$\infty-$groupoid of $A_{\infty}$ algebras 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
 A: Notice that you are looking at the 2-category of algebras, bimodules and bimodule homomorphism.
Because of this: if you regard a morphism $f : A \to B$ of algebras as an $A$-$B$ bimodule $B_f$ ($B$ equipped with the obvious right $B$-action and with left $A$-action induced by $f$) then the 2-morphisms that you are looking at are bimodule homomorphism $B_g \to B_f$ given on $B$ by left multiplication with $b \in B$ (this trivially respects the right $B$-action and the equation $f(a)b = b g(a)$ is precisely the condition that it also respects the left $A$-action.)
So you are looking for the $A_\infty$-version of (the maximal higher groupoid inside) the 2-category of algebras, bimodules and bimodule homomorphisms.
Now, in 
Berger, Moerdijk, Resolution of coloured operads and rectification of homotopy algebras http://arxiv.org/PS_cache/math/pdf/0512/0512576v2.pdf
there is described in section 6 a model-category theoretic construction of a simplicial category whose objects are $A_\infty$-algebras, morphisms are bimodules of $A_\infty$-algebras, 2-morphisms are bimodule homomorphisms, and so on. 
This simplicial category you may think of as presenting an $\infty$-category of $A_\infty$-algebras.
(Notice that this applies to $A_\infty$-algebras over any suitable enriching category, say tor $A_\infty$-spaces You are probably thinking of the standard dg-case, enriched over chain complexes, to which it applies in particular.)
Some paragraphs on this you can also find here:
http://ncatlab.org/nlab/show/model+structure+on+algebras+over+an+operad#InfCatOfMods
A: Seidel's book Fukaya categories and Picard-Lefschetz theory, sections (1a-e), describes in explicit terms the $A_\infty$-category of non-unital functors between two fixed (small) $A_\infty$-categories, as well as the functors between functor-categories obtained by composing with a fixed functor on the left or right. This is not quite the whole story concerning composition. 
Unsurprisingly, you can formulate everything as sums over trees, but the signs are nasty.
