I am in a reading group studying Seidel's book (*Fukaya Categories and Picard-Lefschetz Theory*). All of the participants have backgrounds in symplectic topology/pseudoholomorphic curve methods. We are stuck in trying to understand the chapter presenting the algebraic background for Fukaya Categories.


Seidel makes the following claim: Non-unital $A\_\infty$ functors $\mathcal F: \mathcal A \rightarrow \mathcal B$ are themselves the objects of a non-unital $A\_\infty$ category. The morphisms $\mathrm{hom}( \mathcal F\_0, \mathcal F\_1)$ are something he calls (following Fukaya) pre-natural transformations. (The morphisms $T$ for which $\mu\_1(T) = 0$ are the natural transformations.) Seidel then provides the formulae for the compositions $\mu\_d$.
(This is discussed in Section (1d) of the book [page 10].)


In our working group, we tried to check that these formulae for the compositions satisfied the $A\_\infty$ associativity equations, but were unable to do so beyond $\mu\_1$. 


I have two questions (that may be the same question):



> 
> Why do these composition maps satisfy the $A\_\infty$ associativity equations? Is there a way of understanding this geometrically?
> 
> 
> 



Why do A_\infty functors form an A_\infty category?
I can explain the pictures I usually draw to think of $A\_\infty$ functors,
but I don't know if they're standard. Anyway, I'll describe what is
just a rubric for ingesting the long formulas, nothing more.


Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$
in an $A\_\infty$-category $A$ as an $A$-module, or functor from $A^{op}$ to
chain complexes. So $Y\_L(M) = hom\_A(M,L).$


I confess that when I confront these formulas/concepts, I always think
in terms of the Fukaya category, which is very amenable to pictures
and for which the $A\_\infty$ structures are geometric.


So I draw a curve on a piece of paper and label it $L$. (The curve
is literally a Lagrangian submanifold of my ${\mathbb R}^2$ piece of
paper.) When I want
to think of $L$ in terms of its Yoneda image, I draw the SAME curve, but as a
squiggly line.


So what is the data that the squiggly line gives us? For each object $M$
(a regular curve on my paper), we have the intersection points, which form a
graded vector space $hom\_A^\*(M,L).$ This vector space has the
structure of a chain complex (Floer), with differential given by
football-shaped bi-gons with one regular side and one squiggly side. 
For a pair of other objects, $M\_1, M\_2,$ we get a map
$$\mu^2: hom\_A(M\_2,L)\otimes hom\_A(M\_1,M\_2) \rightarrow hom\_A(M\_1,L),$$
and so on for all the structure of a module (section 1j, p. 19).


For the Fukaya category,
the equations 1.19 follow (for non-squiggly lines) from studying degenerations
of 1-parameter families of holomorphic polygons. Now squigglifying those same pictures
gives 1.19 for an arbitrary module, and the equations are similar for
not just modules but arbitrary functor between two $A\_\infty$-categories.


What data do we have if we have *two* squiggly lines $L\_1$ and $L\_2$? 
They should intersect at a morphism between functors (and it should have a degree).
This morphism of functores gives more data, using the Fukaya perspective.
If we added one normal line $M$, we'd have the spaces $Y\_{L\_1}(M)$ and $Y\_{\_2}(M)$,
and have a triangle which is a map between them. Higher polygons and the
relations between them (by considering one-parameter families) should
give you all the equations and give you a hint as to verify them.
(But no promises!)


Hope that lengthy and pretty vague description was worth our time. 
(Oh, geez, this was a March 11 question? Probably stale by now!)


