I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations.


In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ satisfies Leibniz's rule if we regard the second operation as multiplication. As $N=3$, it means that the multiplication satisfies associativity at the homology level.


Questions
---------


1. These are "classical" interpretations. Though, I still wish to understand the higher operations. What are their meanings? Are there ways to view it as natural in a pure operad-ic view point?
2. As mentioned above, the second operation can be viewed as ordinary multiplication at the homology level. How about the higher ones? How do they behave at the homology level (or perhaps this is not the right question to ask..) ?


I have tried some papers on Fukaya category, wikipedia, and n-lab. Any other references will be appreciated. Thank you very much.



Meaning of A-infinity relations
For your first question. Suppose $(A,d,m,m\_3,m\_4\dots)$ is an $A\_\infty$-algebra. The operation $m\_3$ gives a homotopy between $m(-,m(-,-))$ and $m(m(-,-),-)$, which I will abusively denote as $a(bc)$ and $(ab)c$.


Now consider the two operations $A^{\otimes 4} \to A$ given by $a(b(cd))$ and $((ab)c)d$. Using $m\_3$, you have two ways of going from the first to the second:


* Use $m\_3$ three times to create a homotopy:
$$a(b(cd)) \to a((bc)d) \to (a(bc))d \to ((ab)c)d.$$
* Use $m\_3$ twice to create a homotopy:
$$a(b(cd)) \to (ab)(cd) \to ((ab)c)d.$$


If you combine these two homotopies, you get two classes of degree $2$ maps $A^{\otimes 4} \to A$. These two maps have no reason to be homotopic. Well, the higher operation $m\_4$ gives a homotopy between these two homotopies!


The even higher operations $m\_5$, $m\_6$ and so on work the same way. This is very nicely encoded in Stasheff's associahedra. The first two associahedra is just a point, representing the identity and $m\_2$; the next one is a segment, representing $m\_3$, a homotopy between $a(bc)$ and $(ab)c$; the next one is a pentagon, whose edges are the five arrows I drew above; and so on.


For your second question, the answer is Massey products. Very briefly, suppose that you have a differential graded algebra $A$ and three cycles $a,b,c$ such that $ab = d\alpha$ and $bc = d\beta$. Then the class $abc$ vanishes "in two different ways", because $abc = d(\alpha c) = d(a \beta)$. It follows that $\alpha c - a \beta$ is a homology class, called the triple Massey product $\langle a,b,c \rangle$. The operation $m\_3$ can be used to represent this triple Massey product on homology. It's a bit technical to explain how, and the explanation involves the Homotopy Transfer Theorem.


For both answers, I think a good reference that cites pretty much all the other possible ones is the book *Algebraic Operads* by Loday and Vallette.


