From a map of operads η : O → O , we introduce a cofibrant replacement of O in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules Bimod h O (O ; O )is an algebra over the one dimensional little cubes operad C 1 . In the present work, we also build an explicit weak equivalence of C 1 -algebras from the loop spaceIn particular, if we apply the identifications (3) to the map η : C d → C n , then we get an explicit description of the iterated loop space associated to the space of long embeddings and their polynomial approximations:Applications to the Swiss-Cheese operad. Theorem 3.1 is also used by the author [10] in order to extend the previous results to the coloured case using the Swiss-Cheese operad SC d which is a relative version of the little cubes operad C d . In that case, a typical example of SC d -algebra is a pair of topological spaces (since the operad has two colours S = {o, ; c}) of the formwhere f : Y → X is a continuous map between pointed spaces. In particular, if (A ; B) is an SC d -algebra, then A is a C d -algebra, B is a C d−1 -algebra and there is a map τ : A → B which is more or less central up to homotopy (i.e. τ preserves the product and τ(a) × b b × τ(a) with a ∈ A and b ∈ B). In [10], we give a relative version of the delooping (2) such that, together with Theorem 3.1, we are able to identify explicit SC d+1 -algebras. In particular, if η 1 : C d → O is a map of operads and η 2 : O → M is a map of bimodules over O, then, under technical conditions, the pair of spacesis proved to be weakly equivalent to a typical SC d+1 -algebra using the identifications

In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad SC 1 . Thereafter, we show that the pair formed by the space of long embeddings and the manifold calculus limit of (l)-immersions from R d to R n is an SC d+1 -algebra assuming the Dwyer-Hess' conjecture.are proved to be weakly equivalent to explicit SC d+1 -algebras.Organization of the paper. The paper is divided into 4 sections. The first section gives an introduction on coloured operads and (infinitesimal) bimodules over coloured operads as well as the truncated versions of these notions. In particular, the little cubes operad, the Swiss-Cheese operad and the non-(l)-overlapping little cubes bimodule are defined.In the second section, we give a presentation of the left adjoint functor to the forgetful functor from the category of (P-Q) bimodules to the category of S-sequences. This presentation is used to endow Bimod P-Q with a cofibrantly generated model category structure. Thereafter, we prove that a Boardman-Vogt type resolution yields explicit and functorial cofibrant replacements in the model category of (P-Q) bimodules. We also show that similar statements hold true for truncated bimodules.The third section is devoted to the proof of the main theorem 3.20. For this purpose, we give a presentation of the functor L and prove that the adjunction (4) is a Quillen adjunction. Then we change slightly the Boardman-Vogt resolution introduced in Section 2 in order to obtain explicit cofibrant replacements in the category Op[O ; ∅]. Finally, by using Theorem 3.20 and the Dwyer-Hess' conjecture, we identify SC d+1algebras from maps of operads η : CC d → O .In the last section we give an application of our results to the space of long embeddings in higher dimension. We introduce quickly the Goodwillie calculus as well as the relation between the manifold calculus tower and the mapping space of infinitesimal bimodules. Then we show that the pairs (5) are weakly equivalent to explicit typical SC d+1 -algebras.Convention. By a space we mean a compactly generated Hausdorff space and by abuse of notation we denote by Top this category (see e.g. [18, section 2.4]). If X, Y and Z are spaces, then Top(X; Y) is equipped with the compact-open topology in order to have a homeomorphism Top(X; Top(Y; Z)) Top(X × Y; Z). By using the Serre fibrations, the category Top is endowed with a cofibrantly generated monoidal model structure. In the paper the categories considered are enriched over Top.By convention C ∞ d (0) is the one point topological space and the operadic composition • i with this point consists in forgetting the i-th little cube. Definition 1.5. Let S be a set and O be an S-operad. An algebra over the operad O, or O-algebra, is given by a family of topological spaces X := {X s } s∈S endowed with operations µ : O(s 1 , . . . , s n ; s n+1 ) × X s 1 × · · · × X sn −→ X s n+1 , compatible with the operadic composit...

We prove that if a pair of semi-cosimplicial spaces (X • c ; X • o ) arise from a coloured operad then the semitotalization sTot(X • o ) has the homotopy type of a relative double loop space and the pair (sTot(X • c ) ; sTot(X • o )) is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad SC 2 . Ω d X ; Ω d (X ; A) := Ω d X ; ho f ib(Ω d−1 A → Ω d−1 X) .In this paper we make great use of the operad π 0 (SC 1 ) which is the operad of monoid actions Act: it is a 2-coloured operad whose algebras are the pairs of spaces (X ; A) where X is a monoid and A a left X-module. The operad Act >0 is the non-unital version of Act. Similarly to the uncoloured case there is a notion of Act >0 -bimodule and Act >0 -infinitesimal bimodule. We prove that if O is an operad under Act then it gives rise to a pair of semi-cosimplicial spaces (O c ; O o ) such that the pair (sTot(O c ) ; sTot(O o )) is weakly equivalent to:that is, an SC 2 -space.Université Paris 13, Organization of the paper. The paper is divided into six sections. The first one is an introduction. It describes the categories of coloured operads, bimodules and infinitesimal bimodules over an operad. An explicit description of a point X in Bimod Act >0 and Ibimod Act >0 in terms of pairs of semi-cosimplicial spaces (X c ; X o ) is given. We insist on the link between bimodule structures over Act >0 and monoidal structures on semi-cosimplicial spaces introduced by McClure and Smith in [15].The second section introduces the left adjoint functors to the forgetful functors from the categories of bimodules and infinitesimal bimodules over an S-coloured operad to the category of S-sequences. These adjunctions will be used in the third section in order to define a model category structure on Bimod O and Ibimod O . We also determine an explicit cofibrant replacement of Act (resp. Act >0 ) in the model category Ibimod Act >0 (resp. Bimod Act >0 ) and prove the weak equivalence:where M is an Act >0 -infinitesimal bimodule and M c is its closed part.In section four we prove the first relative delooping theorem. From an Act >0 -bimodule map η : Act → M we extract two semi-cosimplicial spaces (M c ; M o ). We prove, under some conditions, the weak equivalence of pairs:Section five consists in considering a particular case where a double relative delooping theorem holds. Namely, let α : As → O be a map of operads and β : O → B be a map of O-bimodules. The two objects O and B are equipped with semi-cosimplicial structures. Under some conditions, we prove the following weak equivalence of pairs:where X is a coloured operad build out of O and B. The last section is devoted to the proof of the main theorem: if O is an {o ; c}-operad under Act such that O(0 ; c) O(1 ; c) O(1 ; o) * then we have the weak equivalence of pairs: sTot(O c ) ; sTot(O o ) Ω 2 Operad h (As >0 ; O c ) ; Ω 2 Operad h (As >0 ; O c ) ; Operad h {o ; c} (Act >0 ; O) .Convention. By space we mean compactly generated Hausdorff space and by abuse of notation we denote by Top this category...

We study projective and Reedy model category structures for bimodules and infinitesimal bimodules over a topological operad. In both cases, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We study the Extension/Restriction adjunctions. We give also a characterisation of Reedy cofibrations and we check that the two model structures produce compatible homotopy categories. In the case of bimodules the homotopy category induced by the Reedy model structure is a subcategory of the projective one. In the case of infinitesimal bimodules the Reedy and projective homotopy categories are the same.