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Register a new userDelooping derived mapping spaces of bimodules over an operad
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Authors
Julien Ducoulombier
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From a map of operads $eta : Orightarrow O$, we introduce a cofibrant replacement of the operad $O$ in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules $Bimod_{O}^{h}(O;O)$ is an algebra over the one dimensional little cubes operad $mathcal{C}_{1}$. In the present work, we also build an explicit weak equivalence of $mathcal{C}_{1}$-algebras from the loop space $Omega Operad^{h}(O;O)$ to $Bimod_{O}^{h}(O;O)$.
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We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.
We extend some classical results - such as Quillens Theorem A, the Grothendieck construction, Thomasons Theorem and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer-Hess and Turchin results concerning the explicit double delooping of spaces of long knots.
It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.
We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid of the homotopy type of a CW complex and $G$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G$. This fact has several applications. As the first application, we show that the connecting map $Grightarrowmathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $mathrm{Map}_0(B_nG,BG)rightarrowmathrm{Map}(B_nG,BG)rightarrow BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups, $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon and homotopy pullback of $A_n$-maps. In particular, we show that the $T_k^f$-space and the $C_k^f$-space are exactly the same concept and give some new examples of $T_k^f$-spaces.
We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps $D^{m}rightarrow D^{n}$ of discs, $ngeq m$, avoiding any given multisingularity and coinciding with the standard inclusion near $partial D^{m}$. In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces.
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