Let C, D be monoidal infinity categories that admit geometric realizations.
Let $F: C \to D$ be a monoidal functor and A an augmented associative algebra of C.
Denote $Bar(A)= \mathbb{1} \otimes_A \mathbb{1} = colim_{\Delta^{op}} A^{\otimes n} $ the bar-construction of A.
We want to study when the canonical map $\alpha: Bar (F(A)) \to F(Bar(A)) $ is an equivalence. Certainly this is satisfied if F preserves geometric realizations.
Are there weaker conditions that guarantee that $\alpha$ is an equivalence?
I' am especially interested in the following situation:
Let D be a stable presentably symmetric monoidal infinity category and O an operad in D.
Let C be the category of O-algebras with divided powers in D.
The forgetful functor $C \to D$ preserves finite products and so can be considered as a symmetric monoidal functor $F: C \to D$, when C and D are endowed with cartesian symmetric monoidal structures.
In general the forgetful functor $C \to D$ does not preserve geometric realizations and I want to know when F preserves the bar-construction with respect to the cartesian symmetric monoidal structures, i.e. F sends the bar-construction to the suspension.
Equivalently I would like to know if the bar-construction with respect to the cartesian symmetric monoidal structure defines an equivalence $ Mon(C)\to C$, where Mon(C) denotes the category of monoids in C.