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras[END_REF]). For k ≥ 0, we set M B (k) = B ⊗k .

The partial composition maps

We denote by γ B the corresponding structural map of this operad. The multiplication µ

Let us define a wide left action of M B on Z B,C . For k ≥ 1, 1 ≤ s ≤ k, l i ≥ 0 and an injective order preserving map β :

β (f, (g 1 ; c 1 ), . . . , (g s ; c s )) = (γ(f; 1 β(1)-1 , g 1 , z (1) c 1 , . . . , z (β(2)-β(1)-1) c 1 , g 2 ⊲ z (β(2)-β(1)) ),

where 1 β(1)-1 := 1, . . . , 1

β(1)-1

.

For s = 1 above, we get an infinitesimal left action λ i B : M B (k) ⊗ Z B,C (l) → Z B,C (k + l -1) (8.9)