Projective and Reedy Model Category Structures for (Infinitesimal) Bimodules over an Operad

Abstract

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.

Delooping the functor calculus tower

“…Having both structures available could be handy for a possible future work. A detailed study of the projective and Reedy model structures for (infinitesimal) bimodules is done in the work [21] by Fresse and the authors.…”

“…For any topological operad $$O$$, the categories $$\Sigma {\mathrm{Bimod}}_O$$ and $${\mathrm{T}}_k\Sigma {\mathrm{Bimod}}_O$$ admit a cofibrantly generated model structure transferred from $$\Sigma {\mathrm{Seq}}$$, $${\mathrm{T}}_k\Sigma {\mathrm{Seq}}$$ along the adjunctions () and (), respectively, see [19] and also [24, Section 14.3]. We use the following results from [21]. Theorem (i)For a reduced and well‐pointed topological operad $$O$$, the categories $$\Lambda {\mathrm{Bimod}}_O$$ and $${\mathrm{T}}_k\Lambda {\mathrm{Bimod}}_O$$, $$k\geqslant 0$$, admit a cofibrantly generated model structure transferred from $$\Lambda _{>0}{\mathrm{Seq}}$$ and $${\mathrm{T}}_k\Lambda _{>0}{\mathrm{Seq}}$$, respectively, along the adjunctions () and (), respectively.…”

“…One can easily check that for a $$\Lambda$$‐sequence $$X$$ (respectively, $${\mathrm{T}}_k\Lambda$$‐sequence $$X$$), the sequence $${\mathcal {F}}_{Ib}^\Lambda (X)$$ (respectively, $${\mathcal {F}}_{Ib}^{{\mathrm{T}}_k\Lambda }(X)$$) as a ($$k$$‐truncated) $$O_{>0}$$‐Ibimodule is freely generated by the ($$k$$‐truncated) $$\Sigma$$‐sequence $$X$$. We use the following results from [21]. Theorem (i)For any reduced and well‐pointed topological operad $$O$$, the category of $$O$$‐Ibimodules (respectively, $$k$$‐truncated $$O$$‐Ibimodules, $$k\geqslant 0$$) admits a cofibrantly generated model structure transferred from …”