Suppose we have a map of $E\_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there should be a lax $E\_{n-1}$-monoidal functor of quasicategories (where the domain is clearly a Kan complex): $$BX\to Top$$


which maps the unique point of $BX$ to $Y\in Top$ and each morphism of $BX$ (corresponding to the points of $X$) to an automorphism of $Y$ in $Top$ corresponding that element of $X$ acting on $Y$. Moreover, the colimit of this morphism, denote it $Y/X$, should be an $E\_{n-1}$-algebra in $Top$ (by [recent work](http://arxiv.org/abs/1411.7988) of Antonlin-Camarena and Barthel), since $BX$ is an $E\_{n-1}$-algebra and the morphism is lax monoidal. 


My question is: how do I produce the functor $BX\to Top$? Also, is it clearly $E\_{n-1}$-monoidal? 



Construction of Highly Structured Quotient Groups in Quasicategories
Your functor $BX \to \mathrm{Top}$ factors through the inclusion of the full subcategory of $\mathrm{Top}$ generated by the single object $Y$, which is just the data of $\mathrm{End}(Y)$ (an $E\_1$-algebra) -- you might call this subcategory $B(\mathrm{End}(Y))$. By adjunction, the multiplication on $Y$ gives a map $Y \to \mathrm{End}(Y)$, and your map (of one-point categories, which is equivalent to a map of $E\_1$-algebras) is just the composite $X \to Y \to \mathrm{End}(Y)$.


