Hello,

Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ is an element of $B^{\times}$ such that $f(a) b = b g(a)$ for all $a \in A$.

This is a certain sub$2$groupoid of the $2-$category of categories where the objects are the linear categories $A-$mod, the $1-$morphisms are (certain) functors (I think preserving the tensor structure), and the $2-$morphisms are natural transformations. Alternatively, take linear categories with a point as object and morphisms $A$ and consider functors and natural tranformations of those.

Is there a natural analogue of this in the case of $A_{\infty}$ algebras? I was hoping to understand some systematic way of writing down all the higher morphisms in an $\infty-$groupoid of $A_{\infty}$ algebras and hoping that they are all non-trivial. However, I don't even understand the analogue of the elements $b$ from above.

Is this done in the literature somewhere where one can extract explicit formulae? One guess would be to consider $A_{\infty}-$algebras as $A_{\infty}-$categories with one object, but this does not help unless one can explicitly write down some explicit $\infty-$groupoid structure on $A_{\infty}-$categories which I guess would require one to first convert these $A_{\infty}-$categories to $\infty-$categories

--Oren