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

Abstract

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.

On the rational homotopy type of embedding spaces of manifolds in $R^n$

“…Furthermore, note that U R (x) is obtained by composing copies of x with Rlinear extensions of the structure maps (product, Γ structure and differential) of M and Graphs n . Hence it is clear that U R (x) is indeed of the form (18), obtained using the R-linear extensions of k-linear maps ℓ k on HGC M,n , and hence we obtain indeed an L ∞ -structure on HGC M,n . Furthermore, all structure maps used in the definition of ℓ k either leave the number of edges of graphs in Graphs n constant, or decrease the number of edges.…”

“…Implicitly this result appeared in the proof of [61,Theorem 6.5]. Recall that, by [18,Theorem 5.4], an infinitesimal bimodule over a reduced operad P is cofibrant in the Reedy model structure if and only it is cofibrant as an infinitesimal bimodule over P >0 in the projective model structure, where P >0 is the operad such that P >0 (0) = ∅ and P >0 (k…”

“…The category IBimod P can also be equipped with the Reedy model structure, which is transferred by adjunction from the Reedy model structure on Top Λ op . (By [18,Theorem 5.1], this construction returns a valid model structure as soon as the operad P is well-pointed in the sense that the operadic unit defines a cofibration of topological spaces in arity one * → P(1).) Note that the derived mapping spaces IBimod h P (• • • ) coming from the projective and Reedy model structures are equivalent (see [18,Theorem 5.9], or use the general result of [19,Proposition 4.4]).…”

“…and to infinitesimal bimodules is the subject of the memoir [48] in preparation. The theory is very similar in this case, using concepts introduced in [24] for the study of the homotopy of reduced operads and in [18] for the study of the homotopy of infinitesimal bimodules. Notably, we use that every reduced operad P has a decomposition P = lim s cosk Λ s P such that Mor Op (−, cosk ≤s P) = Mor Op ≤s (−, P),…”

“…where Op ≤s denotes the category of s-truncated (reduced) operads, the category of operads that are defined up to arity ≤ s, and we consider the image of the object P under the obvious forgetful functor (−) ≤s : Op → Op ≤s . The coskeletal decomposition P = lim s cosk Λ s P was introduced in [24, Section II.8.4] in the context of reduced operads and was used in [18] in the context of infinitesimal bimodules. In both settings cosk Λ s P is determined, as a symmetric sequence, by the Λ-structure of P:…”