From maps between coloured operads to Swiss-Cheese algebras

Abstract

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...

Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

“…The next subsection is devoted to adapt the Boardman-Vogt resolution (well known for operads, see [BM2]) to the context of (reduced) bimodules. We refer the reader to [Duc1] for a detailed account of the following constructions.…”

Delooping derived mapping spaces of bimodules over an operad

“…Example 2.3. The free bimodule functor F B We recall quickly a presentation of the free bimodule functor introduced by the author in [10,11]. Since we have to deal with the symmetric group action and the map γ : O(0) → M(0), we define an adjunction between the category of bimodules over O and the under category O 0 ↓ Seq where O 0 is the sequence given by O 0 (0) = O(0) and the empty set otherwise:…”

“…where the equivalence relation is generated by the compatibility with the symmetric group action, the compatibility with the unit of the operad O and the compatibility with the map γ : O(0) → M(0) (we refer the reader to [10]).…”

“…In [10], we introduce a functorial way to build a cofibrant replacement in the category of bimodules over an operad O. Unfortunately, in the special case where the bimodule is the operad O itself, this resolution forgets all the information coming from the operadic structure and we cannot expect to get an explicit C 1algebra structure on the derived mapping space of bimodules.…”

Princípio de reconhecimento de espaços de laços relativos

“…Idrissi encontrou um modelo de SC 2 na categoria de grupóides [Id17], e no geral Quesney encontrou modelos para SC N na categoria de conjuntos e os usou para exibir modelos de espaços de laços relativos em dimensão 2 [Qu15]. Ducoulombier provou um teorema de reconhecimento relativo no contexto de espaços de imersões e de espaços cosimpliciais em [Du16] e [Du17].…”