Simple examples of A-infinity categories/algebras

Question

I'm looking for simple examples of non-trivial $A_\infty$-algebras/$A_\infty$-categoies which ideally can be understood in an introductory talk in a grad student seminar.

A few examples I know are Fukaya-type categories of a surface, Massey product and an $A_\infty$ algebra given by multiplication table in the book Algebraic Operads by Loday and Vallette's book.

The first two are a little involved in my opinion ($J$-holomorphic disks or Kadeishvili's theorem) and the third is kind of unintuitive to me. I'm wondering if there's some intuitive construction which can produce $A_\infty$-algebras/$A_\infty$-categoies directly.

Answer 1

Loop spaces are a good motivating example. It is important to notice however that that they are only homotopic to $A_\infty$-spaces, because of the failure of the concatenation map to be strictly unital.

Answer 2

There are a few nice small examples for initial illustration in Lu-Palmieri-Wu-Zhang's paper "A-infinity algebras for ring theorists".

One of them is the $A_\infty$-Koszul dual of $k[t]/(t^n)$ for $n\geq 3$ (this computation can also be found in Dag Madsen's PhD thesis). It is given as an algebra by $k[x,y]/(x^2)$ with higher multiplication given by $m_n(xy^{t_1},\dots,xy^{t_n})=y^{1+\sum_{i=1}^n t_i}$ and all other higher multiplications vanish, i.e. $m_k=0$ for $k\neq 2,n$ and $m_n$ applied to some expression not involving $x$ is also $0$.

Another nice example one can find there explicitly is that a (non-unital) $A_\infty$-algebra on an $A_\infty$-algebra $A$ concentrated in degrees $1$ and $2$ is given by arbitrary $\Bbbk$-linear maps $m_n\colon (A^1)^{\otimes n}\to A^2$. Which is of course seems a bit artificial but gives you some impression how many possibilities you actually have.