A dendroidal approach to operadic right modules and manifold calculus

In this work we study the homotopy theory of the category $\mathsf{RMod}_{\mathcal{P}}$ of right modules over a simplicial operad $\mathcal{P}$ via the formalism of forest spaces $\mathsf{fSpaces}$, as introduced by Heuts, Hinich and Moerdijk. In particular, we show that, for $\mathcal{P}$ a simplicial closed $\Sigma$-free operad, there exists a Quillen equivalence between the projective model structure on $\mathsf{RMod}_{\mathcal{P}}$, and the contravariant model structure on the slice category $\mathsf{fSpaces}_{/N\mathcal{P}}$ over the dendroidal nerve of $\mathcal{P}$. As an application, we comment on how this result can be used to simplify the computation of derived mapping spaces between operadic right modules, and use this formalism to analyse the components and layers of the Goodwillie--Weiss tower coming from embedding calculus.