Cofibrancy of a right module over an operad

Question

If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module is cofibrant as a module over the commutative operad, and the $\mathscr{O}$-module structure is pulled back by the unique map $\mathscr{O} \rightarrow \operatorname{Com}$?

My motivational example is the Fulton-MacPherson module $\mathscr{F}_M$ over the Fulton-MacPherson operad $\mathscr{F}_n$ associated to any parallelizable n-manifold $M$. It is somewhat involved to completely describe these, but the idea is that they consist of configurations of points where some are allowed to be infinitesimally close. Cofibrancy of these are important in lots of recent work, notably by Ching-Salvatore and Fresse-Turchin-Willwacher among others.

I think the idea is that $\mathscr{F}_M$ is freely generated as a $\mathscr{F}_n$ module by the ordered configurations of points in $M$, and this freeness is enough to make it cofibrant. I am not sure to what extent this freeness is a necessary condition.

Answer 1

If $O$ is a reduced operad in $(C,\otimes)$, i.e. one with $O(\{*\})=1$ and $O(\emptyset)=*$, then under mild assumption on $(C,\otimes)$, the projective model structure exists on $\mathrm{RMod}_O$, i.e. a model structure with levelwise equivalences and fibrations. We let $\mathrm{Decom}(-)$ denote the submodule of decomposable elements. In this case, for a right module $R \in \mathrm{RMod}_O$ to be cofibrant, it is sufficient that the (i) the inclusion $\mathrm{Decom}(R)(I) \rightarrow R(I)$ is a $\Sigma_I$-cofibration for all $I$ and (ii) that any right module map between the trunctations $R^{\leq i-1} \rightarrow R'^{\leq i-1}$ extends to a well defined map $\mathrm{Decom}(R)(I) \rightarrow R'(I)$.

In the case of $\mathcal{F}_M$, $\mathrm{Decom}(\mathcal{F}_M)(I) \rightarrow \mathcal{F}_M(I)$ is the inclusion of the actual manifold boundary of a free $\Sigma_I$-manifold. It is not difficult to see that this implies it is a $\Sigma_I$-cofibration (i.e. a cofibration in the projective model structure on symmetric sequences). The freeness mentioned in the question comes when verifying the second condition and is obvious once one has put the effort into formally defining $\mathcal{F}_M$.

I should mention that cofibrancy in the projective model structure implies cofibrancy in the Reedy model structure and both of these model structures are used in the literature. Finally, one would never expect the restriction of a cofibrant $\mathrm{com}$-module to be a cofibrant $\mathcal{F}_n$ module since the condition (ii) will usually not hold.

Answer 2

I think the OP does a good job answering his own question, in the context he had in mind. But, the original question asked "are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure?" So I wanted to mention that Luis Pereira did work related to Reedy cofibrancy in operadic contexts, and has a general proof strategy that explains the connection to freeness, applied to the bar construction. The relevant paper is Cofibrancy of operadic constructions in positive symmetric spectra, published in AGT. I previously wrote about this paper in another MO answer. Pereira's Theorem 1.6 proves a Reedy cofibrancy result, and points to the paper Homotopy completion and topological Quillen homology of structured ring spectra by Harper and Hess (published in G & T), Theorems 4.19 and 6.26, which also show how to prove Reedy cofibrancy.