Operadic right modules via the dendroidal formalism

Abstract

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