Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes\_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in geometry of iterated loop spaces) to be the geometric realization of a simplicial space with space of $n$-simplicies given by $RO^nA$. Does anyone know a reference in the literature for sufficient conditions for when these two constructions yield weakly homotopy equivalent spaces?



Tensor products over operads and bar constructions
The space $B(R,O,A)$ is the tensor product $B(R,O,O)\otimes\_OA$. It is a standard fact that the right module $B(R,O,O)$ is cofibrant in the projective model structure of right modules whenever $R$ is levelwise cofibrant.


By Theorem 15.1.A.(a) of Fresse's "Modules over operads and functors", if $A$ is cofibrant (as a space) and $O$ is levelwise cofibrant, then the functor $-\otimes\_OA$ preserves weak equivalences between cofibrant (in the projective model structure) right modules. In particular the two constructions you want to consider coincide up to weak equivalence if $R$ is cofibrant in the projective model structure.


Using Theorem 15.1.A.(b) of the same book, you can also show that the two constructions coincide up to weak equivalence if $A$ is cofibrant as an algebra, $O$ is levelwise cofibrant and $R$ is levelwise cofibrant.


